One Hat Cyber Team
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216.73.216.115
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194.44.31.54
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Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
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Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
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5.6.40
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book-newprint
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View File Name :
ostr-sam.ps
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b(.)g(.)f(.)h(.)f(.)h(.)70 b(91)434 2286 y(2.1.3)93 b(In\014nite-dimensional)28 b(represen)n(tations)81 b(.)42 b(.)f(.)h(.)f(.)h(.)70 b(96)243 2391 y(2.2)84 b(Some)27 b(classes)f(of)i FQ(\003)p FP(-algebras)d(with)j(3)f(and)g(4)g (generators)62 b(.)42 b(.)28 b(106)434 2497 y(2.2.1)93 b(Represen)n(tations)25 b(of)h(graded)f FO(so)p FP(\(3\))i(and)f (four-tup-)699 2596 y(les)i(of)f(pro)5 b(jections)27 b(satisfying)f(a)i(linear)e(relation)45 b(.)d(.)28 b(106)434 2702 y(2.2.2)93 b(Represen)n(tations)37 b(of)i(a)e(class)h(of)g (quadratic)f(alge-)699 2801 y(bras)27 b(with)h(three)f(generators)k(.) 41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(113)434 2907 y(2.2.3)93 b(Op)r(erator)31 b(relations)h(connected)g(with)h(a)g (dynami-)699 3006 y(cal)27 b(system)h(on)f(a)g(plane)77 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(116)434 3111 y(2.2.4)93 b(Represen)n(tation)32 b(of)g(real)g(forms)g (of)g(Witten's)h(\014rst)699 3211 y(deformation)j(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f (.)h(.)28 b(119)434 3316 y(2.2.5)93 b(Represen)n(tations)38 b(of)h(the)h(Skly)n(anin)f(algebra)e(and)699 3416 y FO(U)756 3428 y FM(q)793 3416 y FP(\()p FO(sl)r FP(\(2\)\))78 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.) f(.)h(.)f(.)h(.)28 b(123)243 3521 y(2.3)84 b(Represen)n(tations)26 b(of)h FO(q)s FP(-deformed)g FO(U)9 b FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)h FP(\))q(\))42 b(.)g(.)g(.)f(.)h(.)f(.)h(.)28 b(133)434 3627 y(2.3.1)93 b(Real)28 b(forms)f(of)g FO(U)1270 3639 y FM(q)1306 3627 y FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)i FP(\)\))42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(133)434 3732 y(2.3.2)93 b(Represen)n(tations)26 b(of)i FO(U)1455 3744 y FM(q)1491 3732 y FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)h FP(\))q(\))50 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.) 28 b(135)243 3837 y(2.4)84 b(Man)n(y-dimensional)26 b(dynamical)h (systems)78 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(151)434 3942 y(2.4.1)93 b(\\Direct)29 b(pro)r(ducts")f(of)h (one-dimensional)f(dynam-)699 4042 y(ical)f(systems)40 b(.)i(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h (.)f(.)h(.)28 b(152)434 4147 y(2.4.2)93 b(\\T)-7 b(riangular")25 b(dynamical)i(systems.)53 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(155)p eop %%Page: 3 3 3 2 bop 118 100 a FK(Con)n(ten)n(ts)2064 b FP(iii)434 333 y(2.4.3)85 b(Op)r(erator)17 b(relations)g(connected)h(with)h(man)n (y)-8 b(-dimen-)699 432 y(sional)27 b(dynamical)g(systems)74 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(161)434 532 y(2.4.4)93 b(Represen)n(tations)17 b(of)i(the)f (non-standard)f(real)h(quan-)699 632 y(tum)29 b(sphere)65 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.) h(.)f(.)h(.)28 b(166)434 731 y(2.4.5)93 b(Heisen)n(b)r(erg)49 b(relations)g(for)g(the)h(quan)n(tum)g FO(E)5 b FP(\(2\))699 831 y(group)70 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h (.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(169)434 930 y(2.4.6)93 b(Wic)n(k)28 b(algebras)d(related)i(to)h(dynamical)f (systems)38 b(.)k(.)28 b(173)243 1030 y(2.5)84 b(On)27 b(represen)n(tations)f(of)h(some)g(n)n(uclear)g(algebras)39 b(.)j(.)f(.)h(.)f(.)h(.)28 b(180)434 1130 y(2.5.1)93 b(Comm)n(utativ)n(e)27 b(mo)r(dels)e(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f (.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(180)434 1229 y(2.5.2)93 b(Cen)n(tered)27 b(op)r(erators)36 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(185)434 1329 y(2.5.3)93 b(Represen)n(tations)26 b(of)i(Cun)n(tz)g(algebras)68 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)28 b(189)243 1429 y(Commen)n(ts)f(to)g (Chapter)g(2)50 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.) f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(198)118 1611 y FR(3)77 b(On)39 b(the)g(complexit)m(y)f(of)i(the)f(description)f(of)i(represen) m(ta-)243 1711 y(tions)31 b(of)g FQ(\003)p FR(-algebras)1432 b(203)243 1810 y FP(3.1)84 b FQ(\003)p FP(-Wild)27 b(algebras)f(and)h (relations)55 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)28 b(203)434 1910 y(3.1.1)93 b(Ma)5 b(jorization)25 b(of)h FQ(\003)p FP(-algebras)e(with)i(resp)r(ect)h(to)f(the)699 2010 y(complexit)n(y)h(of)h(their)f(represen)n(tations)47 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)28 b(203)434 2109 y(3.1.2)93 b FQ(\003)p FP(-Wildness)27 b(of)h FQ(\003)p FP(-algebras)39 b(.)j(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(212)434 2209 y(3.1.3)93 b FQ(\003)p FP(-Wild)29 b(algebras)f (generated)g(b)n(y)h(orthogonal)e(pro-)699 2309 y(jections)h(and)f (idemp)r(oten)n(ts)83 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)28 b(214)434 2408 y(3.1.4)93 b FQ(\003)p FP(-Wild)28 b(semilinear)e(relations)76 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)28 b(221)434 2508 y(3.1.5)93 b FQ(\003)p FP(-Wild)28 b(quadratic)e(and)i(cubic)f(relations)42 b(.)g(.)f(.)h(.)f(.)h(.)28 b(222)434 2607 y(3.1.6)93 b FQ(\003)p FP(-Wild)28 b(groups.)35 b(P)n(erio)r(dic)26 b(groups)h(are)f(not)i FQ(\003)p FP(-wild)c(.)k(227)243 2707 y(3.2)75 b(On)23 b(the)g(complexit)n(y)f (of)h(the)g(description)g(of)f(classes)g(of)h(non-)434 2807 y(self-adjoin)n(t)k(op)r(erators)34 b(.)41 b(.)h(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(229)434 2906 y(3.2.1)93 b(Classes)46 b(of)g(non-self-adjoin)n(t)g(op) r(erators)f(singled)699 3006 y(out)28 b(b)n(y)f(a)g(quadratic)g(or)g(a) g(cubic)h(relation)72 b(.)41 b(.)h(.)f(.)h(.)28 b(230)434 3106 y(3.2.2)93 b(P)n(artial)34 b(isometries,)i(w)n(eakly)e(cen)n (tered)g(op)r(erators)699 3205 y(and)28 b(algebraic)d(op)r(erators)71 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(234)434 3305 y(3.2.3)93 b(Hyp)r(onormal)18 b(op)r(erators)f(and)h (pairs)g(of)h(comm)n(uting)699 3404 y(completely)28 b(non-unitary)e (isometries)78 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)28 b(236)243 3504 y(Commen)n(ts)f(to)g(Chapter)g(3)50 b(.)42 b(.)f(.)h(.)f(.)h(.)f (.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)28 b(238)118 3687 y FR(Bibliograph)m(y)1778 b(243)118 3869 y(Index)2078 b(259)p eop %%Page: 4 4 4 3 bop 118 100 a FP(iv)p eop %%Page: 1 5 1 4 bop 118 100 a FI(Rev)22 b(Math.)27 b(&)20 b(Math.)28 b(Phys.,)21 b(1999,)1373 98 y(c)1353 100 y FN(\015)q FI(1999)f(OP)-5 b(A)20 b(\(Overseas)h(Publishers)h(Asso)r(ciation\))118 166 y(vol.)29 b(11,)20 b(pp.)28 b(1{261)854 b(Amsterdam)19 b(B.V.)i(Published)h(under)f(license)i(in)118 232 y(Rep)n(rints)f (available)g(directly)i(from)c(the)h(publisher)89 b(The)21 b(Netherlands)g(under)g(the)g(Ha)n(rw)n(o)r(o)r(d)f(Academic)118 299 y(Photo)r(cop)n(ying)i(p)r(ermitted)g(b)n(y)f(license)i(only)323 b(Publishers)22 b(imp)n(rint,)g(pa)n(rt)e(of)h(The)g(Go)n(rdon)f(and) 1915 365 y(Breach)h(Publishing)i(Group.)2076 432 y(Printed)f(in)f(Mala) n(ysia)139 1030 y FH(In)m(tro)s(duction)37 b(to)f(the)i(Theory)g(of)f (Represen)m(tations)g(of)603 1146 y(Finitely)e(Presen)m(ted)i FG(\003)p FH(-Algebras.)342 1262 y(I.)g(Represen)m(tations)h(b)m(y)f(b) s(ounded)i(op)s(erators)498 1598 y FF(V)-10 b(asyl)31 b(Ostr)n(o)n(vsky)1209 1591 y(\025)1219 1598 y(\020)g(and)g(Yuri)1644 1591 y(\025)1654 1598 y(\020)g(Samo)1917 1591 y(\025)1927 1598 y(\020lenk)n(o)130 1723 y FP(Institute)e(of)e(Mathematics,)h (Ukrainian)e(National)h(Academ)n(y)g(of)h(Sciences)1164 2142 y FR(Abstract)432 2295 y FE(This)34 b(review)h(giv)n(es)h (fundamen)n(tals)f(of)g(represen)n(tations)i(of)e(\014nitely)326 2374 y(presen)n(ted)e FD(\003)p FE(-algebras)e(b)n(y)h(b)r(ounded)h(op) r(erators.)56 b(The)32 b(theory)g(is)f(illus-)326 2453 y(trated)h(with)f(n)n(umerous)f(examples)h(of)g FD(\003)p FE(-algebras.)53 b(The)31 b(examples,)h(in)326 2532 y(particular,)22 b(include)h FD(\003)p FE(-algebras)f(with)h(t)n(w)n(o)g(self-adjoin)n (t)f(generators)h(that)326 2611 y(satisfy)18 b(a)h(quadratic)h(or)e(a)h (more)f(general)h(relation,)h FD(\003)p FE(-algebras)e(with)h(three)326 2690 y(and)34 b(four)g(generators,)j FD(\003)p FE(-algebras)d(that)h (arise)e(from)f(one-)j(and)f(man)n(y-)326 2769 y(dimensional)25 b(discrete)i(dynamical)g(systems,)f(Wic)n(k)h FD(\003)p FE(-algebras,)g(v)l(arious)326 2847 y FD(\003)p FE(-wild)22 b(algebras.)432 2926 y(This)17 b(b)r(o)r(ok)h(is)g(in)n(tended)h(for)e (graduate)i(studen)n(ts)g(as)f(w)n(ell)f(as)h(for)f(the)i(re-)326 3005 y(searc)n(hers)g(who)h(sp)r(ecialize)g(in)f(the)h(theory)g(of)f (represen)n(tations)i(of)e FD(\003)p FE(-algeb-)326 3084 y(ras)k(and)h(related)g(areas.)p eop %%Page: 2 6 2 5 bop 118 100 a FP(2)p eop %%Page: 3 7 3 6 bop 118 906 a FS(Preface)118 1346 y FR(1.)78 b FP(The)42 b(w)n(ord)f(algebra)e(in)j(this)g(b)r(o)r(ok)f(means)h(an)f(asso)r (ciativ)n(e)f(algebra)118 1446 y(o)n(v)n(er)28 b(the)i(\014eld)h(of)e (complex)h(n)n(um)n(b)r(ers)f FJ(C)15 b FP(.)50 b(The)30 b(terms)f FQ(\003)p FP(-algebra,)f(Banac)n(h)118 1546 y FQ(\003)p FP(-algebra,)34 b FO(C)575 1516 y FN(\003)613 1546 y FP(-algebra,)h FO(W)1054 1516 y FN(\003)1092 1546 y FP(-algebra,)f(as)g(w)n(ell)h(as)f(their)g(prop)r(erties,)i(are)118 1645 y(used)28 b(in)g(this)f(b)r(o)r(ok,)h(as)f(a)g(rule,)g(without)h (sp)r(ecial)g(references.)243 1754 y(This)22 b(b)r(o)r(ok)h(is)f(dev)n (oted)h(to)f FC(r)l(epr)l(esentations)30 b FP(of)23 b(\014nitely)g (presen)n(ted)f FQ(\003)p FP(-al-)118 1853 y(gebras)33 b(\(de\014ned)h(b)n(y)g(a)g(\014nite)g(n)n(um)n(b)r(er)g(of)g (generators)e(and)i(relations\))f(b)n(y)118 1953 y(b)r(ounded)28 b(op)r(erators.)118 2129 y FR(2.)43 b FP(There)29 b(is)g(a)h(whole)f (domain)g(of)h(Algebra)e(called)i(\\Represen)n(tation)e(the-)118 2228 y(ory)38 b(of)i(algebras".)70 b(If)39 b(one)g(in)n(tro)r(duces)g (an)g(in)n(v)n(olution)g FQ(\003)g FP(in)n(to)g(an)g(alge-)118 2328 y(bra)e FB(A)h FP(and)g(considers)f(only)g(those)h(represen)n (tations)e(b)n(y)h(op)r(erators)f(in)i(a)118 2427 y(Hilb)r(ert)d(space) e FO(H)41 b FP(whic)n(h)35 b(preserv)n(e)d(the)j(in)n(v)n(olution)e(\() p FQ(\003)p FP(-represen)n(tations\),)118 2527 y(then)j(these)f (represen)n(tations)e(mak)n(e)h(just)h(an)g(island)g(among)f(all)g(the) i(rep-)118 2627 y(resen)n(tations)c(of)i FB(A)p FP(.)54 b(Moreo)n(v)n(er,)33 b(indecomp)r(osable)f FQ(\003)p FP(-represen)n(tations,)h(so)118 2726 y(dear)25 b(to)g(the)h (algebraist's)e(heart,)i(coincide,)f(in)h(this)g(case,)f(with)i (irreducible)118 2826 y(represen)n(tations,)c(and)g(t)n(w)n(o)g FQ(\003)p FP(-represen)n(tations)e(are)i(equiv)-5 b(alen)n(t)23 b(if)i(and)e(only)118 2926 y(if)d(they)g(are)f(unitarily)g(equiv)-5 b(alen)n(t.)34 b(And)20 b(hence,)i(the)e(problem)f(of)h(describing)118 3025 y FQ(\003)p FP(-represen)n(tations)j(of)j FB(A)p FP(,)h(up)f(to)g(a)f(unitary)h(equiv)-5 b(alence,)26 b(is)g(a)f(subproblem)118 3125 y(\(a)33 b(particular)g(case\))f(of)i (the)g(problem)f(of)g(describing)g(all)g(represen)n(tations)118 3224 y(up)28 b(to)g(an)f(equiv)-5 b(alence.)243 3333 y(But:)367 3441 y(1\))29 b(Mathematical)f(problems)f(related)h(to)g FQ(\003)p FP(-represen)n(tations)e(could)243 3541 y(turn)h(out)h(to)f (b)r(e)h(pith)n(y)g(and)g(in)n(teresting.)367 3649 y(2\))d(Considering) f FQ(\003)p FP(-represen)n(tations)e(allo)n(ws)i(one)g(to)h(c)n(hange)f (the)i(ac-)243 3749 y(cen)n(t)e(sharply)-7 b(,)24 b(from)g(algebra)f (to)h(functional)h(analysis,)e(and)i(to)f(consider)243 3848 y(not)41 b(only)g(represen)n(tations)f(b)n(y)h(b)r(ounded)h(op)r (erators)d(in)j(an)f(in\014nite-)243 3948 y(dimensional)23 b(space)h FO(H)7 b FP(,)24 b(but)h(also)e(represen)n(tations)f(b)n(y)i (un)n(b)r(ounded)g(op-)243 4048 y(erators.)57 b(Represen)n(tations)33 b(of)i(Lie)g(algebras)e(and)i(their)g(applications)243 4147 y(sho)n(w)26 b(ho)n(w)h(imp)r(ortan)n(t)h(and)f(useful)h(suc)n(h)f (represen)n(tations)f(are.)1326 4357 y(3)p eop %%Page: 4 8 4 7 bop 118 100 a FP(4)2148 b FK(Preface)367 333 y FP(3\))34 b(Moreo)n(v)n(er,)e(kno)n(wing)g(only)h FQ(\003)p FP(-represen)n (tations)e(can)i(sometimes)243 432 y(b)r(e)23 b(satisfactory)e(to)h (consumers)g(of)h(represen)n(tation)e(theory)-7 b(.)34 b(The)23 b(repre-)243 532 y(sen)n(tation)29 b(theory)h(of)g FQ(\003)p FP(-algebras)e(suggests)h(some)h(applications)f(of)h(the)243 632 y(theory)c(to:)492 738 y(a\))50 b(the)g(construction)g(and)g(study) g(of)h(mo)r(dels)f(of)g(quan)n(tum)367 838 y(ph)n(ysics,)30 b(in)g(particular,)f(b)n(y)g(using)h(Wic)n(k)g(algebras)d(and)j(their)g (rep-)367 938 y(resen)n(tations;)492 1044 y(b\))d(a)f(study)h(of)g (represen)n(tations)e(of)i FQ(\003)p FP(-algebras)d(whic)n(h)j(are)f (gen-)367 1144 y(erated)38 b(b)n(y)h(idemp)r(oten)n(ts)g(and)g(the)g (corresp)r(onding)e(resolution)h(of)367 1244 y(the)28 b(iden)n(tit)n(y;)492 1350 y(c\))c(a)g(study)g(of)g(op)r(erator)e (Banac)n(h)h(algebras)e(con)n(taining)i(a)h(dense)367 1450 y FQ(\003)p FP(-subalgebra,)c(and)i(construction)e(of)i(in)n(v)n (ertibilit)n(y)f(sym)n(b)r(ols)g(for)f(op-)367 1550 y(erators)26 b(in)i(the)g(algebra,)e(etc.;)492 1656 y(d\))33 b(structure)g(theorems) f(for)h(algebraically)e(de\014ned)i(classes)f(of)367 1756 y(not)c(self-adjoin)n(t)f(op)r(erators.;)492 1863 y(e\))k(the)h(theory)e(of)h(algebras)f(and)h(their)g(represen)n (tations,)f(since)367 1962 y(the)e(island)f(of)g FQ(\003)p FP(-represen)n(tations)d(could)j(turn)h(out)f(to)g(b)r(e)g(an)g(arc)n (hi-)367 2062 y(p)r(elago,)38 b(and)e(the)h(facts)g(ab)r(out)f(it)h (could)f(b)r(e)h(useful)g(for)f(studying)367 2162 y(b)r(oth)30 b(the)f(algebra)f FB(A)h FP(itself)h(and)f(its)g(represen)n(tations)f (but)h(already)367 2261 y(without)21 b(the)g(in)n(v)n(olution)f(in)h (the)g(algebra.)33 b(In)21 b(particular,)g(ev)n(en)f(suc)n(h)367 2361 y(a)f(traditional)f(part)h(of)g(algebra)e(as)i(the)g(theory)f(of)h (groups)f(\(esp)r(ecially)367 2461 y(coun)n(table)k(groups\))f(has)h (long)g(ago)f(included,)i(in)g(its)g(sto)r(c)n(k-in-trade,)367 2560 y(the)28 b(metho)r(ds)g(of)g(theory)e(of)i FQ(\003)p FP(-represen)n(tations;)492 2667 y(f)6 b(\))28 b(other)g(applications.) 118 2838 y FR(3.)35 b FP(In)24 b(the)f(course)g(of)g(some)g(time,)i (the)f(authors)e(ha)n(v)n(e)h(accum)n(ulated)f(a)h(fairly)118 2937 y(large)k(n)n(um)n(b)r(er)h(of)f(examples,)h(and)g(dev)n(elop)r (ed)f(tec)n(hniques)h(for)g(calculating)118 3037 y FQ(\003)p FP(-represen)n(tations)33 b(of)i(classes)f(of)h(\014nitely)h(presen)n (ted)f FQ(\003)p FP(-algebras.)57 b(These)118 3137 y(classes)25 b(include)h(certain)f(curv)n(es)g(in)h(the)h(real)e(plane,)h FQ(\003)p FP(-algebras)d(generated)118 3236 y(b)n(y)k(idemp)r(oten)n (ts,)h(Wic)n(k)g(algebras,)d(and)j(others.)243 3343 y(There)h(came)h (an)g(idea)g(to)g(presen)n(t)g(these)g(examples,)h(classes)d(of)j (exam-)118 3443 y(ples,)h(and)f(metho)r(ds)g(used)g(to)g(describ)r(e)g (their)g(represen)n(tations)e(gradually)-7 b(,)118 3542 y(with)29 b(an)g(increase)e(of)i(complexit)n(y)f(of)h(the)g(problem.)39 b(Actually)-7 b(,)30 b(the)f(c)n(hoice)118 3642 y(of)22 b(examples)f(and)g(metho)r(ds)h(w)n(as)e(determined)i(b)n(y)g(authors') e(taste)h(and)h(their)118 3742 y(exp)r(erience)27 b(in)h(the)g(sub)5 b(ject.)243 3848 y(T)-7 b(rying)22 b(to)h(carry)e(out)j(this)f(idea)g (systematically)-7 b(,)23 b(w)n(e)g(split)g(it)h(in)n(to)f(infor-)118 3948 y(mation)29 b(ab)r(out)f FQ(\003)p FP(-represen)n(tations)e(of)j (algebras)e(considered)h(in)h(the)g(exam-)118 4048 y(ples)k(b)n(y)g(b)r (ounded)g(op)r(erators)e(\(I\))j(and)f(un)n(b)r(ounded)g(op)r(erators)e (\(I)r(I\).)j(This)118 4147 y(b)r(o)r(ok)i(is)g(based)f(on)h(a)g (su\016cien)n(tly)g(large)e(\\zo)r(o")g(of)i(examples)g(that)g(illus-)p eop %%Page: 5 9 5 8 bop 118 100 a FK(Preface)2147 b FP(5)118 333 y(trate)30 b(the)g(notions)g(and)f(metho)r(ds)i(that)f(app)r(ear)f(in)h(studying)g (b)r(ounded)h FQ(\003)p FP(-)118 432 y(represen)n(tations.)i(A)21 b(more)f(accurate)f(title)i(of)g(this)g(b)r(o)r(ok)f(w)n(ould)g(p)r (ossibly)g(b)r(e)118 532 y(\\Represen)n(tations)25 b(of)h FQ(\003)p FP(-algebras)e(b)n(y)i(b)r(ounded)g(op)r(erators)f(b)n(y)h (examples",)118 632 y(but,)i(a)g(similar)e(title)j(has)e(already)f(b)r (een)i(tak)n(en)f(\(see)g([65)o(]\).)118 789 y FR(4.)50 b FP(A)32 b(starting)g(p)r(oin)n(t)g(for)g(the)g(exp)r(osition)g(in)g (this)h(review)e(is)h(represen)n(ta-)118 888 y(tions)f(of)g FQ(\003)p FP(-algebras)d(generated)i(b)n(y)g(t)n(w)n(o)h(self-adjoin)n (t)f(generators)f(satisfy-)118 988 y(ing)e(a)h(quadratic)e(relation)h (\(a)g(\\noncomm)n(utativ)n(e)f(curv)n(e)h(of)g(degree)g(t)n(w)n(o)g (in)118 1088 y(the)38 b(real)e(plane"\).)64 b(But)38 b(w)n(e)e(also)g(giv)n(e)g(far)h(reac)n(hing)e(generalizations)g(of)118 1187 y(suc)n(h)f(\\noncomm)n(utativ)n(e)f(curv)n(es":)48 b(a)34 b(theory)g(of)g(represen)n(tations)e(of)i(op-)118 1287 y(erators)c(satisfying)g(a)h(semilinear)g(relation)f(\(Sections)i (1.3.2{1.3.5,)d(3.1.4\),)118 1387 y(an)e(accoun)n(t)g(of)g(noncomm)n (utativ)n(e)f(dynamical)h(systems,)g(one-dimensional)118 1486 y(\(Section)c(2.1\))g(and)g(man)n(y-dimensional)e(\(Section)i (2.4\),)h(represen)n(tations)d(of)118 1586 y(algebras)31 b(with)j(three)f(and)h(four)f(generators,)f(whic)n(h)h(app)r(ear)g(in)h (theoreti-)118 1685 y(cal)23 b(ph)n(ysics)g(\(Sections)h(2.2,)f(2.3\),) h(v)-5 b(arious)22 b FQ(\003)p FP(-wild)h(problems)g(\(Sections)h(3.1,) 118 1785 y(3.2\).)118 1942 y FR(5.)63 b FP(In)36 b(order)f(to)i(read)e (this)i(b)r(o)r(ok,)h(it)f(is)g(enough)e(to)i(b)r(e)g(familiar)e(with)i (a)118 2042 y(basic)g(univ)n(ersit)n(y)g(course)g(of)h(op)r(erator)e (theory)h(and)h(in)n(v)n(olutiv)n(e)f(algebras)118 2142 y(\()p FQ(\003)p FP(-algebras\).)c(Of)23 b(course,)g(a)g(part)f(dev)n (oted)h(to)g(a)f(description)h(of)g FQ(\003)p FP(-algebras)118 2241 y(and)33 b(their)g(prop)r(erties)f(w)n(ould)h(b)r(e)g(useful)h(in) f(an)g(enlarged)f(edition,)i(where)118 2341 y(\014nitely)44 b(generated)f(and)h(\014nitely)g(presen)n(ted)f(algebras)f(and)i FQ(\003)p FP(-algebras,)118 2441 y(prop)r(erties)27 b(of)g(suc)n(h)h (algebras)d(and)j(examples)e(could)i(b)r(e)g(presen)n(ted.)243 2543 y(W)-7 b(e)31 b(w)n(ould)f(lik)n(e)g(to)h(giv)n(e)f(a)g(list)h(of) g(some)f(related)g(monographs,)g(whic)n(h)118 2642 y(are)d(close,)g(in) g(con)n(ten)n(ts,)g(to)h(this)g(b)r(o)r(ok.)243 2745 y(1\))35 b(Asso)r(ciativ)n(e)g(algebras)f(\(see,)k(e.g.,)f([118)o(,)f (113)n(,)g(209)o(],)i(and)e(the)g(bib-)118 2844 y(liograph)n(y)e (therein\),)k(coun)n(table)d(groups)g(\(see,)i(e.g.,)h([41)o(,)e(189)o (,)f(108)o(],)j(and)118 2944 y(the)d(bibliograph)n(y)d(therein\),)37 b(and)d(represen)n(tations)e(of)j(coun)n(table)e(groups)118 3044 y(and)26 b(asso)r(ciativ)n(e)f(algebras)f(\(see,)j(e.g.,)f([60)o (,)h(13)o(,)g(88)o(],)f(and)h(the)g(bibliograph)n(y)118 3143 y(therein\).)243 3245 y(2\))22 b(Dynamical)f(systems,)i(esp)r (ecially)e(one-dimensional)g(\(see,)i(e.g.,)g([260)o(,)118 3345 y(261)o(,)k(268)o(],)h(etc.\).)243 3447 y(3\))39 b(F)-7 b(unctional)40 b(analysis)f(and)h(op)r(erator)e(theory)-7 b(,)42 b(including)e(sp)r(ectral)118 3547 y(theory)h(\(see,)k(e.g.,)g ([4)o(,)d(104)n(,)g(235)o(,)f(241)o(,)g(37,)g(29)o(],)k(and)d (others\),)i(unitary)118 3647 y(represen)n(tations)36 b(of)i(groups)f(\(see,)j(e.g.,)g([135)o(,)e(312)o(,)g(20)o(],)i (etc.\),)i(op)r(erator)118 3746 y FQ(\003)p FP(-algebras)30 b(and)j(their)g(represen)n(tations)e(\(see,)k(e.g.,)f([69)o(,)f(279)o (,)g(73)o(,)g(9,)g(274)o(,)118 3846 y(206)o(,)24 b(130)n(,)g(133)o(,)g (178)o(,)g(65)o(],)h(and)e(others\),)i(in)f(particular,)f(represen)n (tations)f(b)n(y)118 3945 y(un)n(b)r(ounded)28 b(op)r(erators)e(\(see)h (e.g.,)g([124)o(,)h(243)n(,)g(123)o(,)f(249)o(],)h(etc.\))243 4048 y(4\))18 b(Quan)n(tum)h(groups)e(and)h(homgeneous)g(spaces,)h (their)g(represen)n(tations)118 4147 y(\(esp)r(ecially)j FQ(\003)p FP(-represen)n(tations\),)f(and)g(their)h(applications)f(to)h (the)h(theory)e(of)p eop %%Page: 6 10 6 9 bop 118 100 a FP(6)2148 b FK(Preface)118 333 y FP(in)n(tegrable)26 b(mo)r(dels)i(\(see,)g(e.g.,)f([129)n(,)h(163)o(,)f(237)o(,)h(55)o(,)f (168)o(,)h(119)n(,)g(140)o(,)f(143)o(]\).)243 432 y(5\))34 b(Applications)g(of)h(the)g(theory)f(of)g FQ(\003)p FP(-represen)n (tations)e(to)i(mo)r(dels)h(of)118 532 y(mathematical)22 b(ph)n(ysics)g(\(see,)h(e.g.,)g([80,)f(49)o(,)h(242)n(,)g(300)n(],)h (etc.\),)g(non-comm)n(u-)118 632 y(tativ)n(e)31 b(geometry)f(\(see,)j (e.g.,)f([58)o(,)f(169)o(],)h(etc.\),)h(non-comm)n(utativ)n(e)d(proba-) 118 731 y(bilit)n(y)24 b(theory)g(\([114)o(,)g(203)o(,)g(42)o(],)h (etc.\),)g(to)g(the)f(construction)f(of)i(in)n(v)n(ertibilit)n(y)118 831 y(sym)n(b)r(ols)j(\([153)o(,)h(40)o(],)g(etc.\),)h(to)e(the)h (theory)f(of)h(non)g(self-adjoin)n(t)f(op)r(erators)118 930 y(\(see,)g(e.g.,)f([104)o(,)g(81)o(,)h(178)o(],)f(etc.\).)118 1080 y FR(6.)74 b FP(References)40 b(to)g(the)g(literature)f(often)i (con)n(tained)e(in)i(the)f(commen)n(ts)118 1179 y(to)c(c)n(hapters)e (do)i(not)g(claim)f(to)h(b)r(e)g(complete)g(and,)h(presumably)-7 b(,)37 b(do)f(not)118 1279 y(con)n(tain)k(a)g(full)h(bibliograph)n(y)e (on)h(b)r(o)r(oks)g(and)h(articles)e(directly)i(related)118 1379 y(to)32 b(the)h(questions)f(touc)n(hed)g(up)r(on)h(in)f(this)h(b)r (o)r(ok.)51 b(Sometimes,)33 b(the)g(refer-)118 1478 y(ences)23 b(to)g(original)f(sources)f(are)i(replaced)f(with)i(the)f(references)f (to)h(a)n(v)-5 b(ailable)118 1578 y(monographs)30 b(or)i(reviews)f(con) n(taining)h(additional)g(bibliographical)e(mate-)118 1678 y(rial;)d(probably)-7 b(,)27 b(the)h(authors)e(to)r(o)i(often)f (refer)g(to)h(sources)e(in)i(Russian)f(and)118 1777 y(their)h (translations.)243 1877 y(W)-7 b(e)26 b(also)g(included)h(some)e (references)h(related)f(to)i FQ(\003)p FP(-represen)n(tations)c(b)n(y) 118 1976 y(un)n(b)r(ounded)42 b(op)r(erators,)g(k)n(eeping)f(in)g(mind) h(the)f(future)h(second)f(v)n(olume)118 2076 y(of)32 b(this)g(b)r(o)r(ok)g(that)g(will)g(b)r(e)g(dev)n(oted)f(to)h(represen) n(tations)e(b)n(y)i(un)n(b)r(ounded)118 2176 y(op)r(erators.)118 2325 y FR(7.)40 b FP(The)29 b(authors)f(are)g(sincerely)g(grateful)g (to)h(man)n(y)f(mathematicians)g(who)118 2425 y(con)n(tributed)33 b(to)g(this)g(w)n(ork:)47 b(their)33 b(teac)n(her,)g(professor)e(Y)-7 b(u.)34 b(M.)f(Berezan-)118 2524 y(sky)-7 b(,)30 b(for)g(his)g(kind)g (atten)n(tion)g(and)g(useful)h(advice,)f(all)f(participan)n(ts)g(of)h (the)118 2624 y(seminars)k(on)g(algebraic)f(problems)h(of)h(functional) f(analysis)g(in)h(the)g(Insti-)118 2724 y(tute)43 b(of)g(Mathematics)g (of)f(the)h(Ukrainian)f(National)g(Academ)n(y)g(of)h(Sci-)118 2823 y(ences,)33 b(colleagues)e(Stanisla)n(v)g(Krugly)n(ak,)h(Konrad)e (Sc)n(hm)r(\177)-44 b(udgen)33 b(and)f(Vic-)118 2923 y(tor)41 b(Sh)n(ul'man,)k(studen)n(ts)d(Lyudm)n(yla)f(T)-7 b(uro)n(wsk)i(a,)43 b(Alexandra)e(Piry)n(atin-)118 3023 y(sk)-5 b(a)n(y)n(a,)21 b(Eduard)f(V)-7 b(a)n(ysleb,)22 b(Y)-7 b(ury)21 b(Chap)r(o)n(vsky)-7 b(,)21 b(Stanisla)n(v)f(P)n(op)r (o)n(vyc)n(h,)h(Daniil)118 3122 y(Proskurin,)h(Sla)n(vik)f(Rabanovic)n (h)g(for)h(their)g(v)-5 b(aluable)22 b(con)n(tributions)f(to)i(this)118 3222 y(b)r(o)r(ok.)243 3321 y(W)-7 b(e)37 b(also)e(gratefully)h(ac)n (kno)n(wlege)f(\014nancial)h(supp)r(ort)g(from)h(the)g(join)n(t)118 3421 y(gran)n(t)26 b(from)i(the)g(CRDF)g(and)f(Ukrainian)g(Go)n(v)n (ernmen)n(t)f(no.)37 b(UM1-311.)p eop %%Page: 7 11 7 10 bop 118 900 a FS(Chapter)46 b(1)118 1218 y(P)l(airs)g(of)f (self-adjoin)l(t)i(op)t(erators)e(connected)118 1368 y(b)l(y)g(quadratic)g(relations)i(and)e(some)118 1517 y(generalizations)118 1968 y FH(1.1)112 b(In)m(tro)s(duction)37 b(to)g(represen)m(tations)g(of)g FG(\003)p FH(-algebras)118 2155 y FR(1.1.1)94 b FQ(\003)p FR(-Represen)m(tations:)40 b(k)m(ey)33 b(w)m(ords)118 2313 y(1.)38 b FP(A)29 b(represen)n(tation)d (of)j(an)f(algebra)e FB(A)j FP(on)e(a)h(\014nite-dimensional)g(Hilb)r (ert)118 2413 y(\(unitary\))d(space)g FO(H)32 b FP(is)25 b(a)g(homomorphism)f FO(\031)29 b FP(of)c FB(A)g FP(in)n(to)g(the)h (algebra)d FO(L)p FP(\()p FO(H)7 b FP(\))118 2512 y(of)27 b(linear)g(transformations)e(on)i FO(H)7 b FP(.)36 b(A)28 b FQ(\003)p FP(-represen)n(tation)d(of)i(a)g FQ(\003)p FP(-algebra)d FA(A)118 2612 y FP(is)i(a)f FQ(\003)p FP(-homomorphism)e FO(\031)29 b FP(from)c(the)h(algebra)e(in)n(to)i(the)f FQ(\003)p FP(-algebra)e FO(L)p FP(\()p FO(H)7 b FP(\))26 b(of)118 2712 y(b)r(ounded)k(op)r(erators)e(on)h(a)g(separable)f(Hilb)r (ert)i(space)f FO(H)7 b FP(.)42 b(The)30 b(dimension)118 2811 y(of)e(the)g(represen)n(tation)e(is)h(the)h(dimension)f(of)h FO(H)7 b FP(.)243 2914 y(W)-7 b(e)22 b(emphasize)f(that)g(in)h(this)g (c)n(hapter)f(w)n(e)g(restrict)g(ourselv)n(es)e(to)j(consid-)118 3013 y(ering)j(only)h(\014nite-dimensional)f(represen)n(tations)f(of)i FB(A)p FP(,)g(if)g FB(A)g FP(is)g(an)g(algebra)118 3113 y(without)31 b(in)n(v)n(olution,)f(and)g FQ(\003)p FP(-represen)n (tations)d(b)n(y)j(b)r(ounded)h(op)r(erators)d(on)118 3212 y(a)f(separable)f(Hilb)r(ert)j(space)d(\(dim)15 b FO(H)30 b FQ(\024)22 b(1)p FP(\))28 b(if)g FA(A)g FP(is)f(a)g FQ(\003)p FP(-algebra.)164 3370 y FR(2.)71 b FP(In)40 b(the)g(represen)n(tation)e(theory)g(of)i(algebras,)g(represen)n (tations)e(are)118 3470 y(studied)h(up)g(to)g(some)f(equiv)-5 b(alence.)69 b(W)-7 b(e)39 b(call)g(represen)n(tations)d(of)j FB(A)p FP(,)i FO(\031)118 3569 y FP(on)32 b FO(H)39 b FP(and)d(~)-46 b FO(\031)35 b FP(on)736 3548 y(~)714 3569 y FO(H)7 b FP(,)33 b(equiv)-5 b(alen)n(t)32 b(if)h(there)f(exists) f(an)h(in)n(v)n(ertible)g(op)r(erator)118 3669 y FO(C)16 b FP(:)27 b FO(H)j FQ(7!)470 3648 y FP(~)448 3669 y FO(H)35 b FP(that)28 b(in)n(tert)n(wines)f(the)h(represen)n(tations)d FO(\031)31 b FP(and)h(~)-46 b FO(\031)s FP(,)28 b(i.e.,)823 3857 y FO(C)6 b(\031)s FP(\()p FO(x)p FP(\))25 b(=)i(~)-46 b FO(\031)s FP(\()p FO(x)p FP(\))p FO(C)q(;)182 b FQ(8)p FO(x)22 b FQ(2)i FB(A)p FO(:)243 4048 y FP(In)39 b(the)h(represen)n (tation)d(theory)i(of)g FQ(\003)p FP(-algebras,)h(represen)n(tations)d (are)118 4147 y(studied)i(up)g(to)g(a)f(unitary)h(equiv)-5 b(alence.)70 b(Represen)n(tations)37 b(of)i FA(A)p FP(,)i FO(\031)h FP(on)1326 4357 y(7)p eop %%Page: 8 12 8 11 bop 118 100 a FP(8)917 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FO(H)38 b FP(and)e(~)-46 b FO(\031)35 b FP(on)613 312 y(~)592 333 y FO(H)6 b FP(,)33 b(are)d(said)h(to)h(b)r(e)g(unitarily)f(equiv)-5 b(alen)n(t)31 b(if)h(there)f(exists)g(a)118 432 y(unitary)c(op)r (erator)f FO(U)18 b FP(:)28 b FO(H)i FQ(7!)1098 411 y FP(~)1077 432 y FO(H)k FP(suc)n(h)27 b(that)827 591 y FO(U)9 b(\031)s FP(\()p FO(x)p FP(\))24 b(=)k(~)-47 b FO(\031)s FP(\()p FO(x)p FP(\))p FO(U;)181 b FQ(8)p FO(x)23 b FQ(2)g FA(A)p FO(:)243 750 y FP(T)-7 b(o)43 b(ev)n(ery)g FQ(\003)p FP(-algebra)f FA(A)p FP(,)48 b(one)c(can)f(asso)r(ciate)g(a)h (category)e FQ(\003)p FP(-Rep)13 b FA(A)p FP(,)118 850 y(whose)27 b(ob)5 b(jects)27 b(are)g FQ(\003)p FP(-represen)n(tations)d (of)k FA(A)f FP(considered)g(up)h(to)f(a)g(unitary)118 949 y(equiv)-5 b(alence)27 b(and)h(its)f(morphisms)g(are)g(in)n(tert)n (wining)g(op)r(erators.)243 1049 y(W)-7 b(e)28 b(ha)n(v)n(e)e(the)i (follo)n(wing)f(simple)g(prop)r(osition.)118 1196 y FR(Prop)s(osition) 34 b(1.)43 b FC(A)n(ny)33 b(two)g(\014nite-dimensional)42 b FQ(\003)p FC(-r)l(epr)l(esentations)33 b(of)g(a)118 1296 y FQ(\003)p FC(-algebr)l(a)25 b FA(A)g FC(ar)l(e)g(e)l(quivalent)g (if)h(and)f(only)g(if)h(they)f(ar)l(e)g(unitarily)h(e)l(quivalent.)118 1443 y(Pr)l(o)l(of.)43 b FP(One)31 b(has)g(to)h(pro)n(v)n(e)d(only)i (that)h(the)g(equiv)-5 b(alence)31 b(of)38 b FQ(\003)p FP(-represen)n(ta-)118 1542 y(tions)28 b FO(\031)i FP(on)e FO(H)34 b FP(and)e(~)-46 b FO(\031)31 b FP(on)995 1521 y(~)973 1542 y FO(H)j FP(implies)28 b(their)g(unitary)f(equiv)-5 b(alence.)243 1642 y(Let)27 b FO(C)16 b FP(:)28 b FO(H)h FQ(7!)743 1621 y FP(~)722 1642 y FO(H)34 b FP(b)r(e)28 b(an)f(in)n(v)n(ertible)g(op)r(erator)f(suc)n(h)h(that)826 1801 y FO(C)6 b(\031)s FP(\()p FO(x)p FP(\))24 b(=)j(~)-46 b FO(\031)s FP(\()p FO(x)p FP(\))p FO(C)q(;)182 b FQ(8)p FO(x)22 b FQ(2)i FA(A)p FO(:)536 b FP(\(1.1\))243 1960 y(Let)28 b(us)g(consider)f(the)i(p)r(olar)e(decomp)r(osition)h(of)g (the)h(op)r(erator)d FO(C)6 b FP(,)29 b FO(C)h FP(=)118 2060 y FO(U)9 b(A)p FP(,)23 b(where)e FO(A)i FP(=)g(\()p FO(C)6 b(C)861 2030 y FN(\003)900 2060 y FP(\))932 2030 y FL(1)p FM(=)p FL(2)1058 2060 y FP(is)21 b(an)g(in)n(v)n(ertible)g(p)r (ositiv)n(e)g(op)r(erator)f(on)h FO(H)28 b FP(and)118 2159 y FO(U)39 b FP(is)30 b(a)f(unitary)g(op)r(erator)g(from)g FO(H)37 b FP(to)1433 2138 y(~)1411 2159 y FO(H)g FP(\()p FO(U)1615 2129 y FN(\000)p FL(1)1731 2159 y FP(=)27 b FO(U)1889 2129 y FN(\003)1926 2159 y FP(\).)45 b(Then)30 b(it)g(follo)n(ws)118 2259 y(from)d(\(1.1\))h(that)671 2418 y FO(\031)s FP(\()p FO(x)p FP(\))d(=)d FO(A)1006 2384 y FN(\000)p FL(1)1096 2418 y FO(U)1162 2384 y FN(\000)p FL(1)1255 2418 y FP(~)-46 b FO(\031)s FP(\()p FO(x)p FP(\))p FO(U)9 b(A;)181 b FQ(8)p FO(x)23 b FQ(2)g FA(A)p FO(:)382 b FP(\(1.2\))118 2577 y(T)-7 b(aking)27 b(adjoin)n(ts)g(of)h (b)r(oth)g(sides,)f(w)n(e)g(obtain)676 2736 y(~)-47 b FO(\031)t FP(\()p FO(x)p FP(\))24 b(=)e FO(U)9 b(A)1072 2701 y FN(\000)p FL(1)1161 2736 y FO(\031)s FP(\()p FO(x)p FP(\))p FO(AU)1450 2701 y FN(\000)p FL(1)1541 2736 y FO(;)180 b FQ(8)p FO(x)23 b FQ(2)g FA(A)p FO(:)382 b FP(\(1.3\))118 2895 y(Consequen)n(tly)-7 b(,)182 3054 y FO(A)244 3019 y FL(2)282 3054 y FO(\031)s FP(\()p FO(x)p FP(\))24 b(=)f FO(AU)683 3019 y FN(\000)p FL(1)777 3054 y FP(~)-47 b FO(\031)s FP(\()p FO(x)p FP(\))p FO(U)9 b(A)25 b FP(=)d FO(AU)1301 3019 y FN(\000)p FL(1)1390 3054 y FO(U)9 b(A)1518 3019 y FN(\000)p FL(1)1607 3054 y FO(\031)s FP(\()p FO(x)p FP(\))p FO(AU)1896 3019 y FN(\000)p FL(1)1987 3054 y FO(U)g(A)23 b FP(=)g FO(\031)s FP(\()p FO(x)p FP(\))p FO(A)2449 3019 y FL(2)2487 3054 y FO(:)118 3213 y FP(Since)34 b FO(A)g FP(is)f(a)h(p)r(ositiv)n(e)f(op) r(erator,)g(w)n(e)h(ha)n(v)n(e)e FO(A)1668 3182 y FL(2)1706 3213 y FO(\031)s FP(\()p FO(x)p FP(\))i(=)f FO(\031)s FP(\()p FO(x)p FP(\))p FO(A)2222 3182 y FL(2)2260 3213 y FP(,)j FO(x)d FQ(2)h FB(A)p FP(,)118 3312 y(whic)n(h)i(implies)g (that)g FO(A\031)s FP(\()p FO(x)p FP(\))i(=)e FO(\031)s FP(\()p FO(x)p FP(\))p FO(A)h FP(for)e(an)n(y)g FO(x)i FQ(2)g FA(A)p FP(.)61 b(F)-7 b(rom)35 b(this)h(w)n(e)118 3412 y(obtain)789 3571 y FO(U)9 b(\031)s FP(\()p FO(x)p FP(\))p FO(A)24 b FP(=)f FO(U)9 b(A\031)s FP(\()p FO(x)p FP(\))24 b(=)j(~)-46 b FO(\031)s FP(\()p FO(x)p FP(\))p FO(U)9 b(A;)118 3730 y FP(and)28 b(since)f FO(A)h FP(is)f(in)n(v)n (ertible,)827 3889 y FO(U)9 b(\031)s FP(\()p FO(x)p FP(\))24 b(=)k(~)-47 b FO(\031)s FP(\()p FO(x)p FP(\))p FO(U;)181 b FQ(8)p FO(x)23 b FQ(2)g FA(A)p FO(:)118 4048 y FP(Hence)29 b(w)n(e)g(ha)n(v)n(e)e(a)i(unitary)f(equiv)-5 b(alence)28 b(of)h(the)g(represen)n(tations)e FO(\031)32 b FP(on)d FO(H)118 4147 y FP(and)j(~)-46 b FO(\031)31 b FP(on)495 4126 y(~)473 4147 y FO(H)7 b FP(.)p 2514 4147 4 57 v 2518 4095 50 4 v 2518 4147 V 2567 4147 4 57 v eop %%Page: 9 13 9 12 bop 118 100 a FK(1.1.)36 b(In)n(tro)r(duction)27 b(to)h(represen)n(tations)d(of)j FQ(\003)p FK(-algebras)627 b FP(9)118 333 y FR(3.)35 b FP(In)26 b(the)g(general)e(represen)n (tation)f(theory)i(one)g(distinguishes)g(irreducible)118 432 y(and)37 b(indecomp)r(osable)g(represen)n(tations)e(in)j(the)f(set) h(of)f(all)g FC(\014nite-dimen-)118 532 y(sional)42 b FP(represen)n(tations.)76 b(A)41 b(represen)n(tation)f FO(\031)12 b FP(:)32 b FB(A)46 b FQ(7!)f FO(L)p FP(\()p FO(H)7 b FP(\))41 b(is)g(called)118 632 y(irreducible)22 b(if)h(there)f(exists)h(no)f(non-trivial)f(subspace)h(of)h FO(H)29 b FP(in)n(v)-5 b(arian)n(t)21 b(with)118 731 y(resp)r(ect)i(to)g(all)g(op)r(erators)e FO(\031)s FP(\()p FO(x)p FP(\),)26 b FO(x)e FQ(2)f FB(A)p FP(.)35 b(A)24 b(represen)n(tation)e FO(\031)12 b FP(:)28 b FB(A)23 b FQ(7!)g FO(L)p FP(\()p FO(H)7 b FP(\))118 831 y(is)26 b(called)g(indecomp)r(osable)f(if)h(there)g(exists)g(no)g(decomp)r (osition)f FO(H)30 b FP(=)23 b FO(H)2458 843 y FL(1)2510 831 y FP(+)118 930 y FO(H)187 942 y FL(2)260 930 y FP(in)n(to)35 b(a)h(sum)f(of)h(the)g(t)n(w)n(o)f(non-trivial)f(subspaces)h(that)g (are)g(in)n(v)-5 b(arian)n(t)118 1030 y(with)27 b(resp)r(ect)e(to)h (all)g(the)g(op)r(erators)e FO(\031)s FP(\()p FO(x)p FP(\))k FO(x)23 b FQ(2)g FB(A)p FP(,)k(and)f FO(H)1989 1042 y FL(1)2041 1030 y FQ(\\)16 b FO(H)2181 1042 y FL(2)2241 1030 y FP(=)23 b FQ(f)p FP(0)p FQ(g)p FP(.)35 b(It)118 1130 y(is)26 b(clear)f(that)h(an)n(y)f(irreducible)h(represen)n(tation) e(is)i(indecomp)r(osable.)35 b(Th)n(us)118 1229 y(the)40 b(set)g(of)g(irreducible)f(represen)n(tations)f(is)i(a)f(subset)h(of)f (the)i(set)e(of)h(all)118 1329 y(indecomp)r(osable)23 b(represen)n(tations.)33 b(The)24 b(size)f(of)g(this)h(subset)f(in)h (the)f(whole)118 1429 y(set)e(of)g(indecomp)r(osable)g(represen)n (tations)e(dep)r(ends)j(on)e(the)i(structure)f(of)g FB(A)p FP(.)243 1528 y(A)e(description)g(of)g(all)g(indecomp)r(osable)f (\(particularly)g(irreducible\))h(rep-)118 1628 y(resen)n(tations)j(is) h(one)g(of)h(the)f(most)h(imp)r(ortan)n(t)f(problems)f(of)i(represen)n (tation)118 1727 y(theory)-7 b(.)243 1827 y(In)21 b(the)h(case)e(where) h FA(A)g FP(is)g(an)g(algebra)e(with)j(an)f(in)n(v)n(olution,)h(one)f (can)g(con-)118 1927 y(sider)g(b)r(oth)h(irreducible)g FQ(\003)p FP(-represen)n(tations)d(and)i(indecomp)r(osable)g FQ(\003)p FP(-repre-)118 2026 y(sentations.)34 b(Ho)n(w)n(ev)n(er,)20 b(in)h(this)g(case)f(these)g(notions)g(coincide.)35 b(Namely)-7 b(,)21 b(the)118 2126 y(follo)n(wing)27 b(simple)g(prop)r(osition)g (holds.)118 2288 y FR(Prop)s(osition)h(2.)39 b FC(A)27 b FQ(\003)p FC(-r)l(epr)l(esentation)h FO(\031)j FC(is)d(inde)l(c)l (omp)l(osable)i(if)f(and)f(only)118 2388 y(if)j(it)e(is)i(irr)l(e)l (ducible.)118 2550 y(Pr)l(o)l(of.)43 b FP(It)i(is)f(su\016cien)n(t)g (to)g(pro)n(v)n(e)f(that)h(an)n(y)g(indecomp)r(osable)f FQ(\003)p FP(-repre-)118 2649 y(sen)n(tation)c(is)h(irreducible.)73 b(Assume)40 b(the)h(con)n(trary)-7 b(,)41 b(that)f(is,)j(let)d(an)g (in-)118 2749 y(decomp)r(osable)25 b(represen)n(tation)f(b)r(e)j (reducible,)f(i.e.,)h(there)f(exists)f(a)h(prop)r(er)118 2849 y(subspace)h FO(H)534 2861 y FL(1)599 2849 y FP(in)h FO(H)34 b FP(in)n(v)-5 b(arian)n(t)27 b(with)h(resp)r(ect)f(to)h(all)f FO(\031)s FP(\()p FO(x)p FP(\),)i FO(x)23 b FQ(2)h FA(A)p FP(.)118 2998 y FR(Lemma)35 b(1.)43 b FC(The)35 b(subsp)l(ac)l(e)g FO(H)1168 2968 y FN(?)1161 3019 y FL(1)1254 2998 y FP(=)30 b FQ(f)p FO(y)j FQ(2)e FO(H)16 b FP(:)29 b(\()p FO(y)s(;)14 b(f)9 b FP(\))31 b(=)f(0)p FO(;)47 b FQ(8)p FO(f)39 b FQ(2)31 b FO(H)2403 3010 y FL(1)2440 2998 y FQ(g)i FC(is)118 3098 y(non-trivial)e(and)f(invariant)h(with)f(r)l(esp)l(e)l(ct)f(to)h (al)t(l)h FO(\031)s FP(\()p FO(x)p FP(\))p FC(,)g FO(x)24 b FQ(2)f FA(A)p FC(.)118 3263 y(Pr)l(o)l(of.)43 b FP(If)28 b FO(y)e FQ(2)d FO(H)682 3232 y FN(?)675 3283 y FL(1)738 3263 y FP(,)28 b(then)839 3440 y(\()p FO(\031)s FP(\()p FO(x)p FP(\))p FO(y)s(;)14 b(f)9 b FP(\))24 b(=)f(\()p FO(y)s(;)14 b(\031)s FP(\()p FO(x)1549 3406 y FN(\003)1588 3440 y FP(\))p FO(f)9 b FP(\))23 b(=)g(0)118 3618 y(for)k(all)g FO(f)37 b FP(from)27 b(the)h(in)n(v)-5 b(arian)n(t)26 b(subspace)h FO(H)1541 3630 y FL(1)1579 3618 y FP(,)g(i.e.)37 b FO(\031)s FP(\()p FO(x)p FP(\))p FO(y)27 b FQ(2)d FO(H)2156 3588 y FN(?)2149 3639 y FL(1)2212 3618 y FP(.)p 2514 3618 4 57 v 2518 3565 50 4 v 2518 3618 V 2567 3618 4 57 v 243 3783 a(The)j(con)n(tradiction)g(immediately)g(follo)n(ws)g (from)g(the)h(lemma.)p 2514 3783 V 2518 3730 50 4 v 2518 3783 V 2567 3783 4 57 v 118 3948 a FR(4.)52 b FP(Let)33 b FO(H)39 b FP(b)r(e)33 b(a)g(separable)e(\(generally)g(sp)r(eaking,)j (in\014nite-dimensional\))118 4048 y(Hilb)r(ert)c(space.)40 b(F)-7 b(ollo)n(wing)27 b(the)j(general)d(strategy)h(of)h(represen)n (tation)e(the-)118 4147 y(ory)-7 b(,)27 b(w)n(e)h(tak)n(e)f (irreducible)g(represen)n(tations)f(to)h(b)r(e)i(the)f(\\simplest")f (among)p eop %%Page: 10 14 10 13 bop 118 100 a FP(10)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(all)j FQ(\003)p FP(-represen)n(tations.)43 b(A)31 b FQ(\003)p FP(-represen)n(tation)d FO(\031)12 b FP(:)29 b FA(A)f FQ(7!)g FO(L)p FP(\()p FO(H)7 b FP(\))30 b(is)g(called)h(ir-) 118 432 y(reducible,)26 b(if)g(there)f(exists)g(no)h(non-trivial)e (subspace)h(in)h FO(H)32 b FP(in)n(v)-5 b(arian)n(t)24 b(with)118 532 y(resp)r(ect)36 b(to)h(all)f(the)h(op)r(erators)e FO(\031)s FP(\()p FO(x)p FP(\))j(\()p FO(x)g FQ(2)h FA(A)p FP(\).)64 b(The)36 b(follo)n(wing)g(form)g(of)118 632 y(Sc)n(h)n(ur's)27 b(lemma,)g(giv)n(es)g(an)g(equiv)-5 b(alen)n(t)27 b(condition)h(of)f(irreducibilit)n(y)-7 b(.)118 800 y FR(Prop)s(osition)29 b(3.)39 b FC(A)29 b FQ(\003)p FC(-r)l(epr)l(esentation)f FO(\031)s FP(\()p FQ(\001)p FP(\))h FC(is)g(irr)l(e)l(ducible)h(if)g(and)f(only)g(if)118 900 y(any)h(b)l(ounde)l(d)g(op)l(er)l(ator)h FO(C)e FQ(2)24 b FO(L)p FP(\()p FO(H)7 b FP(\))29 b FC(such)h(that)824 1084 y FO(C)6 b(\031)s FP(\()p FO(x)p FP(\))24 b(=)f FO(\031)s FP(\()p FO(x)p FP(\))p FO(C)q(;)186 b FQ(8)p FO(x)22 b FQ(2)i FA(A)p FO(;)118 1269 y FC(is)30 b(a)g(multiple)h(of)f (the)g(identity,)h(i.e.,)h FO(C)d FP(=)23 b FO(cI)7 b FC(,)30 b(with)g FO(c)23 b FQ(2)h FJ(C)14 b FC(.)118 1437 y(Pr)l(o)l(of.)43 b FP(If)30 b FO(A)d FP(=)e FO(A)704 1407 y FN(\003)772 1437 y FP(comm)n(utes)k(with)h FO(\031)s FP(\()p FQ(\001)p FP(\),)h(i.e.)43 b FO(A\031)s FP(\()p FO(x)p FP(\))28 b(=)d FO(\031)s FP(\()p FO(x)p FP(\))p FO(A)p FP(,)32 b FQ(8)p FO(x)26 b FQ(2)g FA(A)118 1537 y FP(then)866 1721 y FO(E)927 1733 y FM(A)981 1721 y FP(\(\001\))14 b FO(\031)s FP(\()p FO(x)p FP(\))25 b(=)e FO(\031)s FP(\()p FO(x)p FP(\))14 b FO(E)1638 1733 y FM(A)1693 1721 y FP(\(\001\))118 1906 y(for)39 b(all)f FO(x)43 b FQ(2)g FA(A)38 b FP(and)h(Borel)f(sets)h(\001)k FQ(\032)e FJ(R)1520 1875 y FL(1)1603 1906 y FP(\(here)e FO(E)1888 1918 y FM(A)1942 1906 y FP(\(\001\))h(is)f(a)f(sp)r(ectral) 118 2005 y(pro)5 b(jector)35 b(of)i(the)h(op)r(erator)d FO(A)p FP(\).)65 b(In)37 b(this)h(case,)g FO(H)1832 2017 y FL(\001)1929 2005 y FP(=)g FO(E)2093 2017 y FM(A)2148 2005 y FP(\(\001\))p FO(H)44 b FP(is)37 b(an)118 2105 y(in)n(v)-5 b(arian)n(t)27 b(subspace)f(in)i FO(H)7 b FP(.)243 2205 y(If)25 b(the)h(represen)n(tation)d FO(\031)29 b FP(is)c(irreducible,)g(then)h(all)f(suc)n(h)g FO(H)2147 2217 y FL(\001)2231 2205 y FP(are)f(either)118 2305 y FQ(f)p FP(0)p FQ(g)36 b FP(or)g FO(H)7 b FP(,)39 b(i.e.,)h(the)e(sp)r (ectral)e(measure)g FO(E)1566 2317 y FM(A)1621 2305 y FP(\()p FQ(\001)p FP(\))h(is)g(concen)n(trated)f(at)h(one)118 2405 y(p)r(oin)n(t)28 b FO(a)23 b FQ(2)g FJ(R)534 2374 y FL(1)577 2405 y FP(,)28 b(and)g FO(A)23 b FP(=)g FO(aI)7 b FP(.)243 2505 y(If)33 b FO(C)39 b FP(=)33 b FO(A)22 b FP(+)g FO(iB)37 b FP(\()p FO(A)d FP(=)e FO(A)1114 2475 y FN(\003)1152 2505 y FP(,)j FO(B)i FP(=)c FO(B)1475 2475 y FN(\003)1546 2505 y FQ(2)g FO(L)p FP(\()p FO(H)7 b FP(\)\))33 b(comm)n(utes)g(with)h(an)118 2605 y(irreducible)22 b(represen)n(tation)f FO(\031)s FP(\()p FQ(\001)p FP(\))j(of)f(the)g FQ(\003)p FP(-algebra)d FA(A)p FP(,)j(then)h(the)f(op)r(erators)118 2704 y FO(A)p FP(,)36 b FO(B)i FP(also)33 b(comm)n(ute)g(with)i FO(\031)s FP(\()p FQ(\001)p FP(\),)h(and,)f(consequen)n(tly)-7 b(,)35 b FO(C)40 b FP(=)33 b FO(aI)c FP(+)23 b FO(ibI)39 b FP(=)118 2804 y(\()p FO(a)19 b FP(+)f FO(ib)p FP(\))p FO(I)7 b FP(;)27 b FO(a)p FP(,)h FO(b)22 b FQ(2)i FJ(R)p FP(.)243 2904 y(Con)n(v)n(ersely)-7 b(,)25 b(if)k(a)e(represen)n (tation)f FO(\031)s FP(\()p FQ(\001)p FP(\))j(is)f(reducible)f(and)h FO(H)2204 2916 y FL(1)2269 2904 y FP(is)g(a)f(sub-)118 3004 y(space)34 b(in)n(v)-5 b(arian)n(t)34 b(with)i(resp)r(ect)f(to)g FO(\031)s FP(\()p FO(x)p FP(\),)j FO(x)e FQ(2)g FA(A)p FP(,)h(then,)h(b)n(y)c(the)i(lemma,)118 3104 y FO(H)194 3074 y FN(?)187 3124 y FL(1)278 3104 y FP(is)27 b(also)g(in)n(v)-5 b(arian)n(t.)35 b(Then)28 b(the)g(op)r(erator)381 3342 y FO(C)h FP(=)557 3225 y Fz(\022)660 3287 y FO(c)696 3299 y FL(1)733 3287 y FO(I)769 3299 y FM(H)823 3307 y Fy(1)1033 3287 y FP(0)739 3387 y(0)162 b FO(c)979 3399 y FL(2)1016 3387 y FO(I)1052 3407 y FM(H)1110 3387 y Fx(?)1106 3425 y Fy(1)1206 3225 y Fz(\023)1281 3342 y FO(;)180 b(c)1520 3354 y FL(1)1581 3342 y FQ(6)p FP(=)22 b FO(c)1704 3354 y FL(2)1741 3342 y FO(;)180 b(c)1980 3354 y FL(1)2017 3342 y FO(;)14 b(c)2090 3354 y FL(2)2151 3342 y FQ(2)23 b FJ(C)15 b FO(;)118 3576 y FP(comm)n(utes)26 b(with)h(the)g(represen)n(tation)e(and)h(is)g(not)h(a)f(m)n(ultiple)h (of)f(the)h(iden-)118 3675 y(tit)n(y)-7 b(.)p 2514 3675 4 57 v 2518 3623 50 4 v 2518 3675 V 2567 3675 4 57 v 118 3848 a FC(R)l(emark)51 b FP(1)p FC(.)f FP(It)42 b(is)f(p)r(ossible) f(to)i(de\014ne)f(the)h(notion)f(of)g(an)g(indecomp)r(os-)118 3948 y(able)26 b(represen)n(tation)e(in)i(the)g(case)f(where)h FO(H)32 b FP(is)26 b(a)g(separable)e(Hilb)r(ert)j(space)118 4048 y(\(dim)15 b FO(H)36 b FP(=)28 b FQ(1)p FP(\))k(and)f(to)g(pro)n (v)n(e)f(an)h(analog)e(of)i(the)h(previous)e(prop)r(ositions.)118 4147 y(Ho)n(w)n(ev)n(er)c(w)n(e)h(are)g(not)g(going)g(to)g(do)g(it)h (here.)p eop %%Page: 11 15 11 14 bop 118 100 a FK(1.1.)36 b(In)n(tro)r(duction)27 b(to)h(represen)n(tations)d(of)j FQ(\003)p FK(-algebras)586 b FP(11)243 333 y(Irreducible)23 b(represen)n(tations)e(and)j(their)f (in)n(tert)n(wining)h(op)r(erators)d(form)118 432 y(a)f(full)g (sub-category)-7 b(,)19 b FQ(\003)p FP(-Irrep)13 b FA(A)p FP(,)21 b(in)f(the)g(category)e FQ(\003)p FP(-Rep)13 b FA(A)p FP(.)34 b(The)20 b(condition)118 532 y(for)33 b(a)g(sub-category)e(to)i(b)r(e)h(full)g(means)f(that)g(the)h(em)n(b)r (edding)g(functor)f FO(F)118 632 y FP(from)25 b FQ(\003)p FP(-Irrep)12 b FA(A)25 b FP(in)n(to)h FQ(\003)p FP(-Rep)13 b FA(A)25 b FP(is)g(an)g(isomorphism)g(on)g(the)h(corresp)r(onding)118 731 y(morphisms.)35 b(In)22 b(what)g(follo)n(ws,)h(w)n(e)f(will)g (mainly)h(deal)f(with)h FQ(\003)p FP(-algebras)c(and)118 831 y(their)31 b FQ(\003)p FP(-represen)n(tations;)f(th)n(us)h(w)n(e)f (will)h(sometimes)g(omit)g(the)g(in)n(v)n(olution)118 930 y(sign)19 b(with)h(the)g(w)n(ords)e(algebra,)h(morphism,)i (category)-7 b(,)19 b(and)h(represen)n(tation,)118 1030 y(if)28 b(no)g(am)n(biguit)n(y)e(can)h(arise.)118 1243 y FR(1.1.2)94 b FO(C)475 1213 y FN(\003)513 1243 y FR(-represen)m (table)32 b FQ(\003)p FR(-algebras)118 1396 y(1.)j FP(An)25 b(imp)r(ortan)n(t)e(class)g(of)h FQ(\003)p FP(-algebras)d(is)j(the)h (class)e(of)h(all)f FQ(\003)p FP(-algebras)f(that)118 1496 y(ha)n(v)n(e)33 b(\\su\016cien)n(tly)h(man)n(y")f FQ(\003)p FP(-represen)n(tations.)54 b(The)35 b(latter)f(means)g(that) 118 1596 y(there)28 b(exists)f(a)g(residual)g(family)h(\(r.f.\))38 b(of)27 b FQ(\003)p FP(-represen)n(tations,)e(i.e.,)j(for)g(an)n(y)118 1695 y FO(x)e FQ(2)g FA(A)p FP(,)j FO(x)d FQ(6)p FP(=)f(0,)k(there)g (exists)f(a)h FQ(\003)p FP(-represen)n(tation)e FO(\031)32 b FP(\(it)e(can)e(b)r(e)i(c)n(hosen)e(to)118 1795 y(b)r(e)e (irreducible\))f(suc)n(h)g(that)g FO(\031)s FP(\()p FO(x)p FP(\))g FQ(6)p FP(=)d(0.)36 b(F)-7 b(or)25 b(an)n(y)f FO(C)1772 1765 y FN(\003)1811 1795 y FP(-algebra)f(there)i(alw)n(a)n (ys)118 1894 y(exists)i(a)h(r.f.)36 b(of)28 b FQ(\003)p FP(-represen)n(tations.)243 1994 y(If)40 b(a)f FQ(\003)p FP(-algebra)e FA(A)j FP(is)f FO(C)1051 1964 y FN(\003)1090 1994 y FP(-represen)n(table,)h(i.e.,)j(there)d(exists)f(a)h FQ(\003)p FP(-iso-)118 2094 y(morphism)34 b(of)g FA(A)g FP(on)g(a)g FQ(\003)p FP(-subalgebra)e(of)i(a)g FO(C)1640 2064 y FN(\003)1678 2094 y FP(-algebra,)g(then)h(it)g(is)f(clear)118 2193 y(that)28 b FA(A)f FP(has)g(a)g(r.f.)118 2320 y FC(R)l(emark)38 b FP(2)p FC(.)i FP(1\).)d(A)27 b FO(C)829 2290 y FN(\003)867 2320 y FP(-algebra)e(con)n(taining)g(a)h(dense)h FQ(\003)p FP(-subalgebra)d(whic)n(h)118 2419 y(is)g FQ(\003)p FP(-isomorphic)e(to)i(a)f(giv)n(en)g(one)h(is)f(not)h(unique)h(in)f (general.)34 b(F)-7 b(or)23 b(example,)118 2519 y(the)31 b FQ(\003)p FP(-algebra)d FJ(C)15 b FP([)p FO(a)34 b FP(=)28 b FO(a)919 2489 y FN(\003)957 2519 y FP(])j(is)f(isomorphic)g (to)g(a)g(dense)h(subalgebra)e(in)i(an)n(y)118 2619 y FO(C)183 2588 y FN(\003)222 2619 y FP(-algebra)24 b FO(C)6 b FP(\()p FO(K)g FP(\))26 b(of)g(con)n(tin)n(uous)f(functions)i(on)f (an)f(in\014nite)i(compact)f(set)118 2718 y FO(K)k FQ(\032)24 b FJ(R)362 2688 y FL(1)406 2718 y FP(.)39 b(Ho)n(w)n(ev)n(er,)27 b FO(C)6 b FP(\()p FO(K)995 2730 y FL(1)1032 2718 y FP(\))29 b(is)g(isomorphic)e(to)h FO(C)6 b FP(\()p FO(K)1869 2730 y FL(2)1907 2718 y FP(\))29 b(if)g(and)f(only)g(if)h FO(K)2538 2730 y FL(1)118 2818 y FP(and)f FO(K)351 2830 y FL(2)415 2818 y FP(are)f(homeomorphic.)243 2917 y(2\).)61 b(Not)36 b(an)n(y)f FQ(\003)p FP(-algebra)e(is)j FO(C)1263 2887 y FN(\003)1301 2917 y FP(-represen)n(table.)60 b(Moreo)n(v)n(er,) 35 b(not)h(an)n(y)118 3017 y FQ(\003)p FP(-algebra)30 b(p)r(ossesses)i(a)h(r.f.)53 b(of)33 b FQ(\003)p FP(-represen)n (tations.)50 b(F)-7 b(or)32 b(example,)i(if)f(the)118 3117 y(in)n(v)n(olution)j(in)h FB(A)g FP(is)g(not)g(prop)r(er)f(\(an)h (in)n(v)n(olution)f(is)h(prop)r(er)f(if)h FO(xx)2352 3087 y FN(\003)2430 3117 y FP(=)h(0)118 3216 y(implies)c FO(x)f FP(=)g(0\),)i(then)f(suc)n(h)f(a)g FQ(\003)p FP(-algebra)e (cannot)i(b)r(e)h FO(C)2007 3186 y FN(\003)2046 3216 y FP(-represen)n(table;)118 3316 y(moreo)n(v)n(er)25 b(it)j(do)r(es)f(not)h(p)r(ossess)e(r.f.)37 b(of)28 b FQ(\003)p FP(-represen)n(tations.)243 3442 y(Nev)n(ertheless,)33 b(if)h FA(A)e FP(=)g FJ(C)14 b FP([)q FO(G)p FP(])38 b(=)32 b FJ(C)15 b FQ(h)p FO(g)41 b FQ(j)32 b FO(g)j FQ(2)e FO(G)p FQ(i)h FP(is)f(a)f(group)g FQ(\003)p FP(-algebra)118 3542 y(of)c(a)g(coun)n(table)g(discrete)g(group)f FO(G)i FP(with)f(a)g(natural)g(in)n(v)n(olution)f(\(giv)n(en)h(on)118 3642 y(basis)35 b(v)n(ectors)f FO(g)39 b FQ(2)e FO(G)f FP(b)n(y)g FO(g)1059 3612 y FN(\003)1133 3642 y FP(=)g FO(g)1277 3612 y FN(\000)p FL(1)1366 3642 y FP(\),)i(then)e(the)g (follo)n(wing)f(prop)r(osition)118 3741 y(holds.)118 3894 y FR(Prop)s(osition)30 b(4.)41 b FC(The)31 b(involutive)f(algebr)l (a)i FJ(C)15 b FP([)p FO(G)p FP(])36 b FC(is)30 b FO(C)1934 3864 y FN(\003)1972 3894 y FC(-r)l(epr)l(esentable.)118 4048 y(Pr)l(o)l(of.)43 b FP(Indeed,)23 b(an)n(y)e(op)r(erator)f FO(\031)1193 4060 y FM(r)1230 4048 y FP(\()p FO(x)p FP(\))j(of)e(the)h (righ)n(t)f(regular)f(represen)n(tation)118 4147 y FO(\031)165 4159 y FM(r)239 4147 y FP(on)37 b FO(L)421 4159 y FL(2)457 4147 y FP(\()p FO(G)p FP(\))h(is)f(non-zero)e(for)h(an)n(y)g(non-zero)g (elemen)n(t)h FO(x)i FQ(2)f FJ(C)15 b FP([)p FO(G)q FP(],)45 b(and)p eop %%Page: 12 16 12 15 bop 118 100 a FP(12)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(hence)f(the)g FQ(\003)p FP(-algebra)e FJ(C)15 b FP([)p FO(G)p FP(])32 b(is)26 b(isomorphic)f(to)h FO(\031)1692 345 y FM(r)1729 333 y FP(\()p FJ(C)15 b FP([)p FO(G)p FP(]\))33 b(whic)n(h)26 b(is)f(a)h(dense)118 432 y FQ(\003)p FP(-subalgebra)34 b(of)j(the)h FO(C)934 402 y FN(\003)972 432 y FP(-algebra)d FO(C)1365 402 y FN(\003)1359 453 y FM(r)1403 432 y FP(\()p FO(G)p FP(\))j(generated)e(b)n(y)h(the)g(op)r (erators)118 532 y FO(\031)s FP(\()p FO(x)p FP(\),)29 b FO(x)24 b FQ(2)f FJ(C)15 b FP([)p FO(G)p FP(].)p 2514 532 4 57 v 2518 479 50 4 v 2518 532 V 2567 532 4 57 v 118 697 a FR(2.)36 b FP(The)28 b(follo)n(wing)e(prop)r(osition)h (holds.)118 860 y FR(Prop)s(osition)j(5.)41 b FC(Consider)31 b(the)f(fol)t(lowing)i(pr)l(op)l(erties)f(of)g(a)f FQ(\003)p FC(-algebr)l(a)6 b FP(:)191 1038 y(\()p FO(i)p FP(\))42 b FC(the)30 b(algebr)l(a)h(is)f FO(C)899 1008 y FN(\003)937 1038 y FC(-r)l(epr)l(esentable)6 b FP(;)162 1203 y(\()p FO(ii)p FP(\))42 b FC(ther)l(e)30 b(exists)f(a)h(r)l(esidual)h(family)g (of)g(its)e(r)l(epr)l(esentations)7 b FP(;)134 1367 y(\()p FO(iii)p FP(\))41 b FC(the)i(involution)h(on)f(the)g(algebr)l(a)i(is)e (c)l(ompletely)i(pr)l(op)l(er)f FP(\()p FC(i.e.,)49 b(the)326 1467 y(e)l(quality)625 1405 y Fz(P)713 1425 y FM(n)713 1492 y(k)q FL(=1)852 1467 y FO(x)899 1479 y FM(k)940 1467 y FO(x)987 1437 y FN(\003)987 1490 y FM(k)1051 1467 y FP(=)23 b(0)i FC(implies)j FO(x)1534 1479 y FM(k)1598 1467 y FP(=)23 b(0)p FC(,)k FO(k)e FP(=)e(1)p FC(,)j FO(:)14 b(:)g(:)28 b FC(,)f FO(n)p FC(;)g FO(n)c FQ(2)g FJ(N)t FP(\))q(;)148 1631 y(\()p FO(iv)s FP(\))42 b FC(the)36 b(involution)h(in)f(the)g(algebr)l(a)i(is)e(pr)l(op)l(er)i FP(\()p FC(i.e.,)i FO(xx)2070 1601 y FN(\003)2143 1631 y FP(=)35 b(0)g FC(implies)326 1731 y FO(x)23 b FP(=)g(0\))p FC(.)243 1909 y(Then)30 b FP(\()p FO(i)p FP(\))23 b FQ(\))g FP(\()p FO(ii)p FP(\))g FQ(\))g FP(\()p FO(iii)p FP(\))g FQ(\))g FP(\()p FO(iv)s FP(\))p FC(.)243 2009 y(Neither)30 b(of)h(the)e(inverse)i(implic)l(ations)g(holds.)118 2172 y(Pr)l(o)l(of.)43 b FP(The)35 b(direct)f(implications)g(are)g(easily)f (v)n(eri\014ed.)57 b(T)-7 b(o)34 b(see)g(that)h(\()p FO(ii)p FP(\))118 2271 y(do)r(es)24 b(not)g(imply)h(\()p FO(i)p FP(\),)g(one)f(has)g(to)g(consider)f(the)h FQ(\003)p FP(-algebra)e FO(C)6 b FP(\([0)p FO(;)14 b FQ(1)p FP(\)\))25 b(of)f(all)118 2371 y(con)n(tin)n(uous)k(functions)h FO(f)9 b FP(\()p FQ(\001)p FP(\))30 b(on)e([0)p FO(;)14 b FQ(1)p FP(\))29 b(with)g(the)h(natural)e(in)n(v)n(olution)g(and)118 2471 y(p)r(oin)n(t)n(wise)f(m)n(ultiplication.)38 b(It)28 b(is)g(ob)n(vious)e(that)i(this)g(algebra)e(has)i(a)f(r.f.)h(of)118 2570 y(one-dimensional)c(represen)n(tations,)f(ho)n(w)n(ev)n(er)g(it)i (is)g(not)f FO(C)2007 2540 y FN(\003)2046 2570 y FP(-represen)n(table.) 118 2670 y(Indeed,)34 b(assume)e(that)h FO(C)6 b FP(\([0)p FO(;)14 b FQ(1)p FP(\)\))34 b(is)f(em)n(b)r(edded)g(as)f(a)g FQ(\003)p FP(-subalgebra)e(in)j(a)118 2769 y FO(C)183 2739 y FN(\003)222 2769 y FP(-algebra)f FA(A)p FP(,)j(then)g(there)f (exists)g(an)g(in)n(teger)g FO(N)43 b FP(suc)n(h)34 b(that)g FQ(k)p FO(f)9 b FQ(k)33 b FO(<)h(N)9 b FP(,)118 2869 y(where)466 3102 y FO(f)g FP(\()p FO(x)p FP(\))23 b(=)738 2960 y Fz(\()805 3045 y FO(x)c FQ(\000)f FO(n;)83 b FP(if)28 b(2)p FO(n)22 b(<)h(x)g(<)g FP(2)p FO(n)18 b FP(+)g(1)o FO(;)805 3165 y(n;)232 b FP(if)28 b(2)p FO(n)17 b FQ(\000)h FP(1)23 b FQ(\024)g FO(x)g FQ(\024)g FP(2)p FO(n)o(;)1993 3102 y(n)g FQ(2)g FJ(N)t FO(:)243 3350 y FP(This)j(implies)h(that)g (the)g(elemen)n(t)g FO(N)9 b(I)23 b FQ(\000)16 b FO(f)36 b FP(is)26 b(in)n(v)n(ertible)g(in)h FA(A)p FP(,)g(but)g(it)g(is)118 3450 y(a)g(zero)g(divisor)f(in)i FO(C)6 b FP(\([0)p FO(;)14 b FQ(1)p FP(\)\),)29 b(whic)n(h)e(giv)n(es)f(a)i(con)n(tradiction.)243 3550 y(In)33 b(order)f(to)h(sho)n(w)g(that)g(\()p FO(iii)p FP(\))g(do)r(es)g(not)g(imply)h(\()p FO(ii)p FP(\),)h(consider)d(the)i FQ(\003)p FP(-)118 3649 y(algebra)k(with)j(unit)f FO(e)p FP(,)j FA(A)g FP(=)h FJ(C)15 b FQ(h)p FO(a;)f(x)50 b FQ(j)43 b FO(a)1501 3619 y FN(\003)1540 3649 y FO(a)g FP(=)g FO(q)s(aa)1863 3619 y FN(\003)1901 3649 y FO(;)28 b(xx)2046 3619 y FN(\003)2112 3649 y FP(+)e FO(aa)2291 3619 y FN(\003)2373 3649 y FP(=)43 b FO(e)p FQ(i)p FP(,)118 3749 y(where)f(0)k FO(<)h(q)j(<)d FP(1.)80 b(The)42 b(set)g(of)g(w)n (ords)f(whic)n(h)h(do)g(not)g(con)n(tain)f(the)118 3848 y(sub-w)n(ords)32 b FO(a)559 3818 y FN(\003)597 3848 y FO(a)h FP(and)g FO(xx)935 3818 y FN(\003)1007 3848 y FP(forms)g(a)f(linear)h(basis)f(for)h(this)g(algebra.)52 b(T)-7 b(ak)n(e)118 3948 y(an)35 b(arbitrary)e FO(z)40 b FP(=)781 3886 y Fz(P)882 3948 y FO(\013)935 3960 y FM(i)963 3948 y FO(u)1011 3960 y FM(i)1052 3948 y FO(x)1099 3918 y FM(k)1134 3926 y Fw(i)1201 3948 y FQ(2)d FA(A)p FP(,)f(where)f(the)h(w)n(ords)e FO(u)2105 3960 y FM(i)2167 3948 y FP(do)h(not)h(end)118 4048 y(with)41 b FO(x)p FP(,)k(and)c FO(k)653 4060 y FM(i)725 4048 y FQ(\025)k FP(0.)76 b(Denote)41 b(b)n(y)f FO(F)12 b FP(\()p FO(z)t FP(\))41 b(the)g(sum)g(of)g(co)r(e\016cien)n(ts)f(at)118 4147 y(those)27 b(w)n(ords)f FO(u)621 4159 y FM(i)649 4147 y FO(x)696 4117 y FM(k)731 4125 y Fw(i)789 4147 y FP(of)i(minimal)g(length)f(that)h(can)f(also)g(b)r(e)h(written)g(in)f (the)p eop %%Page: 13 17 13 16 bop 118 100 a FK(1.1.)36 b(In)n(tro)r(duction)27 b(to)h(represen)n(tations)d(of)j FQ(\003)p FK(-algebras)586 b FP(13)118 333 y(form)35 b FO(w)r(w)444 303 y FN(\003)484 333 y FP(.)60 b(De\014ne)36 b FO(J)45 b FP(=)35 b FQ(f)p FO(j)14 b FP(:)30 b FO(l)r FP(\()p FO(u)1273 345 y FM(j)1308 333 y FP(\))36 b FQ(\024)g FO(l)r FP(\()p FO(u)1584 345 y FM(i)1611 333 y FP(\))p FO(;)28 b FQ(8)p FO(i)p FQ(g)p FP(.)58 b(W)-7 b(e)36 b(will)g(sho)n(w)f(that)118 432 y FO(F)12 b FP(\()p FO(z)t(z)301 402 y FN(\003)338 432 y FP(\))28 b(=)491 370 y Fz(P)579 457 y FM(j)s FN(2)p FM(J)715 432 y FQ(j)p FO(\013)791 444 y FM(j)826 432 y FQ(j)849 402 y FL(2)886 432 y FP(.)46 b(Indeed,)32 b(the)f(canonical)e(form)i(of)f FO(u)2110 444 y FM(i)2137 432 y FO(x)2184 402 y FM(k)2219 410 y Fw(i)2251 432 y FO(x)2298 402 y FN(\003)p FM(k)2367 410 y Fw(j)2403 432 y FO(u)2451 402 y FN(\003)2451 454 y FM(j)2519 432 y FP(is)118 547 y FQ(\000)p FO(u)231 559 y FM(i)258 480 y Fz(\000)296 485 y(P)384 572 y FL(1)p FN(\024)p FM(s)p FN(\024)p FL(min)o(\()p FM(k)723 580 y Fw(i)749 572 y FM(;k)804 580 y Fw(j)835 572 y FL(\))879 547 y FO(x)926 517 y FM(k)961 525 y Fw(i)988 517 y FN(\000)p FM(s)1076 547 y FP(\()p FO(x)1155 517 y FN(\003)1194 547 y FP(\))1226 517 y FM(k)1261 525 y Fw(j)1292 517 y FN(\000)p FM(s)1380 480 y Fz(\001)1431 547 y FO(u)1479 517 y FN(\003)1479 569 y FM(j)1531 547 y FP(+)14 b(CF)o(\()p FO(u)1803 559 y FM(i)1831 547 y FO(u)1879 517 y FN(\003)1879 569 y FM(j)1916 547 y FP(\),)27 b(where)d(CF\()p FO(u)2429 559 y FM(i)2457 547 y FO(u)2505 517 y FN(\003)2505 569 y FM(i)2543 547 y FP(\))118 647 y(denotes)h(the)g(canonical)e(form)i (of)f FO(u)1250 659 y FM(i)1278 647 y FO(u)1326 617 y FN(\003)1326 669 y FM(i)1363 647 y FP(.)36 b(Since)25 b FO(u)1684 659 y FM(i)1736 647 y FP(and)g FO(u)1943 659 y FM(j)2002 647 y FP(do)g(not)f(end)h(with)118 747 y FO(x)p FP(,)33 b(CF\()p FO(u)415 759 y FM(i)442 747 y FO(u)490 716 y FN(\003)490 768 y FM(j)528 747 y FP(\))f(is)f(a)g(w)n (ord)f(of)h(length)g FQ(j)p FO(u)1387 759 y FM(i)1414 747 y FQ(j)21 b FP(+)g FQ(j)p FO(u)1615 759 y FM(j)1649 747 y FQ(j)p FP(.)48 b(So)31 b(if)h(the)g(unique)f(short-)118 846 y(est)h(w)n(ord,)f(CF\()p FO(u)679 858 y FM(i)707 846 y FO(u)755 816 y FN(\003)755 868 y FM(j)792 846 y FP(\),)i(in)f(CF\()p FO(z)t(z)1213 816 y FN(\003)1250 846 y FP(\))g(has)f(minimal)h(length)g(in)g FO(z)t(z)2238 816 y FN(\003)2274 846 y FP(,)h(then)f FO(i)p FP(,)118 946 y FO(j)40 b FQ(2)c FO(J)43 b FP(\(hence)35 b FQ(j)p FO(u)713 958 y FM(i)741 946 y FQ(j)g FP(=)g FQ(j)p FO(u)970 958 y FM(j)1005 946 y FQ(j)p FP(\).)59 b(Let)35 b(us)g(sho)n(w)f(that)h (if)h FO(u)1944 958 y FM(i)1971 946 y FO(u)2019 916 y FN(\003)2019 967 y FM(j)2092 946 y FP(=)f FO(w)r(w)2314 916 y FN(\003)2353 946 y FP(,)i(then)118 1045 y FO(u)166 1057 y FM(i)225 1045 y FP(=)31 b FO(u)369 1057 y FM(j)404 1045 y FP(.)52 b(Let)33 b FO(u)681 1057 y FM(i)741 1045 y FP(b)r(e)g(a)f(w)n(ord)g(in)h FO(a)p FP(,)h FO(a)1391 1015 y FN(\003)1429 1045 y FP(.)53 b(If)33 b FO(u)1641 1057 y FM(i)1701 1045 y FP(ends)f(with)i FO(a)p FP(,)g(or)e FO(u)2346 1057 y FM(j)2413 1045 y FP(ends)118 1145 y(with)k FO(a)359 1115 y FN(\003)397 1145 y FP(,)i(then)f(CF\()p FO(u)850 1157 y FM(i)877 1145 y FO(u)925 1115 y FN(\003)925 1167 y FM(j)963 1145 y FP(\))f(is)g FO(u)1171 1157 y FM(i)1198 1145 y FO(u)1246 1115 y FN(\003)1246 1167 y FM(j)1319 1145 y FP(\(as)g(in)g(the)g(free)f FQ(\003)p FP(-algebra\),)h(and)f(w)n(e)118 1245 y(conclude)40 b(from)f FO(u)728 1257 y FM(i)755 1245 y FO(u)803 1215 y FN(\003)803 1266 y FM(j)884 1245 y FP(=)k FO(w)r(w)1114 1215 y FN(\003)1193 1245 y FP(that)d FO(u)1433 1257 y FM(i)1504 1245 y FP(=)i FO(u)1659 1257 y FM(j)1694 1245 y FP(.)73 b(In)40 b(the)g(opp)r(osite)g (case,)118 1356 y(write)f FO(u)390 1368 y FM(i)460 1356 y FP(=)j FO(v)607 1368 y FM(i)634 1356 y FO(a)678 1326 y FN(\003)p FM(k)792 1356 y FP(and)d FO(u)1013 1368 y FM(j)1090 1356 y FP(=)j FO(v)1237 1368 y FM(j)1272 1356 y FO(a)1316 1326 y FM(m)1379 1356 y FP(,)h(where)38 b FO(v)1736 1368 y FM(i)1803 1356 y FP(ends)i(with)f FO(a)2249 1326 y FN(\003)2327 1356 y FP(and)g FO(v)2540 1368 y FM(j)118 1456 y FP(ends)f(with)g FO(a)p FP(.)68 b(Then)38 b(CF\()p FO(u)1073 1468 y FM(i)1101 1456 y FO(u)1149 1426 y FN(\003)1149 1478 y FM(j)1187 1456 y FP(\))i(=)g FO(q)1404 1426 y FM(k)q(m)1504 1456 y FO(v)1544 1468 y FM(i)1572 1456 y FO(a)1616 1426 y FM(m)1679 1456 y FO(a)1723 1426 y FN(\003)p FM(k)1797 1456 y FO(v)1840 1426 y FN(\003)1837 1478 y FM(j)1879 1456 y FP(.)68 b(If)38 b FO(m)i(>)g(k)s FP(,)g(then,)118 1568 y(since)d FO(u)379 1580 y FM(i)406 1568 y FO(u)454 1538 y FN(\003)454 1589 y FM(j)532 1568 y FP(=)i FO(w)r(w)758 1538 y FN(\003)797 1568 y FP(,)h(w)n(e)d(ha)n(v)n(e)f(that)i FO(v)1423 1580 y FM(i)1451 1568 y FO(a)1495 1538 y FM(m)1554 1546 y Fy(1)1630 1568 y FP(=)h FO(w)r FP(,)h FO(a)1902 1538 y FM(m)p FN(\000)p FM(m)2072 1546 y Fy(1)2108 1568 y FO(a)2152 1538 y FN(\003)p FM(k)2227 1568 y FO(v)2270 1538 y FN(\003)2267 1589 y FM(j)2348 1568 y FP(=)f FO(w)2513 1538 y FN(\003)2552 1568 y FP(,)118 1667 y(whic)n(h)29 b(is)g(imp)r(ossible,)g(since)g FO(w)i FP(ends)e(with)h FO(a)e FP(and)h FO(a)1827 1637 y FN(\003)1894 1667 y FP(sim)n(ultaneously)-7 b(.)40 b(So)118 1767 y FO(m)34 b FP(=)f FO(k)k FP(and)d FO(w)j FP(=)c FO(v)805 1779 y FM(i)833 1767 y FO(a)877 1737 y FM(k)951 1767 y FP(=)h FO(v)1090 1779 y FM(j)1125 1767 y FO(a)1169 1737 y FM(k)1210 1767 y FP(.)56 b(Hence,)36 b FO(v)1607 1779 y FM(i)1669 1767 y FP(=)d FO(v)1807 1779 y FM(j)1876 1767 y FP(and)h FO(u)2092 1779 y FM(i)2153 1767 y FP(=)g FO(u)2300 1779 y FM(j)2334 1767 y FP(.)56 b(No)n(w)118 1867 y(let)32 b FO(u)290 1879 y FM(k)359 1867 y FP(=)d FO(u)501 1879 y FM(k)q(;)p FL(1)594 1867 y FO(x)641 1837 y FL(#)700 1867 y FO(u)748 1879 y FM(k)q(;)p FL(2)842 1867 y FP(,)j FO(k)g FP(=)d FO(i)p FP(,)j FO(j)5 b FP(,)32 b(where)f FO(u)1536 1879 y FM(k)q(;)p FL(1)1660 1867 y FP(do)r(es)g(not)g(con)n (tain)g FO(x)2346 1837 y FL(#)2436 1867 y FP(\()p FO(x)2515 1837 y FL(#)118 1966 y FP(stands)i(for)g(either)h FO(x)g FP(or)f FO(x)994 1936 y FN(\003)1033 1966 y FP(\).)55 b(Then)34 b(it)g(follo)n(ws)e(from)i FO(u)1984 1978 y FM(i)2011 1966 y FO(u)2059 1936 y FN(\003)2059 1988 y FM(j)2130 1966 y FP(=)e FO(w)r(w)2349 1936 y FN(\003)2423 1966 y FP(that)118 2066 y FO(u)166 2078 y FL(1)p FM(;)p FL(1)284 2066 y FP(=)c FO(u)425 2078 y FL(2)p FM(;)p FL(1)546 2066 y FP(and)i FO(u)758 2078 y FL(1)p FM(;)p FL(2)848 2066 y FO(u)896 2036 y FN(\003)896 2086 y FL(2)p FM(;)p FL(2)1014 2066 y FP(=)e FO(w)1166 2078 y FL(1)1204 2066 y FO(w)1265 2036 y FN(\003)1263 2086 y FL(1)1304 2066 y FP(.)46 b(By)31 b(induction)g(on)g FO(l)r FP(\()p FO(u)2105 2078 y FM(i)2132 2066 y FP(\),)h(w)n(e)e(obtain)118 2166 y(the)i(desired)g(result.)49 b(This)32 b(pro)n(v)n(es)e(that,)k (if)e FO(u)1620 2178 y FM(i)1647 2166 y FO(u)1695 2135 y FN(\003)1695 2187 y FM(j)1763 2166 y FP(=)e FO(w)r(w)1980 2135 y FN(\003)2019 2166 y FP(,)k(then)e FO(u)2317 2178 y FM(i)2374 2166 y FP(=)e FO(u)2517 2178 y FM(j)2552 2166 y FP(.)118 2274 y(F)-7 b(rom)29 b(this)g(it)h(follo)n(ws)e(that)i FO(F)12 b FP(\()p FO(z)t(z)1225 2244 y FN(\003)1262 2274 y FP(\))26 b(=)1410 2212 y Fz(P)1498 2299 y FM(j)s FN(2)p FM(J)1633 2274 y FQ(j)p FO(\013)1709 2286 y FM(j)1744 2274 y FQ(j)1767 2244 y FL(2)1805 2274 y FP(.)42 b(Using)29 b(the)g(existence)118 2374 y(of)j FO(F)12 b FP(\()p FQ(\001)p FP(\),)35 b(it)d(is)g(easy)g(to)g(sho)n(w)f(that)i(the)g FQ(\003)p FP(-algebra)c(is)k(completely)f(prop)r(er.)118 2474 y(But)27 b(in)f(ev)n(ery)f(represen)n(tation,)g(w)n(e)h(ha)n(v)n (e)f(that)i FQ(k)p FO(aa)1780 2443 y FN(\003)1817 2474 y FQ(k)c FP(=)f FQ(k)p FO(a)2055 2443 y FN(\003)2093 2474 y FO(a)p FQ(k)g FP(=)h FO(q)17 b FQ(k)p FO(aa)2473 2443 y FN(\003)2510 2474 y FQ(k)p FP(,)118 2573 y(hence)28 b FQ(k)p FO(aa)479 2543 y FN(\003)516 2573 y FQ(k)23 b FP(=)f(0,)28 b(and)f(so)g FO(A)h FP(is)f(not)h FO(C)1410 2543 y FN(\003)1448 2573 y FP(-represen)n(table.)243 2678 y(T)-7 b(o)25 b(see)h(that)g(\()p FO(iv)s FP(\))h(do)r(es)e(not)h (imply)h(\()p FO(iii)p FP(\),)e(w)n(e)h(refer)f(the)i(reader)d(to)i ([299)o(])118 2778 y(\(see)i(also)e([72)o(])i(and)f(the)h(references)f (therein\).)p 2514 2778 4 57 v 2518 2725 50 4 v 2518 2778 V 2567 2778 4 57 v 243 2986 a FR(3.)33 b FP(It)20 b(is)f(natural)f(to)i(consider,)g(among)e(all)h FQ(\003)p FP(-algebras)d(whic)n(h)k(ha)n(v)n(e)e(a)h(r.f.,)118 3086 y(those)30 b(whic)n(h)h(p)r(osses)f(a)g(residual)g(family)h(of)f (\014nite-dimensional)g(represen-)118 3186 y(tations;)d(w)n(e)h(call)f (these)g(algebras)f(residually)h(\014nite)h(dimensional)f(\(r.f.d.\).) 118 3368 y FR(Prop)s(osition)33 b(6.)42 b FC(If)32 b FO(G)g FC(is)h(a)f(r)l(esidual)t(ly)h(\014nite)f(gr)l(oup)g FP(\()p FC(i.e.)46 b FQ(8)p FO(g)29 b FQ(6)p FP(=)d FO(e)32 b FC(ther)l(e)118 3467 y(exists)f(a)g(normal)h(sub)l(gr)l(oup)e FO(G)1113 3479 y FM(g)1178 3467 y FQ(63)25 b FO(g)34 b FC(such)d(that)g FO(G=G)1865 3479 y FM(g)1934 3467 y FC(is)h(a)f(\014nite)f(gr)l(oup)5 b FP(\))p FC(,)118 3567 y(then)30 b FJ(C)15 b FP([)p FO(G)p FP(])36 b FC(is)30 b(r)l(esidual)t(ly)h(\014nite)f(dimensional.)118 3749 y(Pr)l(o)l(of.)43 b FP(Let)19 b(us)g(\014rst)f(recall)g(an)g(equiv)-5 b(alen)n(t)18 b(de\014nition)h(of)g(the)g(residual)f(\014nite-)118 3848 y(ness)38 b(of)g(a)f(group)g FO(G)p FP(:)58 b(for)38 b(an)n(y)f(\014nite)h(set)g FQ(f)p FO(g)1630 3860 y FL(1)1667 3848 y FO(;)14 b(:)g(:)g(:)27 b(;)14 b(g)1905 3860 y FM(n)1950 3848 y FQ(g)37 b FP(of)h(non-iden)n(tit)n(y)118 3948 y(elemen)n(ts)27 b(of)h FO(G)f FP(there)g(exists)g(a)g(normal)g (subgroup)f(that)h(do)r(es)g(not)h(con)n(tain)118 4048 y(an)n(y)e(of)g(these)g(elemen)n(ts)g(and)g(suc)n(h)g(that)g(the)h (quotien)n(t)f(group)f(of)h FO(G)h FP(b)n(y)f(this)118 4147 y(subgroup)h(is)g(\014nite.)p eop %%Page: 14 18 14 17 bop 118 100 a FP(14)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)243 333 y FP(Let)33 b FO(\013)g FP(=)580 270 y Fz(P)668 358 y FM(k)723 333 y FO(c)759 345 y FM(k)799 333 y FO(g)839 345 y FM(k)880 333 y FP(,)i FO(c)974 345 y FM(k)1047 333 y FQ(6)p FP(=)d(0,)i FO(k)i FP(=)c(1,)h FO(:)14 b(:)g(:)28 b FP(,)34 b FO(n)p FP(,)h(b)r(e)f(an)f(elemen)n(t)g(of)g(the)118 432 y(group)20 b(algebra.)33 b(Cho)r(osing)20 b(a)h(normal)g(subgroup)f FO(N)30 b FP(that)21 b(do)r(es)g(not)h(con)n(tain)118 532 y(the)d(elemen)n(ts)f FO(g)622 544 y FM(i)649 532 y FO(g)692 496 y FN(\000)p FL(1)689 555 y FM(j)781 532 y FP(,)i FO(i)j FQ(6)p FP(=)g FO(j)5 b FP(;)21 b FO(i)p FP(,)f FO(j)28 b FP(=)23 b(1,)18 b FO(:)c(:)g(:)27 b FP(,)21 b FO(n)p FP(,)f(w)n(e)e(conclude)g(that)h(the)g(image)118 632 y(of)27 b FO(\013)d FQ(2)f FJ(C)15 b FP([)p FO(G)p FP(])34 b(under)27 b(the)g(homomorphism)g FJ(C)14 b FP([)q FO(G)p FP(])29 b FQ(7!)23 b FJ(C)15 b FP([)p FO(G=)-5 b(N)9 b FP(])33 b(of)27 b FJ(C)15 b FP([)q FO(G)p FP(])33 b(in)n(to)118 731 y(the)25 b(group)d(algebra)h(of)h(\014nite)g(group)f (is)h(non-zero)e(and)i(hence)g(it)h(is)f(non-zero)118 831 y(in)k(the)g(\014nite-dimensional)f(regular)f(represen)n(tation)g (of)h FJ(C)15 b FP([)p FO(G)q(=)-5 b(N)9 b FP(].)p 2514 831 4 57 v 2518 778 50 4 v 2518 831 V 2567 831 4 57 v 118 994 a FR(4.)56 b FP(The)34 b(follo)n(wing)f(list)h(of)g FO(C)1094 964 y FN(\003)1132 994 y FP(-algebras)e(is)i(naturally)f (connected)h(with)g(a)118 1094 y(group)27 b FO(G)p FP(:)220 1249 y(1.)41 b FO(C)391 1219 y FN(\003)385 1272 y FM(f)429 1249 y FP(\()p FO(G)p FP(\).)c(This)24 b(algebra)e(is)i(the)h (completion)f(of)g FJ(C)15 b FP([)p FO(G)q FP(])30 b(in)24 b(the)h(follo)n(wing)326 1348 y(norm:)571 1517 y FQ(k)p FO(\013)p FQ(k)708 1532 y FM(C)760 1512 y Fx(\003)756 1552 y Fw(f)794 1532 y FL(\()p FM(G)p FL(\))924 1517 y FP(=)211 b(sup)1012 1591 y FM(\031)r FN(2\003)p FL(-f)5 b FM(:)p FL(d)p FM(:)p FL(Rep)k Fv(C)t FL([)p FM(G)p FL(])1526 1517 y FQ(k)p FO(\031)s FP(\()p FO(\013)p FP(\))p FQ(k)p FO(;)180 b(\013)24 b FQ(2)f FJ(C)15 b FP([)p FO(G)q FP(])p FO(;)326 1749 y FP(where)22 b(sup)h(is)g(tak)n(en)f(o)n(v)n(er)f (all)i(\014nite-dimensional)g FQ(\003)p FP(-represen)n(tations)326 1849 y(of)k FJ(C)15 b FP([)p FO(G)q FP(].)220 2009 y(2.)41 b FO(C)391 1979 y FN(\003)385 2030 y FM(r)429 2009 y FP(\()p FO(G)p FP(\).)67 b(This)38 b(algebra)d(is)j(the)g(completion)f (of)g FJ(C)15 b FP([)p FO(G)q FP(])43 b(in)38 b(the)g(righ)n(t)326 2109 y(regular)25 b(norm)814 2278 y FQ(k)p FO(\013)p FQ(k)951 2293 y FM(C)999 2301 y Fw(r)1031 2293 y FL(\()p FM(G)p FL(\))1162 2278 y FP(=)e FQ(k)p FO(\031)1339 2290 y FM(r)1375 2278 y FP(\()p FO(\013)p FP(\))p FQ(k)p FO(;)181 b(\013)23 b FQ(2)g FJ(C)15 b FP([)p FO(G)q FP(])p FO(;)326 2446 y FP(where)25 b FO(\031)611 2458 y FM(r)648 2446 y FP(\()p FO(\013)p FP(\))i(is)e(the)h(op)r(erator)e(of)i(the)g(righ)n (t)f(regular)f(represen)n(tation)326 2546 y(of)j FO(G)h FP(on)f FO(L)685 2558 y FL(2)722 2546 y FP(\()p FO(G)p FP(\).)220 2707 y(3.)41 b FO(C)391 2676 y FN(\003)429 2707 y FP(\()p FO(G)p FP(\).)c(This)24 b(algebra)e(is)i(the)h (completion)f(of)g FJ(C)15 b FP([)p FO(G)q FP(])30 b(in)24 b(the)h(follo)n(wing)326 2806 y(norm:)621 2975 y FQ(k)p FO(\013)p FQ(k)758 2990 y FM(C)810 2974 y Fx(\003)844 2990 y FL(\()p FM(G)p FL(\))975 2975 y FP(=)160 b(sup)1063 3049 y FM(\031)r FN(2\003)p FL(-Rep)10 b Fv(C)t FL([)p FM(G)p FL(])1476 2975 y FQ(k)p FO(\031)s FP(\()p FO(\013)p FP(\))p FQ(k)p FO(;)180 b(\013)23 b FQ(2)h FJ(C)15 b FP([)p FO(G)p FP(])p FO(:)118 3242 y FP(The)35 b FQ(\003)p FP(-algebra)d FJ(C)15 b FP([)p FO(G)p FP(])41 b(is)35 b FQ(\003)p FP(-b)r(ounded,)h(i.e.,)h FQ(k)p FO(\031)s FP(\()p FO(\013)p FP(\))p FQ(k)f(\024)e FO(C)2011 3254 y FM(\013)2094 3242 y FO(<)g FQ(1)h FP(for)g(an)n(y)118 3341 y FO(\013)e FQ(2)g FJ(C)15 b FP([)p FO(G)q FP(],)41 b(and)33 b(an)n(y)f FO(\031)k FQ(2)d(\003)p FP(-Rep)14 b FJ(C)h FP([)p FO(G)p FP(])39 b(\(for)34 b(more)e(detailed)i (information)118 3441 y(ab)r(out)c FQ(\003)p FP(-b)r(oundedness)e(see)h (1.1.3\).)42 b(Hence)30 b(all)f(norms)g(de\014ned)h(ab)r(o)n(v)n(e)e (are)118 3541 y(\014nite)j(for)g(an)n(y)f FO(\013)f FQ(2)f FJ(C)15 b FP([)p FO(G)q FP(].)52 b(The)31 b(follo)n(wing)f(prop)r (osition)g(can)g(b)r(e)h(found)h(in)118 3640 y([57)o(].)118 3795 y FR(Prop)s(osition)e(7.)41 b FC(If)30 b FO(G)g FC(is)g(a)g(r)l(esidual)t(ly)i(\014nite)d(gr)l(oup,)i(then)307 3964 y FQ(k)p FO(\013)p FQ(k)444 3979 y FM(L)490 3987 y Fy(1)521 3979 y FL(\()p FM(G)p FL(\))652 3964 y FQ(\025)23 b(k)p FO(\013)p FQ(k)877 3979 y FM(C)929 3963 y Fx(\003)963 3979 y FL(\()p FM(G)p FL(\))1094 3964 y FQ(\025)f(k)p FO(\013)p FQ(k)1318 3979 y FM(C)1370 3959 y Fx(\003)1366 3999 y Fw(f)1404 3979 y FL(\()p FM(G)p FL(\))1535 3964 y FQ(\025)g(k)p FO(\013)p FQ(k)1759 3979 y FM(C)1811 3963 y Fx(\003)1807 3996 y Fw(r)1845 3979 y FL(\()p FM(G)p FL(\))1976 3964 y FQ(\025)h(k)p FO(\013)p FQ(k)2201 3979 y FM(L)2247 3987 y Fy(2)2278 3979 y FL(\()p FM(G)p FL(\))118 4147 y FC(for)31 b(any)f FO(\013)23 b FQ(2)h FJ(C)15 b FP([)p FO(G)p FP(])p FC(.)p eop %%Page: 15 19 15 18 bop 118 100 a FK(1.1.)36 b(In)n(tro)r(duction)27 b(to)h(represen)n(tations)d(of)j FQ(\003)p FK(-algebras)586 b FP(15)118 333 y FC(Pr)l(o)l(of.)43 b FP(The)28 b(only)f(non-trivial)g (part)g(is)g(the)h(pro)r(of)f(of)h(the)g(inequalit)n(y)661 513 y FQ(k)p FO(\013)p FQ(k)798 528 y FM(C)850 508 y Fx(\003)846 548 y Fw(f)884 528 y FL(\()p FM(G)p FL(\))1015 513 y FQ(\025)23 b(k)p FO(\013)p FQ(k)1240 528 y FM(C)1292 511 y Fx(\003)1288 544 y Fw(r)1325 528 y FL(\()p FM(G)p FL(\))1433 513 y FO(;)180 b FQ(8)p FO(\013)23 b FQ(2)g FJ(C)15 b FP([)p FO(G)p FP(])p FO(:)118 711 y FP(Let)21 b FO(\013)i FP(=)424 649 y Fz(P)511 670 y FM(m)511 736 y(k)q FL(=1)650 711 y FO(c)686 723 y FM(k)727 711 y FO(g)767 723 y FM(k)830 711 y FQ(6)p FP(=)g(0,)e(then)g FQ(k)p FO(\013)p FQ(k)1323 726 y FM(C)1375 710 y Fx(\003)1371 743 y Fw(r)1409 726 y FL(\()p FM(G)p FL(\))1540 711 y FP(=)h FO(d)i(>)e FP(0,)g(and)e(there)g(exists)g FO(\015)27 b FQ(2)118 811 y FO(L)175 823 y FL(2)212 811 y FP(\()p FO(G)p FP(\))20 b(suc)n(h)g(that)g FQ(k)p FO(\015)5 b FQ(k)845 826 y FM(L)891 834 y Fy(2)921 826 y FL(\()p FM(G)p FL(\))1052 811 y FP(=)23 b(1)c(and)g FQ(k)p FO(\013\015)5 b FQ(k)1539 826 y FM(L)1585 834 y Fy(2)1616 826 y FL(\()p FM(G)p FL(\))1747 811 y FQ(\025)23 b FO(d)s FQ(\000)s FO("=)p FP(2.)32 b(Hence,)21 b(there)118 911 y(exists)j FO(\016)i FP(=)494 848 y Fz(P)582 869 y FM(n)582 936 y FL(1)641 911 y FO(\016)678 923 y FM(k)719 911 y FO(h)767 923 y FM(k)831 911 y FP(suc)n(h)e(that)g FQ(k)p FO(\016)s FQ(k)1315 926 y FM(L)1361 934 y Fy(2)1392 926 y FL(\()p FM(G)p FL(\))1523 911 y FP(=)e(1)i(and)f FQ(k)p FO(\015)15 b FQ(\000)c FO(\016)s FQ(k)2091 926 y FM(L)2137 934 y Fy(2)2168 926 y FL(\()p FM(G)p FL(\))2299 911 y FQ(\024)23 b FO("=)p FP(2)p FO(d)o FP(.)118 1010 y(This)i(implies)f FQ(k)p FO(\013\015)17 b FQ(\000)12 b FO(\013\016)s FQ(k)950 1025 y FM(L)996 1033 y Fy(2)1028 1025 y FL(\()p FM(G)p FL(\))1159 1010 y FQ(\024)23 b FO("=)p FP(2,)h(and)g FQ(k)p FO(\013\016)s FQ(k)1752 1025 y FM(L)1798 1033 y Fy(2)1829 1025 y FL(\()p FM(G)p FL(\))1960 1010 y FQ(\025)f FO(d)12 b FQ(\000)g FO(")p FP(.)36 b(F)-7 b(urther,)118 1110 y(let)31 b(us)g(c)n(ho)r(ose)f(a)g(normal)g(subgroup)f FO(N)40 b FP(of)31 b(the)g(group)f FO(G)h FP(whic)n(h)f(do)r(es)h(not) 118 1210 y(con)n(tain)36 b(non-trivial)g(elemen)n(ts)h(among)f FO(g)1504 1222 y FM(k)1544 1210 y FO(g)1587 1174 y FN(\000)p FL(1)1584 1235 y FM(l)1676 1210 y FP(,)j FO(h)1786 1222 y FM(t)1816 1210 y FO(h)1864 1179 y FN(\000)p FL(1)1864 1230 y FM(s)1952 1210 y FP(,)h FO(g)2055 1222 y FM(k)2095 1210 y FO(h)2143 1222 y FM(t)2172 1210 y FO(h)2220 1179 y FN(\000)p FL(1)2220 1230 y FM(s)2309 1210 y FO(g)2352 1174 y FN(\000)p FL(1)2349 1235 y FM(l)2441 1210 y FP(;)i FO(k)s FP(,)118 1309 y FO(l)27 b FP(=)e(1,)j FO(:)14 b(:)g(:)28 b FP(,)h FO(m)p FP(,)h FO(t)p FP(,)f FO(s)c FP(=)g(1,)k FO(:)14 b(:)g(:)27 b FP(,)j FO(n)p FP(,)f(and)g(suc)n(h)g (that)g(the)g(quotien)n(t)g(group)f(of)118 1409 y FO(G)i FP(b)n(y)f(this)h(subgroup)f(is)g(\014nite.)44 b(Then,)30 b(in)g(the)g(regular)e(represen)n(tation)g(of)118 1508 y FO(G=)-5 b(N)9 b FP(,)28 b(w)n(e)f(ha)n(v)n(e)726 1688 y FQ(k)p FO(\013\016)s FQ(k)903 1703 y FM(L)949 1711 y Fy(2)980 1703 y FL(\()p FM(G=)l(N)6 b FL(\))1200 1688 y FP(=)23 b FQ(k)p FO(\013\016)s FQ(k)1465 1703 y FM(L)1511 1711 y Fy(2)1542 1703 y FL(\()p FM(G)p FL(\))1673 1688 y FQ(\025)g FO(d)18 b FQ(\000)g FO(":)118 1869 y FP(Hence,)31 b(for)f(an)n(y)f FO(")h FP(w)n(e)g(ha)n(v)n(e)f FQ(k)p FO(\013)p FQ(k)1205 1884 y FM(C)1257 1864 y Fx(\003)1253 1904 y Fw(f)1291 1884 y FL(\()p FM(G)p FL(\))1426 1869 y FQ(\025)e FO(d)21 b FQ(\000)f FO(")p FP(.)44 b(Therefore)29 b FQ(k)p FO(\013)p FQ(k)2289 1884 y FM(C)2341 1864 y Fx(\003)2337 1904 y Fw(f)2375 1884 y FL(\()p FM(G)p FL(\))2510 1869 y FQ(\025)118 1982 y(k)p FO(\013)p FQ(k)255 1997 y FM(C)307 1981 y Fx(\003)303 2014 y Fw(r)341 1997 y FL(\()p FM(G)p FL(\))449 1982 y FP(.)p 2514 1982 4 57 v 2518 1930 50 4 v 2518 1982 V 2567 1982 4 57 v 118 2148 a FC(R)l(emark)i FP(3)p FC(.)i FP(If)19 b FO(G)g FP(is)g(a)f (residually)f(\014nite)j(group,)f(then)g(w)n(e)f(ha)n(v)n(e)g(the)h (follo)n(wing)118 2247 y(sequence)27 b(of)h FQ(\003)p FP(-homomorphisms)856 2427 y FO(C)921 2393 y FN(\003)959 2427 y FP(\()p FO(G)p FP(\))c FQ(7!)f FO(C)1283 2393 y FN(\003)1277 2448 y FM(f)1322 2427 y FP(\()p FO(G)p FP(\))h FQ(7!)f FO(C)1646 2393 y FN(\003)1640 2448 y FM(r)1684 2427 y FP(\()p FO(G)p FP(\))p FO(;)118 2608 y FP(whic)n(h)g(are)f(iden)n(tical)h(on)g(the)g(dense)g FQ(\003)p FP(-subalgebra)e FJ(C)15 b FP([)p FO(G)p FP(].)41 b(Let)24 b(us)f(note)g(that)118 2707 y(these)28 b(homomorphisms)e(are)h (epimorphisms.)243 2807 y(Ho)n(w)n(ev)n(er,)h(in)i(general,)f(neither)h FO(C)1363 2777 y FN(\003)1402 2807 y FP(\()p FO(G)p FP(\))g(nor)f FO(C)1776 2777 y FN(\003)1770 2827 y FM(r)1815 2807 y FP(\()p FO(G)p FP(\))i(is)e(a)h(r.f.d.)g FQ(\003)p FP(-alge-)118 2906 y(bra.)35 b(Indeed,)23 b FO(C)650 2876 y FN(\003)689 2906 y FP(\()p FB(F)777 2918 y FL(2)812 2906 y FP(\),)h(where)e FB(F)1182 2918 y FL(2)1240 2906 y FP(is)h(the)g(free)f(group)f(with)i (t)n(w)n(o)f(generators,)118 3006 y(is)34 b(a)g(r.f.d.)h FQ(\003)p FP(-algebra)c([56)o(];)38 b(ho)n(w)n(ev)n(er,)c FO(C)1463 2976 y FN(\003)1457 3027 y FM(r)1502 3006 y FP(\()p FB(F)1590 3018 y FL(2)1625 3006 y FP(\))h(is)f(not)g(a)g (r.f.d.)g FQ(\003)p FP(-algebra,)118 3106 y(since)28 b(it)h(is)f(simple)g([219)n(].)39 b(The)28 b(residually)f(\014nite)i (group)e FO(S)5 b(L)p FP(\(2)p FO(;)14 b FJ(Z)n FP([)2296 3073 y FL(1)p 2289 3087 35 4 v 2289 3134 a FM(p)2333 3106 y FP(]\))29 b(\()p FO(p)f FP(is)118 3216 y(a)e(prime)h(n)n(um)n(b) r(er\))g(is)g(an)f(example)g(of)h(a)f(r.f.)h(group)f FO(G)h FP(for)f(whic)n(h)h FO(C)2324 3186 y FN(\003)2362 3216 y FP(\()p FO(G)p FP(\))h(is)118 3316 y(not)g(r.f.d.)g(see)f([107)n (].)118 3531 y FR(1.1.3)94 b(En)m(v)m(eloping)31 b FQ(\003)p FR(-algebras)g(and)h FO(C)1596 3501 y FN(\003)1635 3531 y FR(-algebras)118 3684 y(1.)54 b FP(Sometimes)34 b(it)g(is)g(p)r (ossible)f(to)g(reduce)g(the)h(study)g(of)g(represen)n(tations)118 3784 y(of)e(a)f FQ(\003)p FP(-algebra)d FA(A)j FP(to)h(the)g(study)f (of)h FQ(\003)p FP(-represen)n(tations)c(of)k(its)f(en)n(v)n(eloping) 118 3884 y FQ(\003)p FP(-algebra,)25 b FO(\033)s FP(-)p FO(C)644 3854 y FN(\003)683 3884 y FP(-algebra,)g(or)i FO(C)1191 3854 y FN(\003)1229 3884 y FP(-algebra.)35 b(Let)28 b(us)f(recall)g(the)h(de\014nition.)118 4048 y FR(De\014nition)g(1.)38 b FC(L)l(et)c FA(A)27 b FC(b)l(e)g(a)h FQ(\003)p FC(-algebr)l(a.)38 b(The)28 b(p)l(air)g FP(\()1839 4026 y(~)1830 4048 y FA(A)p FP(;)14 b FO(\036)9 b FP(:)28 b FA(A)23 b FQ(7!)2234 4026 y FP(~)2225 4048 y FA(A)p FP(\))p FC(,)28 b(wher)l(e)127 4126 y FP(~)118 4147 y FA(A)e FC(is)g(a)h FQ(\003)p FC(-algebr)l(a)g(and)g FO(\036)f FC(is)h(a)f FQ(\003)p FC(-homomorphism,)k(is)c(c)l(al)t(le)l(d)i(an)e (enveloping)p eop %%Page: 16 20 16 19 bop 118 100 a FP(16)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FQ(\003)p FC(-algebr)l(a)32 b(of)g(the)f(algebr)l(a)i FA(A)d FC(if)i(for)g(any)g FQ(\003)p FC(-r)l(epr)l(esentation)e FO(\031)12 b FP(:)29 b FA(A)c FQ(7!)h FO(L)p FP(\()p FO(H)7 b FP(\))118 432 y FC(of)27 b(the)g(algebr)l(a)g FA(A)f FC(ther)l(e)h(exists)f(a)g(unique)g FQ(\003)p FC(-r)l(epr)l(esentation)k FP(~)-46 b FO(\031)13 b FP(:)2198 411 y(~)2189 432 y FA(A)23 b FQ(7!)g FO(L)p FP(\()p FO(H)7 b FP(\))118 532 y FC(such)30 b(that)g(the)g(fol)t(lowing)i(diagr)l(am)f (is)f(c)l(ommutative.)1110 1065 y FP(~)1101 1087 y FA(A)287 b FO(L)p FP(\()p FO(H)7 b FP(\))p 1185 1061 239 4 v 1340 1059 a Fu(-)1283 1130 y FP(~)-46 b FO(\031)1390 869 y(\031)1189 785 y Fu(@)1272 868 y(@)1355 951 y(@)1405 1000 y(@)-83 b(R)1101 672 y FA(A)p 1129 1000 4 299 v 1131 1000 a Fu(?)1048 872 y FO(\036)118 1255 y FC(Example)36 b FP(1)p FC(.)41 b FP(Let)27 b(\006)g(b)r(e)g(an)n(y)f(family)h(of)g(elemen)n(ts)g(of)g (a)f FQ(\003)p FP(-algebra)e FA(A)j FP(whic)n(h)118 1355 y(are)i(in)n(v)n(ertible)h(in)g(an)n(y)f FQ(\003)p FP(-represen)n (tation)f FO(\031)12 b FP(:)29 b FA(A)e FQ(7!)g FO(L)p FP(\()p FO(H)7 b FP(\).)45 b(Let)30 b(us)g(denote)118 1454 y(the)g(algebra)d(of)i(quotien)n(ts)g(of)g FA(A)f FP(with)i(resp)r(ect)f(to)g(\006)g(b)n(y)1989 1433 y(~)1983 1454 y FA(U)c FP(=)g FA(A)p FP([\006)2295 1424 y FN(\000)p FL(1)2384 1454 y FP(])k(\(see)118 1554 y([89)o(]\).)36 b(Let)24 b FO(\036)9 b FP(:)29 b FA(A)22 b FQ(7!)788 1532 y FP(~)782 1554 y FA(U)h FP(b)r(e)h(the)h(natural)e(homomorphism.) 34 b(Then)25 b(w)n(e)e(obtain)118 1654 y(an)k(en)n(v)n(eloping)e FQ(\003)p FP(-algebra)f(for)j(the)g(the)g FQ(\003)p FP(-algebra)e FA(A)p FP(.)36 b(Let)27 b(us)g(note)f(that)i(in)118 1753 y(the)g(case)f(where)g FA(A)g FP(is)h FO(C)913 1723 y FN(\003)951 1753 y FP(-represen)n(table,)e FO(\036)i FP(is)f(an)h(injection.)118 1882 y FR(2.)36 b FP(If)321 1861 y(~)312 1882 y FA(A)27 b FP(carries)e(the)j(structure)f(of)g(a)g FO(C)1389 1852 y FN(\003)1427 1882 y FP(-algebra,)e(then)33 b(~)-47 b FO(\031)31 b FP(is)c(a)g(con)n(tin)n(uous)118 1982 y FQ(\003)p FP(-homomorphism)34 b(from)i(the)g FO(C)1203 1952 y FN(\003)1241 1982 y FP(-algebra)1577 1960 y(~)1568 1982 y FA(A)g FP(to)f(the)i FO(C)1990 1952 y FN(\003)2028 1982 y FP(-algebra)d FO(L)p FP(\()p FO(H)7 b FP(\).)118 2082 y(In)40 b(this)g(case)f(the)h(pair)e(\()974 2060 y(~)965 2082 y FA(A)p FO(;)14 b(\036)p FP(\))41 b(is)e(called)g(an)h (en)n(v)n(eloping)e FO(C)2139 2051 y FN(\003)2177 2082 y FP(-algebra)g(of)118 2181 y(the)32 b(algebra)d FA(A)p FP(.)48 b(The)31 b(en)n(v)n(eloping)f FO(C)1341 2151 y FN(\003)1380 2181 y FP(-algebra)f(is)i(unique)g(in)h(the)g(class)e (of)118 2281 y FO(C)183 2251 y FN(\003)222 2281 y FP(-algebras,)h(if)i (it)f(exists)g(\(see)h(Theorem)e(1\).)51 b(Indeed,)34 b(in)f(this)f(case)2436 2259 y(~)2427 2281 y FA(A)g FP(is)118 2380 y(an)c(en)n(v)n(eloping)e FO(\033)s FP(-)p FO(C)785 2350 y FN(\003)824 2380 y FP(-algebra)f(whic)n(h)j(is)g(unique)g(b)n(y) f(Theorem)g(1.)37 b(W)-7 b(e)29 b(will)118 2480 y(denote)f(the)g(en)n (v)n(eloping)e FO(C)1002 2450 y FN(\003)1040 2480 y FP(-algebra)g(of)h (algebra)f FA(A)h FP(b)n(y)h FO(C)2012 2450 y FN(\003)2050 2480 y FP(\()p FA(A)p FP(\).)243 2580 y(Let)34 b FA(A)g FP(=)g FJ(C)15 b FQ(h)q FO(x)725 2592 y FL(1)768 2580 y FO(;)f(:)g(:)g(:)g(;)g(x)1000 2592 y FM(n)1045 2580 y FO(;)g(x)1129 2550 y FN(\003)1129 2600 y FL(1)1168 2580 y FO(;)g(:)g(:)g(:)f(;)h(x)1399 2550 y FN(\003)1399 2600 y FM(n)1480 2580 y FQ(j)34 b FO(P)1590 2592 y FM(k)1631 2580 y FP(\()p FQ(\001)p FP(\))h(=)f(0)p FO(;)48 b(k)38 b FP(=)c(1)p FO(;)14 b(:)g(:)g(:)f(;)h(m)p FQ(i)34 b FP(=)118 2679 y FJ(C)15 b FQ(h)p FO(x)252 2691 y FL(1)295 2679 y FO(;)f(:)g(:)g(:)27 b(;)14 b(x)540 2691 y FM(n)586 2679 y FO(;)g(x)670 2649 y FN(\003)670 2700 y FL(1)709 2679 y FO(;)g(:)g(:)g(:)27 b(;)14 b(x)954 2649 y FN(\003)954 2700 y FM(n)1034 2679 y FQ(j)35 b FO(J)8 b FQ(i)35 b FP(denote)f(the)h FQ(\003)p FP(-algebra)d(with)j(generators)118 2779 y FO(x)165 2791 y FM(j)201 2779 y FP(,)41 b FO(x)312 2749 y FN(\003)312 2801 y FM(j)350 2779 y FP(,)h FO(j)j FP(=)c(1,)d FO(:)14 b(:)g(:)27 b FP(,)42 b FO(m)p FP(,)f(and)d (relations)f FO(P)1602 2791 y FM(k)1643 2779 y FP(\()p FO(x)1722 2791 y FL(1)1760 2779 y FO(;)14 b(:)g(:)g(:)g(;)g(x)1992 2791 y FM(n)2037 2779 y FO(;)g(x)2121 2749 y FN(\003)2121 2800 y FL(1)2160 2779 y FO(;)g(:)g(:)g(:)f(;)h(x)2391 2749 y FN(\003)2391 2800 y FM(n)2437 2779 y FP(\))41 b(=)118 2879 y(0,)j(\(where)c FO(P)565 2891 y FM(k)606 2879 y FP(\()p FQ(\001)p FP(\))h(are)f(non-comm)n(utativ)n(e)f(p)r (olynomials\),)44 b(i.e.,)g FA(A)c FP(is)g(the)118 2978 y(quotien)n(t)29 b(of)h(the)g(free)f FQ(\003)p FP(-algebra)e FJ(C)15 b FQ(h)p FO(x)1345 2990 y FL(1)1389 2978 y FO(;)f(:)g(:)g(:)f (;)h(x)1620 2990 y FM(n)1666 2978 y FO(;)g(x)1750 2948 y FN(\003)1750 2999 y FL(1)1788 2978 y FO(;)g(:)g(:)g(:)g(;)g(x)2020 2948 y FN(\003)2020 2999 y FM(n)2065 2978 y FQ(i)30 b FP(with)g(resp)r(ect)118 3078 y(to)d(the)h(t)n(w)n(o-sided)e FQ(\003)p FP(-ideal)h FO(J)35 b FP(generated)26 b(b)n(y)h(the)h (relations.)36 b(In)27 b(the)h(sequel,)118 3178 y(w)n(e)22 b(will)h(sometimes)g(omit)f(the)h(sym)n(b)r(ols)f FO(x)1470 3147 y FN(\003)1470 3198 y FL(1)1509 3178 y FO(;)14 b(:)g(:)g(:)27 b(;)14 b(x)1754 3147 y FN(\003)1754 3198 y FM(n)1800 3178 y FP(,)24 b(if)f(it)g(is)g(clear)e(from)h(the)118 3277 y(con)n(text,)27 b(that)h(the)g(considered)f(algebra)f(is)h(a)g FQ(\003)p FP(-algebra.)243 3377 y(If)k(a)f FQ(\003)p FP(-algebra)e FA(A)p FP(,)k(generated)d(b)n(y)i(generators)d FO(x)1831 3389 y FL(1)1869 3377 y FP(,)j FO(:)14 b(:)g(:)27 b FP(,)32 b FO(x)2149 3389 y FM(n)2226 3377 y FP(and)e(p)r(oly-)118 3476 y(nomial)h(relations)g FO(P)789 3488 y FM(k)830 3476 y FP(\()p FO(x)909 3488 y FL(1)947 3476 y FO(;)14 b(:)g(:)g(:)g(;)g(x)1179 3488 y FM(n)1224 3476 y FO(;)g(x)1308 3446 y FN(\003)1308 3497 y FL(1)1347 3476 y FO(;)g(:)g(:)g(:)f(;)h(x) 1578 3446 y FN(\003)1578 3497 y FM(n)1624 3476 y FP(\))30 b(=)g(0,)i FO(k)h FP(=)d(1,)h FO(:)14 b(:)g(:)28 b FP(,)33 b FO(m)p FP(,)f(has)118 3576 y(the)c(en)n(v)n(eloping)e FO(C)734 3546 y FN(\003)773 3576 y FP(-algebra)1100 3554 y(~)1091 3576 y FA(A)p FP(,)i(then)g(w)n(e)f(will)h(denote)f(it)h(b)n (y)481 3750 y FO(C)546 3715 y FN(\003)584 3750 y FP(\()p FA(A)p FP(\))23 b(=)g FO(C)884 3715 y FN(\003)923 3750 y FP(\()p FO(x)1002 3762 y FL(1)1039 3750 y FO(;)14 b(:)g(:)g(:)g(;)g (x)1271 3762 y FM(n)1317 3750 y FP(;)g FO(P)1407 3762 y FM(k)1448 3750 y FP(\()p FQ(\001)p FP(\))23 b(=)g(0)p FO(;)14 b(k)25 b FP(=)e(1)p FO(;)14 b(:)g(:)g(:)f(;)h(m)p FP(\))731 3874 y(=)23 b FO(C)884 3840 y FN(\003)923 3874 y FP(\()p FO(x)1002 3886 y FL(1)1039 3874 y FO(;)14 b(:)g(:)g(:)g(;)g (x)1271 3886 y FM(n)1317 3874 y FP(;)g FO(J)8 b FP(\))p FO(:)243 4048 y FP(A)27 b FQ(\003)p FP(-algebra)d FA(A)i FP(is)h(called)f FQ(\003)p FP(-b)r(ounded)g(if)h(for)f(an)n(y)g FO(x)e FQ(2)f FA(A)k FP(there)f(exists)g(a)118 4147 y(n)n(um)n(b)r(er)j FO(C)481 4159 y FM(x)547 4147 y FO(<)c FQ(1)k FP(suc)n(h)f(that)h FQ(k)p FO(\031)s FP(\()p FO(x)p FP(\))p FQ(k)c(\024)f FO(C)1536 4159 y FM(x)1607 4147 y FP(for)k(an)n(y)g(of)h(its)g FQ(\003)p FP(-represen)n(ta-)p eop %%Page: 17 21 17 20 bop 118 100 a FK(1.1.)36 b(In)n(tro)r(duction)27 b(to)h(represen)n(tations)d(of)j FQ(\003)p FK(-algebras)586 b FP(17)118 333 y(tions)28 b FO(\031)12 b FP(:)28 b FA(A)22 b FQ(7!)h FO(L)p FP(\()p FO(H)7 b FP(\).)37 b(F)-7 b(or)27 b(a)g(\014nitely-generated)g FQ(\003)p FP(-algebra)957 507 y FA(A)22 b FP(=)h FJ(C)15 b FQ(h)p FO(x)1261 519 y FL(1)1304 507 y FO(;)f(:)g(:)g(:)f(;)h(x)1535 519 y FM(n)1604 507 y FQ(j)23 b FO(J)8 b FQ(i)118 682 y FP(to)30 b(b)r(e)h FQ(\003)p FP(-b)r(ounded)e(it)i(is)f(su\016cien)n(t)g(that)g (for)g(an)n(y)f(of)h(its)h FQ(\003)p FP(-represen)n(tations)118 781 y FO(\031)12 b FP(:)32 b FA(A)40 b FQ(7!)g FO(L)p FP(\()p FO(H)7 b FP(\))38 b(and)g(an)n(y)f FO(k)43 b FP(=)d(1,)e FO(:)14 b(:)g(:)27 b FP(,)41 b FO(n)p FP(,)f(there)e (exists)g FO(C)2146 793 y FM(k)2225 781 y FP(suc)n(h)g(that)118 881 y FQ(k)p FO(\031)s FP(\()p FO(x)289 893 y FM(k)331 881 y FP(\))p FQ(k)22 b(\024)h FO(C)574 893 y FM(k)615 881 y FP(.)243 981 y(A)30 b FQ(\003)p FP(-algebra)e FA(A)i FP(has)f(the)i(en)n(v)n(eloping)e FO(C)1560 951 y FN(\003)1598 981 y FP(-algebra)g FO(C)1985 951 y FN(\003)2023 981 y FP(\()p FA(A)p FP(\))h(if)h(and)f(only)118 1080 y(if)24 b(it)h(is)e FQ(\003)p FP(-b)r(ounded.)36 b(In)24 b(this)g(case)f(the)h (en)n(v)n(eloping)e FO(C)1827 1050 y FN(\003)1866 1080 y FP(-algebra)g(is)h(the)i(com-)118 1180 y(pletion)i(of)f FA(A)g FP(with)g(resp)r(ect)g(to)h(the)f(norm)g FQ(k)p FO(x)p FQ(k)c FP(=)h(sup)1867 1200 y FM(\031)r FN(2\003)p FL(-Rep)o(\()p Ft(A)p FL(\))2242 1180 y FQ(k)p FO(\031)s FP(\()p FO(x)p FP(\))p FQ(k)g(\024)118 1280 y FO(C)177 1292 y FM(x)242 1280 y FO(<)g FQ(1)28 b FP(\(sup)g(is)f(tak)n(en)g(o)n (v)n(er)f(all)h FQ(\003)p FP(-represen)n(tations)e FO(\031)s FP(\).)118 1409 y FC(Example)43 b FP(2)p FC(.)j FP(The)34 b(group)f FQ(\003)p FP(-algebra)f FJ(C)15 b FP([)p FO(G)p FP(])41 b(is)34 b FQ(\003)p FP(-b)r(ounded,)i(since)e(for)g(an)n(y)118 1509 y(generator)c FO(g)i FQ(2)e FO(G)j FP(and)e(an)n(y)g FO(\031)i FQ(2)e(\003)p FP(-Rep)13 b FJ(C)i FP([)p FO(G)p FP(])38 b(the)32 b(op)r(erator)e FO(\031)s FP(\()p FO(g)s FP(\))i(is)g(uni-)118 1608 y(tary)d(and)h FQ(k)p FO(\031)s FP(\()p FO(g)s FP(\))p FQ(k)c FP(=)h(1.)43 b(Hence)30 b(there)g(exists)g(the)g(en)n(v)n(eloping)f FO(C)2246 1578 y FN(\003)2284 1608 y FP(-algebra)118 1708 y FO(C)183 1678 y FN(\003)222 1708 y FP(\()p FO(G)p FP(\))23 b(=)g FO(C)527 1678 y FN(\003)566 1708 y FP(\()p FJ(C)15 b FP([)p FO(G)p FP(]\).)118 1838 y FC(Example)37 b FP(3)p FC(.)42 b FP(The)27 b FQ(\003)p FP(-algebra)479 2012 y FB(P)534 2024 y FM(n)602 2012 y FP(=)c FJ(C)744 1945 y Fz(\012)789 2012 y FO(p)831 2024 y FL(1)868 2012 y FO(;)14 b(:)g(:)g(:)f(;)h(p)1094 2024 y FM(n)1162 2012 y FQ(j)24 b FO(p)1251 1978 y FL(2)1251 2033 y FM(k)1314 2012 y FP(=)f FO(p)1444 2024 y FM(k)1508 2012 y FP(=)f FO(p)1637 1978 y FN(\003)1637 2033 y FM(k)1678 2012 y FO(;)41 b(k)26 b FP(=)d(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n)2175 1945 y Fz(\013)118 2187 y FP(is)34 b FQ(\003)p FP(-b)r(ounded,)h(since) f FQ(k)p FO(P)952 2199 y FM(k)993 2187 y FQ(k)f(\024)g FP(1,)j FO(k)g FP(=)e(1,)f FO(:)14 b(:)g(:)28 b FP(,)36 b FO(n)p FP(.)56 b(Then)34 b(there)g(exists)f(a)118 2286 y(unique)e FO(C)456 2256 y FN(\003)495 2286 y FP(\()p FB(P)582 2298 y FM(n)627 2286 y FP(\))e(=)g FO(C)847 2256 y FN(\003)885 2286 y FP(\()p FO(p)959 2298 y FL(1)996 2286 y FO(;)14 b(:)g(:)g(:)g(;)g(p)1223 2298 y FM(n)1268 2286 y FP(;)g FO(p)1347 2256 y FL(2)1347 2310 y FM(k)1416 2286 y FP(=)29 b FO(p)1552 2298 y FM(k)1621 2286 y FP(=)g FO(p)1757 2256 y FN(\003)1757 2310 y FM(k)1797 2286 y FO(;)45 b(k)32 b FP(=)c(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(n)p FP(\).)48 b(It)31 b(is)118 2386 y(kno)n(wn)26 b(that)i(the)f FO(C)767 2356 y FN(\003)805 2386 y FP(-algebra)e FO(C)1188 2356 y FN(\003)1227 2386 y FP(\()p FB(P)1314 2398 y FL(2)1351 2386 y FP(\))j(can)e(also)g(b)r(e)h(de\014ned)h(as)e FO(C)2292 2356 y FN(\003)2331 2386 y FP(\()p FB(P)2418 2398 y FL(2)2455 2386 y FP(\))d(=)118 2486 y FQ(f)p FO(f)40 b FQ(2)31 b FO(C)6 b FP(\([0)p FO(;)14 b FP(1])p FO(;)g(M)709 2498 y FL(2)745 2486 y FP(\()p FJ(C)i FP(\)\))38 b FQ(j)31 b FO(f)9 b FP(\(0\))p FO(;)14 b(f)9 b FP(\(1\))32 b(are)27 b(diagonal)n FQ(g)32 b FP(\(see)h([286)n(,)g(234)n(])g(and)118 2585 y(others\).)118 2715 y FR(3.)j FP(If)29 b(a)e FQ(\003)p FP(-algebra)e FA(A)e FP(=)f FJ(C)15 b FQ(h)q FO(x)1046 2727 y FL(1)1089 2715 y FO(;)f(:)g(:)g(:)g(;)g(x)1321 2727 y FM(n)1366 2715 y FQ(i)p FO(=J)36 b FP(is)27 b(not)h FQ(\003)p FP(-b)r(ounded,)f(it)i(do)r(es)e(not)118 2814 y(ha)n(v)n(e)h(an)g(en)n(v)n(eloping)g FO(C)902 2784 y FN(\003)940 2814 y FP(-algebra,)f FO(C)1348 2784 y FN(\003)1387 2814 y FP(\()p FO(x)1466 2826 y FL(1)1504 2814 y FO(;)14 b(:)g(:)g(:)f(;)h(x)1735 2826 y FM(n)1781 2814 y FP(;)g FO(J)8 b FP(\).)41 b(Ho)n(w)n(ev)n(er,)27 b(w)n(e)h(can)118 2914 y(alw)n(a)n(ys)22 b(\014nd)i(an)f(en)n(v)n (eloping)g FO(\033)s FP(-)p FO(C)1201 2884 y FN(\003)1239 2914 y FP(-algebra.)34 b(Let)23 b(us)h(sk)n(etc)n(h)f(this)h(construc-) 118 3014 y(tion.)45 b(If)31 b(in)f(the)h(de\014nition)f(of)h(the)f(en)n (v)n(eloping)f FO(C)1750 2984 y FN(\003)1788 3014 y FP(-algebra,)g(one) h(replaces)118 3113 y(the)e(condition)f(that)h(the)g(diagram)1108 3665 y Fz(e)1101 3686 y FA(A)287 b FO(L)p FP(\()p FO(H)7 b FP(\))p 1185 3658 239 4 v 1340 3656 a Fu(-)1281 3732 y Fz(e)-48 b FO(\031)1390 3466 y(\031)1189 3382 y Fu(@)1272 3465 y(@)1355 3548 y(@)1405 3598 y(@)-83 b(R)1101 3270 y FA(A)p 1129 3598 4 299 v 1131 3598 a Fu(?)1048 3469 y FO(\036)118 3873 y FP(is)23 b(comm)n(utativ)n(e)f(for)h(all)g FQ(\003)p FP(-represen)n(tations)d(of)j FA(A)g FP(b)n(y)g(the)g (condition)g(that)g(it)118 3973 y(is)f(comm)n(utativ)n(e)f(only)g(for)g (the)h(represen)n(tations)e(sub)5 b(ject)21 b(to)h(the)g(restriction) 557 4147 y FQ(k)p FO(\031)s FP(\()p FO(x)728 4159 y FM(k)770 4147 y FP(\))p FQ(k)g(\024)h FO(d)997 4159 y FM(k)1038 4147 y FO(;)180 b(d)1284 4159 y FM(k)1348 4147 y FO(>)23 b FP(0)p FO(;)179 b(k)26 b FP(=)d(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n;)p eop %%Page: 18 22 18 21 bop 118 100 a FP(18)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(then)f(there)f(exists)g(a)f FO(C)873 303 y FN(\003)912 333 y FP(-algebra)1237 311 y(~)1228 333 y FA(A)h FP(making)f(this)i (diagram)e(comm)n(utativ)n(e.)118 432 y(W)-7 b(e)28 b(will)g(denote)f (this)h FO(C)912 402 y FN(\003)951 432 y FP(-algebra)d(b)n(y)568 619 y FO(C)633 585 y FN(\003)671 619 y FP(\()p FO(x)750 631 y FL(1)788 619 y FO(;)14 b(:)g(:)g(:)28 b(;)14 b(x)1034 631 y FM(n)1079 619 y FP(;)28 b FQ(k)p FO(x)1219 631 y FM(k)1260 619 y FQ(k)22 b(\024)h FO(d)1455 631 y FM(k)1496 619 y FO(;)28 b(k)e FP(=)c(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n)p FP(;)g FO(J)8 b FP(\))p FO(:)118 808 y FC(Example)43 b FP(4)p FC(.)i FP(The)34 b(algebra)e FJ(C)15 b FP([)p FO(a)39 b FP(=)33 b FO(a)1355 778 y FN(\003)1393 808 y FP(])h(of)f(complex)h(p)r(olynomials)f(of)g(one)118 907 y(real)j(v)-5 b(ariable)36 b FO(t)i FQ(2)h FJ(R)826 877 y FL(1)906 907 y FP(is)d(not)h FQ(\003)p FP(-b)r(ounded,)i(since)e (for)f(an)n(y)g FO(\025)j FQ(2)f FJ(R)43 b FP(there)118 1007 y(exists)27 b(a)h(represen)n(tation)d FO(\031)1006 1019 y FM(\025)1050 1007 y FP(\()p FO(a)p FP(\))f(=)e FO(\025I)36 b FP(with)28 b FQ(k)p FO(\031)1667 1019 y FM(\025)1710 1007 y FP(\()p FO(a)p FP(\))p FQ(k)23 b FP(=)p FQ(j)g FO(\025)g FQ(j)p FP(.)243 1109 y(Ho)n(w)n(ev)n(er,)k (there)i(exists)g(a)g FO(C)1183 1078 y FN(\003)1221 1109 y FP(-algebra)f FA(A)1602 1121 y FM(d)1666 1109 y FP(=)d FO(C)6 b FP([)p FQ(\000)p FO(d;)14 b(d)p FP(])30 b(with)f(one)g(self-) 118 1208 y(adjoin)n(t)c(generator)e FO(a)p FP(\()p FO(t)p FP(\))h(=)f FO(t)i FP(and)g(a)g(homomorphism)f FO(\022)12 b FP(:)28 b FJ(C)15 b FP([)p FO(a)29 b FP(=)22 b FO(a)2262 1178 y FN(\003)2300 1208 y FP(])h FQ(3)h FO(a)f FQ(7!)118 1308 y FO(a)p FP(\()p FQ(\001)p FP(\))h FQ(2)f FO(C)6 b FP([)p FQ(\000)p FO(d;)14 b(d)p FP(],)28 b(whic)n(h)g(satis\014es)e (the)i(follo)n(wing)f(conditions:)220 1480 y(1.)41 b FQ(k)p FO(a)p FP(\()p FQ(\001)p FP(\))p FQ(k)22 b(\024)h FO(d)p FP(;)220 1654 y(2.)41 b(for)j(an)n(y)g FQ(\003)p FP(-represen)n(tation)e FO(\031)12 b FP(:)34 b FJ(C)15 b FP([)p FO(a)58 b FP(=)52 b FO(a)1729 1624 y FN(\003)1767 1654 y FP(])f FQ(7!)h FO(L)p FP(\()p FO(H)7 b FP(\))45 b(suc)n(h)g(that)326 1754 y FQ(k)p FO(\031)s FP(\()p FO(a)p FP(\))p FQ(k)c(\024)g FO(d)e FP(there)f(exists)g(a)h(unique)f (represen)n(tation)k(~)-47 b FO(\031)13 b FP(:)31 b FO(C)2354 1724 y FN(\003)2393 1754 y FP(\()p FO(a)41 b FP(=)326 1853 y FO(a)370 1823 y FN(\003)408 1853 y FP(;)f FQ(k)p FO(a)p FQ(k)22 b(\024)h FO(d)p FP(\))h FQ(7!)f FO(L)p FP(\()p FO(H)7 b FP(\))26 b(suc)n(h)h(that)g(the)g(follo)n(wing)f (diagram)f(is)i(com-)326 1953 y(m)n(utativ)n(e)1230 2546 y FA(A)1290 2558 y FM(d)1596 2546 y FO(L)p FP(\()p FO(H)7 b FP(\))p 1353 2527 219 4 v 1488 2525 a Fu(-)1441 2597 y FP(~)-46 b FO(\031)1538 2336 y(\031)1337 2252 y Fu(@)1420 2335 y(@)1503 2418 y(@)1553 2467 y(@)-83 b(R)1108 2131 y FJ(C)15 b FP([)p FO(a)29 b FP(=)23 b FO(a)1390 2101 y FN(\003)1428 2131 y FP(])p 1278 2467 4 299 v 1279 2467 a Fu(?)1205 2347 y FO(\022)118 2780 y FC(Example)39 b FP(5)p FC(.)k FP(The)31 b FQ(\003)p FP(-algebra)c FB(Q)1164 2792 y FL(1)1229 2780 y FP(=)g FQ(h)p FO(q)s(;)14 b(q)1470 2750 y FN(\003)1535 2780 y FQ(j)28 b FO(q)1626 2750 y FL(2)1690 2780 y FP(=)f FO(q)s(;)44 b FP(\()p FO(q)1961 2750 y FN(\003)2000 2780 y FP(\))2032 2750 y FL(2)2097 2780 y FP(=)27 b FO(q)2229 2750 y FN(\003)2267 2780 y FQ(i)p FP(,)k(gener-)118 2880 y(ated)c(b)n(y)g(one)g(idemp)r(oten)n(t)g (is)g(not)g FQ(\003)p FP(-b)r(ounded,)g(since)g(for)g(an)n(y)f FO(\025)e FQ(2)f FJ(R)33 b FP(there)118 2980 y(exists)27 b(the)h(represen)n(tation)1004 3212 y FO(\031)1051 3224 y FM(\025)1095 3212 y FP(\()p FO(q)s FP(\))c(=)1310 3094 y Fz(\022)1413 3161 y FP(1)83 b FO(\025)1413 3261 y FP(0)j(0)1628 3094 y Fz(\023)118 3448 y FP(with)28 b FQ(k)p FO(\031)396 3460 y FM(\025)440 3448 y FP(\()p FO(q)s FP(\))p FQ(k)23 b(!)g(1)28 b FO(;)41 b(\025)24 b FQ(!)f(1)p FP(.)37 b(Ho)n(w)n(ev)n(er) 25 b(there)j(exists)876 3635 y FO(C)941 3600 y FN(\003)979 3568 y Fz(\000)1017 3635 y FO(q)s(;)14 b(q)1134 3600 y FN(\003)1173 3635 y FP(;)g FQ(k)p FO(q)s FQ(k)22 b(\024)g FO(d)p FP(;)42 b FO(q)1591 3600 y FL(2)1651 3635 y FP(=)23 b FO(q)1779 3568 y Fz(\001)570 3759 y FP(=)g FQ(f)p FO(f)31 b FQ(2)23 b FO(C)6 b FP(\([0)p FO(;)14 b(d)p FP(])p FO(;)g(M)1233 3771 y FL(2)1270 3759 y FP(\()p FJ(C)i FP(\)\))9 b(:)34 b FO(f)9 b FP(\(0\))27 b(is)h(diagonal)n FQ(g)p FO(:)243 3948 y FP(F)-7 b(or)38 b(ev)n(ery)f FQ(\003)p FP(-algebra)g FA(A)k FP(=)g FJ(C)15 b FQ(h)p FO(x)1346 3960 y FL(1)1389 3948 y FO(;)f(:)g(:)g(:)28 b(;)14 b(x)1635 3960 y FM(m)1698 3948 y FP(;)g FO(J)8 b FQ(i)39 b FP(w)n(e)g(can)f(construct)g(a)118 4048 y(top)r(ological)30 b FQ(\003)p FP(-algebra)917 4026 y(~)908 4048 y FA(A)h FP(whic)n(h)h(is)f(also)f(an)i(en)n(v)n (eloping)e FQ(\003)p FP(-algebra)f(of)i FA(A)p FP(.)118 4147 y(Indeed,)44 b(in)c(the)g(previous)f(section)h(w)n(e)g(ha)n(v)n(e) f(constructed)g(the)i(algebras)p eop %%Page: 19 23 19 22 bop 118 100 a FK(1.1.)36 b(In)n(tro)r(duction)27 b(to)h(represen)n(tations)d(of)j FQ(\003)p FK(-algebras)586 b FP(19)118 333 y FA(A)178 345 y FM(n)266 333 y FP(=)43 b FO(C)439 303 y FN(\003)478 333 y FP(\()p FO(x)557 345 y FL(1)595 333 y FO(;)14 b(:)g(:)g(:)f(;)h(x)826 345 y FM(m)890 333 y FP(;)g FQ(k)p FO(x)1016 345 y FM(j)1050 333 y FQ(k)43 b(\024)g FO(n;)28 b FP(1)43 b FQ(\024)g FO(j)48 b FQ(\024)43 b FO(m)p FP(;)14 b FO(J)8 b FP(\))40 b(and)f FQ(\003)p FP(-homomor-)118 432 y(phisms)c FO(\036)452 444 y FM(n)507 432 y FP(:)30 b FA(A)35 b FQ(\000)-48 b(!)35 b FA(A)850 444 y FM(n)930 432 y FP(with)h(appropriate)d(univ)n (ersal)h(prop)r(erties.)58 b(Since)118 532 y(k)n(er)13 b FO(\036)292 544 y FM(n)368 532 y FQ(\023)30 b FP(k)n(er)13 b FO(\036)637 544 y FM(n)p FL(+1)767 532 y FP(,)33 b(there)f(exists)g (a)g FQ(\003)p FP(-homomorphism)e FO( )2065 502 y FM(n)p FL(+1)2062 552 y FM(n)2203 532 y FP(:)g FA(A)2316 544 y FM(n)p FL(+1)2475 532 y FQ(\000)-48 b(!)118 632 y FA(A)178 644 y FM(n)251 632 y FP(suc)n(h)27 b(that)h(the)g(follo)n(wing)e (diagrams)g(comm)n(ute)858 1205 y FA(A)918 1217 y FM(n)p FL(+1)1730 1205 y FA(A)1790 1217 y FM(n)p 1071 1188 634 4 v 1622 1186 a Fu(-)1295 1270 y FO( )1352 1240 y FM(n)p FL(+1)1349 1290 y FM(n)1337 800 y FA(A)930 997 y FO(\036)979 1009 y FM(n)p FL(+1)1226 912 y Fu(\000)1143 995 y(\000)1060 1078 y(\000)1010 1128 y(\000)-83 b(\011)1626 999 y FO(\036)1675 1011 y FM(n)1425 912 y Fu(@)1508 995 y(@)1591 1078 y(@)1641 1128 y(@)g(R)243 1462 y FP(Consider)24 b(a)h(subalgebra)1076 1440 y(~)1067 1462 y FA(A)g FP(in)h(the)f(Cartesian)f(pro)r(duct)2068 1399 y Fz(Q)2147 1486 y FM(n)p FN(2)p Fv(N)2293 1462 y FA(A)2353 1474 y FM(n)2423 1462 y FP(con-)118 1561 y(sisting)36 b(of)h(elemen)n(ts)g FO(f)18 b FP(:)30 b FJ(N)49 b FQ(\000)-49 b(!)39 b([)1243 1573 y FM(n)p FN(2)p Fv(N)1375 1561 y FA(A)1435 1573 y FM(n)1516 1561 y FP(suc)n(h)e(that)g FO( )1959 1531 y FM(n)p FL(+1)1956 1582 y FM(n)2088 1561 y FP(\()p FO(f)9 b FP(\()p FO(n)25 b FP(+)f(1\)\))38 b(=)118 1661 y FO(f)9 b FP(\()p FO(n)p FP(\),)35 b FO(n)e FQ(2)g FJ(N)t FP(.)60 b(Then)880 1639 y(~)871 1661 y FA(A)33 b FP(is)g(a)g(top)r(ological)f FQ(\003)p FP(-algebra)f(endo)n (w)n(ed)i(with)g(the)118 1760 y(w)n(eak)n(est)f(top)r(ology)g(suc)n(h)h (that)g(the)h(maps)f FO(\031)1575 1772 y FM(n)1630 1760 y FP(:)1691 1739 y(~)1682 1760 y FA(A)f FQ(\000)-48 b(!)32 b FA(A)1966 1772 y FM(n)2011 1760 y FP(,)j FO(f)41 b FQ(7!)33 b FO(f)9 b FP(\()p FO(n)p FP(\))33 b(are)118 1860 y(con)n(tin)n(uous.)67 b(W)-7 b(e)38 b(will)g(denote)1203 1838 y(~)1194 1860 y FA(A)g FP(b)n(y)f(lim)1417 1899 y FQ( )-32 b(\000)1546 1860 y FA(A)1606 1872 y FM(n)1651 1860 y FP(.)68 b(W)-7 b(e)38 b(need)g(the)g(follo)n(wing)118 1968 y(statemen)n(t)28 b([156)n(,)g(Lemma)f(3.9].)118 2136 y FR(Lemma)33 b(2.)43 b FC(L)l(et)32 b FO(A)g FC(b)l(e)h(a)g FO(C)1063 2106 y FN(\003)1101 2136 y FC(-algebr)l(a)h(and)e FO(B)37 b FC(b)l(e)c(a)f FO(C)1922 2106 y FN(\003)1961 2136 y FC(-sub)l(algebr)l(a)h(of)g FO(A)p FC(.)118 2236 y(Assume)d(that)h FO(\031)e Fs(\026)c FO(B)k FP(=)c FO(\031)968 2206 y FN(0)1017 2236 y Fs(\026)g FO(B)35 b FC(for)d(any)g(two)f(r)l (epr)l(esentations)g FO(\031)s FC(,)h FO(\031)2359 2206 y FN(0)2414 2236 y FC(of)g FO(A)118 2335 y FC(implies)f(that)f FO(\031)d FP(=)22 b FO(\031)783 2305 y FN(0)807 2335 y FC(.)38 b(In)30 b(this)g(c)l(ase)g FO(B)d FP(=)c FO(A)p FC(.)243 2504 y FP(The)k(follo)n(wing)g(theorem)g(holds.)118 2672 y FR(Theorem)38 b(1.)45 b FC(The)37 b(p)l(air)g FP(\()1045 2650 y(~)1036 2672 y FA(A)o FO(;)14 b(\036)p FP(\))37 b FC(is)f(a)g(unique)g(enveloping)h FO(\033)s FC(-)p FO(C)2255 2642 y FN(\003)2294 2672 y FC(-algebr)l(a)118 2772 y(for)f FA(A)p FC(.)52 b(The)36 b(homomorphism)h FO(\036)e FC(has)g(dense)g(r)l(ange.)53 b(Mor)l(e)l(over)36 b FP(\()2311 2750 y(~)2302 2772 y FA(A)p FO(;)14 b(\036)p FP(\))35 b FC(is)118 2871 y(also)c(an)f(enveloping)h FQ(\003)p FC(-algebr)l(a.)118 3039 y(Pr)l(o)l(of.)43 b FP(Let)37 b(us)g(notice)g(that)h(the)f(homomorphisms)f FO(\031)1924 3051 y FM(n)1978 3039 y FP(:)2041 3018 y(~)2032 3039 y FA(A)j FQ(\000)-49 b(!)39 b FA(A)2329 3051 y FM(n)2411 3039 y FP(ha)n(v)n(e)118 3139 y(dense)24 b(ranges.)34 b(The)24 b(top)r(ology)f(of)h(the)g FO(\033)s FP(-)p FO(C)1509 3109 y FN(\003)1548 3139 y FP(-algebra)1872 3117 y(~)1863 3139 y FA(A)g FP(can)f(b)r(e)i(de\014ned)f(b)n(y)118 3239 y(a)32 b(coun)n(table)g(increasing)f(family)h FO(p)1266 3251 y FM(n)1311 3239 y FP(\()p FQ(\001)p FP(\))h(of)f FO(C)1595 3208 y FN(\003)1634 3239 y FP(-semi-norms.)49 b(Using)32 b(argu-)118 3338 y(men)n(ts)e(similar)g(to)g(Can)n(tor's)f (diagonal)g(metho)r(d,)i(w)n(e)f(can)g(pro)n(v)n(e)f(that)h(it)h(is)118 3438 y(also)26 b(dense)h(in)g(the)g(top)r(ology)f(de\014ned)h(b)n(y)g (the)g(family)g FO(p)1926 3450 y FM(n)1971 3438 y FP(\()p FQ(\001)p FP(\),)h(whic)n(h)f(pro)n(v)n(es)118 3537 y(that)36 b(the)f(homomorphism)f FO(\036)i FP(has)f(dense)g(range.)59 b(Let)35 b FO(\031)k FP(b)r(e)d(a)e(represen-)118 3637 y(tation)h(of)g FA(A)g FP(in)g FO(L)p FP(\()p FO(H)7 b FP(\).)59 b(If)36 b FO(\031)j FQ(2)c FB(R)1277 3649 y FM(n)1323 3637 y FP(,)i(de\014ne)i(~)-46 b FO(\031)39 b FP(=)c FO(F)12 b FP(\()p FO(\031)s FP(\)\()p FO(\031)2074 3649 y FM(n)2121 3637 y FP(\).)59 b(Since)35 b(the)118 3737 y(algebra)25 b FO(\036)p FP(\()p FO(A)p FP(\))k(is)e(dense)f(in)i FB(A)p FP(,)j(~)-46 b FO(\031)30 b FP(is)d(uniquely)g(de\014ned.)37 b(Let)27 b(us)g(pro)n(v)n(e)f(that)118 3836 y(the)i(en)n(v)n(eloping)e FO(\033)s FP(-)p FO(C)812 3806 y FN(\003)851 3836 y FP(-algebra)f(is)j (unique.)243 3937 y(1.)45 b(W)-7 b(e)31 b(sa)n(y)e(that)i FO(\031)875 3949 y FL(1)940 3937 y FQ(\024)c FO(\031)1079 3949 y FL(2)1148 3937 y FP(for)i FO(\031)1324 3949 y FL(1)1362 3937 y FP(,)i FO(\031)1463 3949 y FL(2)1529 3937 y FQ(2)d FP(Rep\()p FA(A)p FP(\))j(i\013)g(k)n(er)13 b FO(\031)2185 3949 y FL(2)2250 3937 y FQ(\022)27 b FP(k)n(er)13 b FO(\031)2514 3949 y FL(1)2552 3937 y FP(.)118 4036 y(Then)25 b(the)h(set)f(Rep)14 b FA(A)24 b FP(is)h(a)g(net)g(since)g FO(\031)1378 4048 y FM(j)1437 4036 y FQ(\024)d FO(\031)1571 4048 y FL(1)1622 4036 y FQ(\010)13 b FO(\031)1747 4048 y FL(2)1785 4036 y FP(.)36 b(Cho)r(ose)24 b(an)n(y)g(co-\014nite)118 4147 y(subnet)k FO(\031)433 4159 y FM(n)479 4147 y FP(,)f(and)h (de\014ne)940 4126 y(~)931 4147 y FA(A)991 4159 y FM(p)1052 4147 y FP(=)22 b(lim)1139 4186 y FQ( )-32 b(\000)p 1268 4075 217 4 v 1268 4147 a FO(\031)1315 4159 y FM(n)1361 4147 y FP(\()p FA(A)p FP(\).)p eop %%Page: 20 24 20 23 bop 118 100 a FP(20)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)243 333 y FP(2.)36 b(Let)27 b(\()533 311 y(~)524 333 y FA(A)p FO(;)14 b(\036)p FP(\))28 b(b)r(e)g(an)f(en)n(v)n(eloping)f FO(\033)s FP(-)p FO(C)1509 303 y FN(\003)1547 333 y FP(-algebra)g(of)h FA(A)p FP(.)36 b(W)-7 b(e)28 b(will)g(pro)n(v)n(e)118 432 y(that)g FO(\036)p FP(\()p FA(A)p FP(\))f(is)h(dense)f(in)913 411 y(~)904 432 y FA(A)p FP(.)36 b(Let)28 b FB(B)f FP(b)r(e)h(a)f FO(C)1509 402 y FN(\003)1547 432 y FP(-algebra)e(and)i FO(j)14 b FP(:)28 b FA(A)23 b FQ(\000)-49 b(!)23 b FB(B)28 b FP(b)r(e)f(a)118 532 y(con)n(tin)n(uous)20 b FQ(\003)p FP(-surjection.)33 b(If)22 b FO(\031)s FP(,)g FO(\031)1229 502 y FN(0)1276 532 y FQ(2)h FP(Rep)14 b FB(B)21 b FP(and)g FO(\031)8 b FQ(\016)d FO(j)10 b FQ(\016)5 b FO(\036)23 b FP(=)f FO(\031)2153 502 y FN(0)2182 532 y FQ(\016)5 b FO(j)10 b FQ(\016)5 b FO(\036)p FP(,)21 b(then)118 632 y(b)n(y)i(the)g(uniqueness)g(of)k(~)-47 b FO(\031)27 b FP(in)c(the)g(de\014nition)h(of)e(the)i(en)n(v)n(eloping)d(algebra,)h (w)n(e)118 731 y(ha)n(v)n(e)h FO(\031)15 b FQ(\016)d FO(j)28 b FP(=)22 b FO(\031)621 701 y FN(0)657 731 y FQ(\016)12 b FO(j)5 b FP(.)35 b(Then)24 b FO(\031)j FP(=)22 b FO(\031)1232 701 y FN(0)1280 731 y FP(since)i FO(j)30 b FP(is)24 b(surjectiv)n(e.)35 b(Then)24 b(Lemma)g(2)118 831 y(implies)i(that)g FO(j)19 b FQ(\016)14 b FO(\036)p FP(\()p FA(A)p FP(\))26 b(is)f(dense)h(in)f FB(B)p FP(.)37 b(Represen)n(t)1801 809 y(~)1792 831 y FA(A)25 b FP(as)g(lim)1977 870 y FQ( )-32 b(\000)2106 831 y FB(B)2169 843 y FM(n)2240 831 y FP(\(see)25 b([208)o(,)118 939 y(Prop)r(osition)i(1.2])h(or)g ([10)o(,)h(Prop)r(osition)e(5.6]\),)i(where)f FB(B)1951 951 y FM(n)2026 939 y FP(is)g(a)h FO(C)2246 909 y FN(\003)2284 939 y FP(-algebra)118 1038 y(and)k FO(j)319 1050 y FM(n)396 1038 y FP(:)460 1017 y(~)451 1038 y FA(A)f FQ(\000)-49 b(!)32 b FB(B)737 1050 y FM(n)816 1038 y FP(is)h(the)g(canonical)f (surjection.)52 b(Previous)31 b(argumen)n(ts)118 1138 y(sho)n(w)c(that)p 504 1066 318 4 v 28 w FO(j)538 1150 y FM(n)584 1138 y FP(\()p FO(\036)p FP(\()p FA(A)p FP(\)\))e(=)e FB(B)997 1150 y FM(n)1042 1138 y FP(.)38 b(F)-7 b(rom)28 b(this)g(it)h(follo)n(ws)e(that)h FO(\036)p FP(\()p FA(A)p FP(\))h(is)e(a)h(dense)118 1238 y FQ(\003)p FP(-algebra)d(in)584 1216 y(~)575 1238 y FA(A)p FP(,)i(since)h FO(z)928 1250 y FM(k)991 1238 y FQ(\000)-48 b(!)23 b FO(z)31 b FP(in)1290 1216 y(~)1281 1238 y FA(A)c FP(i\013)h FO(j)1501 1250 y FM(n)1546 1238 y FP(\()p FO(z)1617 1250 y FM(k)1658 1238 y FP(\))c FQ(\000)-49 b(!)23 b FO(j)1870 1250 y FM(n)1916 1238 y FP(\()p FO(z)t FP(\))k(for)g(all)h FO(n)p FP(.)243 1343 y(3.)77 b(W)-7 b(e)42 b(will)g(pro)n(v)n(e)d(that)1153 1322 y(~)1144 1343 y FA(A)46 b FQ(')1369 1322 y FP(~)1360 1343 y FA(A)1420 1355 y FM(p)1458 1343 y FP(.)78 b(P)n(assing)40 b(if)i(necessary)d(to)i(the)118 1443 y(quotien)n(t)d(with)g(resp)r(ect) g(to)g(the)g(radical,)h(w)n(e)f(can)f(assume)h(that)g FO(\036)g FP(is)g(an)118 1543 y(injection,)29 b(and)g(so)f FO(\036)p FP(\()p FA(A)p FP(\))d FQ(')g FA(A)p FP(.)39 b(Since)p 1375 1470 174 4 v 29 w FO(\036)p FP(\()p FA(A)p FP(\))26 b(=)e FA(A)p FP(,)29 b(w)n(e)f(only)g(need)h(to)g(pro)n(v)n(e) 118 1642 y(that)40 b(the)f(top)r(ology)f FO(\034)853 1654 y FL(1)930 1642 y FP(on)g FO(\036)p FP(\()p FA(A)p FP(\))i(induced)g(from)1807 1621 y(~)1798 1642 y FA(A)e FP(coincides)h(with)g(the)118 1742 y(top)r(ology)30 b FO(\034)499 1754 y FL(2)567 1742 y FP(on)g FO(\036)p FP(\()p FA(A)p FP(\))i(de\014ned)f(b)n(y)g(the)g(semi-norms)e FQ(k)t FP(~)-46 b FO(\031)1975 1754 y FM(n)2020 1742 y FP(\()p FQ(\001)p FP(\))p FQ(k)p FP(.)47 b(Note)31 b(that)118 1842 y(the)21 b(top)r(ology)f FO(\034)625 1854 y FL(1)684 1842 y FP(is)h(de\014ned)g(b)n(y)g(the)g(semi-norms)f FQ(k)5 b(\001)g(k)1835 1854 y FM(n)1901 1842 y FP(induced)21 b(b)n(y)g(faithful)118 1941 y(represen)n(tations)k(of)h FB(B)849 1953 y FM(n)895 1941 y FP(,)h(and)f(since)g(the)h(subnet)g FO(\031)1763 1953 y FM(n)1835 1941 y FP(is)g(co-\014nite)f(in)h(the)g (set)118 2041 y(of)g(all)f(represen)n(tations,)f(the)j(top)r(ology)d FO(\034)1443 2053 y FL(2)1508 2041 y FP(is)h(stronger)f(then)j FO(\034)2138 2053 y FL(1)2175 2041 y FP(.)37 b(But)27 b(since)118 2140 y(ev)n(ery)k(represen)n(tations)f FO(\031)969 2152 y FM(n)1047 2140 y FP(can)i(b)r(e)g(lifted)h(to)1655 2119 y(~)1646 2140 y FA(A)p FP(,)g FO(\034)1798 2152 y FL(1)1868 2140 y FP(is)f(stronger)f(then)h FO(\034)2514 2152 y FL(2)2552 2140 y FP(.)118 2240 y(This)c(pro)n(v)n(es)d(that)j FO(\034)780 2252 y FL(1)841 2240 y FP(=)22 b FO(\034)964 2252 y FL(2)1002 2240 y FP(.)243 2346 y(Since)36 b(the)g(algebra)927 2324 y(~)918 2346 y FA(A)f FP(is)h(a)f(metrizable)g(lo)r(cally)g FO(C)1932 2316 y FN(\003)1971 2346 y FP(-algebra,)g(b)n(y)h([85)o(,)118 2446 y(Corollary)20 b(4.7],)i(ev)n(ery)f FQ(\003)p FP(-represen)n (tation)f(of)1574 2424 y(~)1565 2446 y FA(A)i FP(is)g(con)n(tin)n (uous.)34 b(And)22 b(th)n(us)27 b(~)-47 b FO(\031)118 2545 y FP(is)19 b(uniquely)f(de\014ned)h(ev)n(en)f(without)h(the)g (requiremen)n(t)f(of)g(b)r(eing)h(con)n(tin)n(uous.)118 2645 y(It)41 b(pro)n(v)n(es)d(that)j(the)f FQ(\003)p FP(-algebra)1220 2623 y(~)1211 2645 y FA(A)g FP(is)g(also)f(an)h(en)n (v)n(eloping)f FQ(\003)p FP(-algebra)f(of)118 2744 y FA(A)p FP(.)p 2514 2744 4 57 v 2518 2692 50 4 v 2518 2744 V 2567 2744 4 57 v 243 2960 a(Note)i(that)g(the)h(homomorphism)e FO(\036)i FP(is)f(an)g(injection)h(if)f(and)h(only)e(if)118 3060 y FA(A)34 b FP(has)g(a)f(r.f.)i(of)f(represen)n(tations.)55 b(In)34 b(this)h(case)e(it)i(is)f(natural)f(to)h(call)g FA(A)118 3159 y FO(\033)s FP(-)p FO(C)261 3129 y FN(\003)300 3159 y FP(-represen)n(table.)118 3305 y FC(R)l(emark)i FP(4)p FC(.)i FP(In)24 b(the)h(case)e(where)h FA(A)g FP(is)g(not)g(\014nitely)h(generated,)e(w)n(e)h(can)g(also)118 3404 y(construct)35 b(an)h(en)n(v)n(eloping)f(pro-)p FO(C)1246 3374 y FN(\003)1283 3404 y FP(-algebra)f(\(the)i(index)g FO(n)g FP(in)g(the)g(ab)r(o)n(v)n(e)118 3504 y(construction)d(should)h (b)r(e)g(replaced)e(b)n(y)i(the)g(m)n(ulti-index)g FO(\013)g FQ(2)f FJ(N)2271 3474 y Fv(I)2304 3504 y FP(,)i(where)118 3604 y FQ(f)p FO(x)207 3616 y FM(\013)254 3604 y FP(;)14 b FO(\013)24 b FQ(2)f FJ(I)-7 b FQ(g)14 b FP(is)20 b(the)h(set)f(of)g (generators)e(of)i FA(A)p FP(\).)35 b(In)20 b(suc)n(h)g(a)g(case,)h (all)f(the)g(ab)r(o)n(v)n(e)118 3703 y(statemen)n(ts)41 b(hold)f(true)h(except)f(that)h(the)g FQ(\003)p FP(-algebra)d(is)j(not) g(necessarily)118 3803 y(en)n(v)n(eloping,)22 b(and)g(the)g(range)f(of) h(the)h(homomorphism)e FO(\036)h FP(need)g(not)h(b)r(e)f(dense)118 3902 y(but)42 b(only)f(quasi-dense,)i(i.e.,)i(suc)n(h)c(that,)k(for)c (an)n(y)f(represen)n(tation)f FO(\031)49 b FQ(2)118 4002 y FP(Rep\()p FA(A)p FP(\),)28 b(the)g(set)g FO(\031)s FP(\()p FO(\036)p FP(\()882 3980 y(~)873 4002 y FA(A)q FP(\)\))g(is)f(dense)h(in)g(Im)14 b FO(\031)s FP(.)118 4147 y FR(4.)36 b FP(It)28 b(is)g(con)n(v)n(enien)n(t)e(to)i(adopt)f (the)h(follo)n(wing)e(de\014nition:)p eop %%Page: 21 25 21 24 bop 118 100 a FK(1.1.)36 b(In)n(tro)r(duction)27 b(to)h(represen)n(tations)d(of)j FQ(\003)p FK(-algebras)586 b FP(21)118 333 y FR(De\014nition)28 b(2.)39 b FC(We)27 b(wil)t(l)i(say)f(that)f(a)h FQ(\003)p FC(-algebr)l(a)g(is)g(of)g(typ)l (e)g(I)40 b FP(\()p FC(nucle)l(ar)9 b FP(\))28 b FC(i\013)118 432 y FA(A)178 444 y FM(n)253 432 y FC(is)i(of)g(typ)l(e)g(I)43 b FP(\()p FC(nucle)l(ar)9 b FP(\))30 b FC(for)h(al)t(l)g FO(n)23 b FQ(2)g FJ(N)t FC(.)118 646 y FR(1.1.4)94 b FQ(\003)p FR(-Represen)m(tations)30 b(of)i(generators)f(and)i (relations)118 799 y(1.)j FP(T)-7 b(o)27 b(an)n(y)g FQ(\003)p FP(-represen)n(tation)e(of)j(a)f(\014nitely)h(generated)f FQ(\003)p FP(-algebra)376 970 y FA(B)c FP(=)g FJ(C)614 902 y Fz(\012)659 970 y FO(x)706 982 y FL(1)744 970 y FO(;)14 b(:)g(:)g(:)f(;)h(x)975 982 y FM(n)1021 970 y FO(;)g(x)1105 935 y FN(\003)1105 990 y FL(1)1143 970 y FO(;)g(:)g(:)g(:)g(;)g(x)1375 935 y FN(\003)1375 990 y FM(n)1444 970 y FQ(j)698 1103 y FO(P)751 1115 y FM(j)787 1103 y FP(\()p FO(x)866 1115 y FL(1)904 1103 y FO(;)g(:)g(:)g(:)f(;)h (x)1135 1115 y FM(n)1181 1103 y FO(;)g(x)1265 1069 y FN(\003)1265 1124 y FL(1)1303 1103 y FO(;)g(:)g(:)g(:)g(;)g(x)1535 1069 y FN(\003)1535 1124 y FM(n)1580 1103 y FP(\))24 b(=)e(0)p FO(;)41 b(j)28 b FP(=)23 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(m) 2278 1036 y Fz(\013)118 1274 y FP(b)n(y)25 b(b)r(ounded)h(op)r(erators) e(there)h(corresp)r(onds)f(a)h(family)h(of)f(b)r(ounded)h(op)r(era-)118 1373 y(tors)h FQ(f)p FO(X)396 1385 y FM(i)446 1373 y FP(=)22 b FO(\031)s FP(\()p FO(x)662 1385 y FM(i)691 1373 y FP(\))p FO(;)14 b(X)836 1343 y FN(\003)829 1395 y FM(i)897 1373 y FP(=)22 b FO(\031)s FP(\()p FO(x)1113 1385 y FM(i)1142 1373 y FP(\))1174 1343 y FN(\003)1236 1373 y FP(=)g FO(\031)s FP(\()p FO(x)1452 1343 y FN(\003)1452 1395 y FM(i)1492 1373 y FP(\))p FQ(g)1566 1343 y FM(n)1566 1395 y(i)p FL(=1)1705 1373 y FP(suc)n(h)27 b(that)345 1544 y FO(P)398 1556 y FM(j)433 1544 y FP(\()p FO(X)534 1556 y FL(1)571 1544 y FO(;)14 b(:)g(:)g(:)g(;)g(X)825 1556 y FM(n)870 1544 y FO(;)g(X)983 1510 y FN(\003)976 1564 y FL(1)1020 1544 y FO(;)g(:)g(:)g(:)g(;)g(X)1281 1510 y FN(\003)1274 1564 y FM(n)1319 1544 y FP(\))23 b(=)g(0)p FO(;)179 b(j)28 b FP(=)23 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(m:) 226 b FP(\(1.4\))118 1714 y(Con)n(v)n(ersely)-7 b(,)36 b(a)f(family)h(of)g(b)r(ounded)g(op)r(erators)e FQ(f)p FO(X)1841 1726 y FM(i)1868 1714 y FO(;)14 b(X)1981 1684 y FN(\003)1974 1736 y FM(i)2018 1714 y FQ(g)2060 1684 y FM(n)2060 1736 y(i)p FL(=1)2172 1714 y FP(,)38 b(satisfying)118 1814 y(\(1.4\),)k(can)d(b)r(e)h(uniquely)g(extended)g(to)f(a)g (represen)n(tation)f(of)h(the)h(whole)118 1913 y FQ(\003)p FP(-algebra)33 b FA(B)p FP(.)59 b(F)-7 b(or)34 b(an)n(y)h(\014nitely)g (presen)n(ted)g FQ(\003)p FP(-algebra,)f(one)g(can)h(c)n(ho)r(ose)118 2013 y(self-adjoin)n(t)30 b(generators)f FO(a)998 2025 y FM(i)1054 2013 y FP(=)f FO(a)1191 1983 y FN(\003)1191 2035 y FM(i)1229 2013 y FP(,)k FO(i)c FP(=)g(1,)j FO(:)14 b(:)g(:)27 b FP(,)32 b FO(l)h FP(\(their)e(n)n(um)n(b)r(er)f(ma)n(y)g (b)r(e)118 2113 y(larger)f(than)h FO(n)p FP(\),)i(connected)e(b)n(y)g (self-adjoin)n(t)g(relations)f FO(Q)2034 2125 y FM(j)2069 2113 y FP(\()p FO(a)2145 2125 y FL(1)2182 2113 y FO(;)14 b(:)g(:)g(:)28 b(;)14 b(a)2425 2125 y FM(l)2450 2113 y FP(\))28 b(=)118 2212 y FO(Q)184 2182 y FN(\003)184 2234 y FM(j)222 2212 y FP(\()p FO(a)298 2224 y FL(1)335 2212 y FO(;)14 b(:)g(:)g(:)g(;)g(a)564 2224 y FM(l)589 2212 y FP(\),)25 b FO(l)g FP(=)d(1,)i FO(:)14 b(:)g(:)27 b FP(,)e FO(r)i FP(\(their)d(n)n(um)n(b)r(er)f(ma)n(y)g(also)g (increase\);)h(there-)118 2312 y(fore,)37 b(an)n(y)e(represen)n(tation) f FO(\031)39 b FP(of)c(the)h(algebra)e FA(B)j FP(=)e FJ(C)15 b FQ(h)q FO(a)2010 2324 y FL(1)2053 2312 y FO(;)f(:)g(:)g(:)f (;)h(a)2281 2324 y FM(l)2343 2312 y FQ(j)36 b FO(a)2446 2324 y FM(i)2510 2312 y FP(=)118 2412 y FO(a)162 2381 y FN(\003)162 2433 y FM(i)200 2412 y FO(;)14 b(i)34 b FP(=)f(1)p FO(;)14 b(:)g(:)g(:)f(;)h(l)r FP(;)27 b FO(Q)767 2424 y FM(j)802 2412 y FP(\()p FO(a)878 2424 y FL(1)915 2412 y FO(;)14 b(:)g(:)g(:)g(;)g(a)1144 2424 y FM(l)1169 2412 y FP(\))34 b(=)g(0)p FO(;)27 b(j)39 b FP(=)33 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(r)r FQ(i)35 b FP(is)f(uniquely)g(deter-)118 2511 y(mined)c(b)n(y)e(a)h(family)g(of)g(self-adjoin)n(t)g(op)r (erators)e FO(A)1767 2523 y FM(i)1821 2511 y FP(=)e FO(A)1973 2481 y FN(\003)1973 2533 y FM(i)2037 2511 y FP(=)g FO(\031)s FP(\()p FO(a)2253 2523 y FM(i)2281 2511 y FP(\),)30 b FO(i)25 b FP(=)g(1,)118 2611 y FO(:)14 b(:)g(:)28 b FP(,)f FO(l)r FP(,)h(suc)n(h)f(that)683 2781 y FO(Q)749 2793 y FM(j)784 2781 y FP(\()p FO(A)878 2793 y FL(1)916 2781 y FO(;)14 b(:)g(:)g(:)f(;)h(A)1162 2793 y FM(l)1188 2781 y FP(\))23 b(=)g(0)p FO(;)180 b(j)28 b FP(=)22 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(r)n(:)395 b FP(\(1.5\))158 2952 y FR(2.)59 b FP(Since)35 b(the)g(prop)r(erties)f(of)h(the)h (represen)n(tation)d(of)i(an)g(algebra)e(\(irre-)118 3051 y(ducibilit)n(y)e(etc.\))47 b(are)30 b(completely)g(determined)h (b)n(y)g(the)g(represen)n(tation)e(of)118 3151 y(its)c(generators,)e (in)i(what)g(follo)n(ws,)f(w)n(e)h(will)g(use)g(an)f(equiv)-5 b(alen)n(t)25 b(language)e(of)118 3251 y(represen)n(tations)j(of)h (relation)g(\(1.5\))g(b)n(y)h(b)r(ounded)g(self-adjoin)n(t)f(op)r (erators.)243 3350 y(In)42 b(studying)g(families)g(of)h(self-adjoin)n (t)e(op)r(erators)g FO(A)2028 3362 y FL(1)2065 3350 y FP(,)h FO(:)14 b(:)g(:)28 b FP(,)46 b FO(A)2386 3362 y FM(n)2432 3350 y FP(,)g(as)118 3450 y(usual,)26 b(the)f(role)g(of)g (the)h(simplest)f(families)h(of)f(op)r(erators)e(is)j(pla)n(y)n(ed)e(b) n(y)h(irre-)118 3550 y(ducible)g(ones.)36 b(A)25 b(family)g(of)f (self-adjoin)n(t)h(op)r(erators)d FO(A)1898 3562 y FM(k)1963 3550 y FP(=)2050 3483 y Fz(R)2089 3579 y Fv(R)2149 3550 y FO(\025)2197 3562 y FM(k)2253 3550 y FO(dE)2357 3562 y FM(k)2398 3550 y FP(\()p FO(\025)2478 3562 y FM(k)2519 3550 y FP(\),)118 3649 y FO(k)k FP(=)d(1,)j FO(:)14 b(:)g(:)27 b FP(,)g FO(n)p FP(,)g(is)g(irreducible,)f(if)h(there)g(is)f(no)h (non-trivial)e(\(di\013eren)n(t)i(from)118 3749 y FO(H)42 b FP(and)35 b FQ(f)p FP(0)p FQ(g)p FP(\))f(subspace)g(in)h FO(H)42 b FP(in)n(v)-5 b(arian)n(t)34 b(with)h(resp)r(ect)g(to)g(all)g (op)r(erators)118 3848 y FO(E)179 3860 y FM(k)220 3848 y FP(\(\001\),)i FO(k)f FP(=)d(1,)g FO(:)14 b(:)g(:)28 b FP(,)35 b FO(n)p FP(;)i(\001)d FQ(2)g FA(B)p FP(\()p FJ(R)1332 3818 y FL(1)1376 3848 y FP(\).)56 b(If)34 b(the)g(op)r (erators)e(of)i(the)g(family)118 3948 y(are)28 b(b)r(ounded,)h(the)g (irreducibilit)n(y)f(of)g(the)h(family)g(means)f(that)g(there)h(is)f (no)118 4048 y(non-trivial)f(subspace)g(in)i FO(H)7 b FP(,)28 b(in)n(v)-5 b(arian)n(t)27 b(with)h(resp)r(ect)g(to)g(all)g(op) r(erators)e(of)118 4147 y(the)i(family)g(\()p FO(A)609 4159 y FM(k)650 4147 y FP(\))682 4117 y FM(n)682 4171 y(k)q FL(=1)807 4147 y FP(.)p eop %%Page: 22 26 22 25 bop 118 100 a FP(22)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)243 333 y FP(The)32 b(follo)n(wing)g(condition)g(is)h(equiv)-5 b(alen)n(t)33 b(to)f(irreducibilit)n(y:)47 b(a)32 b(collec-)118 432 y(tion)g(of)g(self-adjoin)n(t)f(op)r(erators)f(\()p FO(A)1289 444 y FM(k)1330 432 y FP(\))1362 402 y FM(n)1362 456 y(k)q FL(=1)1519 432 y FP(is)i(irreducible)f(if)h(an)n(y)f(b)r (ounded)118 532 y(op)r(erator)24 b FO(C)31 b FP(comm)n(uting)25 b(with)h(all)f FO(A)1332 544 y FM(k)1373 532 y FP(,)h FO(k)g FP(=)c(1,)j FO(:)14 b(:)g(:)28 b FP(,)e FO(n)f FP(\(i.e.,)h(with)g(all)f(their)118 632 y(sp)r(ectral)i(pro)5 b(jections\),)27 b(is)g(a)g(m)n(ultiple)i(of)e(the)h(iden)n(tit)n(y)g (op)r(erator.)141 768 y FR(3.)33 b FP(F)-7 b(or)19 b(a)h(single)f(b)r (ounded)h(self-adjoin)n(t)f(op)r(erator)f FO(A)23 b FP(=)g FO(A)1987 738 y FN(\003)2025 768 y FP(,)f(its)e(irreducibil-)118 867 y(it)n(y)30 b(means)g(that)h(dim)14 b FO(H)34 b FP(=)27 b(1,)k(and)f(this)g(op)r(erator)f(is)h(a)g(m)n(ultiplication)g(b)n(y) 118 967 y(a)g(constan)n(t,)h FO(A)d FP(=)f FO(\025)p FP(,)32 b FO(\025)c FQ(2)g FJ(R)p FP(,)38 b(and)30 b(the)h(sp)r(ectral) f(theorem)g(for)f(a)h(b)r(ounded)118 1067 y(self-adjoin)n(t)37 b(op)r(erator)f(giv)n(es)g(its)i(decomp)r(osition)e(in)n(to)i (irreducible)e(ones:)118 1181 y FO(A)42 b FP(=)328 1114 y Fz(R)384 1135 y FN(k)p FM(A)p FN(k)368 1211 y(\000k)p FM(A)p FN(k)555 1181 y FO(\025)14 b(dE)721 1193 y FM(A)776 1181 y FP(\()p FO(\025)p FP(\),)43 b(where)38 b FO(E)1266 1193 y FM(A)1321 1181 y FP(\()p FQ(\001)p FP(\))h(is)g(the)g(sp)r (ectral)f(measure)g(of)g(the)118 1290 y(op)r(erator)26 b FO(A)p FP(.)166 1426 y FR(4.)77 b FP(Irreducible)40 b(represen)n(tations)g(of)h(a)g(pair)f(of)i(b)r(ounded)f(self-adjoin)n (t)118 1526 y(op)r(erators)26 b(exist)h(in)h(a)f(Hilb)r(ert)h(space)f (of)h(arbitrary)d(dimension.)243 1626 y(Let)i(dim)14 b FO(H)30 b FP(=)22 b FO(n)p FP(,)28 b FO(e)869 1638 y FL(1)905 1626 y FP(,)g FO(:)14 b(:)g(:)27 b FP(,)g FO(e)1169 1638 y FM(n)1214 1626 y FP(,)h(b)r(e)f(an)g(orthonormal)e (basis)h(in)h FO(H)7 b FP(.)37 b(T)-7 b(ak)n(e)118 1725 y(op)r(erators)22 b FO(A)j FP(and)f FO(B)29 b FP(suc)n(h)24 b(that,)h(in)g(the)g(basis)e(\()p FO(e)1708 1737 y FM(k)1749 1725 y FP(\))1781 1695 y FM(n)1781 1749 y(k)q FL(=1)1906 1725 y FP(,)j(they)e(are)g(giv)n(en)f(b)n(y)118 1825 y(the)28 b(matrices)314 2081 y FO(A)23 b FP(=)487 1889 y Fz(0)487 2036 y(B)487 2089 y(@)560 1953 y FO(\025)608 1965 y FL(1)934 1953 y FP(0)733 2050 y(.)765 2075 y(.)797 2100 y(.)582 2208 y(0)284 b FO(\025)956 2220 y FM(n)1002 1889 y Fz(1)1002 2036 y(C)1002 2089 y(A)1088 2081 y FO(;)97 b(B)27 b FP(=)c(\()p FO(b)1454 2093 y FM(ij)1512 2081 y FP(\))p FO(;)181 b(b)1784 2093 y FM(ij)1865 2081 y FP(=)1949 2059 y(\026)1952 2081 y FO(b)1988 2093 y FM(j)s(i)2046 2081 y FO(;)42 b(\025)2159 2093 y FM(j)2218 2081 y FQ(2)23 b FJ(R)p FO(;)118 2354 y FP(where)k FO(\025)406 2366 y FM(i)457 2354 y FQ(6)p FP(=)c FO(\025)593 2366 y FM(j)628 2354 y FP(,)28 b FO(i)23 b FQ(6)p FP(=)f FO(j)5 b FP(,)28 b(and)g FQ(8)p FO(i)e FP(there)h(exists)h FO(j)5 b FP(,)27 b FO(i)c FQ(6)p FP(=)g FO(j)5 b FP(,)27 b(suc)n(h)g(that)h FO(b)2335 2366 y FM(ij)2416 2354 y FQ(6)p FP(=)23 b(0.)243 2453 y(The)k(pair)h(of)f(self-adjoin)n(t)h(op)r(erators)d FO(A)p FP(,)k FO(B)t FP(,)f(is)f(irreducible.)37 b(Indeed,)28 b(if)118 2553 y FO(C)34 b FP(=)27 b(\()p FO(c)371 2565 y FM(ij)430 2553 y FP(\))462 2523 y FM(n)462 2575 y(i;j)s FL(=1)655 2553 y FP(is)j(a)g(matrix)g(comm)n(uting)g(with)h(the)f(op)r (erators)f FO(A)h FP(and)h FO(B)t FP(,)118 2653 y(then)d(the)g (condition)g([)p FO(A;)14 b(C)6 b FP(])23 b(=)g(0)k(giv)n(es)922 2930 y FO(C)i FP(=)1098 2738 y Fz(0)1098 2884 y(B)1098 2937 y(@)1170 2802 y FO(c)1206 2814 y FL(11)1580 2802 y FP(0)1364 2898 y(.)1397 2923 y(.)1429 2948 y(.)1203 3056 y(0)294 b FO(c)1575 3068 y FM(nn)1662 2738 y Fz(1)1662 2884 y(C)1662 2937 y(A)1748 2930 y FO(;)118 3207 y FP(and)40 b([)p FO(C)q(;)14 b(B)t FP(])44 b(=)e(0)e(implies)f FO(c)1064 3219 y FL(11)1178 3207 y FP(=)k FQ(\001)14 b(\001)g(\001)43 b FP(=)f FO(c)1569 3219 y FM(nn)1699 3207 y FP(=)g FO(c)p FP(,)h(i.e.,)g FO(C)50 b FP(=)42 b FO(cI)7 b FP(,)43 b(and)118 3306 y(therefore,)27 b(the)h(pair)f FO(A)p FP(,)h FO(B)k FP(is)27 b(irreducible.)243 3406 y(No)n(w)c(let)i FO(H)31 b FP(b)r(e)24 b(a)g(separable)e(in\014nite-dimensional)i(Hilb)r (ert)h(space,)f(and)118 3506 y(let)32 b(\()p FO(e)313 3518 y FM(k)354 3506 y FP(\))386 3475 y FN(1)386 3529 y FM(k)q FL(=1)543 3506 y FP(b)r(e)g(an)g(orthonormal)e(basis)h(in)h FO(H)7 b FP(.)50 b(A)32 b(pair)f(of)h(b)r(ounded)g(self-)118 3605 y(adjoin)n(t)27 b(op)r(erators)f(ha)n(ving)h(the)h(follo)n(wing)e (matrix)h(represen)n(tation)630 3955 y FO(A)c FP(=)803 3689 y Fz(0)803 3835 y(B)803 3884 y(B)803 3934 y(B)803 3984 y(B)803 4037 y(@)875 3749 y FO(\025)923 3761 y FL(1)1428 3749 y FP(0)1049 3846 y(.)1081 3871 y(.)1113 3896 y(.)1224 4004 y FO(\025)1272 4016 y FM(n)897 4159 y FP(0)1405 4101 y(.)1437 4126 y(.)1470 4151 y(.)1497 3689 y Fz(1)1497 3835 y(C)1497 3884 y(C)1497 3934 y(C)1497 3984 y(C)1497 4037 y(A)1584 3955 y FO(;)97 b(B)27 b FP(=)c(\()p FO(b)1950 3967 y FM(ij)2008 3955 y FP(\))p FO(;)p eop %%Page: 23 27 23 26 bop 118 100 a FK(1.1.)36 b(In)n(tro)r(duction)27 b(to)h(represen)n(tations)d(of)j FQ(\003)p FK(-algebras)586 b FP(23)118 333 y(where)34 b FO(\025)413 345 y FM(i)474 333 y FQ(6)p FP(=)g FO(\025)621 345 y FM(j)656 333 y FP(,)i FO(i)d FQ(6)p FP(=)h FO(j)5 b FP(;)37 b FQ(j)p FO(\025)1046 345 y FM(k)1088 333 y FQ(j)c(\024)h FO(C)40 b(<)33 b FQ(1)p FP(,)j FO(k)h FP(=)c(1,)i(2,)f FO(:)14 b(:)g(:)28 b FP(;)37 b FO(b)2180 345 y FM(ij)2272 333 y FP(=)2367 311 y(\026)2370 333 y FO(b)2406 345 y FM(j)s(i)2464 333 y FP(,)f FO(i)p FP(,)118 432 y FO(j)30 b FP(=)24 b(1,)k(2,)h FO(:)14 b(:)g(:)27 b FP(;)i FQ(8)p FO(i)24 b FQ(6)p FP(=)g FO(j)33 b FQ(9)p FO(b)972 444 y FM(ij)1055 432 y FQ(6)p FP(=)24 b(0;)1238 370 y Fz(P)1326 391 y FN(1)1326 457 y FM(j)s FL(=1)1459 432 y FQ(j)p FO(b)1518 444 y FM(ij)1576 432 y FQ(j)1599 402 y FL(2)1661 432 y FQ(\024)g FO(K)30 b(<)24 b FQ(1)29 b(8)p FO(i)23 b FP(=)i(1,)j(2,)g FO(:)14 b(:)g(:)28 b FP(,)118 532 y(is)g(irreducible.) 118 669 y FR(5.)36 b FP(In)26 b(general,)e(it)i(is)g(not)g(necessary)e (that)i(irreducible)f(pairs)g(connected)g(b)n(y)118 769 y(the)j(relation)f(\(1.5\))g(exist)h(in)f(ev)n(ery)g(dimension.)243 868 y(P)n(airs)c(of)i(comm)n(uting)f(b)r(ounded)i(self-adjoin)n(t)e(op) r(erators)f FO(A)g FP(=)g FO(A)2333 838 y FN(\003)2371 868 y FP(,)j FO(B)h FP(=)118 968 y FO(B)185 938 y FN(\003)223 968 y FP(,)39 b FO(AB)j FP(=)c FO(B)t(A)p FP(,)h(ha)n(v)n(e)c(only)h (one-dimensional)f(irreducible)h(represen)n(ta-)118 1068 y(tions,)45 b(dim)14 b FO(H)53 b FP(=)45 b(1,)g FO(A)h FP(=)g FO(\025)1123 1080 y FL(1)1160 1068 y FP(,)f FO(B)50 b FP(=)c FO(\025)1500 1080 y FL(2)1538 1068 y FP(,)f(\()p FO(\025)1686 1080 y FL(1)1724 1068 y FO(;)14 b(\025)1809 1080 y FL(2)1846 1068 y FP(\))46 b FQ(2)h FJ(R)2080 1080 y FL(2)2123 1068 y FP(.)78 b(The)42 b(join)n(t)118 1167 y(sp)r(ectral)35 b(measure)f FO(E)830 1182 y FL(\()p FM(A)906 1190 y Fy(1)939 1182 y FM(;A)1009 1190 y Fy(2)1041 1182 y FL(\))1071 1167 y FP(\()p FQ(\001)p FO(;)14 b FQ(\001)p FP(\))37 b(=)f FO(E)1417 1179 y FM(A)1467 1187 y Fy(1)1504 1167 y FP(\()p FQ(\001)p FP(\))24 b FQ(\012)f FO(E)1764 1179 y FM(A)1814 1187 y Fy(2)1851 1167 y FP(\()p FQ(\001)p FP(\))36 b(on)f(the)h(plane)g FJ(R)2532 1137 y FL(2)118 1267 y FP(giv)n(es)42 b(a)h(decomp)r(osition)g(of)g(the)h (pair)e FO(A)1502 1279 y FL(1)1589 1267 y FP(=)1703 1200 y Fz(R)1742 1296 y Fv(R)1789 1280 y Fy(2)1834 1267 y FO(\025)1882 1279 y FL(1)1934 1267 y FO(dE)2038 1282 y FL(\()p FM(A)2114 1290 y Fy(1)2147 1282 y FM(;A)2217 1290 y Fy(2)2249 1282 y FL(\))2279 1267 y FP(\()p FO(\025)2359 1279 y FL(1)2397 1267 y FO(;)14 b(\025)2482 1279 y FL(2)2519 1267 y FP(\),)118 1367 y FO(A)180 1379 y FL(2)241 1367 y FP(=)328 1300 y Fz(R)368 1396 y Fv(R)415 1379 y Fy(2)460 1367 y FO(\025)508 1379 y FL(2)559 1367 y FO(dE)663 1382 y FL(\()p FM(A)739 1390 y Fy(1)772 1382 y FM(;A)842 1390 y Fy(2)874 1382 y FL(\))904 1367 y FP(\()p FO(\025)984 1379 y FL(1)1022 1367 y FO(;)g(\025)1107 1379 y FL(2)1145 1367 y FP(\))28 b(in)n(to)f(irreducible)g(ones.)118 1504 y FR(6.)59 b FP(It)36 b(ma)n(y)f(happ)r(en)g(that)h(there)f(are)f(no)h (pairs)f(of)i(b)r(ounded)f(self-adjoin)n(t)118 1603 y(op)r(erators)26 b FO(A)p FP(,)i FO(B)t FP(,)g(connected)f(b)n(y)g(relation)g(\(1.5\))g (at)h(all.)243 1703 y(F)-7 b(or)23 b(example,)h(there)f(are)f(no)i(b)r (ounded)g(pairs)e(of)i(self-adjoin)n(t)f(op)r(erators)118 1803 y FO(A)p FP(,)33 b FO(B)j FP(\(in)c(particular,)f(no)h (irreducible)e(pairs\),)i(connected)g(b)n(y)f(the)h(canon-)118 1902 y(ical)g(comm)n(utation)g(relations)f(\(CCR\),)h([)p FO(A;)14 b(B)t FP(])32 b(=)e FO(iI)7 b FP(.)50 b(Indeed,)34 b(otherwise,)118 2002 y(follo)n(wing,)27 b(e.g.,)g([235)o(],)g(w)n(e)h (w)n(ould)f(ha)n(v)n(e)912 2154 y FO(A)974 2120 y FM(n)1019 2154 y FO(B)c FQ(\000)18 b FO(B)t(A)1317 2120 y FM(n)1386 2154 y FP(=)k FO(i)14 b(nA)1628 2120 y FM(n)p FN(\000)p FL(1)1758 2154 y FO(;)118 2306 y FP(and)687 2458 y FO(n)g FQ(k)p FO(A)855 2424 y FM(n)p FN(\000)p FL(1)985 2458 y FQ(k)22 b FP(=)h FO(n)14 b FQ(k)p FO(A)p FQ(k)1347 2424 y FM(n)p FN(\000)p FL(1)1499 2458 y FQ(\024)23 b FP(2)14 b FQ(k)p FO(A)p FQ(k)1789 2424 y FM(n)1833 2458 y FQ(k)p FO(B)t FQ(k)p FO(:)118 2610 y FP(Since)35 b FQ(k)p FO(A)p FQ(k)e(6)p FP(=)h(0,)h(the)g(latter)f(implies)g FQ(k)p FO(A)p FQ(k)14 b(k)p FO(B)t FQ(k)33 b(\025)g FO(n=)p FP(2)g(for)h(all)g FO(n)p FP(,)i(whic)n(h)118 2710 y(con)n(tradicts)26 b(the)i(assumption)g(that)f FO(A)h FP(and)g FO(B)k FP(are)26 b(b)r(ounded.)243 2810 y(The)c(fact)g(that)h(pairs)e(of)h(op)r(erators) f(satisfying)g(the)i(CCR)f(pla)n(y)g(a)g(crucial)118 2909 y(role)35 b(in)h(mo)r(dels)f(of)g(mathematical)h(ph)n(ysics)e (stresses)h(the)h(need)f(to)h(study)118 3009 y(b)r(oth)i(b)r(ounded)g (and)g(un)n(b)r(ounded)g(families)g(of)g(op)r(erators)d(satisfying)i (the)118 3109 y(relations.)118 3246 y FR(7.)68 b FP(As)38 b(is)g(commonly)g(accepted)g(in)g(represen)n(tation)f(theory)-7 b(,)40 b(collections)118 3345 y(of)27 b(op)r(erators)e(are)h(studied)i (up)f(to)g(unitary)g(equiv)-5 b(alence.)36 b(Tw)n(o)27 b(collections,)118 3445 y(\()p FO(A)212 3457 y FM(k)254 3445 y FP(\))286 3415 y FM(n)286 3469 y(k)q FL(=1)431 3445 y FP(on)20 b(a)f(Hilb)r(ert)i(space)f FO(H)7 b FP(,)21 b(and)f(\()1422 3424 y(~)1400 3445 y FO(A)1462 3457 y FM(k)1503 3445 y FP(\))1535 3415 y FM(n)1535 3469 y(k)q FL(=1)1681 3445 y FP(on)g(a)f(Hilb)r(ert)i(space)2365 3424 y(~)2343 3445 y FO(H)7 b FP(,)22 b(are)118 3553 y(unitarily)33 b(equiv)-5 b(alen)n(t,)34 b(if)f(there)f(exists)h(a)f (unitary)h(op)r(erator)e FO(U)18 b FP(:)29 b FO(H)39 b FQ(\000)-49 b(!)2521 3532 y FP(~)2499 3553 y FO(H)118 3653 y FP(suc)n(h)27 b(that)h(the)g(diagrams)1105 3826 y FO(H)1303 3778 y FM(A)1353 3787 y Fw(k)1222 3826 y FQ(\000)-28 b(\000)-19 b(\000)g(\000)-28 b(!)42 b FO(H)1063 3991 y FM(U)1115 3897 y Fz(?)1115 3947 y(?)1115 3997 y(y)1523 3897 y(?)1523 3947 y(?)1523 3997 y(y)1578 3991 y FM(U)1126 4161 y FP(~)1105 4182 y FO(H)1321 4118 y FL(~)1303 4133 y FM(A)1353 4142 y Fw(k)1222 4182 y FQ(\000)-28 b(\000)-19 b(\000)g(\000)-28 b(!)1534 4161 y FP(~)1513 4182 y FO(H)p eop %%Page: 24 28 24 27 bop 118 100 a FP(24)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(are)g(comm)n(utativ)n(e)g(for)g(all)g FO(k)f FP(=)c(1,)28 b FO(:)14 b(:)g(:)27 b FP(,)h FO(n)p FP(,)g(i.e.,)f FO(U)9 b(A)1804 345 y FM(k)1868 333 y FP(=)1978 312 y(~)1956 333 y FO(A)2018 345 y FM(k)2059 333 y FO(U)g FP(.)243 432 y(The)28 b(description)h(of)f(b)r(ounded)h(represen)n(tations)e(of) i(a)f FQ(\003)p FP(-algebra)e FA(B)k FP(up)118 532 y(to)39 b(unitary)f(equiv)-5 b(alence)39 b(is)g(the)g(same)g(as)f(the)h (description)g(of)f(b)r(ounded)118 632 y(represen)n(tations)26 b(of)h(the)h(generators)d(up)j(to)g(unitary)f(equiv)-5 b(alence.)243 731 y(The)35 b(basic)g(problem)f(of)i FQ(\003)p FP(-represen)n(tation)c(theory)j(is)g(to)g(describ)r(e)g(all)118 831 y(irreducible)f(families)h(of)g(b)r(ounded)g(self-adjoin)n(t)f(op)r (erators)f FO(A)2165 843 y FL(1)2202 831 y FP(,)i FO(:)14 b(:)g(:)28 b FP(,)36 b FO(A)2506 843 y FM(n)2552 831 y FP(,)118 930 y(satisfying)d(the)h(giv)n(en)f(relations)f(\(1.4\),)j (up)f(to)g(unitary)f(equiv)-5 b(alence;)36 b(and)118 1030 y(this)28 b(is)f(the)h(task)g(w)n(e)f(will)h(b)r(e)g(dealing)f (with)h(in)f(the)h(sequel.)118 1245 y FR(1.1.5)94 b(P)m(airs)47 b(of)g(self-adjoin)m(t)f(op)s(erators)h(satisfying)f(quadratic)410 1345 y(relations)118 1498 y FP(In)41 b(this)h(c)n(hapter,)i(w)n(e)c (will)i(study)-7 b(,)44 b(in)e(particular,)h(pairs)d(of)h(self-adjoin)n (t)118 1598 y(b)r(ounded)28 b(op)r(erators)e FO(A)p FP(,)i FO(B)t FP(,)g(whic)n(h)f(satisfy)g(the)h(follo)n(wing)f(relation)137 1777 y FO(P)190 1789 y FL(2)227 1777 y FP(\()p FO(A;)14 b(B)t FP(\))24 b(=)f FO(\013A)684 1743 y FL(2)740 1777 y FP(+)18 b FO(\014)t FQ(f)p FO(A;)c(B)t FQ(g)k FP(+)g FO(i)p FJ(~)p FP([)p FO(B)t(;)c(A)p FP(])19 b(+)f FO(\015)5 b(B)1728 1743 y FL(2)1783 1777 y FP(+)18 b FO(\016)s(A)h FP(+)f FO(\017B)k FP(+)c FO(\037I)30 b FP(=)23 b(0)p FO(;)984 1902 y(\013;)14 b(\014)t(;)g FJ(~)p FO(;)g(\015)5 b(;)14 b(\016)o(;)g(\017;)g(\037)22 b FQ(2)i FJ(R)p FO(:)701 b FP(\(1.6\))118 2081 y FR(1.)36 b FP(Let)28 b(us)g(start)f(with)h(the) g(homogeneous)d(quadratic)i(relation)330 2221 y FO(q)p 330 2258 41 4 v 336 2334 a(i)380 2277 y FP([)p FO(A;)14 b(B)t FP(])24 b(=)e FO(\013A)818 2243 y FL(2)875 2277 y FP(+)c FO(\014)t FQ(f)p FO(A;)c(B)t FQ(g)k FP(+)g FO(\015)5 b(B)1475 2243 y FL(2)1512 2277 y FO(;)180 b(\013;)14 b(\014)t(;)g(\015)5 b(;)14 b(q)26 b FQ(2)d FJ(R)p FO(:)208 b FP(\(1.7\))118 2480 y FR(Prop)s(osition)37 b(8.)44 b FC(By)36 b(using)f(a)h(non-de)l(gener)l(ate)f(line)l(ar)h(tr)l (ansformation,)118 2580 y(r)l(elation)h FP(\(1.7\))29 b FC(c)l(an)h(b)l(e)g(r)l(e)l(duc)l(e)l(d)g(to)f(one)h(of)h(the)f(fol)t (lowing)i(forms)7 b FP(:)p 642 2692 1410 4 v 640 2791 4 100 v 683 2762 a(\(0)757 2774 y FL(0)794 2762 y FP(\))p 1181 2791 V 398 w(\()p FO(I)g(V)1347 2774 y FL(0)1385 2762 y FP(\))p 2050 2791 V 640 2891 V 815 2861 a(0)23 b(=)f(0)p 1181 2891 V 425 w([)p FO(A;)14 b(B)t FP(])24 b(=)f(0)p 2050 2891 V 642 2894 1410 4 v 640 2994 4 100 v 683 2964 a(\()p FO(I)751 2976 y FL(0)789 2964 y FP(\))p 1181 2994 V 403 w(\()p FO(V)1304 2976 y FL(0)1342 2964 y FP(\))p 2050 2994 V 640 3094 4 101 v 786 3064 a FO(A)848 3034 y FL(2)909 3064 y FP(=)f(0)p 1181 3094 V 1389 3032 a FL(1)p 1389 3046 34 4 v 1394 3093 a FM(i)1432 3064 y FP([)p FO(A;)14 b(B)t FP(])23 b(=)g FO(A)1817 3034 y FL(2)p 2050 3094 4 101 v 642 3097 1410 4 v 640 3197 4 100 v 683 3167 a FP(\()p FO(I)7 b(I)794 3179 y FL(0)832 3167 y FP(\))p 1181 3197 V 360 w(\()p FO(V)19 b(I)1359 3179 y FL(0)1397 3167 y FP(\))p 2050 3197 V 640 3297 4 101 v 683 3267 a FO(A)745 3237 y FL(2)801 3267 y FP(+)f FO(B)951 3237 y FL(2)1012 3267 y FP(=)k(0)p 1181 3297 V 1234 3234 a FL(1)p 1234 3248 34 4 v 1239 3296 a FM(i)1277 3267 y FP([)p FO(A;)14 b(B)t FP(])23 b(=)g FO(q)s FP(\()p FO(A)1734 3237 y FL(2)1790 3267 y FP(+)18 b FO(B)1940 3237 y FL(2)1978 3267 y FP(\))p 2050 3297 4 101 v 640 3397 4 100 v 1181 3397 V 1488 3367 a(\()p FO(q)27 b(>)22 b FP(0\))p 2050 3397 V 642 3400 1410 4 v 640 3500 4 100 v 683 3470 a(\()p FO(I)7 b(I)g(I)837 3482 y FL(0)875 3470 y FP(\))p 1181 3500 V 317 w(\()p FO(V)19 b(I)7 b(I)1402 3482 y FL(0)1440 3470 y FP(\))p 2050 3500 V 640 3600 4 101 v 683 3570 a FO(A)745 3540 y FL(2)801 3570 y FQ(\000)18 b FO(B)951 3540 y FL(2)1012 3570 y FP(=)k(0)p 1181 3600 V 1234 3537 a FL(1)p 1234 3551 34 4 v 1239 3598 a FM(i)1277 3570 y FP([)p FO(A;)14 b(B)t FP(])23 b(=)g FO(q)s FP(\()p FO(A)1734 3540 y FL(2)1790 3570 y FQ(\000)18 b FO(B)1940 3540 y FL(2)1978 3570 y FP(\))p 2050 3600 4 101 v 640 3699 4 100 v 1181 3699 V 1488 3669 a(\()p FO(q)27 b(>)22 b FP(0\))p 2050 3699 V 642 3703 1410 4 v 118 3848 a FC(Pr)l(o)l(of.)43 b FP(By)29 b(using)g(a)f(non-degenerate)f(linear)i(transformation,)e(w) n(e)i(can)g(re-)118 3948 y(duce)23 b(the)g(symmetric)f(quadratic)g (form)g FO(\013A)1517 3918 y FL(2)1563 3948 y FP(+)8 b FO(\014)t FQ(f)p FO(A;)14 b(B)t FQ(g)8 b FP(+)g FO(\015)d(B)2133 3918 y FL(2)2193 3948 y FP(to)22 b(a)g(diago-)118 4048 y(nal)j(form,)h(i.e.,)g(w)n(e)f(can)g(assume)g FO(\014)i FP(=)c(0,)j FO(\013)p FP(,)g FO(\015)i FQ(2)23 b(f\000)p FP(1)p FO(;)14 b FP(0)p FO(;)g FP(1)p FQ(g)p FP(.)33 b(If)26 b FO(q)g FP(=)d(0,)i(then)118 4147 y(equation)j(\(1.7\))f(will) i(tak)n(e)e(one)h(of)g(the)g(forms)f(\(0)1688 4159 y FL(0)1725 4147 y FP(\){\()p FO(I)7 b(I)g(I)1953 4159 y FL(0)1991 4147 y FP(\).)39 b(If)28 b FO(q)f FQ(6)p FP(=)d(0,)j(then)p eop %%Page: 25 29 25 28 bop 118 100 a FK(1.1.)36 b(In)n(tro)r(duction)27 b(to)h(represen)n(tations)d(of)j FQ(\003)p FK(-algebras)586 b FP(25)118 333 y(b)n(y)29 b(using)f(the)i(same)e(transformation,)g(w)n (e)g(reduce)h(the)g(righ)n(t-hand)f(side)h(of)118 432 y(equalit)n(y)j(\(1.7\))f(to)h(the)f(corresp)r(onding)f(form)h(and)g (then:)46 b(if)32 b FO(q)s(i)p FP([)p FO(A;)14 b(B)t FP(])29 b(=)g(0,)118 532 y(w)n(e)37 b(get)f(\()p FO(I)7 b(V)520 544 y FL(0)558 532 y FP(\);)42 b(if)37 b FQ(\000)p FO(q)s(i)p FP([)p FO(A;)14 b(B)t FP(])39 b(=)f FO(A)1290 502 y FL(2)1327 532 y FP(,)i(replace)35 b FO(B)41 b FP(b)n(y)c FO(q)s(B)k FP(to)c(get)f(\()p FO(V)2391 544 y FL(0)2429 532 y FP(\);)42 b(if)118 632 y FQ(\000)p FO(q)s(i)p FP([)p FO(A;)14 b(B)t FP(])23 b(=)f FO(A)636 601 y FL(2)687 632 y FQ(\006)13 b FO(B)832 601 y FL(2)870 632 y FP(,)25 b(w)n(e)g(get)g(\()p FO(V)19 b(I)1309 644 y FL(0)1347 632 y FP(\))25 b(and)g(\()p FO(V)19 b(I)7 b(I)1741 644 y FL(0)1779 632 y FP(\))25 b(with)h FO(q)i FP(replaced)c(with)118 731 y FO(q)158 701 y FN(\000)p FL(1)247 731 y FP(;)k(b)n(y)f (substituting)h FO(A)g FP(with)h(sign)13 b FO(q)s(A)28 b FP(w)n(e)f(get)g FO(q)f(>)d FP(0.)p 2514 731 4 57 v 2518 678 50 4 v 2518 731 V 2567 731 4 57 v 118 894 a FR(2.)34 b FP(By)22 b(applying)g(a)f(similar)h(argumen)n(t,)g(w)n(e)g (can)f(pro)n(v)n(e)g(the)h(follo)n(wing)f(state-)118 993 y(men)n(t.)118 1146 y FR(Prop)s(osition)42 b(9.)k FC(By)40 b(using)f(an)g(a\016ne)g(change)h(of)g(variables,)k(e)l (quation)118 1246 y FP(\(1.1.5\))29 b FC(c)l(an)h(b)l(e)g(r)l(e)l(duc)l (e)l(d)f(to)h(one)g(of)h(the)f(fol)t(lowing)i(forms:)p 133 1344 2427 4 v 131 1469 4 125 v 175 1427 a FP(\(0)249 1439 y FL(0)286 1427 y FP(\))e(0)22 b(=)h(0)p 926 1469 V 427 w(\(0)1043 1439 y FL(1)1080 1427 y FP(\))30 b FO(\037I)g FP(=)23 b(0)p FO(;)43 b(\037)23 b FQ(6)p FP(=)g(0)p 1814 1469 V 196 w(\(0)1931 1439 y FL(2)1968 1427 y FP(\))30 b FO(A)23 b FP(=)g(0)p 2558 1469 V 133 1472 2427 4 v 131 1596 4 125 v 175 1555 a(\()p FO(I)243 1567 y FL(0)281 1555 y FP(\))30 b FO(A)405 1525 y FL(2)465 1555 y FP(=)23 b(0)p 926 1596 V 374 w(\()p FO(I)1037 1567 y FL(1)1075 1555 y FP(\))30 b FO(A)1199 1525 y FL(2)1260 1555 y FP(=)23 b FO(I)p 1814 1596 V 473 w FP(\()p FO(I)1925 1567 y FL(2)1963 1555 y FP(\))30 b FO(A)2087 1525 y FL(2)2147 1555 y FP(=)23 b FO(B)p 2558 1596 V 131 1721 V 926 1721 V 969 1679 a FP(\()p FO(I)1044 1649 y FN(0)1037 1700 y FL(1)1075 1679 y FP(\))30 b FO(A)1199 1649 y FL(2)1260 1679 y FP(=)23 b FQ(\000)p FO(I)p 1814 1721 V 2558 1721 V 133 1724 2427 4 v 131 1849 4 125 v 175 1807 a FP(\()p FO(I)7 b(I)286 1819 y FL(0)324 1807 y FP(\))30 b FO(A)448 1777 y FL(2)504 1807 y FP(+)18 b FO(B)654 1777 y FL(2)714 1807 y FP(=)23 b(0)p 926 1849 V 125 w(\()p FO(I)7 b(I)1080 1819 y FL(1)1118 1807 y FP(\))30 b FO(A)1242 1777 y FL(2)1298 1807 y FP(+)18 b FO(B)1448 1777 y FL(2)1509 1807 y FP(=)k FO(I)p 1814 1849 V 2558 1849 V 131 1973 V 926 1973 V 969 1932 a FP(\()p FO(I)7 b(I)1087 1902 y FN(0)1080 1952 y FL(1)1118 1932 y FP(\))30 b FO(A)1242 1902 y FL(2)1298 1932 y FP(+)18 b FO(B)1448 1902 y FL(2)1509 1932 y FP(=)k FQ(\000)p FO(I)p 1814 1973 V 2558 1973 V 133 1977 2427 4 v 131 2101 4 125 v 175 2060 a FP(\()p FO(I)7 b(I)g(I)329 2072 y FL(0)367 2060 y FP(\))30 b FO(A)491 2029 y FL(2)547 2060 y FQ(\000)18 b FO(B)697 2029 y FL(2)757 2060 y FP(=)23 b(0)p 926 2101 V 82 w(\()p FO(I)7 b(I)g(I)1123 2072 y FL(1)1161 2060 y FP(\))30 b FO(A)1285 2029 y FL(2)1341 2060 y FQ(\000)18 b FO(B)1491 2029 y FL(2)1552 2060 y FP(=)k FO(I)p 1814 2101 V 2558 2101 V 131 2226 V 175 2184 a FC(or)30 b FQ(f)345 2163 y FP(~)324 2184 y FO(A;)442 2163 y FP(~)423 2184 y FO(B)t FQ(g)22 b FP(=)h(0)p 926 2226 V 285 w FC(or)30 b FQ(f)1140 2163 y FP(~)1118 2184 y FO(A;)1237 2163 y FP(~)1217 2184 y FO(B)t FQ(g)23 b FP(=)f FO(I)p 1814 2226 V 2558 2226 V 133 2229 2427 4 v 131 2354 4 125 v 175 2312 a FP(\()p FO(I)7 b(V)298 2324 y FL(0)336 2312 y FP(\))30 b([)p FO(A;)14 b(B)t FP(])23 b(=)g(0)p 926 2354 V 206 w(\()p FO(I)7 b(V)1092 2324 y FL(1)1130 2312 y FP(\))1202 2279 y FL(1)p 1202 2293 34 4 v 1207 2341 a FM(i)1245 2312 y FP([)p FO(A;)14 b(B)t FP(])24 b(=)e FO(I)p 1814 2354 4 125 v 253 w FP(\()p FO(I)7 b(V)1980 2324 y FL(2)2018 2312 y FP(\))2090 2279 y FL(1)p 2090 2293 34 4 v 2095 2341 a FM(i)2133 2312 y FP([)p FO(A;)14 b(B)t FP(])23 b(=)g FO(A)p 2558 2354 4 125 v 133 2357 2427 4 v 131 2481 4 125 v 175 2440 a FP(\()p FO(V)255 2452 y FL(0)293 2440 y FP(\))365 2407 y FL(1)p 365 2421 34 4 v 370 2468 a FM(i)408 2440 y FP([)p FO(A;)14 b(B)t FP(])23 b(=)g FO(A)793 2410 y FL(2)p 926 2481 4 125 v 969 2440 a FP(\()p FO(V)1049 2452 y FL(1)1087 2440 y FP(\))1159 2407 y FL(1)p 1159 2421 34 4 v 1164 2468 a FM(i)1202 2440 y FP([)p FO(A;)14 b(B)t FP(])24 b(=)e FO(A)1587 2410 y FL(2)1643 2440 y FP(+)c FO(I)p 1814 2481 4 125 v 95 w FP(\()p FO(V)1937 2452 y FL(2)1975 2440 y FP(\))2047 2407 y FL(1)p 2047 2421 34 4 v 2052 2468 a FM(i)2090 2440 y FP([)p FO(A;)c(B)t FP(])p 2558 2481 4 125 v 131 2606 V 926 2606 V 969 2564 a(\()p FO(V)1068 2534 y FN(0)1049 2585 y FL(1)1092 2564 y FP(\))1164 2532 y FL(1)p 1164 2546 34 4 v 1169 2593 a FM(i)1207 2564 y FP([)p FO(A;)g(B)t FP(])23 b(=)g FO(A)1592 2534 y FL(2)1648 2564 y FQ(\000)18 b FO(I)p 1814 2606 4 125 v 396 w FP(=)k FO(A)2312 2534 y FL(2)2368 2564 y FP(+)c 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y(Considering)e(the)i(solutions)f(of)g(the)h (equations)f(w)n(e)g(ha)n(v)n(e:)243 3749 y(\(0)317 3761 y FL(1)354 3749 y FP(\))h FO(\037I)i FP(=)22 b(0,)28 b FO(\037)23 b FQ(6)p FP(=)f(0.)37 b(There)27 b(are)f(no)i(pairs)e FO(A)p FP(,)i FO(B)k FP(whic)n(h)c(satisfy)f(\(0)2452 3761 y FL(1)2489 3749 y FP(\);)243 3848 y(\(0)317 3860 y FL(2)354 3848 y FP(\))c FO(A)g FP(=)g(0.)35 b(Since)23 b FO(B)k FP(=)c FO(B)1139 3818 y FN(\003)1177 3848 y FP(,)h(it)g(is)e(an)h(arbitrary)e(b)r(ounded)i(self-adjoin)n(t)118 3948 y(op)r(erator;)36 b(the)f(only)f(irreducible)f(represen)n(tations) g(are)g(one-dimensional,)118 4048 y FO(A)24 b FP(=)e(0,)28 b FO(B)f FP(=)c FO(b)p FP(,)k(and)h(their)f(structure)g(is)h(giv)n(en)f (b)n(y)g(the)h(structure)f(theorem)118 4147 y(for)g(a)g(single)g(op)r (erator)f FO(B)t FP(;)p eop %%Page: 26 30 26 29 bop 118 100 a FP(26)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)243 333 y FP(\()p FO(I)311 345 y FL(0)349 333 y FP(\))34 b FO(A)477 303 y FL(2)548 333 y FP(=)f(0.)55 b(Because)32 b FO(A)i FP(=)f FO(A)1345 303 y FN(\003)1383 333 y FP(,)j(w)n(e)d(ha)n (v)n(e)g FO(A)1830 303 y FL(2)1901 333 y FP(=)f FO(A)i FP(=)f(0,)i(and)e(the)118 432 y(case)23 b(\()p FO(I)358 444 y FL(0)396 432 y FP(\))h(is)g(similar)f(to)g(\(0)972 444 y FL(2)1009 432 y FP(\).)36 b(The)24 b(structure)f(of)h(an)n(y)f (solution)g(of)g(equation)118 532 y(\()p FO(I)186 544 y FL(0)224 532 y FP(\))40 b(is)f(the)h(follo)n(wing:)59 b FO(A)43 b FP(=)f(0,)g FO(B)47 b FP(=)1488 465 y Fz(R)1527 561 y Fv(R)1574 545 y Fy(1)1619 532 y FO(\025)14 b(dE)1785 544 y FM(B)1843 532 y FP(\()p FO(\025)p FP(\),)44 b(where)39 b FO(E)2335 544 y FM(B)2392 532 y FP(\()p FQ(\001)p FP(\))h(is)118 632 y(the)32 b(resolution)e(of)h(the)g(iden)n(tit)n(y)g(for)g(the)g(op) r(erator)e FO(B)36 b FP(concen)n(trated)30 b(on)g(a)118 731 y(compact)d(set)h FO(K)g FQ(\032)23 b FJ(R)p FP(;)243 831 y(\()p FO(I)318 801 y FN(0)311 851 y FL(1)349 831 y FP(\))28 b FO(A)471 801 y FL(2)531 831 y FP(=)23 b 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FP(=)22 b FO(I)p 1785 2084 V 2527 2084 V 164 2087 2365 4 v 162 2212 4 125 v 1132 2170 a(F)1185 2182 y FL(4)1223 2170 y FP(-relations)p 2527 2212 V 164 2215 2365 4 v 162 2339 4 125 v 899 2339 V 942 2298 a(\()p FO(I)7 b(I)1053 2310 y FL(1)1091 2298 y FP(\))28 b FO(A)1213 2268 y FL(2)1269 2298 y FP(+)18 b FO(B)1419 2268 y FL(2)1480 2298 y FP(=)k FO(I)p 1785 2339 V 2527 2339 V 164 2343 2365 4 v 162 2467 4 125 v 206 2426 a FP(\()p FO(I)7 b(I)g(I)360 2438 y FL(0)398 2426 y FP(\))28 b FQ(f)p FO(A;)14 b(B)t FQ(g)22 b FP(=)h(0)p 899 2467 V 82 w(\()p FO(I)7 b(I)g(I)1096 2438 y FL(1)1134 2426 y FP(\))28 b FQ(f)p FO(A;)14 b(B)t FQ(g)23 b FP(=)f FO(I)p 1785 2467 V 2527 2467 V 164 2471 2365 4 v 162 2595 4 125 v 467 2554 a FP(Lie)28 b(algebras)d(and)j (their)f(nonlinear)g(transformations)p 2527 2595 V 164 2598 2365 4 v 162 2723 4 125 v 206 2681 a(\()p FO(I)7 b(V)329 2693 y FL(0)367 2681 y FP(\))28 b([)p FO(A;)14 b(B)t FP(])23 b(=)g(0)p 899 2723 V 150 w(\()p FO(I)7 b(V)1065 2693 y FL(1)1103 2681 y 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b(the)f(sequel,)h(w)n(e)f(sa)n(y)f(that)h FO(A)h FP(is)f(a)g FO(F)1623 1889 y FM(n)1669 1877 y FP(-algebra,)e(if)32 b FQ(8)p FO(x)2182 1889 y FL(1)2219 1877 y FO(;)14 b(:)g(:)g(:)g(;)g(x) 2451 1889 y FM(n)2519 1877 y FQ(2)118 1977 y FO(A)p FP(,)28 b(w)n(e)f(ha)n(v)n(e)g(that)h FO(F)778 1989 y FM(n)823 1977 y FP(\()p FO(x)902 1989 y FL(1)940 1977 y FO(;)14 b(:)g(:)g(:)g(;)g(x)1172 1989 y FM(n)1217 1977 y FP(\))24 b(=)e(0.)243 2077 y(The)31 b(follo)n(wing)e(Amitsur{Levitski)i(theorem) f(tak)n(es)g(place)g([5]:)44 b FC(the)32 b(al-)118 2177 y(gebr)l(a)f FO(M)417 2189 y FM(n)461 2177 y FP(\()p FJ(C)16 b FP(\))36 b FC(is)30 b(a)g FO(F)830 2189 y FL(2)p FM(n)908 2177 y FC(-algebr)l(a,)i(but)d(not)g(a)h FO(F)1656 2189 y FL(2)p FM(n)p FN(\000)p FL(1)1820 2177 y FC(-algebr)l(a)p FP(.)243 2277 y FO(F)296 2289 y FM(n)341 2277 y FP(-algebras)23 b(form)h(one)h(of)f(the)i(most)e(simple)h(class)f(of)h(algebras,)e(if)j (con-)118 2376 y(sidered)g(from)h(the)g(standp)r(oin)n(t)g(of)f(the)h (structure)g(of)f(irreducible)g(represen-)118 2476 y(tations.)118 2643 y FR(Theorem)31 b(2.)40 b FC(Consider)32 b(the)d(fol)t(lowing)k (statements:)197 2811 y FP(\(i\))42 b FC(ther)l(e)35 b(exists)h(a)f(r)l(esidual)i(family)g FB(L)f FC(of)h(irr)l(e)l(ducible) g(r)l(epr)l(esentations)326 2910 y FP(Irrep)13 b FO(A)23 b FQ(\033)g FB(L)g FQ(3)g FO(\031)33 b FC(such)d(that)g FP(dim)14 b FO(H)1511 2922 y FM(\031)1579 2910 y FQ(\024)23 b FO(n)29 b FC(for)i(al)t(l)g FO(\031)26 b FQ(2)e FB(L)p FP(;)173 3078 y(\(ii\))43 b FC(ther)l(e)23 b(exists)h(a)g(r)l(esidual)g (family)i FB(L)e FC(of)g(r)l(epr)l(esentations)g FP(Rep)14 b FO(A)23 b FQ(\033)g FB(L)g FQ(3)326 3178 y FO(\031)33 b FC(such)c(that)h FP(dim)14 b FO(H)985 3190 y FM(\031)1054 3178 y FQ(\024)22 b FO(n)30 b FC(for)g(al)t(l)h FO(\031)c FQ(2)c FB(L)p FP(;)150 3346 y(\(iii\))43 b FO(A)30 b FC(is)g(a)g FO(F)632 3358 y FL(2)p FM(n)711 3346 y FC(-algebr)l(a)6 b FP(;)153 3514 y(\(iv\))42 b FC(for)30 b(any)g FO(\031)d FQ(2)c FP(Irrep)13 b FO(A)p FC(,)30 b FP(dim)15 b FO(H)1300 3526 y FM(\031)1368 3514 y FQ(\024)22 b FO(n)p FC(.)118 3681 y(We)31 b(have)g(the)g(fol)t(lowing)i(implic)l(ations:)47 b FP(\(i\))31 b FQ(\))g FP(\(ii\))g FQ(\))f FP(\(iii\))i FQ(\))e FP(\(iv\))p FC(.)42 b(Nei-)118 3781 y(ther)30 b(of)h(the)f(inverse)g(implic)l(ations)i(hold.)118 3948 y(Pr)l(o)l(of.)43 b FP(Here)25 b(w)n(e)f(will)i(only)e(pro)n(v)n(e)g (that)h(\(ii\))h FQ(\))f FP(\(iii\),)h(since)f(it)g(is)g(this)g(state-) 118 4048 y(men)n(t)37 b(that)f(will)g(b)r(e)h(used)f(later)f(in)i (examples)e(to)h(pro)n(v)n(e)f(that)h(the)h(corre-)118 4147 y(sp)r(onding)27 b(algebra)f(is)i(a)f FO(F)968 4159 y FL(2)p FM(n)1047 4147 y FP(-algebra.)p eop %%Page: 28 32 28 31 bop 118 100 a FP(28)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)243 333 y FP(Assume)j(that)h(\(ii\))h(holds,)f(but)g FO(A)g FP(is)f(not)h(a)f FO(F)1732 345 y FL(2)p FM(n)1811 333 y FP(-algebra.)43 b(Then)31 b(there)118 432 y(exist)g FO(x)365 444 y FL(1)403 432 y FP(,)h FO(:)14 b(:)g(:)27 b FP(,)33 b FO(x)685 444 y FL(2)p FM(n)793 432 y FQ(2)c FO(A)j FP(suc)n(h)f(that)h FO(F)1399 444 y FL(2)p FM(n)1477 432 y FP(\()p FO(x)1556 444 y FL(1)1594 432 y FO(;)14 b(:)g(:)g(:)g(;)g(x)1826 444 y FL(2)p FM(n)1904 432 y FP(\))30 b(=)f FO(x)h FQ(6)p FP(=)f(0.)47 b(Let)32 b(us)118 532 y(c)n(ho)r(ose)26 b FO(\031)h FQ(2)c FB(L)28 b FP(suc)n(h)g(that)f FO(\031)s FP(\()p FO(x)p FP(\))e FQ(6)p FP(=)d(0.)37 b(Then)28 b(w)n(e)f(get)303 713 y FO(\031)s FP(\()p FO(F)438 725 y FL(2)p FM(n)518 713 y FP(\()p FO(x)597 725 y FL(1)635 713 y FO(;)14 b(:)g(:)g(:)f(;)h(x)866 725 y FL(2)p FM(n)945 713 y FP(\)\))24 b(=)e FO(F)1173 725 y FL(2)p FM(n)1252 713 y FP(\()p FO(\031)s FP(\()p FO(x)1413 725 y FL(1)1451 713 y FP(\))p FO(;)14 b(:)g(:)g(:)g(;)g(\031)s FP(\()p FO(x)1797 725 y FL(2)p FM(n)1877 713 y FP(\)\))23 b(=)g FO(\031)s FP(\()p FO(x)p FP(\))h FQ(6)p FP(=)f(0)p FO(:)118 894 y FP(But)32 b(since)f(dim)14 b FO(H)715 906 y FM(\031)789 894 y FQ(\024)29 b FO(n)p FP(,)j(this)g(con)n(tradicts)e(the)i (Amitsur{Levitski)e(theo-)118 994 y(rem.)243 1093 y(Neither)g(of)g(the) g(in)n(v)n(erse)f(implications)g(of)h(Theorem)f(1)h(holds.)44 b(Indeed,)118 1193 y(to)39 b(see)g(that)g(\(ii\))h(do)r(es)e(not)h (imply)h(\(i\),)i(and)d(\(iv\))h(do)r(es)f(not)g(imply)g(\(iii\),)118 1293 y(consider)24 b(the)g(nilp)r(oten)n(t)i(algebra)c(of)j(complex)f FO(n)12 b FQ(\002)g FO(n)24 b FP(matrices)g(of)g(the)h(form)989 1596 y FO(X)30 b FP(=)1175 1404 y Fz(0)1175 1551 y(B)1175 1604 y(@)1248 1468 y FP(0)262 b FQ(\003)1377 1565 y FP(.)1409 1590 y(.)1442 1615 y(.)1248 1723 y(0)g(0)1594 1404 y Fz(1)1594 1551 y(C)1594 1604 y(A)1680 1596 y FO(;)118 1900 y FP(whic)n(h)28 b(only)f(has)g(the)h(trivial)f(irreducible)g (represen)n(tation)f(for)h(an)n(y)g FO(n)22 b FQ(2)i FJ(N)t FP(.)243 2000 y(Condition)31 b(\(iii\))h(do)r(es)f(not)h(imply)f (\(ii\).)50 b(F)-7 b(or)30 b(example,)i(the)g(algebra)e(of)118 2099 y(matrices)d FO(X)i FQ(2)24 b FO(M)709 2111 y FM(n)753 2099 y FP(\()p FJ(C)16 b FP(\))34 b(of)27 b(the)h(form)831 2485 y FO(X)i FP(=)1017 2218 y Fz(0)1017 2365 y(B)1017 2414 y(B)1017 2464 y(B)1017 2514 y(B)1017 2567 y(@)1090 2279 y FO(a)1134 2291 y FL(11)1322 2279 y FQ(\003)117 b FO(:)14 b(:)g(:)133 b FQ(\003)1126 2434 y FP(0)1331 2368 y(.)1331 2401 y(.)1331 2434 y(.)1720 2368 y(.)1720 2401 y(.)1720 2434 y(.)1136 2523 y(.)1136 2556 y(.)1136 2589 y(.)163 b FQ(\003)117 b FO(:)14 b(:)g(:)133 b FQ(\003)1126 2689 y FP(0)119 b FO(:)14 b(:)g(:)132 b FP(0)117 b FO(a)1719 2701 y FL(11)1789 2218 y Fz(1)1789 2365 y(C)1789 2414 y(C)1789 2464 y(C)1789 2514 y(C)1789 2567 y(A)118 2866 y FP(is)31 b(a)g FO(F)331 2878 y FL(2)p FM(n)p FN(\000)p FL(2)494 2866 y FP(-algebra.)45 b(But)32 b(for)e(an)n(y)g(represen)n (tation)g FO(\031)s FP(,)i(dim)14 b FO(H)2187 2878 y FM(\031)2261 2866 y FQ(\024)28 b FO(n)21 b FQ(\000)f FP(1,)118 2966 y(and)28 b(the)g(nilp)r(oten)n(t)g(elemen)n(t)909 3319 y FO(S)g FP(=)1076 3078 y Fz(0)1076 3224 y(B)1076 3274 y(B)1076 3323 y(B)1076 3377 y(@)1148 3141 y FP(0)111 b(1)290 b(0)1277 3238 y(.)1310 3263 y(.)1342 3288 y(.)1457 3238 y(.)1490 3263 y(.)1522 3288 y(.)1480 3396 y(0)111 b(1)1148 3495 y(0)443 b(0)1674 3078 y Fz(1)1674 3224 y(C)1674 3274 y(C)1674 3323 y(C)1674 3377 y(A)1761 3319 y FO(;)118 3683 y FP(w)n(e)34 b(ha)n(v)n(e)f(that)i FO(\031)s FP(\()p FO(S)770 3653 y FM(n)p FN(\000)p FL(1)901 3683 y FP(\))f(=)g(\()p FO(\031)s FP(\()p FO(S)5 b FP(\)\))1300 3653 y FM(n)p FN(\000)p FL(1)1466 3683 y FP(=)34 b(0,)i(since)e FO(\031)s FP(\()p FO(S)5 b FP(\))35 b(is)f(a)g(nilp)r(oten)n(t)118 3782 y(elemen)n(t)i(in)h FO(M)620 3794 y FM(n)p FN(\000)p FL(1)749 3782 y FP(\()p FJ(C)16 b FP(\).)68 b(Hence)37 b(suc)n(h)e(represen)n(tations)f(do)i(not)g(separate)118 3882 y FO(S)174 3852 y FM(n)p FN(\000)p FL(1)332 3882 y FP(and)27 b(the)h(zero)f(elemen)n(t)g(of)h(the)g(algebra.)p 2514 3882 4 57 v 2518 3829 50 4 v 2518 3882 V 2567 3882 4 57 v 118 4048 a FR(2.)43 b FP(If)30 b FA(A)g FP(is)f(a)h FQ(\003)p FP(-algebra,)d(and)j(one)f(only)h(considers)e(its)i FQ(\003)p FP(-representations,)118 4147 y(then,)e(eviden)n(tly)-7 b(,)28 b(\(i\))g FQ(,)g FP(\(ii\).)p eop %%Page: 29 33 29 32 bop 118 100 a FK(1.2.)36 b FO(F)337 112 y FM(n)383 100 y FK(-algebras)25 b(and)i(their)h(represen)n(tations)848 b FP(29)243 333 y(\(iii\))33 b(do)r(es)g(not)f(imply)i(\(ii\),)g (since,)g(for)f(example,)g(the)h(algebra)d FO(M)2406 345 y FM(n)2450 333 y FP(\()p FJ(C)16 b FP(\))118 432 y(with)24 b(a)f(non-standard)g(in)n(v)n(olution)f(do)r(es)i(not)f(ha)n (v)n(e)g(non-zero)f FQ(\003)p FP(-represen)n(ta-)118 532 y(tions.)243 632 y(Condition)e(\(iv\))h(do)r(es)f(not)g(imply)h (\(iii\).)35 b(F)-7 b(or)20 b(example,)i(the)e(W)-7 b(eyl)21 b FQ(\003)p FP(-alge-)118 731 y(bra)31 b FJ(C)15 b FQ(h)p FO(P)47 b FP(=)28 b FO(P)614 701 y FN(\003)652 731 y FO(;)14 b(Q)29 b FP(=)g FO(Q)944 701 y FN(\003)1011 731 y FQ(j)g FP([)p FO(P)r(;)14 b(Q)p FP(])29 b(=)g FO(iI)7 b FQ(i)31 b FP(of)g(di\013eren)n(tial)g(op)r(erators)e(with)118 831 y(the)k(co)r(e\016cien)n(ts)g(b)r(eing)g(p)r(olynomials)e(in)j(one) e(v)-5 b(ariable)32 b(do)r(es)g(not)h(ha)n(v)n(e)f FQ(\003)p FP(-)118 930 y(represen)n(tations)26 b(in)i(b)r(ounded)g(op)r(erators,) e(but)j(it)f(is)g(not)g(a)f FO(F)2111 942 y FM(n)2157 930 y FP(-algebra)e(for)118 1030 y(an)n(y)i FO(n)c FQ(2)g FJ(N)t FP(.)118 1178 y FR(3.)43 b FP(If)30 b FA(A)g FP(is)g(a)f FO(C)633 1148 y FN(\003)672 1178 y FP(-algebra)e(then)k(all)e (conditions)h(of)f(the)i(lemma)f(are)e(equiv-)118 1278 y(alen)n(t,)34 b(since)f(for)g(a)g(Banac)n(h)f(semi-simple)h(algebra)e FA(A)p FP(,)j(the)g(set)f(Irrep)13 b FA(A)33 b FP(of)118 1378 y(its)k(irreducible)g(represen)n(tations)e(is)h(a)h(residual)f (family)h(\(see)g([153)o(]\))g(and,)118 1477 y(therefore,)27 b(\()p FO(iv)s FP(\))c FQ(\))g FP(\()p FO(i)p FP(\).)118 1693 y FR(1.2.2)94 b(Examples)37 b(of)i FO(F)1018 1705 y FM(n)1064 1693 y FR(-algebras)f(generated)i(b)m(y)f(idemp)s(oten)m (ts)410 1792 y(and)32 b(their)g(represen)m(tations)118 1946 y FP(Here,)27 b(w)n(e)g(giv)n(e)f(a)h(n)n(um)n(b)r(er)g(of)g (examples)g(of)g(algebras)e(and)i FQ(\003)p FP(-algebras)e(gen-)118 2045 y(erated)20 b(b)n(y)h(idemp)r(oten)n(ts,)h(and)f(construct)f(a)g (residual)g(family)h(of)g(represen)n(ta-)118 2145 y(tions)27 b(or)e FQ(\003)p FP(-represen)n(tations)f FO(\031)30 b FP(for)d(eac)n(h)f(of)g(them)i(suc)n(h)e(that)h(dim)14 b FO(H)2346 2157 y FM(\031)2414 2145 y FQ(\024)23 b FO(n)p FP(,)118 2245 y(and,)28 b(therefore,)e(sho)n(w)h(that)h(these)g (algebras)d(are)i FO(F)1788 2257 y FL(2)p FM(n)1866 2245 y FP(-algebras.)118 2393 y FR(1.)37 b FP(Represen)n(tations)26 b(of)h(the)i FO(F)1125 2405 y FL(4)1162 2393 y FP(-algebra)d(generated) g(b)n(y)i(t)n(w)n(o)f(idemp)r(oten)n(ts)118 2493 y FO(q)155 2505 y FL(1)192 2493 y FP(,)h FO(q)280 2505 y FL(2)317 2493 y FP(,)g(and)g(the)g(unit)g(elemen)n(t,)448 2673 y FO(Q)514 2685 y FL(2)574 2673 y FP(=)23 b FJ(C)15 b FQ(h)p FO(q)785 2685 y FL(1)828 2673 y FO(;)f(q)902 2685 y FL(2)962 2673 y FQ(j)24 b FO(q)1046 2685 y FL(1)1083 2639 y(2)1143 2673 y FP(=)f FO(q)1268 2685 y FL(1)1305 2673 y FO(;)28 b(q)1393 2685 y FL(2)1430 2639 y(2)1491 2673 y FP(=)22 b FO(q)1615 2685 y FL(2)1652 2673 y FQ(i)574 2808 y FP(=)h FJ(C)15 b FQ(h)p FO(u)28 b FP(=)23 b(2)p FO(q)991 2820 y FL(1)1046 2808 y FQ(\000)18 b FO(e;)28 b(v)e FP(=)d(2)p FO(q)1452 2820 y FL(2)1507 2808 y FQ(\000)18 b FO(e)23 b FQ(j)g FO(u)1746 2773 y FL(2)1806 2808 y FP(=)f FO(e;)28 b(v)2026 2773 y FL(2)2086 2808 y FP(=)23 b FO(e)p FQ(i)118 2988 y FP(are)37 b(w)n(ell)g(kno)n(wn;)42 b(nev)n(ertheless)37 b(w)n(e)g(presen)n(t)g(a)g(description)g(of)h(the) g(irre-)118 3087 y(ducible)29 b(represen)n(tation)e(of)h(the)h(algebra) e(to)i(sho)n(w)e(the)i(sc)n(heme)f(of)h(in)n(v)n(esti-)118 3187 y(gations)e(w)n(e)g(will)h(follo)n(w)e(in)i(more)f(complicated)g (examples.)243 3287 y(All)40 b(\014nite)g(dimensional)f(irreducible)g (represen)n(tations)e(of)j FO(Q)2267 3299 y FL(2)2304 3287 y FP(,)i(up)e(to)118 3386 y(equiv)-5 b(alence,)27 b(are:)243 3486 y(a\))f(four)h(one-dimensional)f(represen)n(tations:)34 b FO(\031)1779 3498 y FL(0)p FM(;)p FL(0)1870 3486 y FP(\()p FO(q)1939 3498 y FL(1)1976 3486 y FP(\))24 b(=)e(0,)27 b FO(\031)2258 3498 y FL(0)p FM(;)p FL(0)2348 3486 y FP(\()p FO(q)2417 3498 y FL(2)2455 3486 y FP(\))c(=)118 3586 y(0;)k FO(\031)257 3598 y FL(1)p FM(;)p FL(0)347 3586 y FP(\()p FO(q)416 3598 y FL(1)454 3586 y FP(\))c(=)g(1,)j FO(\031)735 3598 y FL(1)p FM(;)p FL(0)825 3586 y FP(\()p FO(q)894 3598 y FL(2)932 3586 y FP(\))d(=)g(0;)j FO(\031)1213 3598 y FL(0)p FM(;)p FL(1)1304 3586 y FP(\()p FO(q)1373 3598 y FL(1)1410 3586 y FP(\))e(=)e(0,)27 b FO(\031)1692 3598 y FL(0)p FM(;)p FL(1)1782 3586 y FP(\()p FO(q)1851 3598 y FL(2)1889 3586 y FP(\))c(=)g(1;)j FO(\031)2170 3598 y FL(1)p FM(;)p FL(1)2261 3586 y FP(\()p FO(q)2330 3598 y FL(1)2367 3586 y FP(\))e(=)e(1,)118 3685 y FO(\031)165 3697 y FL(1)p FM(;)p FL(1)255 3685 y FP(\()p FO(q)324 3697 y FL(2)362 3685 y FP(\))h(=)g(1;)243 3785 y(b\))h(the)h(family)-7 b(,)25 b(parameterized)e(b)n(y)h FO(z)i FQ(2)e FJ(C)15 b FQ(nf)p FP(0)p FO(;)f FP(1)p FQ(g)p FP(,)28 b(of)d(t)n(w)n (o-dimensional)118 3884 y(represen)n(tations:)475 4110 y FO(\031)522 4122 y FM(z)561 4110 y FP(\()p FO(q)630 4122 y FL(1)667 4110 y FP(\))e(=)810 3993 y Fz(\022)871 4059 y FP(1)83 b(0)871 4159 y(0)g(0)1037 3993 y Fz(\023)1112 4110 y FO(;)97 b(\031)1279 4122 y FM(z)1318 4110 y FP(\()p FO(q)1387 4122 y FL(2)1424 4110 y FP(\))24 b(=)1567 3993 y Fz(\022)1719 4059 y FO(z)249 b FP(1)1628 4159 y FO(z)22 b FQ(\000)c FO(z)1815 4129 y FL(2)1935 4159 y FP(1)g FQ(\000)g FO(z)2120 3993 y Fz(\023)2195 4110 y FO(:)p eop %%Page: 30 34 30 33 bop 118 100 a FP(30)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)243 333 y FP(Ev)n(ery)22 b FO(\031)27 b FQ(2)c FP(Irrep)13 b FO(Q)888 345 y FL(2)950 333 y FP(is)24 b(one-)g(or)f(t)n(w)n (o-dimensional.)34 b(Indeed,)26 b(the)f(space)118 432 y FO(H)43 b FP(=)36 b FJ(C)15 b FQ(h)p FO(e)456 444 y FM(\025)505 432 y FO(;)f(\031)s FP(\()p FO(v)s FP(\))p FO(e)738 444 y FM(\025)782 432 y FQ(i)36 b FP(\()p FO(e)921 444 y FM(\025)1000 432 y FP(is)g(an)f(eigen)n(v)n(ector)e(of)j FO(\031)s FP(\()p FO(u)p FP(\))24 b FQ(\001)f FO(\031)s FP(\()p FO(v)s FP(\),)39 b FQ(k)p FO(e)2289 444 y FM(\025)2332 432 y FQ(k)c FP(=)h(1,)118 532 y FO(\025)24 b FQ(6)p FP(=)e(0\))28 b(is)f(in)n(v)-5 b(arian)n(t)27 b(for)g(the)h(represen)n (tation)e FO(\031)s FP(.)118 672 y FR(2.)35 b FP(In)25 b(the)f(algebra)e FO(Q)821 684 y FL(2)858 672 y FP(,)j(one)f(can)g(in)n (tro)r(duce)g(t)n(w)n(o)f(natural)h(structures)f(of)h(an)118 772 y(algebra)i(with)i(in)n(v)n(olution:)243 871 y(1\))f(the)h FQ(\003)529 883 y FL(1)566 871 y FP(-algebra)541 1031 y FB(P)596 1043 y FL(2)656 1031 y FP(=)22 b FJ(C)15 b FQ(h)q FO(p)872 994 y FN(\003)906 1002 y Fy(1)872 1053 y FL(1)971 1031 y FP(=)23 b FO(p)1101 1043 y FL(1)1138 1031 y FO(;)14 b(p)1217 994 y FN(\003)1251 1002 y Fy(1)1217 1053 y FL(2)1310 1031 y FP(=)23 b FO(p)1440 1043 y FL(2)1500 1031 y FQ(j)g FO(p)1588 996 y FL(2)1588 1051 y(1)1648 1031 y FP(=)f FO(p)1777 1043 y FL(1)1814 1031 y FO(;)14 b(p)1893 996 y FL(2)1893 1051 y(2)1953 1031 y FP(=)23 b FO(p)2083 1043 y FL(2)2120 1031 y FQ(i)656 1165 y FP(=)f FJ(C)15 b FQ(h)q FO(u)878 1131 y FN(\003)912 1139 y Fy(1)977 1165 y FP(=)22 b FO(u;)14 b(v)1192 1131 y FN(\003)1226 1139 y Fy(1)1286 1165 y FP(=)22 b FO(v)27 b FQ(j)c FO(u)1534 1131 y FL(2)1594 1165 y FP(=)f FO(v)1724 1131 y FL(2)1785 1165 y FP(=)g FO(e)p FQ(i)656 1290 y FP(=)g FJ(C)29 b FP([)6 b FJ(Z)902 1302 y FL(2)951 1290 y FQ(\003)18 b FJ(Z)1073 1302 y FL(2)1104 1290 y FP(])118 1449 y(is)32 b(a)f(group)g FQ(\003)561 1461 y FL(1)597 1449 y FP(-algebra)f (generated)h(b)n(y)g(t)n(w)n(o)g(unitary)g(self-adjoin)n(t)h(genera-) 118 1549 y(tors.)58 b(Irreducible)34 b(t)n(w)n(o-dimensional)f FQ(\003)p FP(-represen)n(tations)f(of)i FB(P)2193 1561 y FL(2)2265 1549 y FP(\(up)i(to)e(a)118 1649 y(unitary)27 b(equiv)-5 b(alence\))28 b(are:)315 1859 y FO(\031)362 1871 y FM(\036)407 1859 y FP(\()p FO(p)481 1871 y FL(1)518 1859 y FP(\))23 b(=)661 1742 y Fz(\022)722 1808 y FP(1)82 b(0)722 1908 y(0)g(0)888 1742 y Fz(\023)963 1859 y FO(;)97 b(\031)1130 1871 y FM(\036)1174 1859 y FP(\()p FO(p)1248 1871 y FL(2)1286 1859 y FP(\))23 b(=)1429 1742 y Fz(\022)1561 1807 y FP(cos)1672 1777 y FL(2)1723 1807 y FO(\036)154 b FP(cos)13 b FO(\036)h FP(sin)g FO(\036)1490 1909 y FP(cos)f FO(\036)h FP(sin)g FO(\036)159 b FP(sin)2104 1874 y FL(2)2155 1909 y FO(\036)2280 1742 y Fz(\023)2355 1859 y FO(;)118 2069 y(\036)39 b FQ(2)h FP(\(0)p FO(;)14 b(\031)s(=)p FP(2\).)65 b(They)37 b(are)f(equiv)-5 b(alen)n(t)36 b(to)h(the)h(represen)n(tations)d FO(\031)2337 2081 y FM(z)2376 2069 y FP(,)k FO(z)j FQ(2)118 2169 y FP(\(0)p FO(;)14 b FP(1\))23 b FQ(\032)f FJ(R)p FP(.)243 2269 y(2\))27 b(the)h FQ(\003)529 2281 y FL(2)566 2269 y FP(-algebra)536 2428 y FB(Q)591 2440 y FL(1)651 2428 y FP(=)22 b FJ(C)15 b FQ(h)q FO(q)862 2440 y FL(1)905 2428 y 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y FL(2)1366 3834 y FP(.)582 4071 y FO(x)f FP(=)g FO(\013)793 4083 y FL(0)849 4071 y FP(+)949 3967 y FM(N)1002 3975 y Fy(1)932 3992 y Fz(X)938 4169 y FM(i)p FL(=1)1065 4071 y FO(a)1109 4083 y FM(i)1137 4071 y FP(\()p FO(q)1206 4083 y FL(1)1243 4071 y FO(q)1280 4083 y FL(2)1318 4071 y FP(\))1350 4037 y FM(i)1396 4071 y FP(+)1497 3967 y FM(N)1550 3975 y Fy(2)1479 3992 y Fz(X)1482 4169 y FM(j)s FL(=1)1613 4071 y FO(b)1649 4083 y FM(j)1684 4071 y FP(\()p FO(q)1753 4083 y FL(2)1790 4071 y FO(q)1827 4083 y FL(1)1864 4071 y FP(\))1896 4037 y FM(j)p eop %%Page: 31 35 31 34 bop 118 100 a FK(1.2.)36 b FO(F)337 112 y FM(n)383 100 y FK(-algebras)25 b(and)i(their)h(represen)n(tations)848 b FP(31)849 402 y(+)949 298 y FM(N)1002 306 y Fy(3)932 323 y Fz(X)932 502 y FM(k)q FL(=0)1066 402 y FO(c)1102 414 y FM(k)1143 402 y FP(\()p FO(q)1212 414 y FL(1)1250 402 y FO(q)1287 414 y FL(2)1324 402 y FP(\))1356 368 y FM(k)1397 402 y FO(q)1434 414 y FL(1)1490 402 y FP(+)1590 298 y FM(N)1643 306 y Fy(4)1573 323 y Fz(X)1580 502 y FM(l)p FL(=0)1707 402 y FO(d)1750 414 y FM(l)1775 402 y FP(\()p FO(q)1844 414 y FL(2)1882 402 y FO(q)1919 414 y FL(1)1956 402 y FP(\))1988 368 y FM(l)2014 402 y FO(q)2051 414 y FL(2)2088 402 y FO(;)118 668 y(\013)171 680 y FL(0)208 668 y FP(,)28 b FO(a)303 680 y FM(i)331 668 y FP(,)f FO(b)417 680 y FM(j)452 668 y FP(,)h FO(c)539 680 y FM(k)580 668 y FP(,)f FO(d)673 680 y FM(l)722 668 y FQ(2)c FJ(C)15 b FP(.)43 b(Then)28 b(one)f(has)381 936 y FO(\031)428 948 y FM(z)467 936 y FP(\()p FO(x)p FP(\))d(=)689 795 y Fz( )755 873 y FO(\013)808 885 y FL(0)953 873 y FP(0)780 998 y(0)106 b FO(\013)981 1010 y FL(0)1019 795 y Fz(!)1103 936 y FP(+)1186 795 y Fz( )1252 817 y(P)1339 838 y FM(N)1392 846 y Fy(1)1339 904 y FM(i)p FL(=1)1465 879 y FO(a)1509 891 y FM(i)1536 879 y FO(z)1579 849 y FM(i)1689 817 y Fz(P)1777 838 y FM(N)1830 846 y Fy(1)1777 904 y FM(i)p FL(=1)1903 879 y FO(a)1947 891 y FM(i)1974 879 y FO(z)2017 849 y FM(i)p FN(\000)p FL(1)1408 1004 y FP(0)438 b(0)2129 795 y Fz(!)597 1221 y FP(+)680 1079 y Fz( )871 1093 y(P)958 1113 y FM(N)1011 1121 y Fy(2)958 1180 y FM(j)s FL(=1)1091 1155 y FO(b)1127 1167 y FM(j)1162 1155 y FO(z)1205 1125 y FM(j)1447 1155 y FP(0)746 1229 y Fz(P)833 1249 y FM(N)886 1257 y Fy(2)833 1316 y FM(j)s FL(=1)966 1291 y FO(b)1002 1303 y FM(j)1037 1291 y FO(z)1080 1261 y FM(j)1114 1291 y FP(\(1)18 b FQ(\000)g FO(z)t FP(\))83 b(0)1489 1079 y Fz(!)1573 1221 y FP(+)1656 1079 y Fz( )1721 1102 y(P)1809 1122 y FM(N)1862 1130 y Fy(3)1809 1189 y FM(k)q FL(=0)1948 1164 y FO(c)1984 1176 y FM(k)2025 1164 y FO(z)2068 1134 y FM(k)2191 1164 y FP(0)1894 1288 y(0)255 b(0)2232 1079 y Fz(!)597 1506 y FP(+)680 1364 y Fz( )871 1381 y(P)958 1401 y FM(N)1011 1409 y Fy(4)958 1468 y FM(l)p FL(=0)1082 1443 y FO(d)1125 1455 y FM(l)1150 1443 y FO(z)1193 1413 y FM(l)p FL(+1)1635 1381 y Fz(P)1723 1401 y FM(N)1776 1409 y Fy(4)1723 1468 y FM(l)p FL(=0)1846 1443 y FO(d)1889 1455 y FM(l)1915 1443 y FO(z)1958 1413 y FM(l)746 1517 y Fz(P)833 1537 y FM(N)886 1545 y Fy(4)833 1604 y FM(l)p FL(=0)957 1579 y FO(d)1000 1591 y FM(l)1026 1579 y FO(z)1069 1549 y FM(l)p FL(+1)1177 1579 y FP(\(1)19 b FQ(\000)f FO(z)t FP(\))1510 1517 y Fz(P)1598 1537 y FM(N)1651 1545 y Fy(4)1598 1604 y FM(l)p FL(=0)1721 1579 y FO(d)1764 1591 y FM(l)1790 1579 y FO(z)1833 1549 y FM(l)1858 1579 y FP(\(1)g FQ(\000)g FO(z)t FP(\))2107 1364 y Fz(!)2187 1506 y FO(:)118 1772 y FP(It)i(easily)e(follo)n(ws)g(from)h(the)h (structure)e(of)h(the)h(matrix)f FO(\031)1884 1784 y FM(z)1922 1772 y FP(\()p FO(x)p FP(\))i(that)e FO(\031)2272 1784 y FM(z)2311 1772 y FP(\()p FO(x)p FP(\))24 b(=)e(0)118 1871 y(for)27 b(an)n(y)g FO(z)f FQ(2)e FJ(C)39 b FQ(n)18 b(f)p FP(0)p FO(;)c FP(1)p FQ(g)26 b FP(if)i(and)f(only)g(if)i FO(x)23 b FP(=)g(0.)p 2514 1871 4 57 v 2518 1818 50 4 v 2518 1871 V 2567 1871 4 57 v 118 2071 a FR(4.)51 b FP(The)33 b(structure)f(of)h(indecomp)r(osable)e(represen)n(tations)g (of)i(the)g(algebra)118 2170 y FO(Q)184 2182 y FL(2)242 2170 y FP(is)21 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2339 y FP(,)118 2438 y FO(m)191 2450 y FM(ij)289 2438 y FQ(2)39 b FJ(N)c FQ([)25 b(f1g)p FP(;)41 b FO(m)852 2450 y FM(ii)942 2438 y FP(=)e(1)e FO(m)1198 2450 y FM(ij)1295 2438 y FP(=)i FO(m)1472 2450 y FM(j)s(i)1569 2438 y FO(>)g FP(1,)g FO(i)g FQ(6)p FP(=)f FO(j)5 b FP(;)42 b FO(i)p FP(,)e FO(j)k FP(=)38 b(1,)f FO(:)14 b(:)g(:)28 b FP(,)118 2538 y FO(m)p FP(,)f(whic)n(h)h(is)f(de\014ned)g(in)h(terms)f(of)g (generators)e(\()p FO(w)1762 2550 y FM(i)1790 2538 y FP(\))1822 2508 y FM(m)1822 2559 y(i)p FL(=1)1961 2538 y FP(and)i(the)h(relations)118 2637 y(\()p FO(w)209 2649 y FM(i)237 2637 y FO(w)296 2649 y FM(j)332 2637 y FP(\))364 2607 y FM(m)423 2615 y Fw(ij)523 2637 y FP(=)42 b FO(e)p FP(,)g FO(i)p FP(,)g FO(j)47 b FP(=)42 b(1,)d FO(:)14 b(:)g(:)27 b FP(,)43 b FO(m)p FP(;)i(if)39 b FO(m)1611 2649 y FM(ij)1712 2637 y FP(=)j FQ(1)p FP(,)g(then)e(there)f(is)g(no) 118 2737 y(relation)d(b)r(et)n(w)n(een)g(the)h(generators)d FO(w)1383 2749 y FM(i)1447 2737 y FP(and)j FO(w)1677 2749 y FM(j)1712 2737 y FP(.)63 b(If)37 b(the)g(Cartan)e(matrix)118 2837 y FO(K)29 b FP(=)305 2769 y Fz(\000)350 2837 y FQ(\000)7 b FP(cos)12 b FO(\031)s(=m)711 2849 y FM(ij)769 2769 y Fz(\001)807 2787 y FM(m)807 2870 y(i;j)s FL(=1)970 2837 y FP(,)23 b(whic)n(h)e(corresp)r(onds)f(to)i FO(M)9 b FP(,)22 b(is)g(p)r(ositiv)n(e)f(de\014nite)118 2952 y(\(all)32 b(its)g(principal)f(minors)h(are)e(p)r(ositiv)n(e\),)j(then) g(the)f(group)f FO(G)2174 2964 y FM(M)2279 2952 y FP(is)h(\014nite;)118 3051 y(if)h(det)14 b FO(K)37 b FP(=)31 b(0,)i(but)g(the)g(other)f (principal)g(minors)g(are)f(p)r(ositiv)n(e,)j(then)e(the)118 3151 y(group)26 b FO(G)418 3163 y FM(M)519 3151 y FP(is)h(in\014nite,)i (but)e FO(G)1123 3163 y FM(M)1225 3151 y FP(is)g(a)f(semi-direct)h(pro) r(duct)g(of)g(the)h(lattice)118 3251 y FJ(Z)179 3220 y FM(m)p FN(\000)p FL(1)345 3251 y FP(and)22 b(a)h(\014nite)h(group)e FO(G)1070 3263 y FM(f)1113 3251 y FP(\()p FO(M)9 b FP(\),)24 b FO(G)1379 3263 y FM(M)1476 3251 y FP(=)f FJ(Z)1625 3220 y FM(m)p FN(\000)p FL(1)1777 3251 y FJ(o)9 b FO(G)1916 3263 y FM(f)1959 3251 y FP(\()p FO(M)g FP(\))23 b(see)46 b([41)o(])23 b(and)118 3350 y(others.)243 3450 y(Since)28 b(the)g(Co)n(xeter)f(group)f FO(G)1213 3462 y FM(M)1315 3450 y FP(is)i(generated)f(b)n(y)g(\015ips)h FO(w)2130 3420 y FL(2)2128 3471 y FM(i)2192 3450 y FP(=)23 b FO(e)p FP(,)28 b FO(i)23 b FP(=)g(1,)118 3550 y FO(:)14 b(:)g(:)28 b FP(,)g FO(m)p FP(,)g FJ(C)15 b FP([)q FO(G)561 3562 y FM(M)640 3550 y FP(])29 b(also)e(giv)n(es)g(an)h(example)g(of)g(an)g (algebra)e(generated)h(b)n(y)h FO(m)118 3649 y FP(pro)5 b(jections.)37 b(There)27 b(is)h(a)g(natural)f(in)n(v)n(olution)g(in)h FJ(C)15 b FP([)p FO(G)1894 3661 y FM(M)1973 3649 y FP(])28 b(suc)n(h)g(that)g(all)g(of)118 3749 y(the)g(group)e(elemen)n(ts)i(are) e(unitary)-7 b(,)28 b FO(g)1327 3719 y FN(\003)1388 3749 y FP(=)22 b FO(g)1518 3719 y FN(\000)p FL(1)1607 3749 y FP(.)37 b(\(Generally)27 b(sp)r(eaking,)g(this)118 3848 y(is)h(not)f(the)h(unique)g(in)n(v)n(olution)f(that)h(can)f(b)r(e) h(de\014ned)g(on)f FJ(C)15 b FP([)p FO(G)2140 3860 y FM(M)2220 3848 y FP(].\))243 3948 y(The)41 b(dimensions)g(of)h(the)f (irreducible)g FQ(\003)p FP(-represen)n(tations)e FO(\031)2262 3960 y FM(\013)2351 3948 y FP(of)i(the)118 4048 y(group)18 b FQ(\003)p FP(-al)n(gebra)g(of)g(the)h(Co)n(xeter)e(group)h FO(G)1507 4060 y FM(M)1604 4048 y FP(=)23 b FJ(Z)1753 4018 y FM(m)p FN(\000)p FL(1)1895 4048 y FJ(o)p FO(G)2025 4060 y FM(f)2088 4048 y FP(are)17 b(ma)5 b(jorized)118 4147 y(b)n(y)32 b(the)g(n)n(um)n(b)r(er)g FQ(j)p FO(G)780 4159 y FM(f)823 4147 y FQ(j)p FP(.)51 b(These)31 b(represen)n(tations)f (form)i(a)g(residual)f(family)-7 b(,)p eop %%Page: 33 37 33 36 bop 118 100 a FK(1.2.)36 b FO(F)337 112 y FM(n)383 100 y FK(-algebras)25 b(and)i(their)h(represen)n(tations)848 b FP(33)118 333 y(b)r(ecause)29 b(irreducible)f FQ(\003)p FP(-represen)n(tations)e(of)j FJ(C)15 b FP([)p FO(G)1723 345 y FM(M)1803 333 y FP(],)29 b(with)g(the)h(in)n(v)n(olution)118 432 y FO(g)161 402 y FN(\003)225 432 y FP(=)25 b FO(g)358 402 y FN(\000)p FL(1)447 432 y FP(,)30 b(mak)n(e)f(a)g(residual)f (family)-7 b(.)43 b(Hence)29 b FJ(C)15 b FP([)p FO(G)1776 444 y FM(M)1856 432 y FP(])30 b(is)f(a)g FO(F)2118 447 y FL(2)p FN(j)p FM(G)2223 456 y Fw(f)2260 447 y FN(j)2284 432 y FP(-algebra)118 532 y(whic)n(h)f(is)f(generated)g(b)n(y)g (\015ips.)118 658 y FC(R)l(emark)37 b FP(5)p FC(.)k FP(It)26 b(is)g(a)g(v)n(ery)f(di\016cult)i(problem)f(to)g(describ)r(e)g (indecomp)r(osable)118 757 y(represen)n(tations)38 b(of)i FJ(C)15 b FP([)p FO(G)955 769 y FM(M)1035 757 y FP(])40 b(\(except)g(for)f(the)i(case)e(where)g(the)h(Co)n(xeter)118 857 y(group)27 b FO(G)419 869 y FM(M)520 857 y FP(is)h(a)f(\014nite)h (group)f(or)f(is)i FJ(Z)11 b(o)19 b(Z)1524 869 y FL(2)1555 857 y FP(\))28 b([39)o(].)118 999 y FR(6.)43 b FP(No)n(w)29 b(let)h FA(A)609 1011 y FM(k)677 999 y FP(=)c FJ(C)15 b FQ(h)p FO(u)902 956 y FL(\()p FM(k)q FL(\))902 1021 y(1)1001 999 y FO(;)f(:)g(:)g(:)f(;)h(u)1233 956 y FL(\()p FM(k)q FL(\))1233 1009 y FM(n)1274 1018 y Fw(k)1352 999 y FQ(j)27 b FP(\(\))1466 1011 y FM(k)1507 999 y FQ(i)k FP(b)r(e)f FO(F)1738 1011 y FL(2)p FM(m)1830 1020 y Fw(k)1871 999 y FP(-algebras)d(generated)118 1116 y(b)n(y)40 b(the)h(\015ips)f FO(u)638 1073 y FL(\()p FM(k)q FL(\))638 1138 y(1)730 1116 y FP(,)h FO(:)14 b(:)g(:)27 b FP(,)44 b FO(u)1033 1073 y FL(\()p FM(k)q FL(\))1033 1126 y FM(n)1074 1135 y Fw(k)1165 1116 y FP(and)c(relations)f(\(\))1753 1128 y FM(k)1795 1116 y FP(,)k(suc)n(h)d(that)h FO(\031)2304 1086 y FL(\()p FM(k)q FL(\))2437 1116 y FP(is)f(a)118 1216 y(residual)27 b(family)h(with)g(dim)15 b FO(H)1093 1235 y FM(\031)1134 1218 y Fy(\()p Fw(k)q Fy(\))1242 1216 y FQ(\024)24 b FO(m)1404 1228 y FM(k)1444 1216 y FP(,)k FO(k)f FP(=)c(1,)k FO(:)14 b(:)g(:)28 b FP(,)g FO(n)p FP(\).)38 b(Of)28 b(course,)e(the)118 1315 y(algebra)426 1492 y FJ(C)15 b FQ(h)q FO(u)561 1449 y FL(\(1\))561 1514 y(1)655 1492 y FO(;)f(:)g(:)g(:)g(;)g(u)888 1458 y FL(\()p FM(n)p FL(\))888 1513 y FM(n)929 1521 y Fw(n)1007 1492 y FQ(j)23 b FP(\(\))1117 1504 y FL(1)1155 1492 y FO(;)14 b(:)g(:)g(:)g(;)g FP(\(\))1404 1504 y FM(n)1450 1492 y FO(;)27 b FP([)p FO(u)1571 1449 y FL(\()p FM(k)q FL(\))1571 1515 y FM(i)1664 1492 y FO(;)14 b(u)1749 1449 y FL(\()p FM(l)p FL(\))1749 1515 y FM(j)1825 1492 y FP(])24 b(=)e(0)p FO(;)27 b(k)f FQ(6)p FP(=)d FO(l)r FQ(i)118 1674 y FP(is)28 b(a)f FO(F)324 1686 y FL(2)p FM(m)416 1694 y Fy(1)449 1686 y FN(\001\001\001\001)n(\001)p FM(m)606 1694 y Fw(n)651 1674 y FP(-algebra)e(ha)n(ving)i(the)h(residual)f (family)g FO(\031)1993 1644 y FL(\(1\))2101 1674 y FQ(\012)18 b(\001)c(\001)g(\001)19 b(\012)f FO(\031)2433 1644 y FL(\()p FM(n)p FL(\))2530 1674 y FP(.)118 1817 y FR(7.)37 b FP(Examples)27 b(of)g(algebras)f(that)i(w)n(e)g(will)g(consider)e(in) j(the)f(sequel)f(are)g(also)118 1928 y(de\014ned)d(b)n(y)g(generators)d FO(u)958 1885 y FL(\(1\))958 1950 y(1)1047 1928 y FP(,)i FO(:)14 b(:)g(:)28 b FP(,)d FO(u)1314 1885 y FL(\()p FM(n)p FL(\))1314 1938 y FM(n)1355 1946 y Fw(n)1410 1928 y FP(,)g(but)f(if)g(the)h(upp)r(er)e(indices)h(are)f(not)118 2028 y(equal,)31 b(the)g(generators)d(pairwise)i(comm)n(ute)g(or)g(an)n (ti-comm)n(ute.)45 b(In)31 b(items)118 2139 y(7)f(and)h(8,)g(these)g (relations)f(are)f(as)h(follo)n(ws:)42 b FO(u)1613 2096 y FL(\()p FM(k)q FL(\))1613 2162 y FM(i)1706 2139 y FO(u)1754 2096 y FL(\()p FM(l)p FL(\))1754 2162 y FM(j)1859 2139 y FP(=)28 b FO(\017)1986 2151 y FM(k)q(l)2062 2139 y FO(u)2110 2096 y FL(\()p FM(l)p FL(\))2110 2162 y FM(j)2186 2139 y FO(u)2234 2096 y FL(\()p FM(k)q FL(\))2234 2162 y FM(i)2327 2139 y FP(,)j FO(k)h FQ(6)p FP(=)27 b FO(l)118 2238 y FP(\()p FO(\017)184 2250 y FM(k)q(l)272 2238 y FP(=)e(+1)k(or)f FQ(\000)p FP(1,)h FO(\017)794 2250 y FM(k)q(l)882 2238 y FP(=)c FO(\017)1006 2250 y FM(lk)1068 2238 y FP(\),)30 b FO(k)s FP(,)g FO(l)d FP(=)e(1,)k FO(:)14 b(:)g(:)28 b FP(,)h FO(n)p FP(,)h(and)f(do)g(not)g(dep)r(end)h(on)118 2338 y FO(i)23 b FP(=)f(1,)28 b FO(:)14 b(:)g(:)27 b FP(,)h FO(n)575 2350 y FM(k)643 2338 y FP(and)g FO(j)g FP(=)22 b(1,)28 b FO(:)14 b(:)g(:)27 b FP(,)h FO(n)1272 2350 y FM(l)1297 2338 y FP(.)243 2438 y(Let)f FA(A)451 2450 y FM(n;\017)571 2438 y FP(b)r(e)h(an)g(algebra)d(generated)i(b)n (y)g FO(s)1623 2450 y FL(1)1660 2438 y FP(,)h FO(:)14 b(:)g(:)g FP(,)27 b FO(s)1911 2450 y FM(n)1957 2438 y FP(,)268 2603 y FA(A)328 2615 y FM(n;\017)443 2603 y FP(=)c FJ(C)585 2536 y Fz(\012)630 2603 y FO(s)669 2615 y FL(1)706 2603 y FO(;)14 b(:)g(:)g(:)g(;)g(s)930 2615 y FM(n)998 2603 y FQ(j)23 b FO(s)1083 2569 y FL(2)1083 2623 y FM(i)1143 2603 y FP(=)g(1)p FO(;)k(s)1362 2615 y FM(i)1390 2603 y FO(s)1429 2615 y FM(j)1487 2603 y FP(=)22 b FO(\017)1608 2615 y FM(ij)1667 2603 y FO(s)1706 2615 y FM(j)1740 2603 y FO(s)1779 2615 y FM(i)1807 2603 y FP(;)28 b FO(i;)14 b(j)27 b FP(=)c(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n) 2349 2536 y Fz(\013)2388 2603 y FO(;)964 2727 y(\017)23 b FP(=)g(\()p FO(\017)1175 2739 y FM(ij)1233 2727 y FP(\))p FO(;)180 b(\017)1502 2739 y FM(ii)1576 2727 y FP(=)23 b(1)p FO(:)243 2893 y FP(The)30 b(algebra)f FA(A)770 2905 y FM(n;\017)894 2893 y FP(is)h(\014nite)i(dimensional)e(and)g (semi-simple,)i(it)f(has)f(a)118 2992 y(\014nite)22 b(residual)f (family)g(of)h(irreducible)f FQ(\003)p FP(-represen)n(tations)d FO(\031)2054 3004 y FM(p)2114 2992 y FP(with)k FO(s)2336 2962 y FN(\003)2336 3014 y FM(i)2398 2992 y FP(=)g FO(s)2524 3004 y FM(i)2552 2992 y FP(,)118 3092 y FO(i)h FP(=)f(1,)28 b FO(:)14 b(:)g(:)27 b FP(,)h FO(n)p FP(,)f(and)h(is)f(an)h FO(F)1039 3104 y FM(m)1102 3092 y FP(-algebra,)e(where)h FO(m)c FQ(\025)f FP(2)1909 3062 y FM(n=)p FL(2+1)2105 3092 y FP(.)118 3246 y FR(8.)34 b FP(Let)21 b FA(B)442 3261 y FL(\()p Ft(A)516 3270 y Fw(k)553 3261 y FL(\))p FM(;\017)653 3246 y FP(=)i FJ(C)15 b FQ(h)p FO(u)875 3203 y FL(\(1\))875 3268 y(1)970 3246 y FO(;)f(:)g(:)g(:)f(;)h(u)1202 3203 y FL(\()p FM(n)p FL(\))1202 3256 y FM(n)1243 3264 y Fw(n)1322 3246 y FQ(j)23 b FP(\(\))1432 3258 y FL(1)1470 3246 y FO(;)14 b(:)g(:)g(:)f(;)h FP(\(\))1718 3258 y FM(n)1764 3246 y FP(;)28 b FO(u)1863 3203 y FL(\()p FM(k)q FL(\))1863 3269 y FM(i)1955 3246 y FO(u)2003 3203 y FL(\()p FM(l)p FL(\))2003 3269 y FM(j)2103 3246 y FP(=)23 b FO(\017)2225 3258 y FM(k)q(l)2287 3246 y FO(u)2335 3203 y FL(\()p FM(l)p FL(\))2335 3269 y FM(j)2411 3246 y FO(u)2459 3203 y FL(\()p FM(k)q FL(\))2459 3269 y FM(i)2552 3246 y FO(;)118 3345 y(k)j FQ(6)p FP(=)d FO(l)r(;)k(k)s(;)14 b(l)24 b FP(=)f(1)14 b FO(:)g(:)g(:)f(;)h(n)p FP(;)27 b FO(i)c FP(=)f(1)p FO(;)14 b(:)g(:)g(:)g(;)g(n)1291 3357 y FM(k)1331 3345 y FO(;)28 b(j)g FP(=)22 b(1)p FO(;)14 b(:)g(:)g(:)g(;)g(n)1808 3357 y FM(l)1833 3345 y FQ(i)p FP(.)243 3445 y(This)38 b(is)g(a)f FO(F)669 3461 y FL(2)702 3444 y Fw(n)p Fy(+1)815 3461 y FM(m)874 3469 y Fy(1)906 3461 y FN(\001\001\001\001)o(\001)p FM(m)1064 3469 y Fw(n)1108 3445 y FP(-algebra)f(whic)n(h)i(has)g(a)g (residual)f(family)h(of)118 3545 y FQ(\003)p FP(-rep)o(resentations)24 b FO(\031)813 3515 y FL(\(1\))918 3545 y FQ(\012)14 b(\001)g(\001)g (\001)h(\012)f FO(\031)1238 3515 y FL(\()p FM(n)p FL(\))1350 3545 y FQ(\012)h FO(\031)1477 3557 y FM(p)1541 3545 y FP(with)26 b(dim)15 b FO(H)1950 3564 y FM(\031)1991 3547 y Fy(\(1\))2068 3564 y FN(\012\001\001\001)o(\012)p FM(\031)2272 3547 y Fy(\()p Fw(n)p Fy(\))2357 3564 y FN(\012)p FM(\031)2448 3572 y Fw(p)2510 3545 y FQ(\024)118 3675 y FP(2)160 3645 y FM(n)209 3675 y FQ(\001)t FO(m)309 3687 y FL(1)350 3675 y FQ(\001)t(\001)f(\001)g(\001)t(\001)t FO(m)578 3687 y FM(n)643 3675 y FP(\(here,)22 b FO(\031)923 3645 y FL(\(1\))1016 3675 y FQ(\012)t(\001)14 b(\001)g(\001)t(\012)t FO(\031)1305 3645 y FL(\()p FM(n)p FL(\))1405 3675 y FQ(\012)t FO(\031)1521 3687 y FM(p)1560 3675 y FP(\()p FO(u)1640 3632 y FL(\()p FM(k)q FL(\))1640 3698 y FM(i)1732 3675 y FP(\))23 b(=)g(1)t FQ(\012)t(\001)14 b(\001)g(\001)s(\012)t FO(\031)2209 3645 y FL(\()p FM(k)q FL(\))2301 3675 y FP(\()p FO(u)2381 3632 y FL(\()p FM(k)q FL(\))2381 3698 y FM(i)2474 3675 y FP(\))t FQ(\012)118 3775 y(\001)g(\001)g(\001)k (\012)g FP(1)g FQ(\012)g FO(\031)506 3787 y FM(p)545 3775 y FP(\()p FO(s)616 3787 y FM(k)657 3775 y FP(\)\).)118 3936 y FR(9.)58 b FP(In)35 b(items)h(7)e(and)h(8)g(ab)r(o)n(v)n(e,)g (the)g(generators)e FO(u)1789 3893 y FL(\()p FM(k)q FL(\))1789 3960 y FM(i)1916 3936 y FP(and)i FO(u)2133 3893 y FL(\()p FM(l)p FL(\))2133 3960 y FM(j)2245 3936 y FP(comm)n(ute)118 4036 y(or)26 b(an)n(ticomm)n(ute)h(indep)r(enden)n(tly)h(of)g FO(i)e FP(and)i FO(j)5 b FP(.)36 b(In)28 b(this)f(item,)h(whether)f (the)118 4147 y(generators)e FO(u)568 4104 y FL(\()p FM(k)q FL(\))568 4170 y FM(i)688 4147 y FP(and)j FO(u)898 4104 y FL(\()p FM(l)p FL(\))898 4170 y FM(j)1002 4147 y FP(comm)n(ute)g(or)e(not)i(dep)r(ends)g(on)f FO(i)p FP(,)h FO(j)5 b FP(.)p eop %%Page: 34 38 34 37 bop 118 100 a FP(34)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)243 333 y FP(Let)37 b FO(A)463 345 y FM(k)544 333 y FP(=)i FJ(C)15 b FQ(h)p FO(u;)f(v)s(;)g(s)938 345 y FL(1)981 333 y FO(;)g(:)g(:)g(:)f(;)h(s)1204 345 y FM(k)1284 333 y FQ(j)40 b FO(u)1395 303 y FL(2)1471 333 y FP(=)f FO(v)1618 303 y FL(2)1695 333 y FP(=)g FO(s)1838 303 y FL(2)1838 354 y FM(i)1914 333 y FP(=)g FO(e;)28 b(i)39 b FP(=)g(1)p FO(;)14 b(:)g(:)g(:)f(;)h(k)s(;)118 432 y(us)205 444 y FM(i)257 432 y FP(=)24 b FO(\013)399 444 y FM(i)427 432 y FO(s)466 444 y FM(i)494 432 y FO(u;)j(v)s(s)674 444 y FM(i)726 432 y FP(=)e FO(\014)863 444 y FM(i)890 432 y FO(s)929 444 y FM(i)957 432 y FO(v)s(;)j(s)1090 444 y FM(i)1117 432 y FO(s)1156 444 y FM(j)1216 432 y FP(=)d FO(\017)1340 444 y FM(ij)1398 432 y FO(s)1437 444 y FM(j)1472 432 y FO(s)1511 444 y FM(i)1538 432 y FP(;)j FO(i;)14 b(j)29 b FP(=)24 b(1)p FO(;)14 b(:)g(:)g(:)g(;)g(k)s FQ(i)p FP(,)29 b(where)f FO(\013)2458 444 y FM(i)2510 432 y FP(=)118 532 y FQ(\006)p FP(1,)f FO(\014)322 544 y FM(i)373 532 y FP(=)22 b FQ(\006)p FP(1,)27 b FO(\017)651 544 y FM(ij)732 532 y FP(=)c FO(\017)854 544 y FM(j)s(i)935 532 y FP(=)g FQ(\006)p FP(1;)k FO(i)22 b FQ(6)p FP(=)h FO(j)5 b FP(,)28 b FO(\017)1443 544 y FM(ii)1516 532 y FP(=)23 b(1.)243 636 y(Of)g(course,)f(this)i(algebra)d(should)i(ha)n (v)n(e)e(b)r(een)j(denoted)f(b)n(y)f FO(A)2191 648 y FM(k)q(;\013;\014)s(;\017)2403 636 y FP(,)i(but)118 736 y(w)n(e)j(lea)n(v)n(e)g(out)g(the)h(sym)n(b)r(ols)f FO(\013)p FP(,)h FO(\014)t FP(,)g(and)g FO(\017)f FP(for)g(brevit)n(y)-7 b(.)243 840 y(Let)35 b(us)f(notice)h(that)g(the)g(algebras)e FJ(C)15 b FQ(h)p FO(u;)f(v)s(;)g(s)1722 852 y FL(1)1800 840 y FQ(j)35 b FO(us)1945 852 y FL(1)2017 840 y FP(=)g FQ(\000)p FO(s)2221 852 y FL(1)2257 840 y FO(u;)28 b(v)s(s)2438 852 y FL(1)2510 840 y FP(=)118 940 y FQ(\000)p FO(s)222 952 y FL(1)259 940 y FO(v)s FQ(i)35 b FP(and)f FJ(C)15 b FQ(h)p FO(u;)f(v)s(;)g(s)827 952 y FL(1)904 940 y FQ(j)34 b FO(us)1048 952 y FL(1)1119 940 y FP(=)f FQ(\000)p FO(s)1321 952 y FL(1)1358 940 y FO(u;)27 b(v)s(s)1538 952 y FL(1)1610 940 y FP(=)33 b FO(s)1747 952 y FL(1)1784 940 y FO(v)s FQ(i)i FP(w)n(ere)e(considered)h(in)118 1040 y([239)o(,)27 b(155)o(].)118 1220 y FR(Lemma)37 b(3.)44 b FC(The)36 b(algebr)l(a)h FO(A)1107 1232 y FM(k)1184 1220 y FC(is)e(a)h FO(F)1409 1237 y FL(2)1442 1220 y Fw(k)q Fy(+2)9 b FC(-algebr)l(a,)38 b(and)e(has)g(a)g(r)l(esidual)118 1320 y(family)c FB(L)430 1332 y FM(k)501 1320 y FC(subje)l(ct)d(to)h(the)g(c)l(ondition)6 b FP(:)39 b FQ(8)p FO(\031)26 b FQ(2)d FB(L)1667 1332 y FM(k)1709 1320 y FC(,)30 b FP(dim)14 b FO(H)1985 1332 y FM(\031)2053 1320 y FQ(\024)23 b FP(2)2183 1289 y FM(k)q FL(+1)2307 1320 y FC(.)118 1500 y(Pr)l(o)l(of.)43 b FP(F)-7 b(or)29 b FO(k)g FP(=)c(0,)k FO(A)847 1512 y FL(0)914 1500 y FP(is)g(a)g FO(F)1123 1512 y FL(4)1161 1500 y FP(-algebra,)f(and)h(has)f(a)h(residual)g(family)g FB(L)2514 1512 y FL(0)2552 1500 y FP(,)118 1600 y(since)38 b(the)g(algebra)d FO(A)847 1612 y FL(0)925 1600 y FP(=)k FO(Q)1095 1612 y FL(2)1132 1600 y FP(.)67 b(Let)38 b FO(k)k FP(=)e FO(n)p FP(,)g(and)d(assume)g(that)h(all)f FO(A)2529 1612 y FM(n)118 1699 y FP(are)f FO(F)319 1715 y FL(2)352 1698 y Fw(n)p Fy(+2)469 1699 y FP(-algebras)f(and)i(there)g(exists)g FB(L)1521 1711 y FM(n)1604 1699 y FP(with)h(dim)14 b FO(H)2024 1711 y FM(\031)2108 1699 y FQ(\024)39 b FP(2)2254 1669 y FM(n)p FL(+1)2383 1699 y FP(.)66 b(By)118 1799 y(induction,)41 b(consider)d(the)g(algebra)f FO(A)1375 1811 y FM(n)p FL(+1)1504 1799 y FP(.)69 b(It)39 b(con)n(tains)e(the)i (subalgebra)118 1898 y FO(B)28 b FP(=)23 b FO(C)6 b FQ(h)p FO(u;)14 b(v)s(;)g(s)598 1910 y FL(1)635 1898 y FO(;)g(:)g(:)g(:)g(;)g (s)859 1910 y FM(n)927 1898 y FQ(j)24 b FO(\013)1027 1910 y FM(i)1055 1898 y FO(;)14 b(\014)1139 1910 y FM(i)1166 1898 y FO(;)g(\017)1237 1910 y FM(ij)1295 1898 y FQ(i)p FP(.)38 b(Clearly)-7 b(,)27 b FO(B)32 b FP(is)c(isomorphic)f(to)g(some) 118 1998 y FO(A)180 2010 y FM(n)226 1998 y FP(.)33 b(Therefore,)20 b(the)e(claim)h(is)f(true)h(for)e FO(B)t FP(.)35 b(Let)18 b FB(L)1702 2010 y FM(n)1766 1998 y FP(b)r(e)h(a)f(residual)g(family)g (for)118 2098 y FO(B)31 b FP(with)c(dim)14 b FO(H)621 2110 y FM(\031)689 2098 y FQ(\024)23 b FP(2)819 2068 y FM(n)p FL(+1)948 2098 y FP(.)36 b(W)-7 b(e)27 b(construct)f(a)h (residual)e(family)i(for)f FO(A)2331 2110 y FM(n)p FL(+1)2487 2098 y FP(b)n(y)118 2197 y(applying)e(the)h(follo)n(wing)e(pro)r (cedure:)35 b FQ(8)p FO(\031)25 b FQ(2)e FB(L)1614 2209 y FM(n)1660 2197 y FP(,)i FO(\031)i FP(:)c FO(H)1897 2209 y FM(\031)1965 2197 y FQ(!)g FO(H)2140 2209 y FM(\031)2185 2197 y FP(,)i(in)n(tro)r(duce)123 2297 y(^)-47 b FO(\031)26 b FQ(2)e FB(L)329 2309 y FM(n)p FL(+1)459 2297 y FP(,)32 b(^)-47 b FO(\031)27 b FP(:)c FO(H)698 2309 y FM(\031)761 2297 y FQ(\010)18 b FO(H)913 2309 y FM(\031)981 2297 y FQ(!)24 b FO(H)1157 2309 y FM(\031)1220 2297 y FQ(\010)18 b FO(H)1372 2309 y FM(\031)1417 2297 y FP(,)28 b(b)n(y)283 2538 y(^)-47 b FO(\031)t FP(\()p FO(u)p FP(\))23 b(=)551 2421 y Fz(\022)612 2487 y FO(\031)s FP(\()p FO(u)p FP(\))235 b(0)673 2587 y(0)143 b FO(\013)911 2599 y FM(n)p FL(+1)1040 2587 y FO(\031)s FP(\()p FO(u)p FP(\))1203 2421 y Fz(\023)1278 2538 y FO(;)101 b FP(^)-46 b FO(\031)s FP(\()p FO(v)s FP(\))24 b(=)1666 2421 y Fz(\022)1727 2487 y FO(\031)s FP(\()p FO(v)s FP(\))231 b(0)1786 2587 y(0)141 b FO(\014)2016 2599 y FM(n)p FL(+1)2145 2587 y FO(\031)s FP(\()p FO(v)s FP(\))2303 2421 y Fz(\023)2378 2538 y FO(;)493 2770 y FP(^)-46 b FO(\031)s FP(\()p FO(s)610 2782 y FM(i)638 2770 y FP(\))23 b(=)781 2653 y Fz(\022)842 2719 y FO(\031)s FP(\()p FO(s)963 2731 y FM(i)991 2719 y FP(\))247 b(0)912 2819 y(0)152 b FO(\017)1140 2831 y FM(n)p FL(+1)p FM(i)1293 2819 y FO(\031)s FP(\()p FO(s)1414 2831 y FM(i)1442 2819 y FP(\))1474 2653 y Fz(\023)1549 2770 y FO(;)180 b(i)23 b FP(=)f(1)p FO(;)14 b(:)g(:)g(:)g(;)g(n;)983 3003 y FP(^)-46 b FO(\031)s FP(\()p FO(s)1100 3015 y FM(n)p FL(+1)1229 3003 y FP(\))24 b(=)1372 2885 y Fz(\022)1434 2952 y FP(0)83 b FO(I)1433 3052 y(I)91 b FP(0)1602 2885 y Fz(\023)1677 3003 y FO(;)118 3241 y(I)43 b FP(:)36 b FO(H)325 3253 y FM(\031)407 3241 y FQ(!)g FO(H)595 3253 y FM(\031)675 3241 y FP(is)g(the)f(iden)n(tit)n(y)h(op)r(erator.) 59 b(Let)35 b(us)h(sho)n(w)e(that)i FB(L)2354 3253 y FM(n)p FL(+1)2519 3241 y FP(is)118 3340 y(indeed)c(a)f(residual)g (family)-7 b(.)49 b(F)-7 b(or)31 b FQ(8)p FO(x)f FQ(2)g FO(A)1487 3352 y FM(n)p FL(+1)1648 3340 y FP(there)h(exists)h(an)f (expansion)118 3440 y FO(x)24 b FP(=)e FO(b)312 3452 y FL(1)368 3440 y FP(+)c FO(s)490 3452 y FM(n)p FL(+1)619 3440 y FO(b)655 3452 y FL(2)692 3440 y FP(,)28 b(with)g FO(b)968 3452 y FL(1)1005 3440 y FP(,)f FO(b)1091 3452 y FL(2)1151 3440 y FQ(2)d FO(B)t FP(.)37 b(If)28 b FO(b)1476 3452 y FL(2)1536 3440 y FQ(6)p FP(=)23 b(0,)k(then)h FQ(9)p FO(\031)f FQ(2)c FB(L)2162 3452 y FM(n)2235 3440 y FP(suc)n(h)28 b(that)118 3539 y FO(\031)s FP(\()p FO(b)236 3551 y FL(2)274 3539 y FP(\))23 b FQ(6)p FP(=)g(0)k FQ(\))638 3781 y FP(^)-47 b FO(\031)t FP(\()p FO(x)p FP(\))24 b(=)906 3664 y Fz(\022)968 3731 y FO(\031)s FP(\()p FO(b)1086 3743 y FL(1)1123 3731 y FP(\))83 b FQ(\003)968 3830 y FO(\031)s FP(\()p FO(b)1086 3842 y FL(2)1123 3830 y FP(\))g FQ(\003)1280 3664 y Fz(\023)1364 3781 y FQ(6)p FP(=)23 b(0)p FO(;)183 b FP(^)-46 b FO(\031)26 b FQ(2)e FB(L)1907 3793 y FM(n)p FL(+1)2036 3781 y FO(:)243 4028 y FP(If)31 b FO(b)365 4040 y FL(2)431 4028 y FP(=)d(0,)j FO(b)656 4040 y FL(1)722 4028 y FQ(6)p FP(=)d(0,)j(then)h FQ(9)p FO(\031)g FQ(2)d FB(L)1372 4040 y FM(n)1418 4028 y FP(:)43 b FO(\031)s FP(\()p FO(b)1602 4040 y FL(1)1640 4028 y FP(\))29 b FQ(6)p FP(=)f(0)j FQ(\))k FP(^)-46 b FO(\031)s FP(\()p FO(b)2099 4040 y FL(1)2136 4028 y FP(\))29 b FQ(6)p FP(=)f(0.)47 b(Note)118 4128 y(that)28 b FQ(8)t FP(^)-46 b FO(\031)25 b FQ(2)f FB(L)555 4140 y FM(n)p FL(+1)685 4128 y FP(,)j(dim)19 b(^)-47 b FO(\031)27 b FQ(\024)22 b FP(2)1090 4098 y FM(n)p FL(+2)1219 4128 y FP(.)p 2514 4128 4 57 v 2518 4075 50 4 v 2518 4128 V 2567 4128 4 57 v eop %%Page: 35 39 35 38 bop 118 100 a FK(1.2.)36 b FO(F)337 112 y FM(n)383 100 y FK(-algebras)25 b(and)i(their)h(represen)n(tations)848 b FP(35)118 333 y FC(R)l(emark)48 b FP(6)p FC(.)f FP(There)36 b(is)h(a)g(natural)f(in)n(v)n(olution)g(in)i FO(A)1847 345 y FM(k)1925 333 y FP(giv)n(en)e(b)n(y)h FO(u)2324 303 y FN(\003)2401 333 y FP(=)h FO(u)p FP(,)118 432 y FO(v)161 402 y FN(\003)223 432 y FP(=)22 b FO(v)s FP(,)k FO(s)441 402 y FN(\003)441 454 y FM(i)502 432 y FP(=)c FO(s)628 444 y FM(i)656 432 y FP(.)36 b(Since)25 b(there)f(exists)g(a)g (residual)g(family)g(for)g FO(Q)2177 444 y FL(2)2238 432 y FP(suc)n(h)h(that)118 532 y FQ(8)p FO(\031)h FQ(2)d FB(L)375 544 y FL(0)439 532 y FP(the)j(op)r(erators)e FO(\031)s FP(\()p FO(u)p FP(\),)j FO(\031)s FP(\()p FO(v)s FP(\))h(are)d(self-adjoin)n(t)g(\(see)h(1.2.2,)f(item)i(2\),)118 632 y(there)19 b(exists)f(a)h(residual)f(family)h(for)f FO(A)1329 644 y FM(k)1389 632 y FP(satisfying)g(the)h(condition:)33 b FQ(8)p FO(\031)25 b FQ(2)f FB(L)2534 644 y FM(k)118 731 y FP(the)32 b(op)r(erators)e FO(\031)s FP(\()p FO(u)p FP(\),)j FO(\031)s FP(\()p FO(v)s FP(\),)i FO(\031)s FP(\()p FO(s)1191 743 y FM(i)1219 731 y FP(\))d(are)f(self-adjoin)n(t.) 49 b(Therefore,)31 b(there)h(is)118 831 y(a)38 b(residual)g(family)g (for)g FO(A)983 843 y FM(k)1063 831 y FP(consisting)f(only)h(of)g (irreducible)g FQ(\003)p FP(-rep)o(resen-)118 930 y(tations.)243 1030 y(F)-7 b(or)28 b(a)h(description)g(of)g(irreducible)f FQ(\003)p FP(-rep)o(resentations)g(of)h FO(A)2206 1042 y FM(k)2247 1030 y FP(,)h(see)f(also)118 1130 y([247)o(].)118 1249 y FC(R)l(emark)39 b FP(7)p FC(.)i FQ(8)27 b FO(k)s FP(,)h FO(\013)767 1261 y FM(i)794 1249 y FP(,)g FO(\014)892 1261 y FM(i)920 1249 y FP(,)f FO(\017)1004 1261 y FM(ij)1063 1249 y FP(,)g(the)h(algebra)e FO(A)1609 1261 y FM(k)1678 1249 y FP(is)h(semi-simple.)243 1369 y(Moreo)n(v)n(er,)e(the)j(follo)n (wing)e(theorem)h(holds.)118 1509 y FR(Theorem)41 b(3.)46 b FC(L)l(et)38 b FO(Q)873 1521 y FL(2)p FM(;m)1026 1509 y FP(=)g FO(A)1191 1521 y FM(m)1292 1509 y FC(with)h FO(\013)1534 1521 y FM(i)1600 1509 y FP(=)e(1)p FC(,)k FO(\014)1857 1521 y FM(i)1922 1509 y FP(=)d(1)p FC(,)i FO(\017)2166 1521 y FM(ij)2262 1509 y FP(=)e(1)p FC(,)i(for)118 1609 y(al)t(l)c FO(i)p FC(,)h FO(j)5 b FC(.)54 b(Then)36 b(every)g(algebr)l(a)g FO(A)1246 1621 y FM(k)1322 1609 y FC(is)g(isomorphic)i(to)d FO(M)2034 1621 y FL(2)2067 1605 y Fw(n)2111 1609 y FP(\()p FO(Q)2209 1621 y FL(2)p FM(;m)2325 1609 y FP(\))g FC(or)h(to)118 1708 y FO(M)199 1720 y FL(2)232 1704 y Fw(n)277 1708 y FP(\()p FO(Z)6 b FP(\()p FO(A)466 1720 y FM(k)507 1708 y FP(\)\))p FC(,)31 b(wher)l(e)f FO(Z)6 b FP(\()p FO(A)1018 1720 y FM(k)1059 1708 y FP(\))30 b FC(is)g(the)g(c)l(enter)f(of)i FO(A)1757 1720 y FM(k)1798 1708 y FC(.)118 1848 y(Pr)l(o)l(of.)43 b FP(Let)28 b(us)g(split)f(the)h(pro)r(of)f(in)n(to)h(four)f(steps.)243 1948 y(1\).)39 b(Let)29 b(us)f(de\014ne)h(the)g(algebra)d FO(A)1374 1960 y FM(k)1440 1948 y FP(=)e FJ(C)15 b FQ(h)p FO(u;)f(v)s(;)g(s)1819 1960 y FL(1)1862 1948 y FO(;)g(:)g(:)g(:)f(;)h (s)2085 1960 y FM(k)2151 1948 y FQ(j)24 b FO(\013)2251 1960 y FM(i)2279 1948 y FO(;)14 b(\014)2363 1960 y FM(i)2390 1948 y FO(;)g(\017)2461 1960 y FM(ij)2519 1948 y FQ(i)p FP(.)118 2047 y(Supp)r(ose)23 b(that)h(there)f(exist)g FO(i)p FP(,)h FO(j)k FP(suc)n(h)23 b(that)g FO(\017)1545 2059 y FM(ij)1627 2047 y FP(=)f FQ(\000)p FP(1,)h(for)g(example,)h FO(s)2374 2059 y FL(1)2411 2047 y FO(s)2450 2059 y FL(2)2510 2047 y FP(=)118 2147 y FQ(\000)p FO(s)222 2159 y FL(2)259 2147 y FO(s)298 2159 y FL(1)335 2147 y FP(.)64 b(Then,)39 b(using)d(the)h(follo)n(wing)e(substitution)i(of)g(generators)d FO(s)2435 2117 y FN(0)2435 2168 y FL(1)2510 2147 y FP(=)118 2247 y FO(s)157 2259 y FL(1)194 2247 y FO(;)14 b(s)270 2217 y FN(0)270 2267 y FL(2)330 2247 y FP(=)23 b FO(s)457 2259 y FL(2)494 2247 y FO(;)596 2586 y(s)635 2552 y FN(0)635 2606 y FM(j)693 2586 y FP(=)781 2341 y Fz(8)781 2416 y(>)781 2441 y(>)781 2465 y(>)781 2490 y(<)781 2640 y(>)781 2665 y(>)781 2690 y(>)781 2714 y(:)855 2410 y FO(s)894 2422 y FM(j)928 2410 y FO(;)503 b(\017)1488 2422 y FL(1)p FM(j)1579 2410 y FP(=)23 b FO(\017)1701 2422 y FL(2)p FM(j)1792 2410 y FP(=)f(1)p FO(;)855 2529 y(s)894 2541 y FL(1)931 2529 y FO(s)970 2541 y FM(j)1005 2529 y FO(;)426 b(\017)1488 2541 y FL(1)p FM(j)1579 2529 y FP(=)23 b FQ(\000)p FO(\017)1766 2541 y FL(2)p FM(j)1856 2529 y FP(=)g(1)p FO(;)855 2649 y(s)894 2661 y FL(2)931 2649 y FO(s)970 2661 y FM(j)1005 2649 y FO(;)426 b(\017)1488 2661 y FL(1)p FM(j)1579 2649 y FP(=)23 b FQ(\000)p FO(\017)1766 2661 y FL(2)p FM(j)1856 2649 y FP(=)g FQ(\000)p FP(1)p FO(;)855 2697 y Fz(p)p 938 2697 171 4 v 71 x FP(\()p FQ(\000)p FP(1\))13 b FO(s)1161 2780 y FL(1)1198 2768 y FO(s)1237 2780 y FL(2)1274 2768 y FO(s)1313 2780 y FM(j)1348 2768 y FO(;)83 b(\017)1488 2780 y FL(1)p FM(j)1579 2768 y FP(=)23 b FO(\017)1701 2780 y FL(2)p FM(j)1792 2768 y FP(=)f FQ(\000)p FP(1)p FO(;)599 3097 y(u)647 3063 y FN(0)693 3097 y FP(=)781 2852 y Fz(8)781 2927 y(>)781 2952 y(>)781 2977 y(>)781 3002 y(<)781 3151 y(>)781 3176 y(>)781 3201 y(>)781 3226 y(:)855 2921 y FO(u;)502 b(\013)1481 2933 y FL(1)1541 2921 y FP(=)23 b FO(\013)1682 2933 y FL(2)1742 2921 y FP(=)g(1)p FO(;)855 3041 y(s)894 3053 y FL(1)931 3041 y FO(u;)426 b(\013)1481 3053 y FL(1)1541 3041 y FP(=)23 b FQ(\000)p FO(\013)1747 3053 y FL(2)1807 3041 y FP(=)g(1)p FO(;)855 3160 y(s)894 3172 y FL(2)931 3160 y FO(u;)426 b(\013)1481 3172 y FL(1)1541 3160 y FP(=)23 b FQ(\000)p FO(\013)1747 3172 y FL(2)1807 3160 y FP(=)g FQ(\000)p FP(1)p FO(;)855 3209 y Fz(p)p 938 3209 V 71 x FP(\()p FQ(\000)p FP(1\))13 b FO(s)1161 3292 y FL(1)1198 3280 y FO(s)1237 3292 y FL(2)1274 3280 y FO(u;)83 b(\013)1481 3292 y FL(1)1541 3280 y FP(=)23 b FO(\013)1682 3292 y FL(2)1742 3280 y FP(=)g FQ(\000)p FP(1)p FO(;)604 3609 y(v)647 3574 y FN(0)693 3609 y FP(=)781 3364 y Fz(8)781 3438 y(>)781 3463 y(>)781 3488 y(>)781 3513 y(<)781 3663 y(>)781 3687 y(>)781 3712 y(>)781 3737 y(:)855 3432 y FO(v)s(;)503 b(\014)1471 3444 y FL(1)1531 3432 y FP(=)23 b FO(\014)1666 3444 y FL(2)1726 3432 y FP(=)f(1)p FO(;)855 3552 y(s)894 3564 y FL(1)931 3552 y FO(v)s(;)427 b(\014)1471 3564 y FL(1)1531 3552 y FP(=)23 b FQ(\000)p FO(\014)1731 3564 y FL(2)1790 3552 y FP(=)g(1)p FO(;)855 3672 y(s)894 3684 y FL(2)931 3672 y FO(v)s(;)427 b(\014)1471 3684 y FL(1)1531 3672 y FP(=)23 b FQ(\000)p FO(\014)1731 3684 y FL(2)1790 3672 y FP(=)g FQ(\000)p FP(1)p FO(;)855 3720 y Fz(p)p 938 3720 V 71 x FP(\()p FQ(\000)p FP(1\))13 b FO(s)1161 3803 y FL(1)1198 3791 y FO(s)1237 3803 y FL(2)1274 3791 y FO(v)s(;)84 b(\014)1471 3803 y FL(1)1531 3791 y FP(=)23 b FO(\014)1666 3803 y FL(2)1726 3791 y FP(=)f FQ(\000)p FP(1)p FO(;)118 3948 y FP(w)n(e)41 b(obtain)g(that)g FO(A)781 3960 y FM(k)868 3948 y FP(=)k FJ(C)15 b FQ(h)p FO(u)1112 3918 y FN(0)1141 3948 y FO(;)f(v)1221 3918 y FN(0)1244 3948 y FO(;)g(s)1320 3918 y FN(0)1320 3969 y FL(1)1357 3948 y FO(;)g(:)g(:)g(:)g(;)g(s)1581 3918 y FN(0)1581 3972 y FM(k)1667 3948 y FQ(j)46 b FO(\013)1789 3918 y FN(0)1789 3969 y FL(1)1872 3948 y FP(=)f FO(\013)2035 3918 y FN(0)2035 3969 y FL(2)2118 3948 y FP(=)g FO(\014)2279 3918 y FN(0)2275 3969 y FL(1)2357 3948 y FP(=)h FO(\014)2519 3918 y FN(0)2515 3969 y FL(2)2552 3948 y FO(;)118 4048 y(\017)152 4018 y FN(0)152 4068 y FL(12)255 4048 y FP(=)33 b FQ(\000)p FP(1)p FO(;)27 b(\017)544 4018 y FN(0)544 4069 y FL(1)p FM(j)644 4048 y FP(=)33 b FO(\017)776 4018 y FN(0)776 4069 y FL(2)p FM(j)877 4048 y FP(=)f(1)p FO(;)c(j)38 b(>)32 b FP(2)p FQ(i)p FP(.)55 b(Let)33 b FO(A)1604 4060 y FM(k)q FN(\000)p FL(2)1764 4048 y FP(denote)h(the)g(subalgebra)118 4147 y FJ(C)15 b FQ(h)p FO(s)243 4117 y FN(0)243 4168 y FL(3)287 4147 y FO(;)f(:)g(:)g(:)f(;)h(s)510 4117 y FN(0)510 4171 y FM(k)551 4147 y FO(;)g(u)636 4117 y FN(0)658 4147 y FO(;)g(v)738 4117 y FN(0)762 4147 y FQ(i)p FP(.)p eop %%Page: 36 40 36 39 bop 118 100 a FP(36)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FR(Lemma)j(4.)40 b FC(The)31 b(algebr)l(a)g FO(A)1085 345 y FM(k)1156 333 y FC(is)f(isomorphic)j(to)c FO(M)1851 345 y FL(2)1888 333 y FP(\()p FO(A)1982 345 y FM(k)q FN(\000)p FL(2)2108 333 y FP(\))p FC(.)118 496 y(Pr)l(o)l(of.)43 b FP(Let)35 b FO(A)596 508 y FM(k)637 496 y FP(,)h FO(A)758 508 y FM(k)q FN(\000)p FL(2)919 496 y FP(b)r(e)e(algebras)f(describ)r (ed)h(ab)r(o)n(v)n(e.)56 b(Then)34 b FQ(8)p FO(x)g FQ(2)h FO(A)2534 508 y FM(k)118 596 y FP(there)29 b(exists)f(a)g(unique)h (decomp)r(osition)g FO(x)c FP(=)g FO(s)1653 566 y FN(0)1653 617 y FL(1)1690 596 y FO(a)1734 608 y FL(1)1790 596 y FP(+)19 b FO(s)1913 566 y FN(0)1913 617 y FL(2)1950 596 y FO(a)1994 608 y FL(2)2050 596 y FP(+)g FO(s)2173 566 y FN(0)2173 617 y FL(1)2210 596 y FO(s)2249 566 y FN(0)2249 617 y FL(2)2286 596 y FO(a)2330 608 y FL(3)2387 596 y FP(+)g FO(a)2515 608 y FL(4)2552 596 y FP(,)118 696 y(where)31 b FO(a)406 708 y FM(i)464 696 y FQ(2)f FO(A)611 708 y FM(k)q FN(\000)p FL(2)737 696 y FP(,)j FO(i)d FP(=)f(1)j FO(:)14 b(:)g(:)27 b FP(,)33 b(4.)49 b(This)32 b(implies)g(the)g(follo) n(wing)f(iden)n(tit)n(y)118 795 y FO(x)d FP(=)286 728 y Fz(\000)324 795 y FP(\(1)20 b(+)g FO(s)542 765 y FN(0)542 816 y FL(1)579 795 y FP(\))14 b FO(s)664 807 y FL(2)702 795 y FO(a)746 765 y FN(0)746 816 y FL(1)803 795 y FP(+)20 b(\(1)g FQ(\000)g FO(s)1106 765 y FN(0)1106 816 y FL(1)1143 795 y FP(\))14 b FO(s)1228 807 y FL(2)1266 795 y FO(a)1310 765 y FN(0)1310 816 y FL(2)1367 795 y FP(+)20 b(\(1)g(+)g FO(s)1670 765 y FN(0)1670 816 y FL(1)1708 795 y FP(\))14 b FO(a)1798 765 y FN(0)1798 816 y FL(3)1855 795 y FP(+)20 b(\(1)g FQ(\000)g FO(s)2158 765 y FN(0)2158 816 y FL(1)2195 795 y FP(\))14 b FO(a)2285 765 y FN(0)2285 816 y FL(4)2323 728 y Fz(\001)2361 795 y FO(=)p FP(2.)45 b(It)118 895 y(can)27 b(b)r(e)h(easily)f(v)n(eri\014ed)g(that)h FO( )12 b FP(:)28 b FO(A)1262 907 y FM(k)1326 895 y FQ(!)23 b FO(M)1513 907 y FL(2)1550 895 y FP(\()p FO(A)1644 907 y FM(k)q FN(\000)p FL(2)1770 895 y FP(\),)230 1118 y FO( )s FP(\()p FO(s)358 1084 y FN(0)358 1139 y FL(1)396 1118 y FP(\))g(=)539 1001 y Fz(\022)601 1068 y FO(e)115 b FP(0)600 1167 y(0)82 b FQ(\000)p FO(e)827 1001 y Fz(\023)902 1118 y FO(;)97 b( )s FP(\()p FO(s)1150 1084 y FN(0)1150 1139 y FL(2)1188 1118 y FP(\))23 b(=)1331 1001 y Fz(\022)1392 1068 y FP(0)84 b FO(e)1393 1167 y(e)g FP(0)1558 1001 y Fz(\023)1633 1118 y FO(;)97 b( )s FP(\()p FO(s)1881 1084 y FN(0)1881 1139 y FM(j)1916 1118 y FP(\))23 b(=)2059 1001 y Fz(\022)2120 1064 y FO(s)2159 1034 y FN(0)2159 1086 y FM(j)2293 1064 y FP(0)2136 1167 y(0)99 b FO(s)2316 1137 y FN(0)2316 1189 y FM(j)2351 1001 y Fz(\023)2426 1118 y FO(;)611 1354 y( )s FP(\()p FO(u)748 1319 y FN(0)771 1354 y FP(\))23 b(=)914 1237 y Fz(\022)975 1303 y FO(u)1023 1273 y FN(0)1143 1303 y FP(0)990 1403 y(0)97 b FO(u)1177 1372 y FN(0)1200 1237 y Fz(\023)1275 1354 y FO(;)f( )s FP(\()p FO(v)1526 1319 y FN(0)1550 1354 y FP(\))24 b(=)1693 1237 y Fz(\022)1754 1303 y FO(v)1797 1273 y FN(0)1916 1303 y FP(0)1767 1403 y(0)95 b FO(v)1947 1372 y FN(0)1970 1237 y Fz(\023)2045 1354 y FO(;)118 1579 y FP(is)31 b(an)f (isomorphism.)46 b(The)31 b(in)n(v)n(erse)e(mapping)i FO( )1705 1549 y FN(\000)p FL(1)1823 1579 y FP(:)d FO(M)1955 1591 y FL(2)1992 1579 y FP(\()p FO(A)2086 1591 y FM(k)q FN(\000)p FL(2)2213 1579 y FP(\))g FQ(!)h FO(A)2447 1591 y FM(k)2519 1579 y FP(is)118 1679 y(de\014ned)f(b)n(y)f(the)h(form)n (ula:)118 1895 y FO( )175 1860 y FN(\000)p FL(1)278 1778 y Fz(\022)339 1844 y FO(a)383 1814 y FN(0)383 1865 y FL(3)503 1844 y FO(a)547 1814 y FN(0)547 1865 y FL(1)339 1944 y FO(a)383 1914 y FN(0)383 1964 y FL(2)503 1944 y FO(a)547 1914 y FN(0)547 1964 y FL(4)585 1778 y Fz(\023)669 1895 y FP(=)766 1839 y(1)p 766 1876 42 4 v 766 1952 a(2)818 1827 y Fz(\000)856 1895 y FP(\(1+)p FO(s)1034 1860 y FN(0)1034 1915 y FL(1)1070 1895 y FP(\))p FO(s)1141 1907 y FL(2)1179 1895 y FO(a)1223 1860 y FN(0)1223 1915 y FL(1)1260 1895 y FP(+\(1)p FQ(\000)p FO(s)1503 1860 y FN(0)1503 1915 y FL(1)1539 1895 y FP(\))p FO(s)1610 1907 y FL(2)1648 1895 y FO(a)1692 1860 y FN(0)1692 1915 y FL(2)1729 1895 y FP(+\(1+)p FO(s)1972 1860 y FN(0)1972 1915 y FL(1)2008 1895 y FP(\))p FO(a)2084 1860 y FN(0)2084 1915 y FL(3)2121 1895 y FP(+\(1)p FQ(\000)p FO(s)2364 1860 y FN(0)2364 1915 y FL(1)2400 1895 y FP(\))p FO(a)2476 1860 y FN(0)2476 1915 y FL(4)2514 1827 y Fz(\001)2552 1895 y FO(:)118 2110 y FP(whic)n(h)g(completes)f(the)h(pro)r(of)f(of)g (the)h(lemma.)p 2514 2110 4 57 v 2518 2058 50 4 v 2518 2110 V 2567 2110 4 57 v 243 2274 a(Using)f(this)h(lemma)f(one)h(can)f (obtain)g(the)h(follo)n(wing)f(prop)r(osition:)118 2423 y FR(Prop)s(osition)k(11.)40 b FC(The)31 b(algebr)l(a)h FO(A)1312 2435 y FM(k)1376 2423 y FP(=)23 b FJ(C)15 b FQ(h)q FO(u;)f(v)s(;)g(s)1755 2435 y FL(1)1797 2423 y FO(;)g(:)g(:)g(:)g(;)g(s)2021 2435 y FM(k)2085 2423 y FQ(j)24 b FO(\013)2185 2435 y FM(i)2213 2423 y FO(;)14 b(\014)2297 2435 y FM(i)2324 2423 y FO(;)g(\017)2395 2435 y FM(ij)2453 2423 y FQ(i)30 b FC(is)118 2523 y(isomorphic)j(to)c FO(M)724 2535 y FL(2)757 2519 y Fw(m)816 2523 y FP(\()p FO(A)910 2535 y FM(k)q FN(\000)p FL(2)p FM(m)1095 2523 y FP(\))h FC(for)h(some)f FO(m)p FC(,)g(wher)l(e)475 2694 y FO(A)537 2706 y FM(k)q FN(\000)p FL(2)p FM(m)745 2694 y FP(=)23 b FJ(C)15 b FQ(h)p FO(u)967 2659 y FN(0)996 2694 y FO(;)f(v)1076 2659 y FN(0)1099 2694 y FO(;)g(s)1175 2659 y FN(0)1175 2714 y FL(2)p FM(m)p FL(+1)1355 2694 y FO(;)g(:)g(:)g(:)g(;)g(s)1579 2659 y FN(0)1579 2714 y FM(k)1643 2694 y FQ(j)23 b FO(\013)1742 2659 y FN(0)1742 2714 y FM(i)1770 2694 y FO(;)14 b(\014)1858 2659 y FN(0)1854 2714 y FM(i)1881 2694 y FO(;)g(\017)1952 2659 y FN(0)1952 2714 y FM(ij)2033 2694 y FP(=)23 b(1)p FQ(i)p FO(:)243 2864 y FP(So,)k(w)n(e)f(m)n(ust)i(study)f(the)h(structure)e(of)h(the)h (algebra)d FO(A)2023 2876 y FM(k)2091 2864 y FP(with)j(the)g(con-)118 2964 y(dition)g(that)g FO(\017)572 2976 y FM(ij)653 2964 y FP(=)23 b(1.)243 3063 y(2\))32 b(W)-7 b(e)33 b(further)f(assume,)h (without)g(loss)e(of)i(generalit)n(y)-7 b(,)32 b(that)h(for)e(some)118 3163 y FO(m)25 b FQ(2)h FJ(N)t FP(,)35 b(the)30 b(relations)e FO(\013)946 3175 y FM(i)999 3163 y FP(=)d FO(\014)1136 3175 y FM(i)1163 3163 y FP(,)30 b(1)24 b FQ(\024)h FO(i)g(<)g(m)p FP(,)30 b FO(\013)1695 3175 y FM(i)1748 3163 y FQ(6)p FP(=)25 b FO(\014)1885 3175 y FM(i)1912 3163 y FP(,)30 b FO(m)25 b FQ(\024)g FO(i)g FQ(\024)g FO(k)s FP(,)k(hold.)118 3263 y(Let)f(us)f(in)n(tro)r(duce)h(the)g(new)f(generators)e FO(u)1506 3233 y FN(0)1552 3263 y FP(=)e FO(u)p FP(,)k FO(v)1781 3233 y FN(0)1828 3263 y FP(=)22 b FO(v)s FP(,)885 3503 y FO(s)924 3469 y FN(0)924 3524 y FM(j)982 3503 y FP(=)1069 3361 y Fz(\()1136 3447 y FO(s)1175 3459 y FM(j)1210 3447 y FO(;)163 b FP(1)23 b FQ(\024)f FO(j)28 b(<)23 b(m;)1136 3566 y(s)1175 3578 y FM(j)1210 3566 y FO(s)1249 3578 y FM(k)1290 3566 y FO(;)83 b(m)23 b FQ(\024)g FO(j)28 b(<)22 b(k)s(:)118 3749 y FP(Then)30 b FO(A)399 3761 y FM(k)466 3749 y FP(=)c FJ(C)15 b FQ(h)p FO(u)691 3719 y FN(0)720 3749 y FO(;)f(v)800 3719 y FN(0)823 3749 y FO(;)g(s)899 3719 y FN(0)899 3769 y FL(1)936 3749 y FO(;)g(:)g(:)g(:)g(;)g(s)1160 3719 y FN(0)1160 3772 y FM(k)1227 3749 y FQ(j)26 b FO(\013)1329 3761 y FM(i)1383 3749 y FP(=)g FO(\014)1521 3761 y FM(i)1548 3749 y FO(;)i(i)e(<)f(k)s (;)j(\013)1894 3761 y FM(k)1961 3749 y FP(=)e FQ(\006)p FP(1)p FO(;)h(\014)2256 3761 y FM(k)2323 3749 y FP(=)e FQ(\006)p FP(1)p FQ(i)p FP(,)118 3848 y(and)30 b(there)f(are)g(t)n(w)n (o)g(p)r(ossibilities:)42 b FO(\013)1333 3860 y FM(k)1400 3848 y FP(=)27 b FO(\014)1539 3860 y FM(k)1610 3848 y FP(or)i FO(\013)1767 3860 y FM(k)1834 3848 y FQ(6)p FP(=)e FO(\014)1973 3860 y FM(k)2014 3848 y FP(.)43 b(The)30 b(\014rst)g(case)118 3948 y(is)e(considered)e(in)i(3\),)g(and)f(the)h (second)f(in)h(4\).)243 4048 y(3\))i(Let)g(us)g(arrange)e(the)j(family) f FQ(f)p FO(s)1393 4060 y FM(j)1427 4048 y FQ(g)g FP(so)g(that,)h(for)f (some)f FO(m)p FP(,)i(the)g(con-)118 4147 y(ditions)j FO(\013)450 4159 y FM(i)512 4147 y FP(=)g FO(\014)658 4159 y FM(i)719 4147 y FP(=)g(1,)h(1)f FQ(\024)f FO(i)h(<)g(m)p FP(,)i(and)e FO(\013)1607 4159 y FM(i)1668 4147 y FP(=)g FO(\014)1814 4159 y FM(i)1876 4147 y FP(=)f FQ(\000)p FP(1,)i FO(m)f FQ(\024)g FO(i)f FQ(\024)h FO(k)s FP(,)p eop %%Page: 37 41 37 40 bop 118 100 a FK(1.2.)36 b FO(F)337 112 y FM(n)383 100 y FK(-algebras)25 b(and)i(their)h(represen)n(tations)848 b FP(37)118 333 y(hold.)68 b(Using)37 b(the)i(new)f(generators)d FO(s)1395 303 y FN(0)1395 354 y FM(i)1463 333 y FP(=)k FO(s)1606 345 y FM(i)1634 333 y FP(,)i(1)e FQ(\024)h FO(i)g(<)f(m)p FP(,)i FO(s)2233 303 y FN(0)2233 354 y FM(i)2301 333 y FP(=)e FO(s)2444 345 y FM(i)2472 333 y FO(s)2511 345 y FM(k)2552 333 y FP(,)118 432 y FO(m)29 b FQ(\024)h FO(i)f(<)g(k)23 b FQ(\000)e FP(1,)32 b FO(u)764 402 y FN(0)816 432 y FP(=)d FO(u)p FP(,)j FO(v)1056 402 y FN(0)1109 432 y FP(=)d FO(v)s FP(,)k(w)n(e)e(obtain)g(the)h(algebra)e FO(A)2194 444 y FM(k)2267 432 y FP(with)h(the)118 532 y(co)r(e\016cien)n(ts)d FO(\013)592 502 y FN(0)592 554 y FM(i)644 532 y FP(=)c FO(\014)784 502 y FN(0)780 554 y FM(i)831 532 y FP(=)g(1,)k FO(i)c(<)g(k)s FP(.)38 b(If)29 b FO(\013)1399 502 y FN(0)1399 555 y FM(k)1464 532 y FP(=)24 b FO(\014)1604 502 y FN(0)1600 555 y FM(k)1665 532 y FP(=)g(1,)k(then)h(the)f(algebra)f FO(A)2534 544 y FM(k)118 632 y FP(is)f(isomorphic)e(to)i FO(Q)784 644 y FL(2)p FM(;k)877 632 y FP(.)36 b(In)26 b(the)g(case)f(where)g FO(\013)1644 601 y FN(0)1644 655 y FM(k)1708 632 y FP(=)e FO(\014)1847 601 y FN(0)1843 655 y FM(k)1907 632 y FP(=)f FQ(\000)p FP(1,)k(w)n(e)f(ha)n(v)n(e)f(the)118 731 y(follo)n(wing)j (prop)r(osition:)118 882 y FR(Prop)s(osition)44 b(12.)k FO(A)881 894 y FM(k)965 882 y FP(=)42 b FJ(C)15 b FQ(h)p FO(u;)f(v)s(;)g(s)1362 894 y FL(1)1405 882 y FO(;)g(:)g(:)g(:)g(;)g(s) 1629 894 y FM(k)1712 882 y FQ(j)43 b FO(\013)1831 894 y FM(i)1902 882 y FP(=)g FO(\014)2057 894 y FM(i)2127 882 y FP(=)g(1)p FO(;)27 b(i)42 b(<)h(k)s(;)118 981 y(\013)171 993 y FM(k)235 981 y FP(=)23 b FO(\014)370 993 y FM(k)434 981 y FP(=)f FQ(\000)p FP(1)p FQ(i)683 959 y(\030)683 986 y FP(=)770 981 y FO(M)851 993 y FL(2)888 981 y FP(\()p FO(Z)6 b FP(\()p FO(A)1077 993 y FM(k)1118 981 y FP(\)\))p FC(.)118 1149 y(Pr)l(o)l(of.)43 b FP(Let)29 b(us)g(denote)g FO(B)g FP(=)c FJ(C)15 b FQ(h)p FO(s)1212 1161 y FL(1)1255 1149 y FO(;)f(:)g(:)g(:)g(;)g(s)1479 1161 y FM(k)q FN(\000)p FL(1)1605 1149 y FO(;)g(f)t(;)g(f)1774 1119 y FN(\000)p FL(1)1862 1149 y FQ(i)p FP(,)30 b FO(f)j FP(=)25 b(\(1)19 b(+)g FO(s)2327 1161 y FM(k)2368 1149 y FP(\))p FO(uv)j FP(+)118 1249 y(\(1)c FQ(\000)g FO(s)332 1261 y FM(k)373 1249 y FP(\))p FO(v)s(u)p FP(,)28 b(and)f(write)h(an)n(y)f FO(x)c FQ(2)g FO(A)1288 1261 y FM(k)1357 1249 y FP(in)28 b(the)g(form)234 1471 y FO(x)c FP(=)402 1415 y(1)p 402 1452 42 4 v 402 1528 a(2)454 1403 y Fz(\000)492 1471 y FP(\(1)18 b(+)g FO(s)706 1483 y FM(k)747 1471 y FP(\))c FO(a)837 1483 y FL(1)893 1471 y FP(+)k(\(1)g FQ(\000)g FO(s)1190 1483 y FM(k)1231 1471 y FP(\))c FO(a)1321 1483 y FL(2)1376 1471 y FP(+)k(\(1)h(+)f FO(s)1674 1483 y FM(k)1714 1471 y FP(\))c FO(ua)1852 1483 y FL(3)1908 1471 y FP(+)k(\(1)g FQ(\000)g FO(s)2205 1483 y FM(k)2246 1471 y FP(\))c FO(ua)2384 1483 y FL(4)2421 1403 y Fz(\001)118 1677 y FO(a)162 1689 y FM(i)213 1677 y FQ(2)24 b FO(B)t FP(.)37 b(Note,)28 b(that)g(this)h(decomp)r(osition)e(is)g(unique.)38 b(Indeed,)28 b(if)h(w)n(e)e(ha)n(v)n(e)118 1777 y(another)g(one:)239 1978 y FO(x)d FP(=)407 1922 y(1)p 407 1959 V 407 2035 a(2)459 1911 y Fz(\000)497 1978 y FP(\(1)18 b(+)g FO(s)711 1990 y FM(k)752 1978 y FP(\))c FO(b)834 1990 y FL(1)889 1978 y FP(+)k(\(1)h FQ(\000)f FO(s)1187 1990 y FM(k)1227 1978 y FP(\))c FO(b)1309 1990 y FL(2)1365 1978 y FP(+)k(\(1)g(+)g FO(s)1662 1990 y FM(k)1703 1978 y FP(\))c FO(ub)1833 1990 y FL(3)1888 1978 y FP(+)k(\(1)g FQ(\000)g FO(s)2185 1990 y FM(k)2226 1978 y FP(\))c FO(ub)2356 1990 y FL(4)2393 1911 y Fz(\001)2431 1978 y FO(;)118 2197 y(b)154 2209 y FM(i)204 2197 y FQ(2)24 b FO(B)t FP(,)j(then)f(w)n(e)g(obtain)g(the)h (iden)n(tit)n(y)g(0)22 b(=)h(1)p FO(=)p FP(2)1691 2130 y Fz(\000)1727 2197 y FP(\(1)16 b(+)f FO(s)1936 2209 y FM(k)1977 2197 y FP(\)\()p FO(b)2077 2209 y FL(1)2130 2197 y FQ(\000)h FO(a)2255 2209 y FL(1)2292 2197 y FP(\))g(+)g(\(1)f FQ(\000)118 2305 y FO(s)157 2317 y FM(k)198 2305 y FP(\)\()p FO(b)298 2317 y FL(2)353 2305 y FQ(\000)i FO(a)479 2317 y FL(2)516 2305 y FP(\))h(+)g(\(1)f(+)h FO(s)862 2317 y FM(k)902 2305 y FP(\))p FO(u)p FP(\()p FO(b)1050 2317 y FL(3)1105 2305 y FQ(\000)f FO(a)1231 2317 y FL(3)1268 2305 y FP(\))h(+)g(\(1)f FQ(\000)g FO(s)1613 2317 y FM(k)1654 2305 y FP(\))p FO(u)p FP(\()p FO(b)1802 2317 y FL(4)1857 2305 y FQ(\000)g FO(a)1983 2317 y FL(4)2020 2305 y FP(\))2052 2238 y Fz(\001)2091 2305 y FP(.)36 b(Multiplying)118 2405 y(this)28 b(form)n(ula)f(b)n(y)g(1)18 b(+)g FO(s)882 2417 y FM(k)923 2405 y FP(,)27 b(w)n(e)h(conclude)f(that)934 2588 y(0)c(=)g(2\(1)17 b(+)h FO(s)1342 2600 y FM(k)1383 2588 y FP(\)\()p FO(b)1483 2600 y FL(1)1539 2588 y FQ(\000)g FO(a)1666 2600 y FL(1)1703 2588 y FP(\))p FO(:)118 2771 y FP(Com)n(bining)27 b(it)h(with)g(the)g(iden)n(tit)n(y)549 2954 y FO(u)p FP(\(1)17 b(+)h FO(s)810 2966 y FM(k)851 2954 y FP(\)\()p FO(b)951 2966 y FL(1)1007 2954 y FQ(\000)g FO(a)1134 2966 y FL(1)1171 2954 y FP(\))p FO(u)23 b FP(=)g(\(1)18 b FQ(\000)g FO(s)1576 2966 y FM(k)1617 2954 y FP(\)\()p FO(b)1717 2966 y FL(1)1773 2954 y FQ(\000)g FO(a)1900 2966 y FL(1)1937 2954 y FP(\))23 b(=)g(0)p FO(;)118 3137 y FP(w)n(e)h(obtain)f(that)h(\()p FO(b)735 3149 y FL(1)783 3137 y FQ(\000)11 b FO(a)903 3149 y FL(1)940 3137 y FP(\))23 b(=)g(0.)35 b(In)24 b(the)g(same)f(w)n(a)n(y)g(w)n(e)g(sho)n(w)g(that)h FO(b)2323 3149 y FL(2)2383 3137 y FP(=)f FO(a)2515 3149 y FL(2)2552 3137 y FP(,)118 3236 y FO(b)154 3248 y FL(3)214 3236 y FP(=)g FO(a)346 3248 y FL(3)383 3236 y FP(,)h FO(b)466 3248 y FL(4)526 3236 y FP(=)f FO(a)658 3248 y FL(4)695 3236 y FP(.)36 b(No)n(w)23 b(w)n(e)h(can)f(write)h(the)g (form)n(ula)f(for)g(the)h(isomorphism)118 3336 y FO( )12 b FP(:)28 b FO(A)297 3348 y FM(k)361 3336 y FQ(!)23 b FO(M)548 3348 y FL(2)585 3336 y FP(\()p FO(B)t FP(\),)1004 3486 y FO( )s FP(\()p FO(x)p FP(\))i(=)1284 3369 y Fz(\022)1345 3435 y FO(a)1389 3447 y FL(1)1509 3435 y FO(a)1553 3447 y FL(3)1345 3535 y FO(a)1389 3547 y FL(4)1509 3535 y FO(a)1553 3547 y FL(2)1591 3369 y Fz(\023)1665 3486 y FO(:)118 3681 y FP(A)d(direct)g(v)n(eri\014cation)f(sho)n(ws)g(that)h FO( )j FP(is)d(an)g(epimorphism)f(and)h(the)h(algebra)118 3781 y FO(B)32 b FP(is)27 b(isomorphic)g(to)g(the)h FO(Z)6 b FP(\()p FO(A)1118 3793 y FM(k)1159 3781 y FP(\).)p 2514 3781 4 57 v 2518 3728 50 4 v 2518 3781 V 2567 3781 4 57 v 243 3948 a(4\))33 b(Consider)g(the)h(second)f(p)r(ossibilit)n(y) -7 b(.)55 b(One)33 b(can)g(assume)g(that)h FO(\013)2436 3960 y FM(k)2510 3948 y FP(=)118 4048 y FQ(\000)p FO(\014)230 4060 y FM(k)312 4048 y FP(=)40 b FQ(\000)p FP(1)e(\(otherwise)g(w)n(e)g (replace)f FO(u)h FP(with)h FO(v)j FP(or)37 b(vice)h(v)n(ersa\).)68 b(Using)118 4147 y(the)32 b(previous)e(results)h(w)n(e)f(ha)n(v)n(e)h FO(\013)1241 4159 y FM(j)1305 4147 y FP(=)e FO(\014)1446 4159 y FM(j)1480 4147 y FP(,)k FO(j)h(<)29 b(k)s FP(.)47 b(Then)32 b(b)n(y)f(the)g(metho)r(d)p eop %%Page: 38 42 38 41 bop 118 100 a FP(38)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(describ)r(ed)35 b(in)g(3\),)h(w)n(e)e(obtain)h(the)g(iden)n (tities)g FO(\013)1693 345 y FM(j)1763 333 y FP(=)f FO(\014)1909 345 y FM(j)1979 333 y FP(=)h(1,)h FO(j)k(<)34 b(k)26 b FQ(\000)d FP(1,)118 432 y FO(\013)171 444 y FM(k)q FN(\000)p FL(1)338 432 y FP(=)41 b FO(\014)491 444 y FM(k)q FN(\000)p FL(1)617 432 y FP(.)70 b(Th)n(us,)41 b FO(\013)1010 444 y FM(j)1087 432 y FP(=)g FO(\014)1240 444 y FM(j)1316 432 y FP(=)g(1,)g FO(j)47 b(<)41 b(k)28 b FQ(\000)e FP(1,)41 b FO(\013)2036 444 y FM(k)2118 432 y FP(=)g FQ(\000)p FO(\014)2336 444 y FM(k)2377 432 y FP(,)g(and)118 532 y FO(\013)171 544 y FM(k)q FN(\000)p FL(1)328 532 y FP(equals)30 b(+1)g(or)f FQ(\000)p FP(1.)46 b(These)30 b(cases)g(are)f(considered)h(in)h(the)g(follo)n(wing)118 632 y(prop)r(ositions.)118 781 y FR(Prop)s(osition)g(13.)41 b FO(A)861 793 y FM(k)926 781 y FP(=)23 b FJ(C)15 b FQ(h)p FO(u;)f(v)s(;)g(s)1304 793 y FL(1)1347 781 y FO(;)g(:)g(:)g(:)g(;)g(s) 1571 793 y FM(k)1635 781 y FQ(j)24 b FO(\013)1735 793 y FM(i)1787 781 y FP(=)g FO(\014)1923 793 y FM(i)1974 781 y FP(=)g(1)p FO(;)j(i)c(<)h(k)s(;)j(\013)2445 793 y FM(k)2510 781 y FP(=)118 881 y FQ(\000)p FO(\014)230 893 y FM(k)294 881 y FP(=)22 b FQ(\000)p FP(1)p FQ(i)543 858 y(\030)543 885 y FP(=)630 881 y FO(M)711 893 y FL(2)748 881 y FP(\()p FO(Q)846 893 y FL(2)p FM(;k)q FN(\000)p FL(1)1025 881 y FP(\))p FC(.)118 1041 y(Pr)l(o)l(of.)43 b FP(If)25 b(w)n(e)e(denote)h FO(v)880 1053 y FL(1)941 1041 y FP(=)e(1)p FO(=)p FP(2\(\(1)11 b(+)g FO(s)1386 1053 y FM(k)1426 1041 y FP(\))p FO(v)k FP(+)c(\(1)g FQ(\000)g FO(s)1789 1053 y FM(k)1829 1041 y FP(\))p FO(uv)s(u)p FP(\),)25 b FO(v)2120 1053 y FL(2)2181 1041 y FP(=)d(1)p FO(=)p FP(2\(\(1)11 b FQ(\000)118 1140 y FO(s)157 1152 y FM(k)198 1140 y FP(\))p FO(v)22 b FP(+)c(\(1)g(+)g FO(s)589 1152 y FM(k)630 1140 y FP(\))p FO(uv)s(u)p FP(\),)27 b(then)330 1293 y FO(A)392 1305 y FM(k)456 1271 y FQ(\030)456 1297 y FP(=)544 1293 y FJ(C)14 b FQ(h)q FO(u;)g(v)755 1305 y FL(1)798 1293 y FO(;)g(v)875 1305 y FL(2)912 1293 y FO(;)g(s)988 1305 y FL(1)1025 1293 y FO(;)g(:)g(:)g(:)g(;)g(s)1249 1305 y FM(k)1289 1293 y FQ(i)1345 1271 y(\030)1345 1297 y FP(=)1432 1293 y FO(M)1513 1305 y FL(2)1550 1293 y FP(\()p FO(A)1644 1305 y FM(k)q FN(\000)p FL(1)1771 1293 y FP(\))1826 1271 y FQ(\030)1826 1297 y FP(=)1914 1293 y FO(M)1995 1305 y FL(2)2031 1293 y FP(\()p FO(Q)2129 1305 y FL(2)p FM(;k)q FN(\000)p FL(1)2308 1293 y FP(\))p FO(;)118 1446 y FP(where)29 b FO(A)422 1458 y FM(k)q FN(\000)p FL(1)573 1446 y FP(=)c FJ(C)15 b FQ(h)p FO(v)789 1458 y FL(1)833 1446 y FO(;)f(v)910 1458 y FL(2)947 1446 y FO(;)g(s)1023 1458 y FL(1)1060 1446 y FO(;)g(:)g(:)g(:)g(;)g(s)1284 1458 y FM(k)q FN(\000)p FL(1)1409 1446 y FQ(i)1467 1424 y(\030)1467 1450 y FP(=)1557 1446 y FO(Q)1623 1458 y FL(2)p FM(;k)q FN(\000)p FL(1)1801 1446 y FP(.)41 b(Indeed,)30 b(an)n(y)e FO(x)e FQ(2)g FO(A)2534 1458 y FM(k)118 1546 y FP(can)k(b)r(e)i(represen)n(ted)d(in)i(the)g(form)g FO(x)e 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b(0)680 3960 y(0)82 b(1)115 b(0)147 b(0)680 4059 y(0)82 b(0)h FQ(\000)p FP(1)114 b(0)680 4159 y(0)82 b(0)115 b(0)g FQ(\000)p FP(1)1224 3794 y Fz(1)1224 3940 y(C)1224 3990 y(C)1224 4043 y(A)1311 4010 y FO(;)96 b( )s FP(\()p FO(s)1558 4022 y FM(k)1600 4010 y FP(\))23 b(=)1743 3794 y Fz(0)1743 3940 y(B)1743 3990 y(B)1743 4043 y(@)1815 3860 y FP(1)115 b(0)g(0)g(0)1815 3960 y(0)83 b FQ(\000)p FP(1)f(0)115 b(0)1815 4059 y(0)g(0)g(1)g(0)1815 4159 y(0)g(0)g(0)82 b FQ(\000)p FP(1)2359 3794 y Fz(1)2359 3940 y(C)2359 3990 y(C)2359 4043 y(A)2446 4010 y FO(;)p eop %%Page: 39 43 39 42 bop 118 100 a FK(1.2.)36 b FO(F)337 112 y FM(n)383 100 y FK(-algebras)25 b(and)i(their)h(represen)n(tations)848 b FP(39)118 333 y(determine)28 b(the)g(needed)g(isomorphism.)p 2514 333 4 57 v 2518 280 50 4 v 2518 333 V 2567 333 4 57 v 243 507 a(The)f(pro)r(of)g(of)h(the)g(theorem)f(is)g(completed.)p 2514 507 V 2518 454 50 4 v 2518 507 V 2567 507 4 57 v 243 682 a(By)d(using)g(Theorem)f(2.1,)i(it)f(is)h(p)r(ossible)f(to)g 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FQ(f)643 1645 y Fz(e)624 1666 y FO(A;)740 1645 y Fz(e)723 1666 y FO(B)t FQ(g)23 b FP(=)f(0,)f(and)e(\()p FO(I)7 b(I)g(I)1335 1678 y FL(1)1373 1666 y FP(\))20 b FO(A)1487 1636 y FL(2)1527 1666 y FQ(\000)r FO(B)1661 1636 y FL(2)1721 1666 y FP(=)j FO(I)j FP(\(\\non-comm)n (utativ)n(e)118 1776 y(h)n(yp)r(erb)r(ola"\),)e(whic)n(h)g(is)f(the)i (same)e(as)g(the)i(relation)e FQ(f)1857 1755 y Fz(e)1838 1776 y FO(A)o(;)1954 1755 y Fz(e)1936 1776 y FO(B)t FQ(g)g FP(=)g FO(I)7 b FP(.)35 b(W)-7 b(e)25 b(sho)n(w)118 1876 y(that)g(these)g(relations)f(are)g FO(F)1029 1888 y FL(4)1067 1876 y FP(-relations,)g(i.e.,)i(the)f(corresp)r(onding)e(algebras)118 1976 y(are)k FO(F)310 1988 y FL(4)347 1976 y FP(-algebras.)118 2145 y FR(Prop)s(osition)g(15.)38 b FC(Irr)l(e)l(ducible)28 b(self-adjoint)h(solutions)f FO(A)p FC(,)g FO(B)j FC(of)d(the)f(r)l (ela-)118 2245 y(tions)j FP(\()p FO(I)7 b(I)435 2257 y FL(1)473 2245 y FP(\))p FO(;)44 b FP(\()p FO(I)7 b(I)g(I)726 2257 y FL(0)764 2245 y FP(\))p FO(;)44 b FP(\()p FO(I)7 b(I)g(I)1017 2257 y FL(1)1055 2245 y FP(\))30 b FC(ar)l(e)g(the)g(fol)t (lowing)7 b FP(:)210 2414 y(1\))42 b FC(one-dimensional)52 b FP(\(dim)14 b FO(H)52 b FP(=)44 b(1\))p FC(,)i FO(A)f FP(=)f FO(\025)1789 2426 y FL(1)1827 2414 y FR(1)p FC(,)h FO(B)k FP(=)44 b FO(\025)2214 2426 y FL(2)2252 2414 y FR(1)p FC(,)h(wher)l(e)326 2530 y(the)30 b(p)l(air)h FP(\()p FO(\025)715 2542 y FL(1)753 2530 y FO(;)14 b(\025)838 2542 y FL(2)875 2530 y FP(\))30 b FC(b)l(elongs)h(to)f(the)g(cir)l(cle) g FO(K)1758 2486 y FL(\(1\))1752 2558 y(\()p FM(I)5 b(I)1841 2566 y Fy(1)1874 2558 y FL(\))1927 2530 y FP(=)23 b FQ(f)p FP(\()p FO(\025)2137 2542 y FL(1)2174 2530 y FO(;)14 b(\025)2259 2542 y FL(2)2297 2530 y FP(\))24 b FQ(2)f FJ(R)2485 2499 y FL(2)2552 2530 y FQ(j)326 2667 y FO(\025)374 2637 y FL(2)374 2688 y(1)413 2667 y FP(+)q FO(\025)527 2637 y FL(2)527 2688 y(2)587 2667 y FP(=)g(1)p FQ(g)p FC(,)g(the)f(p)l(air)g(of)h(interse)l(cting)f(lines)g FO(K)1892 2624 y FL(\(1\))1886 2696 y(\()p FM(I)5 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FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FC(Pr)l(o)l(of.)43 b FP(Since)35 b(the)h(op)r(erators)d FO(A)1190 303 y FL(2)1263 333 y FP(and)i FO(B)1499 303 y FL(2)1571 333 y FP(b)r(elong)g(to)g(the)g(cen)n(ter)f(of)h(the)118 432 y(algebra,)29 b(they)i(are)e(scalar)f(in)j(an)f(irreducible)f (represen)n(tation,)g FO(A)2266 402 y FL(2)2331 432 y FP(=)e FO(\025)2471 402 y FL(2)2471 453 y(1)2509 432 y FO(I)7 b FP(,)118 532 y FO(B)185 502 y FL(2)246 532 y FP(=)22 b FO(\025)381 502 y FL(2)381 553 y(2)419 532 y FO(I)7 b FP(.)35 b(If)24 b(the)g(represen)n(tation)e(is)h(not)h (one-dimensional,)f(then)h FO(\025)2362 544 y FL(1)2423 532 y FO(>)e FP(0,)118 632 y FO(\025)166 644 y FL(2)243 632 y FO(>)40 b FP(0,)f(and)f(\()p FO(A=\025)808 644 y FL(1)845 632 y FP(\))877 601 y FL(2)954 632 y FP(=)i(\()p FO(B)t(=\025)1248 644 y FL(2)1285 632 y FP(\))1317 601 y FL(2)1394 632 y FP(=)f FO(I)7 b FP(.)67 b(Then)38 b(the)g(prop)r (osition)e(fol-)118 731 y(lo)n(ws)28 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y FP(\))30 b(and)g(\()p FO(I)7 b(I)g(I)2386 2101 y FL(1)2424 2089 y FP(\))31 b(b)n(y)118 2188 y(un)n(b)r(ounded)21 b(op)r(erators,)e(new)i(represen)n(tations)d (app)r(ear)h(\(see)h([193)o(])g(and)g(oth-)118 2288 y(ers\).)118 2413 y FR(2.)36 b FP(W)-7 b(e)28 b(ha)n(v)n(e)f(the)h(follo)n(wing)e (prop)r(osition.)118 2562 y FR(Prop)s(osition)k(16.)41 b FC(The)31 b(fol)t(lowing)h(algebr)l(as)38 b FP(\()p FC(without)30 b(involution)6 b FP(\))539 2725 y FJ(C)593 2657 y Fz(\012)638 2725 y FO(x;)14 b(y)26 b FQ(j)e FO(x)883 2690 y FL(2)939 2725 y FP(+)18 b FO(y)1066 2690 y FL(2)1126 2725 y FP(=)k FO(e)1252 2657 y Fz(\013)1314 2725 y FP(=)h FJ(C)1456 2657 y Fz(\012)1501 2725 y FO(a;)14 b(b)23 b FQ(j)g FO(a)1731 2690 y FL(2)1786 2725 y FQ(\000)18 b FO(b)1905 2690 y FL(2)1965 2725 y FP(=)23 b FO(e)2092 2657 y Fz(\013)2131 2725 y FO(;)118 2887 y FC(and)636 3049 y FJ(C)690 2982 y Fz(\012)735 3049 y FO(x;)14 b(y)26 b FQ(j)d FO(x)979 3015 y FL(2)1040 3049 y FP(=)g FO(y)1172 3015 y FL(2)1209 2982 y 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FO(:)p eop %%Page: 41 45 41 44 bop 118 100 a FK(1.2.)36 b FO(F)337 112 y FM(n)383 100 y FK(-algebras)25 b(and)i(their)h(represen)n(tations)848 b FP(41)243 333 y(Irreducible)27 b FQ(\003)p FP(-represen)n(tations)d (of)k(the)g FQ(\003)p FP(-algebra)730 521 y FJ(C)784 454 y Fz(\012)829 521 y FO(a;)14 b(b)22 b FQ(j)h FO(a)g FP(=)g FO(a)1213 486 y FN(\003)1251 521 y FO(;)28 b(b)22 b FP(=)h FO(b)1484 486 y FN(\003)1522 521 y FO(;)28 b FQ(f)p FO(a;)14 b(b)p FQ(g)21 b FP(=)i(0)1925 454 y Fz(\013)118 709 y FP(can)k(b)r(e)h(obtained)g(as)f(follo)n(ws:)210 883 y(1\))42 b(one-dimensional)31 b(\(dim)14 b FO(H)39 b FP(=)31 b(1\),)j FO(A)d FP(=)g FO(\025)1696 895 y FL(1)1734 883 y FO(I)7 b FP(,)34 b FO(B)i FP(=)31 b FO(\025)2077 895 y FL(2)2114 883 y FO(I)7 b FP(,)34 b(where)e(the)326 983 y(pair)27 b(\()p FO(\025)577 995 y FL(1)615 983 y FO(;)14 b(\025)700 995 y FL(2)737 983 y FP(\))28 b(b)r(elongs)f(to)g (the)h(set)827 1171 y FO(K)904 1137 y FL(\(1\))1016 1171 y FP(=)22 b FQ(f)p FP(\()p FO(\025)1225 1183 y FL(1)1263 1171 y FO(;)14 b(\025)1348 1183 y FL(2)1385 1171 y FP(\))24 b FQ(2)f FJ(R)1573 1137 y FL(2)1639 1171 y FQ(j)g FO(\025)1733 1183 y FL(1)1771 1171 y FO(\025)1819 1183 y FL(2)1880 1171 y FP(=)f(0)p FQ(g)p FP(;)210 1398 y(2\))42 b(t)n(w)n (o-dimensional)26 b(\(dim)14 b FO(H)30 b FP(=)23 b(2\),)769 1636 y FO(A)h FP(=)e FO(\025)990 1648 y FL(1)1042 1518 y Fz(\022)1103 1585 y FP(1)115 b(0)1103 1685 y(0)83 b FQ(\000)p FP(1)1334 1518 y Fz(\023)1409 1636 y FO(;)97 b(B)27 b FP(=)22 b FO(\025)1754 1648 y FL(2)1806 1518 y Fz(\022)1867 1585 y FP(0)83 b(1)1867 1685 y(1)g(0)2033 1518 y Fz(\023)2108 1636 y FO(;)326 1873 y FP(where)27 b(the)h(pair)f(\()p FO(\025)960 1885 y FL(1)998 1873 y FO(;)14 b(\025)1083 1885 y FL(2)1120 1873 y FP(\))28 b(b)r(elongs)f(to)725 2062 y FO(K)802 2027 y FL(\(2\))914 2062 y FP(=)c FQ(f)p FP(\()p FO(\025)1124 2074 y FL(1)1161 2062 y FO(;)14 b(\025)1246 2074 y FL(2)1284 2062 y FP(\))23 b FQ(2)h FJ(R)1472 2027 y FL(2)1538 2062 y FQ(j)f FO(\025)1632 2074 y FL(1)1693 2062 y FO(>)f FP(0)p FO(;)28 b(\025)1921 2074 y FL(2)1981 2062 y FO(>)23 b FP(0)p FQ(g)p FO(:)243 2288 y FP(Let)30 b(us)h(sho)n(w)e(that)i(these)f(represen)n(tations)f (separate)g(elemen)n(ts)h(of)g(the)118 2388 y(algebra.)35 b(Let)783 2588 y FO(x)24 b FP(=)e FO(\013e)d FP(+)f FO(\014)t(a)h FP(+)f FO(\015)5 b(b)17 b FP(+)1516 2509 y Fz(X)1539 2686 y FM(i;j)1650 2588 y FO(c)1686 2600 y FM(ij)1744 2588 y FO(a)1788 2554 y FM(i)1816 2588 y FO(b)1852 2554 y FM(j)1887 2588 y FO(:)118 2865 y FP(If)33 b FO(\031)s FP(\()p FO(x)p FP(\))g(=)e(0)i(for)f(an)n(y)g(one-dimensional)f (represen)n(tation,)i(then)g FO(\013)f FP(=)f FO(\014)36 b FP(=)118 2964 y FO(\015)29 b FP(=)23 b(0.)39 b(If,)29 b(further,)f FO(\031)s FP(\()p FO(x)p FP(\))d(=)f(0)k(for)g(an)n(y)f(t) n(w)n(o-dimensional)g(represen)n(tation,)118 3064 y(then)h(w)n(e)f(ha)n (v)n(e:)490 3252 y FO(\031)s FP(\()p FO(x)p FP(\))d(=)f FO(\031)813 3160 y Fz(\020)863 3173 y(X)886 3350 y FM(i;j)997 3252 y FO(c)1033 3264 y FM(ij)1091 3252 y FO(a)1135 3218 y FM(i)1163 3252 y FO(b)1199 3218 y FM(j)1233 3160 y Fz(\021)675 3523 y FP(=)763 3431 y Fz(\020)903 3444 y(X)813 3623 y FM(i)p FL(=2)p FM(k)q(;j)s FL(=2)p FM(l)1127 3523 y FO(c)1163 3535 y FM(ij)1221 3523 y FO(\025)1269 3489 y FM(i)1269 3544 y FL(1)1307 3523 y FO(\025)1355 3483 y FM(j)1355 3545 y FL(2)1392 3431 y Fz(\021)1456 3406 y(\022)1517 3473 y FP(1)83 b(0)1517 3572 y(0)g(1)1683 3406 y Fz(\023)754 3796 y FP(+)837 3704 y Fz(\020)1019 3717 y(X)886 3896 y FM(i)p FL(=2)p FM(k)q FL(+1)p FM(;j)s FL(=2)p FM(l)1285 3796 y FO(c)1321 3808 y FM(ij)1379 3796 y FO(\025)1427 3762 y FM(i)1427 3817 y FL(1)1465 3796 y FO(\025)1513 3756 y FM(j)1513 3818 y FL(2)1550 3704 y Fz(\021)1614 3679 y(\022)1675 3746 y FP(1)115 b(0)1675 3845 y(0)82 b FQ(\000)p FP(1)1905 3679 y Fz(\023)754 4069 y FP(+)837 3977 y Fz(\020)1019 3990 y(X)886 4169 y FM(i)p FL(=2)p FM(k)q(;j)s FL(=2)p FM(l)p FL(+1)1285 4069 y FO(c)1321 4081 y FM(ij)1379 4069 y FO(\025)1427 4035 y FM(i)1427 4090 y FL(1)1465 4069 y FO(\025)1513 4029 y FM(j)1513 4091 y FL(2)1550 3977 y Fz(\021)1614 3952 y(\022)1675 4019 y FP(0)g(1)1675 4118 y(1)g(0)1841 3952 y Fz(\023)p eop %%Page: 42 46 42 45 bop 118 100 a FP(42)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)754 370 y FP(+)837 278 y Fz(\020)1061 291 y(X)886 470 y FM(i)p FL(=2)p FM(k)q FL(+1)p FM(;j)s FL(=2)p FM(l)p FL(+1)1369 370 y FO(c)1405 382 y FM(ij)1463 370 y FO(\025)1511 336 y FM(i)1511 391 y FL(1)1549 370 y FO(\025)1597 330 y FM(j)1597 392 y FL(2)1634 278 y Fz(\021)1698 253 y(\022)1791 319 y FP(0)115 b(1)1759 419 y FQ(\000)p FP(1)82 b(0)1990 253 y Fz(\023)2074 370 y FP(=)22 b(0)118 609 y(whic)n(h)28 b(implies)f FO(c)673 621 y FM(ij)755 609 y FP(=)c(0,)k FQ(8)p FO(i)p FP(,)f FO(j)5 b FP(,)28 b(i.e.,)g FO(x)23 b FP(=)g(0.)243 709 y(Then)36 b(b)n(y)f(Theorem)g(2,)i(the)f(algebra)e FJ(C)1556 642 y Fz(\012)1602 709 y FO(x;)14 b(y)39 b FQ(j)e FO(x)1873 679 y FL(2)1947 709 y FP(=)f FO(y)2092 679 y FL(2)2129 642 y Fz(\013)2205 709 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FC(is)h(quasi-)118 532 y(nilp)l(otent,)30 b(i.e.,)980 724 y FP(lim)933 774 y FM(n)p FQ(\000)-46 b(!)p FN(1)1179 688 y Fw(n)1169 649 y Fz(p)p 1252 649 333 4 v 75 x FQ(k)p FP([)p FO(P)r(;)14 b(Q)p FP(])1498 700 y FM(n)1543 724 y FQ(k)23 b FP(=)f(0)p FO(:)118 946 y FC(Pr)l(o)l(of.)43 b FP(The)34 b(Kleinec)n(k)n(e{Shirok) n(o)n(v)29 b(theorem)k(follo)n(ws,)g(for)g(example,)h(from)118 1045 y(the)d(form)n(ula)f(ad)660 1008 y FM(n)660 1066 y(P)729 1045 y FO(Q)795 1015 y FM(n)868 1045 y FP(=)d FO(n)p FP(!)14 b(\(ad)1167 1057 y FM(P)1236 1045 y FO(Q)p FP(\))1334 1015 y FM(n)1379 1045 y FP(,)32 b(where)e(ad)1765 1057 y FM(P)1834 1045 y FO(X)k FP(=)28 b FO(P)12 b(X)27 b FQ(\000)20 b FO(X)7 b(P)42 b FP(is)30 b(a)118 1145 y(b)r(ounded)e(op)r(erator)e(in)i FO(L)p FP(\()p FO(H)7 b FP(\),)27 b(and)h FQ(k)14 b FP(ad)1441 1157 y FM(P)1510 1145 y FQ(k)23 b(\024)f FP(2)p FQ(k)p FO(P)12 b FQ(k)p FP(.)36 b(Then)393 1345 y Fw(n)384 1306 y Fz(p)p 467 1306 V 75 x FQ(k)p FP([)p FO(P)r(;)14 b(Q)p FP(])713 1357 y FM(n)758 1381 y FQ(k)22 b(\024)977 1325 y FP(2)p 920 1362 156 4 v 943 1408 a Fw(n)933 1378 y FQ(p)p 1002 1378 73 4 v 71 x FO(n)p FP(!)1099 1381 y FQ(k)p FO(P)12 b FQ(k)17 b(\001)i(k)p FO(Q)p FQ(k)j(\000)-49 b(!)23 b FP(0)p FO(;)180 b(n)23 b FQ(\000)-49 b(!)23 b(1)p FO(:)p 2514 1381 4 57 v 2518 1328 50 4 v 2518 1381 V 2567 1381 4 57 v 243 1629 a FP(In)30 b(Section)h(1.3.3)e(b)r(elo)n(w)h(w)n(e)g (will)h(giv)n(e)f(man)n(y)g(analogies)e(of)j(this)f(theo-)118 1728 y(rem.)118 1892 y FR(2.)36 b FP(No)n(w)27 b(w)n(e)h(consider)e(b)r (ounded)i(self-adjoin)n(t)f(op)r(erators)f(satisfying)i(\(1.8\).)118 2073 y FR(Prop)s(osition)d(17.)37 b FC(Irr)l(e)l(ducible)26 b(p)l(airs,)i FO(A)p FC(,)f FO(B)j FC(of)d(b)l(ounde)l(d)f (self-adjoint)h(op-)118 2172 y(er)l(ators)c(which)i(satisfy)f(the)f(r)l (elation)30 b FP(\(1.8\))22 b FC(ar)l(e)h(one-dimensional,)k(and)c (they)118 2272 y(ar)l(e)36 b(given)h(by)7 b FP(:)52 b FO(A)34 b FP(=)g FO(\025)p FC(,)k FO(B)h FP(=)34 b FO(\026)p FC(,)k(wher)l(e)e FP(\()p 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b(is)g(quasi-nilp)r(oten)n(t.)44 b(But)30 b(since)118 3324 y([)p FO(A;)14 b(B)t FP(])25 b(is)f(sk)n(ew-adjoin)n(t,)g(it)h(yields)f([)p FO(A;)14 b(B)t FP(])24 b(=)e(0.)36 b(Then)24 b FO(P)1936 3336 y FL(2)1974 3324 y FP(\()p FO(A)p FP(\))g(=)e(0,)j(and)f(the)118 3424 y(statemen)n(t)g(follo)n(ws)e(from)h(the)h(sp)r(ectral)f(theorem)g (for)g(a)g(pair)g(of)g(comm)n(uting)118 3524 y(self-adjoin)n(t)k(op)r (erators.)p 2514 3524 V 2518 3471 50 4 v 2518 3524 V 2567 3524 4 57 v 118 3728 a FC(R)l(emark)50 b FP(9)p FC(.)f FP(The)41 b(prop)r(osition)e(ab)r(o)n(v)n(e)f(implies)j(that)f (if)h(the)g(p)r(olynomial)118 3828 y FO(P)171 3840 y FL(2)209 3828 y FP(\()p FQ(\001)p FP(\))27 b(has)g(no)g(real)f(ro)r (ots,)g(then)i(there)f(are)f(no)h(b)r(ounded)g(self-adjoin)n(t)g(pairs) 118 3927 y(that)37 b(satisfy)h(\(1.8\).)65 b(In)37 b(particular,)h (there)f(are)f(no)g(b)r(ounded)i(pairs)e(that)118 4027 y(satisfy)j(the)g(CCR)g(\(relation)f(\()p FO(I)7 b(V)1234 4039 y FL(1)1272 4027 y FP(\)\),)42 b(or)c([)p FO(A;)14 b(B)t FP(])42 b(=)g FO(i)p FP(\()p FO(A)1998 3997 y FL(2)2061 4027 y FP(+)25 b FO(I)7 b FP(\))40 b(\(relation)118 4127 y(\()p FO(V)198 4139 y FL(1)236 4127 y FP(\)\).)p eop %%Page: 44 48 44 47 bop 118 100 a FP(44)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FC(R)l(emark)32 b FP(10)p FC(.)i FP(Relation)20 b(\()p FO(V)977 345 y FL(2)1015 333 y FP(\))h([)p FO(A;)14 b(B)t FP(])23 b(=)g FO(i)p FP(\()p FO(A)1514 303 y FL(2)1555 333 y FP(+)t FO(B)t FP(\))d(can)g(also)f(b)r(e)i(rewritten)f(in)118 432 y(the)28 b(equiv)-5 b(alen)n(t)28 b(form)f([)p FO(A;)14 b(A)1036 402 y FL(2)1093 432 y FP(+)k FO(B)t FP(])23 b(=)g FO(i)p FP(\()p FO(A)1500 402 y FL(2)1556 432 y FP(+)18 b FO(B)t FP(\),)29 b(whic)n(h)f(can)f(b)r(e)h(reduced)118 532 y(b)n(y)38 b(a)g(non-degenerate)f(nonlinear)h(c)n(hange)f(of)i(v)-5 b(ariables,)2082 511 y(~)2060 532 y FO(A)41 b FP(=)g FO(A)2331 502 y FL(2)2394 532 y FP(+)26 b FO(B)t FP(,)138 611 y(~)118 632 y FO(B)33 b FP(=)28 b FO(B)t FP(,)k(to)f(the)g(form)g ([)925 611 y(~)903 632 y FO(A;)1022 611 y FP(~)1002 632 y FO(B)t FP(])e(=)f FO(i)1265 611 y FP(~)1243 632 y FO(A)p FP(.)47 b(Since)1617 611 y(~)1595 632 y FO(A)32 b FP(and)1873 611 y(~)1853 632 y FO(B)k FP(are)30 b(also)f(b)r(ounded)118 731 y(self-adjoin)n(t)h(op)r(erators,)f(b)n(y)i(the)g(Kleinec)n(k)n (e{Shirok)n(o)n(v)26 b(theorem,)31 b(w)n(e)f(ha)n(v)n(e)118 831 y([)163 810 y(~)141 831 y FO(A;)260 810 y FP(~)240 831 y FO(B)t FP(])24 b(=)e(0,)27 b(and)h([)p FO(A;)14 b(B)t FP(])23 b(=)g(0,)k(whic)n(h)h(yields)f FO(A)1643 801 y FL(2)1704 831 y FP(=)22 b FQ(\000)p FO(B)t FP(.)243 931 y(Irreducible)36 b(represen)n(tations)f(of)h(relation\()p FO(V)1716 943 y FL(2)1753 931 y FP(\))h(are)f(one-dimensional,)118 1031 y FO(A)23 b FP(=)g FO(\025)p FP(,)28 b FO(B)f FP(=)c FO(\026)p FP(,)28 b(\()p FO(\025;)14 b(\026)p FP(\))24 b FQ(2)f FO(M)1051 1046 y FL(\()p FM(V)1116 1054 y Fy(2)1148 1046 y FL(\))1201 1031 y FP(=)g FQ(f)p FP(\()p FO(\025;)14 b(\026)p FP(\))24 b FQ(2)f FJ(R)1686 1000 y FL(2)1752 1031 y FQ(j)g FO(\025)1846 1000 y FL(2)1907 1031 y FP(=)g FQ(\000)p FO(\026)p FQ(g)p FP(.)243 1131 y(An)i(arbitrary)d(pair)i(of)h (b)r(ounded)g(self-adjoin)n(t)f(op)r(erators)f(satisfying)h(re-)118 1230 y(lation)j(\()p FO(V)433 1242 y FL(2)471 1230 y FP(\))h(has)f(the)h(form)487 1457 y FO(A)c FP(=)660 1344 y Fz(Z)706 1532 y FM(M)769 1543 y Fy(\()p Fw(V)825 1555 y Fy(2)859 1543 y(\))903 1457 y FO(\025)14 b(dE)5 b FP(\()p FO(\025;)14 b(\026)p FP(\))p FO(;)99 b(A)23 b FP(=)1568 1344 y Fz(Z)1614 1532 y FM(M)1677 1543 y Fy(\()p Fw(V)1733 1555 y Fy(2)1767 1543 y(\))1811 1457 y FO(\025)14 b(dE)5 b FP(\()p FO(\025;)14 b(\026)p FP(\))p FO(;)118 1710 y FP(where)27 b FO(E)5 b FP(\()p FQ(\001)p FO(;)14 b FQ(\001)p FP(\))29 b(is)e(the)h(resolution)f(of)g(the)h(iden)n(tit)n(y) g(on)f FO(M)1950 1725 y FL(\()p FM(V)2015 1733 y Fy(2)2047 1725 y FL(\))2077 1710 y FP(.)118 1844 y FR(3.)35 b FP(If)22 b(in)h(the)g(study)g(of)f(represen)n(tations)f(of)h(relations)f(giv)n 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g(relations)f(\()p FO(I)7 b(V)19 b FP(\))36 b(and)118 3948 y(\()p FO(V)19 b FP(\))i(studied)f(in)h(previous)e(sections.)33 b(W)-7 b(e)21 b(pro)n(v)n(e)d(man)n(y)i(Kleinec)n(k)n(e{Shirok)n(o)n(v) 118 4048 y(t)n(yp)r(e)32 b(theorems)g(\(Section)g(1.3.3\).)49 b(F)-7 b(or)31 b(general)g(semilinear)g(relations,)h(w)n(e)118 4147 y(describ)r(e)23 b(prop)r(erties)e(of)i(irreducible)f(represen)n (tations)f(\(Section)i(1.3.4\).)34 b(All)p eop %%Page: 45 49 45 48 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(45)118 333 y(irreducible)19 b(represen)n(tations)e (are)h(classi\014ed)g(up)i(to)f(unitary)g(equiv)-5 b(alence)18 b(for)118 432 y(semilinear)27 b FO(F)564 444 y FL(4)601 432 y FP(-relations)g(\(Section)g(1.3.5\).)118 603 y FR(1.)39 b FP(Consider)27 b(b)r(ounded)i(op)r(erators)d FO(A)f FP(=)f FO(A)1525 573 y FN(\003)1563 603 y FP(,)29 b FO(B)g FQ(2)24 b FO(L)p FP(\()p FO(H)7 b FP(\),)29 b(whic)n(h)f(satisfy)g(a)118 703 y(relation)f(of)g(the)h(form:)930 841 y FM(n)890 866 y Fz(X)896 1043 y FM(i)p FL(=1)1024 945 y FO(f)1065 957 y FM(i)1092 945 y FP(\()p FO(A)p FP(\))p FO(B)t(g)1325 957 y FM(i)1354 945 y FP(\()p FO(A)p FP(\))23 b(=)g FO(h)p FP(\()p FO(A)p FP(\))p FO(;)616 b FP(\(1.9\))118 1219 y(where)35 b FO(f)407 1231 y FM(i)434 1219 y FP(\()p FQ(\001)p FP(\),)k FO(g)623 1231 y FM(i)650 1219 y FP(\()p FQ(\001)p FP(\),)f FO(h)p FP(\()p FQ(\001)p FP(\),)g FO(i)e FP(=)g(1,)f FO(:)14 b(:)g(:)27 b FP(,)38 b FO(n)p FP(,)g(are)c(complex)h(b)r(ounded)h(Borel)118 1319 y(mappings)30 b(de\014ned)g(on)g FJ(R)36 b FP(or)29 b(a)g(subset)i FO(D)r FP(,)f FO(\033)s FP(\()p FO(A)p FP(\))f FQ(\032)d FO(D)r FP(.)45 b(Relation)29 b(\(1.9\))h(is)118 1418 y(called)c(semilinear)g(and)h(the)g(op)r(erators)d FO(A)p FP(,)k FO(B)i FP(are)c(called)g(a)h(represen)n(tation)118 1518 y(of)h(\(1.9\).)243 1625 y(If)g FO(f)367 1637 y FM(i)394 1625 y FP(\()p FQ(\001)p FP(\),)g FO(g)572 1637 y FM(i)600 1625 y FP(\()p FQ(\001)p FP(\),)g FO(h)p FP(\()p FQ(\001)p FP(\),)g 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FP(with)h(resp)r(ect)f(to)h(the)g(t)n(w)n(o-sided)e FQ(\003)p FP(-ideal)118 2123 y(generated)27 b(b)n(y)g(the)h(elemen)n(t) 1062 2060 y Fz(P)1149 2081 y FM(n)1149 2148 y(i)p FL(=1)1275 2123 y FO(f)1316 2135 y FM(i)1343 2123 y FP(\()p FO(a)p FP(\))14 b FO(b)g(g)1555 2135 y FM(i)1582 2123 y FP(\()p FO(a)p FP(\))19 b FQ(\000)f FO(h)p FP(\()p FO(a)p FP(\).)243 2229 y(Unless)32 b(otherwise)f(stated,)i(w)n(e)f(assume)g FO(f)1621 2241 y FM(i)1648 2229 y FP(,)h FO(g)1744 2241 y FM(i)1772 2229 y FP(,)g FO(h)f FP(to)g(b)r(e)h(de\014ned)f(here)118 2329 y(on)i(the)h(whole)f FJ(R)p FP(.)63 b(The)35 b(general)e(case)g (can)h(b)r(e)h(easily)f(deriv)n(ed)g(from)g(this)118 2429 y(one.)118 2576 y FC(R)l(emark)41 b FP(12)p FC(.)h FP(As)29 b(b)r(efore,)h(if)g(the)g(op)r(erators)e FO(A)f FP(=)e FO(A)1821 2545 y FN(\003)1860 2576 y FP(,)30 b FO(B)k FP(are)28 b(un)n(b)r(ounded,)118 2675 y(then)33 b(it)g(is)g(necessary)e(to)i(de\014ne)g(the)g(meaning)f(of)g(the)i(op)r (erator)d(equalit)n(y)118 2775 y(\(1.9\).)46 b(W)-7 b(e)31 b(in)n(v)n(estigate)f(the)h(question)f(of)h(the)g(\\correct")e (de\014nition)i(of)g(re-)118 2874 y(lation)f(\(1.9\))g(with)h(un)n(b)r (ounded)g(op)r(erators)e(for)g(some)h(sp)r(ecial)g(relations)g(in)118 2974 y([248)o(].)118 3121 y FR(2.)61 b FP(The)35 b(study)h(of)g(b)r (ounded)g(represen)n(tations)e(of)h(\(1.9\))h(can)f(b)r(e)h(reduced)118 3221 y(to)27 b(the)g(study)g(of)g(op)r(erators)e(satisfying)h(the)i (corresp)r(onding)d(homogeneous)118 3320 y(relation:)1003 3459 y FM(n)964 3484 y Fz(X)970 3661 y FM(i)p FL(=1)1098 3563 y FO(f)1139 3575 y FM(i)1166 3563 y FP(\()p FO(A)p FP(\))p FO(B)t(g)1399 3575 y FM(i)1427 3563 y FP(\()p FO(A)p FP(\))f(=)f(0)p FO(:)633 b FP(\(1.10\))243 3848 y(If)38 b(the)h(function)g FO(\036)p FP(\()p FO(t)p FP(\))j(=)e FO(h)p FP(\()p FO(t)p FP(\))p FO(=)1314 3786 y Fz(P)1401 3807 y FM(n)1401 3873 y(i)p FL(=1)1527 3848 y FO(f)1568 3860 y FM(i)1595 3848 y FP(\()p FO(t)p FP(\))p FO(g)1729 3860 y FM(i)1757 3848 y FP(\()p FO(t)p 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FP(\))14 b FO(g)1293 515 y FM(i)1321 503 y FP(\()p FO(t)p FP(\))p FO(;)84 b FP(if)1612 440 y Fz(P)1699 461 y FM(n)1699 527 y(i)p FL(=1)1825 503 y FO(f)1866 515 y FM(i)1893 503 y FP(\()p FO(t)p FP(\))14 b FO(g)2041 515 y FM(i)2069 503 y FP(\()p FO(t)p FP(\))23 b FQ(6)p FP(=)g(0)p FO(;)666 622 y FP(0)p FO(;)791 b FP(otherwise)o FO(:)118 786 y FR(Prop)s(osition)44 b(18.)k FC(If)58 b FP(\(1.9\))41 b FC(has)g(a)48 b FP(\()p FC(b)l(ounde)l(d)9 b FP(\))41 b FC(solution)g FP(\()p FO(A;)14 b(B)t FP(\))p FC(,)45 b(then)118 885 y(ther)l(e)30 b(exists)f FO(C)h(>)22 b FP(0)29 b FC(such)h(that)638 1086 y FQ(j)p FO(h)p FP(\()p FO(t)p FP(\))p FQ(j)24 b(\024)e FO(C)1002 991 y Fz(\014)1002 1040 y(\014)1002 1090 y(\014)1070 982 y FM(n)1030 1007 y Fz(X)1036 1184 y FM(i)p FL(=1)1164 1086 y FO(f)1205 1098 y FM(i)1232 1086 y FP(\()p FO(t)p FP(\))p FO(g)1366 1098 y FM(i)1394 1086 y FP(\()p FO(t)p FP(\))1488 991 y Fz(\014)1488 1040 y(\014)1488 1090 y(\014)1516 1086 y FO(;)184 b(t)23 b FQ(2)g FO(\033)s FP(\()p 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FO(A)p FP(\))14 b FO( )s FP(\()p FO(B)t FP(\))g FO(g)1407 2027 y FM(i)1435 2015 y FP(\()p FO(A)p FP(\))24 b(=)f FO(h)p FP(\()p FO(A)p FP(\))p FO(:)118 2224 y FP(F)-7 b(rom)27 b(this)h(w)n(e)f(conclude)h(that)g(\()p FO(A;)14 b( )s FP(\()p FO(B)t FP(\)\))29 b(is)e(a)g(solution)h(of)34 b(\(1.9\))27 b(and)823 2442 y FO( )s FP(\()p FO(B)t FP(\))1065 2338 y FM(n)1025 2363 y Fz(X)1031 2540 y FM(i)p FL(=1)1159 2442 y FO(f)1200 2454 y FM(i)1227 2442 y FP(\()p FO(A)p FP(\))14 b FO(g)1407 2454 y FM(i)1435 2442 y FP(\()p FO(A)p FP(\))24 b(=)f FO(h)p FP(\()p FO(A)p FP(\))p FO(:)118 2686 y FP(Th)n(us,)29 b FO( )s FP(\()p FO(B)t FP(\))c(=)666 2664 y(~)655 2686 y FO(\036)p FP(\()p FO(A)p FP(\))30 b(on)e(the)h(image)f(of)g(the)h(op)r(erator)1936 2624 y Fz(P)2023 2644 y FM(n)2023 2711 y(i)p FL(=1)2149 2686 y FO(f)2190 2698 y FM(i)2217 2686 y FP(\()p FO(A)p FP(\))14 b FO(g)2397 2698 y FM(i)2425 2686 y FP(\()p FO(A)p FP(\).)118 2786 y(Since)29 b FO( )s FP(\()p FO(B)t FP(\))h(is)f(a)g(b)r(ounded)g (op)r(erator,)f(there)h(exists)f FO(C)k(>)25 b FP(0)j(suc)n(h)h(that)g (for)118 2885 y(ev)n(ery)d FO(t)d FQ(2)h FO(\033)s FP(\()p FO(A)p FP(\))29 b(and)835 2823 y Fz(P)923 2844 y FM(n)923 2910 y(i)p FL(=1)1048 2885 y FO(f)1089 2897 y FM(i)1117 2885 y FP(\()p FO(t)p FP(\))14 b FO(g)1265 2897 y FM(i)1292 2885 y FP(\()p FO(t)p FP(\))24 b FQ(6)p FP(=)f(0)k(w)n(e)g(ha)n(v)n(e) 896 3108 y FQ(j)p FO(h)p FP(\()p FO(t)p FP(\))p FQ(j)c(\024)g FO(C)1260 3012 y Fz(\014)1260 3062 y(\014)1260 3112 y(\014)1327 3004 y FM(n)1288 3029 y Fz(X)1294 3206 y FM(i)p FL(=1)1422 3108 y FO(f)1463 3120 y FM(i)1490 3108 y FP(\()p FO(t)p FP(\))p FO(g)1624 3120 y FM(i)1652 3108 y FP(\()p FO(t)p FP(\))1746 3012 y Fz(\014)1746 3062 y(\014)1746 3112 y(\014)1774 3108 y FO(:)118 3341 y FP(On)31 b(the)g(other)g(hand,)h(k)n (er)987 3279 y Fz(P)1074 3299 y FM(n)1074 3366 y(i)p FL(=1)1200 3341 y FO(f)1241 3353 y FM(i)1268 3341 y FP(\()p FO(A)p FP(\))14 b FO(g)1448 3353 y FM(i)1476 3341 y FP(\()p FO(A)p FP(\))30 b FQ(\032)e FP(k)n(er)13 b FO(h)p FP(\()p FO(A)p FP(\),)33 b(whic)n(h)e(implies)118 3452 y(that)d(the)g (inequalit)n(y)g(holds)f(for)g(ev)n(ery)g FO(t)c FQ(2)h FO(\033)s FP(\()p FO(A)p FP(\),)29 b(and)f(\()p FO(A;)14 b(B)23 b FQ(\000)2224 3430 y FP(~)2213 3452 y FO(\036)q FP(\()p FO(A)p FP(\)\))29 b(is)e(a)118 3551 y(represen)n(tation)f(of)i (relation)e(\(1.10\).)p 2514 3551 4 57 v 2518 3499 50 4 v 2518 3551 V 2567 3551 4 57 v 243 3711 a(It)32 b(is)h(easy)e(to)i (pro)n(v)n(e)e(that)h(the)h(corresp)r(ondence)e(b)r(et)n(w)n(een)i (irreducible)118 3811 y(represen)n(tations)20 b(of)h(\(1.9\))h(and)f (\(1.10\))g(preserv)n(es)f(unitary)h(equiv)-5 b(alence.)34 b(So,)118 3911 y(in)c(studying)f(b)r(ounded)h(represen)n(tations,)e(w)n (e)h(can)g(restrict)g(ourselv)n(es)e(only)118 4010 y(to)h(the)g (homogeneous)d(relations)i(\(1.10\).)118 4147 y FR(3.)36 b FP(T)-7 b(o)27 b(the)h(semilinear)f(relation)g(\(1.10\))o(,)h(w)n(e)f (asso)r(ciate:)p eop %%Page: 47 51 47 50 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(47)210 333 y(a\))42 b(the)28 b(c)n(haracteristic)d (function:)760 574 y(\010\()p FO(t;)14 b(s)p FP(\))24 b(=)1141 470 y FM(n)1101 495 y Fz(X)1107 672 y FM(i)p FL(=1)1235 574 y FO(f)1276 586 y FM(i)1303 574 y FP(\()p FO(t)p FP(\))14 b FO(g)1451 586 y FM(i)1479 574 y FP(\()p FO(s)p FP(\))p FO(;)180 b FP(\()p FO(t;)14 b(s)24 b FQ(2)f FJ(R)p FP(\);)206 858 y(b\))42 b(the)28 b(c)n(haracteristic)d(binary)i (relation:)901 1033 y(\000)c(=)f FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(j)h FP(\010\()p FO(t;)14 b(s)p FP(\))23 b(=)g(0)p FQ(g)f(\032)g FJ(R)1933 999 y FL(2)1976 1033 y FP(;)215 1240 y(c\))42 b(an)f(orien)n(ted)f(graph)g(\()p FJ(R)q FO(;)14 b FP(\000\),)50 b(where)41 b(the)h(edge)1974 1222 y Fo(r)p 1974 1224 117 4 v 8 w Fu(-)9 b Fo(r)1962 1305 y FM(t)96 b(s)2140 1240 y FP(,)45 b FO(t)p FP(,)g FO(s)h FQ(2)g FJ(R)p FP(,)326 1377 y(b)r(elongs)27 b(to)g(the)h(graph)f (if)h(and)f(only)g(if)h(\010\()p FO(t;)14 b(s)p FP(\))24 b(=)e(0.)243 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y(will)e(consider)e(the)i(graph)f(as)g(non-orien)n(ted.)243 2234 y(In)k(what)h(follo)n(ws)e(\000)890 2246 y FM(s)957 2234 y FP(denote)i(the)g(set)f FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))30 b FQ(j)g FP(\010\()p FO(t;)14 b(s)p FP(\))30 b(=)f(0)p FO(;)e FP(\010\()p FO(s;)14 b(t)p FP(\))30 b(=)118 2334 y(0)p FQ(g)22 b(\032)h FJ(R)366 2304 y FL(2)437 2334 y FP(or)j(the)i(corresp)r(onding)e(non-orien)n(ted)g(graph.)243 2434 y(Consider)g(some)h(examples)g(of)h(semilinear)e(relations.)243 2533 y(1\))f(Relations)g(ad)794 2545 y FM(A)848 2533 y FP(\()p FO(B)t FP(\))f(=)e([)p FO(A;)14 b(B)t FP(])24 b(=)f FO(AB)c FQ(\000)14 b FO(B)t(A)23 b FP(=)g(0)i(and)h(\(ad)2224 2545 y FM(A)2278 2533 y FP(\))2310 2503 y FM(n)2355 2533 y FP(\()p FO(B)t FP(\))e(=)118 2633 y([)p FO(A;)14 b(:)g(:)g(:)g FP([)p FO(A;)g(B)t FP(])g FO(:)g(:)g(:)g FP(])35 b(=)g(0)f(ha)n(v)n(e)g (the)h(c)n(haracteristic)e(functions)i(of)g(the)g(form)118 2733 y(\010\()p FO(t;)14 b(s)p FP(\))37 b(=)e FO(t)24 b FQ(\000)f FO(s)36 b FP(and)f(\010\()p FO(t;)14 b(s)p FP(\))36 b(=)g(\()p FO(t)24 b FQ(\000)f FO(s)p FP(\))1483 2702 y FM(n)1564 2733 y FP(resp)r(ectiv)n(ely)-7 b(,)37 b(and)e(de\014ne)g(the)118 2832 y(same)g(graph,)h(all)f(connected)g (comp)r(onen)n(ts)g(of)g(whic)n(h)g(are)f(the)i(follo)n(wing:)160 2948 y Fo(r)p 160 2950 4 4 v 155 2945 V 150 2940 V 146 2935 V 143 2931 V 139 2926 V 136 2922 V 134 2918 V 132 2914 V 130 2910 V 128 2906 V 127 2902 V 127 2899 V 126 2895 V 126 2892 V 127 2889 V 128 2886 V 129 2883 V 130 2880 V 132 2877 V 135 2875 V 135 2875 V 137 2873 V 140 2870 V 142 2869 V 145 2867 V 147 2866 V 150 2864 V 152 2864 V 155 2863 V 157 2863 V 160 2862 V 162 2863 V 165 2863 V 167 2864 V 170 2864 V 172 2866 V 175 2867 V 177 2869 V 180 2870 V 182 2873 V 185 2875 V 160 2950 V 164 2945 V 169 2940 V 173 2935 V 177 2931 V 180 2926 V 183 2922 V 185 2918 V 188 2914 V 189 2910 V 191 2906 V 192 2902 V 192 2899 V 193 2895 V 193 2892 V 192 2889 V 192 2886 V 190 2883 V 189 2880 V 187 2877 V 185 2875 V 140 3031 a FM(\025)201 2966 y FP(,)28 b FO(\025)23 b FQ(2)h FJ(R)p FP(.)243 3103 y(2\))j(Characteristic)f(functions)i(corresp)r (onding)e(to)h(the)h(relations)666 3278 y(ad)753 3290 y FM(A;)p FN(\000)p FL(1)912 3278 y FP(\()p FO(B)t FP(\))c(=)f FQ(f)p FO(A;)14 b(B)t FQ(g)22 b FP(=)h FO(AB)f FP(+)d FO(B)t(A)k FP(=)g(0)118 3453 y(and)30 b(\(ad)402 3465 y FM(A;)p FN(\000)p FL(1)561 3453 y FP(\))593 3423 y FM(n)638 3453 y FP(\()p FO(B)t FP(\))e(=)f FQ(f)p FO(A;)14 b(:)g(:)g(:)f FQ(f)p FO(A;)h(B)t FQ(g)g FO(:)g(:)g(:)f FQ(g)27 b FP(=)g(0)j(are)f(\010\()p FO(t;)14 b(s)p FP(\))27 b(=)g FO(t)20 b FP(+)g FO(s)30 b FP(and)118 3553 y(\010\()p FO(t;)14 b(s)p FP(\))23 b(=)g(\()p FO(t)5 b FP(+)g FO(s)p FP(\))667 3523 y FM(n)712 3553 y FP(,)23 b(resp)r(ectiv)n(ely)-7 b(.)33 b(As)21 b(b)r(efore,)h(they)f(de\014ne)h(the)f(same)f(graph)118 3652 y(with)28 b(connected)g(comp)r(onen)n(ts)f(of)g(the)h(form:)937 3843 y Fo(r)p 937 3845 V 932 3840 V 928 3835 V 924 3831 V 920 3826 V 917 3822 V 914 3818 V 911 3813 V 909 3809 V 907 3805 V 906 3802 V 905 3798 V 904 3794 V 904 3791 V 904 3788 V 904 3784 V 905 3781 V 906 3778 V 908 3776 V 910 3773 V 912 3770 V 912 3770 V 915 3768 V 917 3766 V 920 3764 V 922 3762 V 925 3761 V 927 3760 V 930 3759 V 932 3758 V 935 3758 V 937 3758 V 940 3758 V 942 3758 V 945 3759 V 947 3760 V 950 3761 V 952 3762 V 955 3764 V 957 3766 V 960 3768 V 962 3770 V 937 3845 V 942 3840 V 946 3835 V 950 3831 V 954 3826 V 957 3822 V 960 3818 V 963 3813 V 965 3809 V 967 3805 V 968 3802 V 969 3798 V 970 3794 V 970 3791 V 970 3788 V 970 3784 V 969 3781 V 968 3778 V 966 3776 V 964 3773 V 962 3770 V 921 3927 a FL(0)979 3861 y FO(;)1140 3843 y Fo(r)p 1140 3845 125 4 v 100 w(r)1120 3927 y FM(\025)39 b FN(\000)p FM(\025)1306 3861 y FO(;)180 b FP(\()p FO(\025)24 b(>)e FP(0\))p FO(:)243 4048 y FP(3\))39 b(Let)h FO(AB)t(A)k FP(=)f FO(\013B)t FP(,)g FO(\013)h FQ(2)f FJ(R)p FP(.)80 b(Then)40 b(\010\()p FO(t;)14 b(s)p FP(\))43 b(=)g FO(ts)26 b FQ(\000)g FO(\013)p FP(,)44 b(and)39 b(all)118 4147 y(connected)28 b(comp)r(onen)n(ts)f(of) g(the)h(corresp)r(onding)e(graph)g(are)h(of)g(the)h(form:)p eop %%Page: 48 52 48 51 bop 118 100 a FP(48)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)243 333 y FP(a\))g FO(\013)d FQ(6)p FP(=)e(0)443 539 y Fo(r)p 443 541 125 4 v 100 w(r)363 622 y FM(\013\025)445 597 y Fx(\000)p Fy(1)548 622 y FM(\025)609 557 y FO(;)180 b FP(\()p FO(\025)892 523 y FL(2)953 557 y FQ(6)p FP(=)23 b FO(\013;)14 b FP(0\))1412 539 y Fo(r)1396 622 y FL(0)1454 557 y FO(;)1698 539 y Fo(r)p 1698 541 4 4 v 1694 536 V 1689 531 V 1685 526 V 1681 522 V 1678 517 V 1675 513 V 1673 509 V 1670 505 V 1669 501 V 1667 497 V 1666 494 V 1665 490 V 1665 487 V 1665 483 V 1666 480 V 1666 477 V 1668 474 V 1669 471 V 1671 469 V 1673 466 V 1673 466 V 1676 464 V 1678 462 V 1681 460 V 1683 458 V 1686 457 V 1688 456 V 1691 455 V 1693 454 V 1696 454 V 1698 454 V 1701 454 V 1703 454 V 1706 455 V 1708 456 V 1711 457 V 1713 458 V 1716 460 V 1718 462 V 1721 464 V 1723 466 V 1698 541 V 1703 536 V 1708 531 V 1712 526 V 1715 522 V 1719 517 V 1721 513 V 1724 509 V 1726 505 V 1728 501 V 1729 497 V 1731 494 V 1731 490 V 1732 487 V 1731 483 V 1731 480 V 1730 477 V 1729 474 V 1727 471 V 1726 469 V 1723 466 V 1624 622 a FN(\006)1676 580 y(p)p 1730 580 44 3 v 1730 622 a FM(\013)1795 557 y FO(;)180 b FP(\()p FO(\013)24 b(>)f FP(0)o(\);)243 756 y(b\))30 b(if)g FO(\013)d FP(=)f(0,)j(then)i(an)n(y)d(v)n(ertex)h FO(\025)e FQ(2)g FJ(R)e FQ(n)20 b(f)p FP(0)p FQ(g)28 b FP(is)h(connected)h(with)g(0)f(b) n(y)118 855 y(the)f(edge)492 837 y Fo(r)p 492 839 125 4 v 99 w(r)472 920 y FM(\025)89 b FL(0)658 855 y FP(.)243 994 y(4\))29 b(An)n(y)h(connected)f(comp)r(onen)n(t)g(of)h(the)g(graph) e(corresp)r(onding)g(to)h(the)118 1093 y(relation)313 1279 y FO(A)375 1245 y FL(2)412 1279 y FO(B)23 b FQ(\000)18 b FP(\()p FO(q)k FP(+)c FO(q)795 1245 y FN(\000)p FL(1)884 1279 y FP(\))p FO(AB)t(A)h FP(+)f FO(B)t(A)1338 1245 y FL(2)1399 1279 y FP(=)23 b(0)p FO(;)179 b(q)26 b FQ(2)e FJ(R)g FQ(n)18 b(f\000)p FP(1)p FO(;)c FP(0)p FO(;)g FP(1)p FQ(g)p FO(;)118 1465 y FP(is)28 b(either)f(an)g(in\014nite)i(c)n (hain:)1101 1633 y Fo(r)p 1101 1634 V 99 w(r)p 1225 1634 V 100 w(r)1033 1716 y FM(q)1065 1691 y Fx(\000)p Fy(1)1143 1716 y FM(t)45 b(t)83 b(q)r(t)962 1637 y FO(:)14 b(:)g(:)327 b(:)14 b(:)g(:)1516 1651 y(;)180 b(t)23 b FQ(6)p FP(=)g(0)p FO(;)118 1894 y FP(or)261 1877 y Fo(r)p 261 1878 4 4 v 257 1873 V 252 1868 V 248 1864 V 244 1859 V 241 1855 V 238 1851 V 236 1846 V 233 1842 V 232 1838 V 230 1835 V 229 1831 V 228 1827 V 228 1824 V 228 1821 V 229 1817 V 229 1814 V 231 1812 V 232 1809 V 234 1806 V 236 1803 V 236 1803 V 239 1801 V 241 1799 V 244 1797 V 246 1796 V 249 1794 V 251 1793 V 254 1792 V 256 1792 V 259 1791 V 261 1791 V 264 1791 V 266 1792 V 269 1792 V 271 1793 V 274 1794 V 276 1796 V 279 1797 V 281 1799 V 284 1801 V 286 1803 V 261 1878 V 266 1873 V 271 1868 V 275 1864 V 278 1859 V 282 1855 V 284 1851 V 287 1846 V 289 1842 V 291 1838 V 292 1835 V 294 1831 V 294 1827 V 295 1824 V 294 1821 V 294 1817 V 293 1814 V 292 1812 V 290 1809 V 289 1806 V 286 1803 V 245 1960 a FL(0)303 1894 y FO(:)118 2151 y FR(1.3.3)94 b(Kleinec)m(k)m(e{Shirok)m(o)m(v)32 b(t)m(yp)s(e)g(theorems)118 2307 y FP(W)-7 b(e)41 b(b)r(egin)g(with)h (the)f(study)g(of)g(the)g(structure)f(of)h(op)r(erators)e FO(A)46 b FP(=)e FO(A)2513 2277 y FN(\003)2552 2307 y FP(,)118 2407 y FO(B)g FP(satisfying)d(\(1.10\))o(;)46 b(in)41 b(particular,)h(w)n(e)d(will)i(lo)r(ok)e(at)h(the)g(connection) 118 2507 y(b)r(et)n(w)n(een)27 b(the)h(sp)r(ectral)e(prop)r(erties)g (of)i(the)f(op)r(erator)e FO(A)j FP(and)f(the)g(structure)118 2606 y(of)j(the)g(op)r(erator)f FO(B)t FP(.)44 b(It)30 b(turns)g(out)g(that)h(for)e(a)h(broad)f(class)g(of)h(semilinear)118 2706 y(relations)38 b(the)i(c)n(haracteristic)d(binary)h(relation)g (corresp)r(onding)f(to)i(them)118 2805 y(completely)30 b(de\014nes)f(the)i(solutions)e(of)36 b(\(1.10\))o(.)43 b(This)30 b(fact)g(pro)n(vides)e(man)n(y)118 2905 y(Kleinec)n(k)n (e{Shirok)n(o)n(v)c(t)n(yp)r(e)k(theorems.)118 3059 y FR(1.)42 b FP(W)-7 b(e)30 b(start)f(b)n(y)g(studying)h (\014nite-dimensional)f(represen)n(tations)f(of)h(semi-)118 3159 y(linear)35 b(relations.)60 b(Note)36 b(that)g(the)g(same)f (argumen)n(ts)f(w)n(ork)h(in)h(the)g(more)118 3258 y(general)c(case)f (where)i(the)g(op)r(erator)e FO(A)i FP(in)g(the)g(represen)n(tation)f (has)g(a)g(dis-)118 3358 y(crete)23 b(sp)r(ectrum.)36 b(So)24 b(let)g FO(\033)s FP(\()p FO(A)p FP(\))g(=)f FQ(f)p FO(\025)1317 3370 y FL(1)1354 3358 y FO(;)14 b(:)g(:)g(:)27 b(;)14 b(\025)1600 3370 y FM(m)1664 3358 y FQ(g)23 b FP(and)h FO(H)1956 3370 y FM(\025)1995 3378 y Fw(j)2054 3358 y FP(b)r(e)g(eigenspaces)118 3458 y(of)31 b FO(A)f FP(corresp)r(onding)f(to)h FO(\025)997 3470 y FM(j)1033 3458 y FP(.)46 b(With)31 b(resp)r(ect)f(to)h(the)g(decomp)r(osition)f FO(H)k FP(=)118 3557 y FO(H)187 3569 y FM(\025)226 3577 y Fy(1)282 3557 y FQ(\010)19 b FO(:)14 b(:)g(:)20 b FQ(\010)e FO(H)635 3569 y FM(\025)674 3577 y Fw(m)734 3557 y FP(,)30 b(the)f(op)r(erator)e FO(B)33 b FP(can)c(b)r(e)g(written)g(in)h(the)f (form)g(of)f(the)118 3657 y(blo)r(c)n(k)f(matrix:)37 b FO(B)27 b FP(=)22 b(\()p FO(B)912 3669 y FM(ln)979 3657 y FP(\))1011 3627 y FM(m)1011 3680 y(l;n)p FL(=1)1182 3657 y FO(:)118 3828 y FR(Prop)s(osition)33 b(19.)42 b FC(F)-6 b(or)33 b(op)l(er)l(ators)g FO(A)p FC(,)g FO(B)k FC(to)32 b(de\014ne)g(a)g(r)l(epr)l(esentation)h(of)118 3927 y(the)24 b(r)l(elation)31 b FP(\(1.10\))o FC(,)25 b(it)f(is)g(ne)l(c)l(essary)g(and)g(su\016cient)f(that)h(the)f(blo)l (ck)i(matrix)118 4027 y FO(B)50 b FP(=)45 b(\()p FO(B)436 4039 y FM(sn)512 4027 y FP(\))544 3997 y FM(m)544 4047 y(s;n)p FL(=1)767 4027 y FC(b)l(e)d(supp)l(orte)l(d)h(by)f FP(\000)k Fs(\026)1517 4042 y FM(\033)r FL(\()p FM(A)p FL(\))1705 4027 y FP(\()p FC(i.e.,)h FO(B)1985 4039 y FM(sn)2107 4027 y FP(=)e(0)c FC(for)i(any)118 4127 y FP(\()p FO(\025)198 4139 y FM(s)234 4127 y FO(;)14 b(\025)319 4139 y FM(n)365 4127 y FP(\))32 b FO(=)-51 b FQ(2)24 b FP(\000\))p FC(.)p eop %%Page: 49 53 49 52 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(49)118 333 y FC(Pr)l(o)l(of.)43 b FP(The)28 b(statemen)n(t)f(follo)n(ws)g(immediately)h(from)f(the)h (equalit)n(y)258 420 y Fz(\020)308 433 y(X)314 610 y FM(i)p FL(=1)442 512 y FO(f)483 524 y FM(i)510 512 y FP(\()p FO(A)p FP(\))14 b FO(B)19 b(g)772 524 y FM(i)799 512 y FP(\()p FO(A)p FP(\))925 420 y Fz(\021)975 570 y FM(k)q(j)1070 512 y FP(=)k(\010\()p FO(\025)1298 524 y FM(k)1339 512 y FO(;)14 b(\025)1424 524 y FM(j)1460 512 y FP(\))g FO(B)1569 524 y FM(k)q(j)1640 512 y FO(;)180 b(k)s(;)14 b(j)28 b FP(=)23 b(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(m:)p 2514 512 4 57 v 2518 459 50 4 v 2518 512 V 2567 512 4 57 v 118 751 a FR(2.)43 b FP(A)30 b(pair)f FO(A)e FP(=)f FO(A)743 720 y FN(\003)782 751 y FP(,)k FO(B)h FP(=)26 b FO(B)1087 720 y FN(\003)1125 751 y FP(,)31 b(is)e(a)h(represen)n(tation)e(of)i(relation)e(\(1.10\))h(if)118 850 y(and)f(only)f(if)h(the)g(blo)r(c)n(k)f(matrix)g FO(B)g FP(=)c(\()p FO(B)1443 862 y FM(ij)1502 850 y FP(\))1534 820 y FM(n)1534 872 y(i;j)s FL(=1)1724 850 y FP(is)k(supp)r(orted)h(b)n (y)349 1009 y(\000)401 1021 y FM(s)437 1009 y FQ(j)460 1024 y FM(\033)r FL(\()p FM(A)p FL(\))629 1009 y FP(=)23 b FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(2)g FO(\033)s FP(\()p FO(A)p FP(\))d FQ(\002)e FO(\033)s FP(\()p FO(A)p FP(\))24 b FQ(j)f FP(\010\()p FO(t;)14 b(s)p FP(\))24 b(=)f(\010\()p FO(s;)14 b(t)p FP(\))23 b(=)g(0)p FQ(g)p FO(:)118 1168 y FP(In)30 b(fact,)f(if)h FO(A)c FP(=)g FO(A)731 1138 y FN(\003)769 1168 y FP(,)k FO(B)g FP(=)25 b FO(B)1072 1138 y FN(\003)1140 1168 y FP(is)k(a)g(represen)n(tation)f (of)h(\(1.10\),)g(then)h FO(A)p FP(,)g FO(B)118 1268 y FP(also)d(satisfy)g(the)h(relation)989 1385 y FM(n)950 1410 y Fz(X)956 1587 y FM(i)p FL(=1)1087 1489 y FP(\026)-45 b FO(g)1124 1501 y FM(i)1151 1489 y FP(\()p FO(A)p FP(\))14 b FO(B)1390 1467 y FP(\026)1373 1489 y FO(f)1414 1501 y FM(i)1441 1489 y FP(\()p FO(A)p FP(\))24 b(=)e(0)p FO(:)118 1725 y FP(Hence)h(the)g(blo)r(c)n(k)g(matrix)f FO(B)27 b FP(=)c(\()p FO(B)1251 1737 y FM(ij)1310 1725 y FP(\))1342 1695 y FM(m)1342 1747 y(i;j)s FL(=1)1527 1725 y FP(is)g(supp)r(orted)g(b)n(y)f(\(\000)9 b FQ(\\)g FP(\000)2311 1695 y FN(\003)2350 1725 y FP(\))p FQ(j)2405 1740 y FM(\033)r FL(\()p FM(A)p FL(\))2552 1725 y FP(,)118 1833 y(where)30 b(\000)413 1803 y FN(\003)479 1833 y FP(=)d FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))27 b FQ(2)h FJ(R)f FQ(\002)20 b FJ(R)33 b FQ(j)28 b FP(\010\()p FO(s;)14 b(t)p FP(\))28 b(=)1547 1771 y Fz(P)1635 1792 y FM(n)1635 1858 y(i)p FL(=1)1760 1833 y FO(g)1800 1845 y FM(i)1828 1833 y FP(\()p FO(t)p FP(\))14 b FO(f)1977 1845 y FM(i)2004 1833 y FP(\()p FO(s)p FP(\))28 b(=)g(0)p FQ(g)p FP(,)i(whic)n(h)118 1933 y(giv)n(es)d(the)g(required)g(statemen)n(t.)118 2073 y FR(3.)45 b FP(No)n(w)30 b(w)n(e)g(will)h(try)g(to)f(remo)n(v)n (e)f(the)i(condition)f(in)h(Prop)r(osition)e(19)h(that)118 2173 y FO(\033)s FP(\()p FO(A)p FP(\))j(is)e(discrete.)47 b(F)-7 b(or)30 b(this)i(purp)r(ose,)f(w)n(e)g(will)g(consider)f(a)h (more)f(general)118 2272 y(situation.)243 2372 y(Let)38 b FO(M)9 b FP(,)40 b FO(N)47 b FP(b)r(e)38 b(normal)f(op)r(erators)f (acting)i(on)f(Hilb)r(ert)i(spaces)e FO(H)2478 2384 y FM(M)2552 2372 y FP(,)118 2471 y FO(H)187 2483 y FM(N)250 2471 y FP(,)28 b(resp)r(ectiv)n(ely)-7 b(,)27 b(and)g(let)h FO(E)1112 2483 y FM(M)1186 2471 y FP(\()p FQ(\001)p FP(\),)g FO(E)1385 2483 y FM(N)1449 2471 y FP(\()p FQ(\001)p FP(\))g(b)r(e)g (their)g(sp)r(ectral)f(measures.)118 2619 y FR(De\014nition)i(3.)39 b FC(We)27 b(say)i(that)f(a)g(subset)f FO(F)35 b FQ(\032)23 b FJ(C)35 b FQ(\002)14 b FJ(C)49 b FP(\()p FO(M)t(;)14 b(N)9 b FP(\))p FC(-supp)l(orts)28 b(an)118 2718 y(op)l(er)l(ator)j FO(B)13 b FP(:)28 b FO(H)641 2730 y FM(N)727 2718 y FQ(!)23 b FO(H)902 2730 y FM(M)1005 2718 y FC(if)976 2877 y FO(E)1037 2889 y FM(M)1111 2877 y FP(\()p FO(\013)p FP(\))14 b FO(B)19 b(E)1385 2889 y FM(N)1448 2877 y FP(\()p FO(\014)t FP(\))24 b(=)f(0)118 3037 y FC(for)31 b(any)f(p)l(air)h FP(\()p FO(\013;)14 b(\014)t FP(\))31 b FC(of)f(Bor)l(el)h(sets)e(such) h(that)g FP(\()p FO(\013)19 b FQ(\002)f FO(\014)t FP(\))h FQ(\\)g FO(F)35 b FP(=)22 b FJ(?)p FC(.)243 3184 y FP(It)30 b(is)f(not)h(di\016cult)g(to)g(pro)n(v)n(e)e(that)i(there)f(exists)g(a) h(smallest)f(closed)g(set)118 3283 y FO(F)39 b FP(supp)r(orting)26 b FO(B)k FP(\(tak)n(e)d(the)f(complemen)n(t)h(to)f(the)h(union)g(of)f (all)h(suc)n(h)f(op)r(en)118 3383 y(sets)38 b FO(\013)25 b FQ(\002)g FO(\014)t FP(\).)68 b(W)-7 b(e)38 b(will)g(denote)g(this)g (set)g(b)n(y)f(supp)1839 3403 y FM(M)s(;N)1988 3383 y FP(\()p FO(B)t FP(\).)68 b(It)38 b(is)g(clear)118 3483 y(that)e(supp)477 3503 y FM(M)s(;N)626 3483 y FP(\()p FO(B)t FP(\))h FQ(\032)f FO(\033)s FP(\()p FO(M)9 b FP(\))24 b FQ(\002)f FO(\033)s FP(\()p FO(N)9 b FP(\);)41 b(in)36 b(particular,)g(it)g(is)f(a)g(subset)h(of)118 3582 y FJ(R)7 b FQ(\002)g FJ(R)34 b FP(when)22 b FO(M)30 b FP(and)22 b FO(N)31 b FP(are)21 b(self-adjoin)n(t.)34 b(W)-7 b(e)22 b(shall)g(also)f(write)g(supp)2369 3603 y FM(M)2443 3582 y FP(\()p FO(B)t FP(\))118 3682 y(instead)28 b(of)f(supp)670 3702 y FM(M)s(;M)830 3682 y FP(\()p FO(B)t FP(\).)118 3829 y FR(Theorem)k(6.)40 b FC(If)30 b FO(A)p FC(,)h FO(B)j FC(is)c(a)g(r)l(epr)l(esentation)g(of)h(r)l(elation)36 b FP(\(1.10\))p FC(,)30 b(then)1075 3988 y FP(supp)1246 4008 y FM(A)1300 3988 y FP(\()p FO(B)t FP(\))24 b FQ(\032)f FP(\000)p FO(;)118 4147 y FC(wher)l(e)31 b FP(\000)e FC(is)h(the)g(binary)h(r)l(elation)f(c)l(orr)l(esp)l(onding)h(to)k FP(\(1.10\))p FC(.)p eop %%Page: 50 54 50 53 bop 118 100 a FP(50)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FC(Pr)l(o)l(of.)43 b FP(Let)34 b FO(\013)23 b FQ(\002)f FO(\014)37 b FP(not)d(in)n(tersect)f(\000.)54 b(Denote)34 b(b)n(y)f FO(\026)h FP(the)g(scalar)d(sp)r(ectral)118 432 y(measure)j(of)h(the)g(op)r(erator)e FO(A)p FP(.)60 b(By)34 b(Luzin's)h(theorem,)h(for)f(ev)n(ery)e FO(")i(>)g FP(0,)118 532 y(there)d(exist)g(compact)f(sets)h FO(\013)1092 502 y FN(0)1146 532 y FQ(\032)e FO(\013)i FP(and)g FO(\014)1543 502 y FN(0)1597 532 y FQ(\032)e FO(\014)36 b FP(suc)n(h)c(that)g FO(\026)p FP(\()p FO(\013)22 b FQ(n)f FO(\013)2424 502 y FN(0)2448 532 y FP(\))30 b FO(<)118 632 y(")p FP(,)41 b FO(\026)p FP(\()p FO(\014)31 b FQ(n)25 b FO(\014)499 601 y FN(0)523 632 y FP(\))41 b FO(<)g(")p FP(,)h(and)c(the)h (functions)g FO(f)1551 601 y FN(0)1542 653 y FM(i)1615 632 y FP(=)i FO(f)1762 644 y FM(i)1831 632 y Fs(\026)1866 644 y FM(\013)1909 627 y Fx(0)1935 632 y FP(,)h FO(g)2043 601 y FN(0)2040 653 y FM(i)2109 632 y FP(=)f FO(g)2255 644 y FM(i)2323 632 y Fs(\026)2358 644 y FM(\014)2399 627 y Fx(0)2464 632 y FP(are)118 731 y(con)n(tin)n(uous.)35 b(Put)23 b FO(P)35 b FP(=)23 b FO(E)960 743 y FM(A)1014 731 y FP(\()p FO(\013)1099 701 y FN(0)1123 731 y FP(\),)i FO(Q)e FP(=)g FO(E)1441 743 y FM(A)1495 731 y FP(\()p FO(\014)1578 701 y FN(0)1602 731 y FP(\),)i FO(B)1749 701 y FN(0)1795 731 y FP(=)e FO(P)12 b(B)t(Q)d FP(:)28 b FO(QH)h FQ(!)23 b FO(P)12 b(H)7 b FP(.)118 831 y(The)28 b(op)r(erator)e(of)h(m)n(ultiplication,)853 1074 y(\001)9 b(:)28 b FO(X)h FQ(7!)1226 970 y FM(n)1186 995 y Fz(X)1193 1172 y FM(i)p FL(=1)1320 1074 y FO(f)1370 1040 y FN(0)1361 1095 y FM(i)1393 1074 y FP(\()p FO(A)p FP(\))14 b FO(X)21 b(g)1666 1040 y FN(0)1663 1095 y FM(i)1690 1074 y FP(\()p FO(A)p FP(\))p FO(;)118 1333 y FP(acts)38 b(on)h(the)g(space)f FO(L)p FP(\()p FO(QH)r(;)14 b(P)e(H)7 b FP(\).)71 b(The)38 b(sp)r(ectrum)h(of)g(the)h(op)r(erator)d(\001)118 1432 y(b)r(elongs)27 b(to)h(the)g(set)494 1584 y Fz(n)589 1572 y FM(n)549 1597 y Fz(X)555 1774 y FM(i)p FL(=1)683 1676 y FO(f)724 1688 y FM(i)751 1676 y FP(\()p FO(\025)p FP(\))14 b FO(g)917 1688 y FM(i)945 1676 y FP(\()p FO(\026)p FP(\))24 b FQ(j)f FO(\025)h FQ(2)f FO(\013)1332 1642 y FN(0)1356 1676 y FO(;)k(\026)d FQ(2)f FO(\014)1609 1642 y FN(0)1633 1584 y Fz(o)1711 1676 y FP(=)g(\010\()p FO(\013)1944 1642 y FN(0)1986 1676 y FQ(\002)18 b FO(\014)2120 1642 y FN(0)2144 1676 y FP(\))p FO(;)118 1935 y FP(see,)29 b(for)f(example)h([241)n(].)41 b(By)29 b(the)g(condition)g(of)g(the)g (theorem,)g(\010\()p FO(\013)2341 1905 y FN(0)2384 1935 y FQ(\002)19 b FO(\014)2519 1905 y FN(0)2543 1935 y FP(\))118 2034 y(do)r(es)40 b(not)g(con)n(tain)f(zero,)k(hence)d(the)g(op)r (erator)f(\001)h(is)g(in)n(v)n(ertible.)74 b(Since)118 2134 y(\001\()p FO(B)286 2104 y FN(0)310 2134 y FP(\))25 b(=)f(0,)k(w)n(e)g(ha)n(v)n(e)g(that)g FO(B)1112 2104 y FN(0)1160 2134 y FP(=)d(0,)j(i.e.,)h FO(E)1562 2146 y FM(A)1616 2134 y FP(\()p FO(\013)1701 2104 y FN(0)1725 2134 y FP(\))14 b FO(B)t(E)1899 2146 y FM(A)1954 2134 y FP(\()p FO(\014)2037 2104 y FN(0)2061 2134 y FP(\))25 b(=)f(0.)39 b(Letting)118 2234 y FO(")23 b FQ(!)g FP(0,)k(w)n(e)g (obtain)h(that)g FO(E)1000 2246 y FM(A)1054 2234 y FP(\()p FO(\013)p FP(\))14 b FO(B)t(E)1313 2246 y FM(A)1368 2234 y FP(\()p FO(\014)t FP(\))24 b(=)f(0.)p 2514 2234 4 57 v 2518 2181 50 4 v 2518 2234 V 2567 2234 4 57 v 243 2399 a(In)k(a)h(similar)e(w)n(a)n(y)-7 b(,)27 b(more)g(general)f(results)h (can)g(b)r(e)h(pro)n(v)n(ed.)118 2565 y FR(Theorem)j(7.)40 b FC(If)30 b FO(M)9 b FC(,)30 b FO(N)39 b FC(ar)l(e)30 b(normal)g(op)l(er)l(ators)h(and)962 2704 y FM(n)923 2729 y Fz(X)929 2906 y FM(i)p FL(=1)1056 2808 y FO(f)1097 2820 y FM(i)1125 2808 y FP(\()p FO(M)9 b FP(\))14 b FO(B)k(g)1414 2820 y FM(i)1441 2808 y FP(\()p FO(N)9 b FP(\))23 b(=)g(0)p FO(;)606 b FP(\(1.11\))118 3062 y FC(then)1028 3162 y FP(supp)1199 3182 y FM(M)s(;N)1348 3162 y FP(\()p FO(B)t FP(\))23 b FQ(\032)g FP(\000)p FO(;)118 3321 y FC(wher)l(e)31 b FP(\000)23 b(=)515 3254 y Fz(\010)564 3321 y FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(2)g FJ(C)40 b FQ(\002)18 b FJ(C)44 b FQ(j)23 b FP(\010\()p FO(t;)14 b(s)p FP(\))23 b(=)1467 3259 y Fz(P)1555 3279 y FM(n)1555 3346 y(i)p FL(=1)1680 3321 y FO(f)1721 3333 y FM(i)1748 3321 y FP(\()p FO(t)p FP(\))14 b FO(g)1896 3333 y FM(i)1924 3321 y FP(\()p FO(s)p FP(\))24 b(=)e(0)2180 3254 y Fz(\011)2228 3321 y FC(.)118 3536 y FR(4.)35 b FP(F)-7 b(or)23 b(an)n(y)g FO(F)35 b FQ(\032)23 b FJ(C)j FQ(\002)11 b FJ(C)50 b FP(let)25 b(us)e(denote)h(b)n(y)g FA(M)p FP(\()p FO(F)12 b FP(\))24 b(the)h(set)e(of)h(all)g(op)r(erators)118 3635 y(supp)r(orted)j(b)n(y)g FO(F)12 b FP(.)37 b(If)28 b(necessary)-7 b(,)25 b(w)n(e)i(will)h(write)f(more)f(explicitly:)37 b FA(M)2391 3647 y FM(A)2445 3635 y FP(\()p FO(F)12 b FP(\))118 3735 y(or)32 b FA(M)312 3747 y FM(M)s(;N)461 3735 y FP(\()p FO(F)12 b FP(\).)53 b(Similarly)-7 b(,)34 b(w)n(e)e(will)h(denote)g(b)n(y)f(\001)h(\(instead)g(of)g(\001)2319 3747 y FM(M)s(;N)2501 3735 y FP(or)118 3835 y(\001)187 3847 y FM(A)241 3835 y FP(\))28 b(the)g(m)n(ultiplication)g(op)r (erator)911 4083 y FO(X)h FQ(7!)1155 3979 y FM(n)1115 4004 y Fz(X)1121 4180 y FM(i)p FL(=1)1249 4083 y FO(f)1290 4095 y FM(i)1317 4083 y FP(\()p FO(M)9 b FP(\))14 b FO(X)20 b(g)1614 4095 y FM(i)1642 4083 y FP(\()p FO(N)9 b FP(\))p eop %%Page: 51 55 51 54 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(51)118 333 y(on)27 b FO(L)p FP(\()p FO(H)391 345 y FM(N)454 333 y FO(;)14 b(H)560 345 y FM(M)634 333 y FP(\).)243 432 y(As)28 b(an)g(analogue)f(of)h(Prop)r(osition)e (19,)i(it)h(w)n(ould)e(b)r(e)i(natural)e(to)i(exp)r(ect)118 532 y(that)f(the)g(equalit)n(y)1092 707 y FA(M)p FP(\(\000\))c(=)f(k)n (er)12 b(\001)763 b(\(1.12\))118 882 y(holds.)43 b(Theorem)28 b(6)h(sho)n(ws)g(that)h FA(M)p FP(\(\000\))d FQ(\033)f FP(k)n(er)12 b(\001.)43 b(W)-7 b(e)30 b(shall)f(see)h(that)f(the)118 982 y(in)n(v)n(erse)h(inclusion)g(is)h(true)g(under)g(some)f (restrictions)g(on)g(the)i(smo)r(othness)118 1081 y(of)c(the)g (functions)g FO(f)755 1093 y FM(i)782 1081 y FP(,)g FO(g)873 1093 y FM(i)900 1081 y FP(,)g(and)f(it)h(is)f(not)h(true)g(in)f(the)h (general)f(case.)243 1181 y(Let)41 b(us)h(establish)f(some)g (additional)f(results.)78 b(Denote)42 b(b)n(y)f(pr)2354 1201 y FL(1)2391 1181 y FP(,)k(pr)2538 1201 y FL(2)118 1280 y FP(the)40 b(pro)5 b(jections)38 b(on)n(to)h(the)h(comp)r(onen)n (ts)e(in)i FO(\033)s FP(\()p FO(M)9 b FP(\))27 b FQ(\002)f FO(\033)s FP(\()p FO(N)9 b FP(\).)73 b(Let)40 b FO(S)47 b FP(=)118 1380 y(\()p FO(\033)s FP(\()p FO(M)9 b FP(\))19 b FQ(\002)f FO(\033)s FP(\()p FO(N)9 b FP(\)\))20 b FQ(\\)f FP(\000.)118 1540 y FR(Lemma)24 b(5.)37 b FC(L)l(et)25 b FO(g)742 1552 y FM(i)792 1540 y FQ(2)e FP(Lip)14 b FO(\033)s FP(\()p FO(N)9 b FP(\))p FC(,)28 b FO(k)e FP(=)c(1)p FC(,)k FO(:)14 b(:)g(:)27 b FC(,)g FO(n)p FC(.)37 b(If)26 b FP(pr)1946 1560 y FL(1)1997 1540 y FO(S)i FP(=)23 b FO(\033)s FP(\()p FO(M)9 b FP(\))p FC(,)27 b(then)713 1781 y FQ(k)p FP(\001)p FQ(k)c(\024)f FP(2)1071 1678 y FM(n)1032 1703 y Fz(X)1038 1879 y FM(i)p FL(=1)1165 1781 y FQ(k)p FO(f)1248 1793 y FM(i)1275 1781 y FQ(k)14 b(k)p FO(g)1413 1793 y FM(i)1439 1781 y FQ(k)1481 1793 y FL(Lip)1595 1781 y FP(diam)g FO(\033)s FP(\()p FO(N)9 b FP(\))118 2034 y FC(wher)l(e)31 b FQ(k)p FO(f)436 2046 y FM(i)462 2034 y FQ(k)23 b FP(=)g(sup)p FQ(f)p FO(f)823 2046 y FM(i)849 2034 y FP(\()p FO(t)p FP(\))h FQ(j)f FO(t)g FQ(2)h FO(\033)s FP(\()p FO(M)9 b FP(\))p FQ(g)p FC(.)118 2194 y(Pr)l(o)l(of.)43 b FP(Set)37 b FO(C)45 b FP(=)37 b(2)792 2131 y Fz(P)879 2152 y FM(n)879 2218 y(i)p FL(=1)1005 2194 y FQ(k)p FO(f)1088 2206 y FM(i)1115 2194 y FQ(k)14 b(k)p FO(g)1253 2206 y FM(i)1279 2194 y FQ(k)1321 2206 y FL(Lip)1435 2194 y FP(diam)g FO(\033)s FP(\()p FO(N)9 b FP(\))37 b(and)g(\014x)f FO(s)2192 2206 y FL(0)2267 2194 y FQ(2)j FO(\033)s FP(\()p FO(N)9 b FP(\).)118 2293 y(Then)261 2514 y(\001\()p FO(B)t FP(\))24 b(=)612 2410 y FM(n)573 2435 y Fz(X)579 2612 y FM(i)p FL(=1)706 2514 y FO(f)747 2526 y FM(i)775 2514 y FP(\()p FO(M)9 b FP(\))14 b FO(B)k(g)1064 2526 y FM(i)1091 2514 y FP(\()p FO(s)1162 2526 y FL(0)1199 2514 y FP(\))h(+)1373 2410 y FM(n)1333 2435 y Fz(X)1339 2612 y FM(i)p FL(=1)1467 2514 y FO(f)1508 2526 y FM(i)1535 2514 y FP(\()p FO(M)9 b FP(\))14 b FO(B)k FP(\()p FO(g)1856 2526 y FM(i)1884 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3232 y(\015)550 3282 y(\015)636 3174 y FM(n)596 3199 y Fz(X)603 3376 y FM(i)p FL(=1)730 3278 y FO(f)771 3290 y FM(i)798 3278 y FP(\()p FO(M)g FP(\))14 b FO(B)k FP(\()p FO(g)1119 3290 y FM(i)1147 3278 y FP(\()p FO(N)9 b FP(\))19 b FQ(\000)f FO(g)1429 3290 y FM(i)1456 3278 y FP(\()p FO(s)1527 3290 y FL(0)1564 3278 y FP(\))p FO(I)7 b FP(\))1671 3182 y Fz(\015)1671 3232 y(\015)1671 3282 y(\015)1741 3278 y FQ(\024)1839 3222 y FP(1)p 1839 3259 42 4 v 1839 3335 a(2)1904 3278 y FO(C)f FQ(k)p FO(B)t FQ(k)p FO(:)118 3508 y FP(F)-7 b(urthermore,)25 b FO(\033)671 3441 y Fz(\000)710 3446 y(P)797 3466 y FM(n)797 3533 y(i)p FL(=1)923 3508 y FO(g)963 3520 y FM(i)990 3508 y FP(\()p FO(s)1061 3520 y FL(0)1099 3508 y FP(\))14 b FO(f)1186 3520 y FM(i)1213 3508 y FP(\()p FO(M)9 b FP(\))1367 3441 y Fz(\001)1428 3508 y FP(=)23 b FQ(f)p FP(\010\()p FO(t;)14 b(s)1756 3520 y FL(0)1793 3508 y FP(\))23 b FQ(j)g FO(t)g FQ(2)h FO(\033)s FP(\()p FO(M)9 b FP(\))p FQ(g)p FP(.)36 b(By)25 b(the)118 3608 y(condition)31 b(of)g(the)g(theorem,)g(for)g(an)n(y)f FO(t)e FQ(2)h FO(\033)s FP(\()p FO(M)9 b FP(\))32 b(there)e(exists)h FO(s)d FP(=)g FO(s)p FP(\()p FO(t)p FP(\))h FQ(2)118 3707 y FO(\033)s FP(\()p FO(N)9 b FP(\))29 b(suc)n(h)e(that)h(\010\()p FO(t;)14 b(s)p FP(\))23 b(=)g(0.)36 b(Therefore,)764 3882 y FQ(j)p FP(\010\()p FO(t;)14 b(s)985 3894 y FL(0)1023 3882 y FP(\))p FQ(j)23 b FP(=)g FQ(j)p FP(\010\()p FO(t;)14 b(s)1410 3894 y FL(0)1447 3882 y FP(\))19 b FQ(\000)f FP(\010\()p FO(t;)c(s)p FP(\()p FO(t)p FP(\)\))p FQ(j)712 3987 y Fz(\014)712 4037 y(\014)712 4087 y(\014)779 3979 y FM(n)739 4004 y Fz(X)745 4180 y FM(i)p FL(=1)859 4083 y FP(\()p FO(g)931 4095 y FM(i)959 4083 y FP(\()p FO(s)1030 4095 y FL(0)1067 4083 y FP(\))19 b FQ(\000)f FO(g)1241 4095 y FM(i)1268 4083 y FP(\()p FO(s)p FP(\()p FO(t)p FP(\)\)\))c FO(f)1552 4095 y FM(i)1581 4083 y FP(\()p FO(s)p FP(\))1684 3987 y Fz(\014)1684 4037 y(\014)1684 4087 y(\014)1735 4083 y FQ(\024)1832 4026 y FP(1)p 1832 4063 V 1832 4139 a(2)1898 4083 y FO(C)q(:)p eop %%Page: 52 56 52 55 bop 118 100 a FP(52)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(Hence,)388 262 y Fz(\015)388 312 y(\015)434 270 y(P)522 291 y FM(n)522 358 y(i)p FL(=1)647 333 y FO(g)687 345 y FM(i)714 333 y FP(\()p FO(s)785 345 y FL(0)823 333 y FP(\))14 b FO(f)910 345 y FM(i)937 333 y FP(\()p FO(M)9 b FP(\))1091 262 y Fz(\015)1091 312 y(\015)1161 333 y FQ(\024)22 b FO(C)t(=)p FP(2.)36 b(Th)n(us,)759 500 y Fz(\015)759 550 y(\015)759 600 y(\015)844 492 y FM(n)805 517 y Fz(X)811 694 y FM(i)p FL(=1)938 596 y FO(f)979 608 y FM(i)1007 596 y FP(\()p FO(M)9 b FP(\))14 b FO(B)k(g)1296 608 y FM(i)1323 596 y FP(\()p FO(s)1394 608 y FL(0)1431 596 y FP(\))1463 500 y Fz(\015)1463 550 y(\015)1463 600 y(\015)1533 596 y FQ(\024)1631 540 y FP(1)p 1631 577 42 4 v 1631 653 a(2)1696 596 y FO(C)6 b FQ(k)p FO(B)t FQ(k)p FO(;)118 862 y FP(and)28 b FQ(k)p FP(\001\()p FO(B)t FP(\))p FQ(k)23 b(\024)f FO(C)6 b FQ(k)p FO(B)t FQ(k)p FP(.)p 2514 862 4 57 v 2518 809 50 4 v 2518 862 V 2567 862 4 57 v 118 1051 a FR(Lemma)22 b(6.)35 b FC(L)l(et)23 b FO(E)5 b FP(\()p FQ(\001)p FP(\))24 b FC(b)l(e)g(a)g(pr)l(oje)l(ction-value)l(d)i(me)l(asur)l(e)d(on)h(a)g (c)l(omp)l(act)g(set)118 1151 y FO(K)6 b FC(,)31 b(let)g FO(\013)422 1163 y FL(1)459 1151 y FC(,)g FO(:)14 b(:)g(:)28 b FC(,)j FO(\013)749 1163 y FM(N)843 1151 y FC(b)l(e)g(Bor)l(el)g (subsets)f(of)i FO(K)k FC(such)31 b(that)g(the)g(interse)l(ction)118 1250 y(of)g(any)f FO(m)18 b FP(+)g(1)29 b FC(subsets)g FO(\013)953 1262 y FM(i)1011 1250 y FC(is)h(empty)g(.)39 b(Then)949 1412 y FM(N)918 1436 y Fz(X)921 1613 y FM(j)s FL(=1)1052 1515 y FQ(k)p FO(E)5 b FP(\()p FO(\013)1245 1527 y FM(j)1280 1515 y FP(\))p FO(\030)t FQ(k)1394 1481 y FL(2)1454 1515 y FQ(\024)23 b FO(m)p FQ(k)p FO(\030)t FQ(k)1739 1481 y FL(2)118 1788 y FC(for)31 b(any)f(ve)l(ctor)g FO(\030)t FC(.)118 1962 y(Pr)l(o)l(of.)43 b FP(Since)28 b FO(E)5 b FP(\()p FO(\013)746 1974 y FM(j)781 1962 y FP(\))23 b(=)924 1896 y Fz(R)963 1992 y FM(K)1041 1962 y FO(\037)1093 1974 y FM(\013)1136 1982 y Fw(j)1171 1962 y FP(\()p FO(t)p FP(\))14 b FO(dE)5 b FP(\()p FO(t)p FP(\),)29 b(where)d FO(\037)1825 1974 y FM(\013)1868 1982 y Fw(j)1931 1962 y FP(is)h(the)g(c)n(haracteris-)118 2062 y(tic)h(function)g(of)g FO(\013)711 2074 y FM(j)746 2062 y FP(,)g(w)n(e)f(ha)n(v)n(e)f(that)365 2231 y FM(N)335 2256 y Fz(X)337 2433 y FM(j)s FL(=1)468 2335 y FO(E)5 b FP(\()p FO(\013)619 2347 y FM(j)655 2335 y FP(\))23 b(=)798 2222 y Fz(Z)844 2410 y FM(K)908 2243 y Fz(\020)958 2256 y(X)1002 2433 y FM(j)1091 2335 y FO(\037)1143 2347 y FM(\013)1186 2355 y Fw(j)1222 2335 y FP(\()p FO(t)p FP(\))1316 2243 y Fz(\021)1380 2335 y FO(dE)5 b FP(\()p FO(t)p FP(\))23 b FQ(\024)g FO(m)1781 2222 y Fz(Z)1827 2410 y FM(K)1905 2335 y FO(dE)5 b FP(\()p FO(t)p FP(\))24 b(=)e FO(mI)7 b(:)118 2631 y FP(Therefore,)518 2569 y Fz(P)605 2589 y FM(N)605 2656 y(j)s FL(=1)738 2631 y FQ(k)p FO(E)e FP(\()p FO(\013)931 2643 y FM(j)966 2631 y FP(\))p FO(\030)t FQ(k)1080 2601 y FL(2)1140 2631 y FP(=)1228 2569 y Fz(P)1316 2589 y FM(N)1316 2656 y(j)s FL(=1)1435 2631 y FP(\()p FO(E)g FP(\()p FO(\013)1618 2643 y FM(j)1653 2631 y FP(\))p FO(\030)t(;)14 b(\030)t FP(\))24 b FQ(\024)f FO(m)14 b FQ(k)p FO(\030)t FQ(k)2157 2601 y FL(2)2193 2631 y FP(.)p 2514 2631 V 2518 2578 50 4 v 2518 2631 V 2567 2631 4 57 v 118 2820 a FR(Theorem)31 b(8.)40 b FC(L)l(et)30 b(one)g(of)g(the)g(fol)t(lowing)i(c)l(onditions) f(hold)9 b FP(:)178 2995 y(\(a\))42 b FO(g)366 3007 y FM(i)393 2995 y FC(,)30 b FO(i)23 b FP(=)f(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(n)p FC(,)30 b(ar)l(e)g(p)l(olynomials)7 b FP(;)173 3173 y(\(b\))43 b FO(g)366 3185 y FM(i)419 3173 y FQ(2)27 b FP(Lip)14 b FO(\033)s FP(\()p FO(N)9 b FP(\))p FC(,)33 b FO(i)26 b FP(=)g(1)p FC(,)32 b FO(:)14 b(:)g(:)27 b FC(,)33 b FO(n)p FC(,)f(and)g(the)g(Hausdor\013)g (dimension)h(of)326 3272 y FO(\033)s FP(\()p FO(M)9 b FP(\))30 b FC(is)g(less)g(then)g FP(2)p FC(.)243 3447 y(Then)g FA(M)p FP(\(\000\))24 b(=)e(k)n(er)13 b(\001)p FC(.)118 3622 y(Pr)l(o)l(of.)43 b FP(Let)29 b FO(B)h FQ(2)c FA(M)p FP(\(\000\),)k(then)g(supp)1320 3642 y FM(M)s(;N)1482 3622 y FO(B)g FQ(\032)25 b FO(S)5 b FP(.)41 b(F)-7 b(or)28 b(an)n(y)h FO(\014)g FQ(\032)c FO(\033)s FP(\()p FO(N)9 b FP(\))30 b(w)n(e)118 3739 y(will)i(denote)g(the)h(set) f(pr)911 3760 y FL(1)949 3739 y FP(\(pr)1060 3704 y FN(\000)p FL(1)1060 3761 y(2)1149 3739 y FP(\()p FO(\014)t FP(\)\))h(b)n(y)1461 3717 y(~)1449 3739 y FO(\014)t FP(.)50 b(It)33 b(is)f(easy)f(to)h(sho)n (w)f(that)h(if)g FO(\014)118 3839 y FP(is)j(closed)g(then)h(so)f(is)872 3817 y(~)860 3839 y FO(\014)5 b FP(,)37 b(and)e(that)h(pr)1408 3859 y FL(1)1445 3839 y FP(\(pr)1556 3803 y FN(\000)p FL(1)1556 3861 y(2)1645 3839 y FP(\()p FQ([)1732 3809 y FN(1)1732 3862 y FM(k)q FL(=1)1858 3839 y FO(\014)1905 3851 y FM(k)1945 3839 y FP(\)\))h(=)f FQ([)2202 3809 y FN(1)2202 3862 y FM(k)q FL(=1)2350 3817 y FP(~)2327 3839 y FO(\014)2374 3851 y FM(k)2415 3839 y FP(,)h(for)118 3948 y(an)n(y)25 b(sequence)g FQ(f)p FO(\014)704 3960 y FM(k)744 3948 y FQ(g)p FP(.)36 b(Hence)1101 3926 y(~)1089 3948 y FO(\014)30 b FP(is)c(a)f(Borel)f(set)h(for)g(an)n(y)g FO(F)1993 3960 y FM(\033)2038 3948 y FP(-set)g FO(\014)30 b FP(\(here)25 b(w)n(e)118 4048 y(consider)30 b(only)h(suc)n(h)f FO(\014)t FP(\).)48 b(Since)31 b(\()p FO(\033)s FP(\()p FO(M)9 b FP(\))21 b FQ(n)1526 4026 y FP(~)1515 4048 y FO(\014)t FP(\))g FQ(\002)f FO(\033)s FP(\()p FO(N)9 b FP(\))32 b(do)r(es)e(not)h(in)n(tersect)118 4147 y FO(S)5 b FP(,)28 b(one)f(has)g(\()p FO(I)f FQ(\000)18 b FO(E)763 4159 y FM(M)837 4147 y FP(\()881 4125 y(~)869 4147 y FO(\014)t FP(\)\))c FO(B)t(E)1126 4159 y FM(N)1190 4147 y FP(\()p FO(\014)t FP(\))24 b(=)f(0.)p eop %%Page: 53 57 53 56 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(53)243 333 y(Let)25 b FO(")e(>)f FP(0)j(and)g(let)g FO(\033)s FP(\()p FO(N)9 b FP(\))24 b(=)e FQ([)1237 298 y FM(K)1237 353 y(j)s FL(=1)1357 333 y FO(\013)1410 345 y FM(j)1445 333 y FP(,)j(where)g FO(\013)1784 345 y FM(i)1825 333 y FQ(\\)13 b FO(\013)1946 345 y FM(j)2005 333 y FP(=)22 b FJ(?)p FP(,)j(diam)14 b FO(\013)2452 345 y FM(j)2510 333 y FO(<)118 432 y(")p FP(,)34 b(and)f FO(K)38 b FP(=)31 b FO(K)6 b FP(\()p FO(")p FP(\))33 b(is)g(the)g(least)g(p)r(ossible)g(for)f(all)h(suc)n(h) g(decomp)r(ositions.)118 532 y(Then)255 766 y(\001\()p FO(B)t FP(\))24 b(=)597 663 y FM(K)567 688 y Fz(X)569 864 y FM(j)s FL(=1)701 766 y FP(\001\()p FO(B)t FP(\))p FO(E)962 778 y FM(N)1026 766 y FP(\()p FO(\013)1111 778 y FM(j)1146 766 y FP(\))g(=)1319 663 y FM(K)1289 688 y Fz(X)1292 864 y FM(j)s FL(=1)1423 766 y FO(E)1484 778 y FM(M)1558 766 y FP(\()8 b(~)-50 b FO(\013)1643 778 y FM(j)1678 766 y FP(\))14 b(\001\()p FO(B)t FP(\))p FO(E)1985 778 y FM(N)2050 766 y FP(\()p FO(\013)2135 778 y FM(j)2170 766 y FP(\))p FO(:)138 b FP(\(1.13\))118 1037 y(Set)28 b FO(\015)g FP(=)419 970 y Fz(\010)468 1037 y FO(t)23 b FQ(2)g FO(\033)s FP(\()p FO(M)9 b FP(\))g(:)878 975 y Fz(P)965 996 y FM(n)965 1062 y(i)p FL(=1)1091 1037 y FO(f)1132 1049 y FM(i)1159 1037 y FP(\()p FO(t)p FP(\))14 b FO(g)1307 1049 y FM(i)1335 1037 y FP(\()p FO(N)9 b FP(\))23 b(=)g(0)1628 970 y Fz(\011)1676 1037 y FP(.)37 b(Then)1101 1212 y FO(E)1162 1224 y FM(M)1236 1212 y FP(\()p FO(\015)5 b FP(\)\001)23 b(=)g(0)p FO(:)770 b FP(\(1.14\))118 1386 y(In)28 b(fact,)g(for)f(an)n(y)g FO(Y)41 b FQ(2)24 b FO(L)p FP(\()p FO(H)1019 1398 y FM(N)1081 1386 y FO(;)14 b(H)1187 1398 y FM(M)1261 1386 y FP(\),)28 b(w)n(e)f(ha)n(v)n(e)603 1626 y FO(E)664 1638 y FM(M)738 1626 y FP(\()p FO(\015)5 b FP(\)\001\()p FO(Y)19 b FP(\))24 b(=)1201 1523 y FM(n)1161 1547 y Fz(X)1168 1724 y FM(i)p FL(=1)1295 1626 y FO(E)1356 1638 y FM(M)1430 1626 y FP(\()p FO(\015)5 b FP(\))14 b FO(f)1597 1638 y FM(i)1624 1626 y FP(\()p FO(M)9 b FP(\))14 b FO(Y)19 b(g)1899 1638 y FM(i)1926 1626 y FP(\()p FO(N)9 b FP(\))p FO(;)118 1873 y FP(and)389 2093 y(\()p FO(E)482 2105 y FM(M)557 2093 y FP(\()p FO(\015)c FP(\)\001\()p FO(Y)19 b FP(\))p FO(\030)t(;)14 b(\027)5 b FP(\))24 b(=)1175 1989 y FM(n)1135 2014 y Fz(X)1142 2191 y FM(i)p FL(=1)1255 2093 y FP(\()p FO(Y)19 b(g)1394 2105 y FM(i)1422 2093 y FP(\()p FO(N)9 b FP(\))p FO(\030)t(;)14 b(E)1700 2105 y FM(M)1774 2093 y FP(\()p FO(\015)5 b FP(\))14 b(\()p FO(f)1973 2105 y FM(i)2000 2093 y FP(\()p FO(M)9 b FP(\)\))2186 2059 y FN(\003)2225 2093 y FO(\027)c FP(\))578 2369 y(=)705 2266 y FM(n)666 2291 y Fz(X)672 2467 y FM(i)p FL(=1)800 2256 y Fz(Z)846 2445 y FM(\015)888 2302 y Fz(\000)927 2369 y FO(Y)18 b(g)1033 2381 y FM(i)1060 2369 y FP(\()p FO(N)9 b FP(\))p FO(\030)t(;)p 1277 2297 163 4 v 14 w(f)1318 2381 y FM(i)1346 2369 y FP(\()p FO(t)p FP(\))14 b FO(dE)1558 2381 y FM(M)1633 2369 y FP(\()p FO(t)p FP(\))p FO(\027)1773 2302 y Fz(\001)578 2646 y FP(=)666 2533 y Fz(Z)712 2722 y FM(\015)755 2554 y Fz(\020)844 2542 y FM(n)804 2567 y Fz(X)810 2744 y FM(i)p FL(=1)938 2646 y FO(Y)19 b(g)1045 2658 y FM(i)1072 2646 y FP(\()p FO(N)9 b FP(\))14 b FO(f)1267 2658 y FM(i)1294 2646 y FP(\()p FO(t)p FP(\))p FO(\030)t(;)g(dE)1569 2658 y FM(M)1644 2646 y FP(\()p FO(t)p FP(\))p FO(\027)1784 2554 y Fz(\021)1858 2646 y FP(=)22 b(0)p FO(;)118 2893 y FP(for)27 b(an)n(y)g FO(\030)g FQ(2)d FO(H)613 2905 y FM(N)676 2893 y FP(,)j FO(\027)i FQ(2)23 b FO(H)943 2905 y FM(M)1017 2893 y FP(.)243 2992 y(Relation)k(\(1.14\))g(implies) 716 3248 y(\001\()p FO(B)t FP(\))d(=)1057 3144 y FM(K)1027 3169 y Fz(X)1030 3346 y FM(j)s FL(=1)1161 3248 y FO(E)1222 3260 y FM(M)1296 3248 y FP(\()8 b(^)-50 b FO(\013)1381 3260 y FM(j)1416 3248 y FP(\))14 b(\001\()p FO(B)t FP(\))g FO(E)1737 3260 y FM(N)1802 3248 y FP(\()p FO(\013)1887 3260 y FM(j)1922 3248 y FP(\))p FO(;)118 3510 y FP(where)35 b(^)-50 b FO(\013)411 3522 y FM(j)469 3510 y FP(=)31 b(~)-50 b FO(\013)610 3522 y FM(j)664 3510 y FQ(n)17 b FO(\015)5 b FP(.)37 b(Hence,)360 3769 y FQ(j)p FP(\(\001\()p FO(B)t FP(\))p FO(\030)t(;)14 b(\027)5 b FP(\))p FQ(j)25 b FP(=)905 3673 y Fz(\014)905 3723 y(\014)905 3773 y(\014)963 3665 y FM(K)933 3690 y Fz(X)936 3867 y FM(j)s FL(=1)1053 3701 y Fz(\000)1091 3769 y FO(E)1152 3781 y FM(M)1226 3769 y FP(\()8 b(^)-50 b FO(\013)1311 3781 y FM(j)1346 3769 y FP(\)\001\()p FO(B)t FP(\))p FO(E)1639 3781 y FM(N)1704 3769 y FP(\()p FO(\013)1789 3781 y FM(j)1824 3769 y FP(\))p FO(\030)t(;)14 b(E)1994 3781 y FM(M)2068 3769 y FP(\()8 b(^)-50 b FO(\013)2153 3781 y FM(j)2189 3769 y FP(\))p FO(\027)2267 3701 y Fz(\001)2306 3673 y(\014)2306 3723 y(\014)2306 3773 y(\014)160 4071 y FQ(\024)248 3979 y Fz(\020)327 3967 y FM(K)297 3992 y Fz(X)300 4169 y FM(j)s FL(=1)417 4001 y Fz(\015)417 4050 y(\015)463 4071 y FO(E)524 4083 y FM(M)598 4071 y FP(\()8 b(^)-50 b FO(\013)683 4083 y FM(j)719 4071 y FP(\)\001\()p FO(B)t FP(\))p FO(E)1012 4083 y FM(N)1076 4071 y FP(\()p FO(\013)1161 4083 y FM(j)1196 4071 y FP(\))p FO(\030)1268 4001 y Fz(\015)1268 4050 y(\015)1315 4021 y FL(2)1352 3979 y Fz(\021)1402 3996 y FL(1)p FM(=)p FL(2)1506 3979 y Fz(\020)1586 3967 y FM(K)1556 3992 y Fz(X)1558 4169 y FM(j)s FL(=1)1690 4071 y FQ(k)p FO(E)1793 4083 y FM(M)1866 4071 y FP(\()8 b(^)-50 b FO(\013)1951 4083 y FM(j)1986 4071 y FP(\))p FO(\027)5 b FQ(k)2106 4037 y FL(2)2144 3979 y Fz(\021)2193 3996 y FL(1)p FM(=)p FL(2)2298 4071 y FO(:)42 b FP(\(1.15\))p eop %%Page: 54 58 54 57 bop 118 100 a FP(54)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(Since)h(^)-50 b FO(\013)380 345 y FM(j)438 333 y FQ(\032)23 b FP(pr)604 353 y FL(1)642 333 y FP(\(\()8 b(^)-50 b FO(\013)759 345 y FM(j)797 333 y FQ(\002)s FO(\013)918 345 y FM(j)952 333 y FP(\))s FQ(\\)s FO(S)5 b FP(\),)22 b(w)n(e)d(can)g(apply)h(Lemma)f(5)g(to)h(the)g(op)r (erators)118 432 y FO(N)9 b(E)255 444 y FM(N)318 432 y FP(\()p FO(\013)403 444 y FM(j)439 432 y FP(\),)28 b FO(M)9 b(E)673 444 y FM(M)746 432 y FP(\()f(^)-50 b FO(\013)831 444 y FM(j)866 432 y FP(\).)38 b(W)-7 b(e)28 b(obtain)404 607 y FQ(k)p FO(E)507 619 y FM(M)580 607 y FP(\()8 b(^)-50 b FO(\013)665 619 y FM(j)700 607 y FP(\)\001\()p FO(B)t FP(\))p FO(E)993 619 y FM(N)1058 607 y FP(\()p FO(\013)1143 619 y FM(j)1178 607 y FP(\))p FQ(k)23 b(\024)g FO(C)6 b FQ(k)p FO(B)t FQ(k)14 b FP(diam)f FO(\013)1839 619 y FM(j)1897 607 y FQ(\024)22 b FO(C)6 b FQ(k)p FO(B)t FQ(k)14 b FO(";)494 756 y FQ(k)p FO(E)597 768 y FM(M)671 756 y FP(\()8 b(^)-50 b FO(\013)756 768 y FM(j)791 756 y FP(\)\001\()p FO(B)t FP(\))p FO(E)1084 768 y FM(N)1149 756 y FP(\()p FO(\013)1234 768 y FM(j)1269 756 y FP(\))p FO(\030)t FQ(k)23 b(\024)g FO(C)6 b FQ(k)p FO(B)t FQ(k)14 b FO(")p FQ(k)p FO(E)1866 768 y FM(N)1927 756 y FP(\()p FO(\013)2012 768 y FM(j)2048 756 y FP(\))p FO(\030)t FQ(k)p FO(;)510 868 y FM(K)480 893 y Fz(X)482 1069 y FM(j)s FL(=1)613 971 y FQ(k)p FO(E)716 983 y FM(M)790 971 y FP(\()8 b(^)-50 b FO(\013)875 983 y FM(j)910 971 y FP(\)\001\()p FO(B)t FP(\))p FO(E)1203 983 y FM(N)1268 971 y FP(\()p FO(\013)1353 983 y FM(j)1388 971 y FP(\))p FO(\030)t FQ(k)1502 937 y FL(2)1562 971 y FQ(\024)23 b FO(C)1715 937 y FL(2)1753 971 y FQ(k)p FO(B)t FQ(k)1904 937 y FL(2)1940 971 y FO(")1979 937 y FL(2)2016 971 y FQ(k)p FO(\030)t FQ(k)2140 937 y FL(2)2176 971 y FO(;)118 1230 y FP(since)28 b(the)f(sets)h FO(\013)680 1242 y FM(j)743 1230 y FP(are)e(m)n(utually)i(disjoin)n(t.)243 1330 y(First)23 b(let)h(all)g FO(g)709 1342 y FM(i)736 1330 y FP(\()p FQ(\001)p FP(\))h(b)r(e)f(p)r(olynomials)f(and)g(let)h FO(m)g FP(b)r(e)g(the)g(greatest)f(degree)118 1429 y(of)31 b(the)g(p)r(olynomials)f FO(g)867 1441 y FM(i)894 1429 y FP(\()p FQ(\001)p FP(\).)47 b(Then,)32 b(for)f(an)n(y)f FO(t)e FQ(2)h FO(\033)s FP(\()p FO(M)9 b FP(\))21 b FQ(n)f FO(\015)5 b FP(,)31 b(the)g(equation)118 1529 y(\010\()p FO(t;)14 b(s)p FP(\))31 b(=)f(0)h(do)r(es)h(not)g(ha)n(v)n(e)e(more)h (then)i FO(m)f FP(ro)r(ots.)49 b(Th)n(us,)32 b FO(t)g FP(can)g(not)g(b)r(e-)118 1629 y(long)j(to)g(more)f(than)i FO(m)f FP(sets)43 b(^)-50 b FO(\013)1163 1641 y FM(j)1198 1629 y FP(,)37 b(b)r(ecause)e(for)g(an)n(y)g FO(t)g FQ(2)45 b FP(^)-50 b FO(\013)2083 1641 y FM(j)2153 1629 y FP(there)35 b(exists)118 1728 y FO(s)p FP(\()p FO(t)p FP(\))26 b FQ(2)g FO(\013)411 1740 y FM(j)475 1728 y FP(suc)n(h)i(that)i(\010\()p FO(t;)14 b(s)p FP(\()p FO(t)p FP(\)\))26 b(=)f(0,)k(and)g(the)g(sets)g FO(\013)1903 1740 y FM(j)1967 1728 y FP(are)f(disjoin)n(t.)41 b(Ap-)118 1842 y(plying)24 b(Lemma)f(6,)h(w)n(e)f(can)h(conclude)f (that)1527 1779 y Fz(P)1615 1800 y FM(K)1615 1867 y(j)s FL(=1)1747 1842 y FQ(k)p FO(E)1850 1854 y FM(M)1924 1842 y FP(\()8 b(^)-50 b FO(\013)2009 1854 y FM(j)2044 1842 y FP(\))p FO(\027)5 b FQ(k)2164 1812 y FL(2)2225 1842 y FQ(\024)22 b FO(m)p FQ(k)p FO(\027)5 b FQ(k)2515 1812 y FL(2)2552 1842 y FP(.)118 1960 y(Hence,)32 b FQ(j)p FP(\(\001\()p FO(B)t FP(\))p FO(\030)t(;)14 b(\027)5 b FP(\))p FQ(j)30 b(\024)e FO(C)6 b FQ(k)p FO(B)t FQ(k)14 b FO(")p FQ(k)p FO(\030)t FQ(k)g FO(m)1428 1930 y FL(1)p FM(=)p FL(2)1530 1960 y FQ(k)p FO(\027)5 b FQ(k)1660 1930 y FL(2)1696 1960 y FP(.)46 b(Letting)31 b FO(")d FQ(!)g FP(0,)j(w)n(e)f(ob-)118 2060 y(tain)e(\001\()p FO(B)t FP(\))c(=)e(0.)243 2159 y(No)n(w)31 b(assume)h(that)h(the)f (second)g(condition)g(holds.)50 b(W)-7 b(e)33 b(can)f(estimate)118 2259 y(the)c(second)f(factor)g(in)h(the)g(righ)n(t-hand)e(side)i(of)f (\(1.15\))g(as)g(follo)n(ws:)467 2423 y Fz(\020)547 2411 y FM(K)517 2436 y Fz(X)520 2613 y FM(j)s FL(=1)651 2515 y FQ(k)p FO(E)754 2527 y FM(M)827 2515 y FP(\()8 b(^)-50 b FO(\013)912 2527 y FM(j)948 2515 y FP(\))p FO(\027)5 b FQ(k)1068 2481 y FL(2)1105 2423 y Fz(\021)1155 2440 y FL(1)p FM(=)p FL(2)1282 2515 y FQ(\024)23 b FP(\()p FO(K)6 b FQ(k)p FO(\027)f FQ(k)1609 2481 y FL(2)1645 2515 y FP(\))1677 2481 y FL(1)p FM(=)p FL(2)1805 2515 y FP(=)22 b FO(K)1969 2481 y FL(1)p FM(=)p FL(2)2073 2515 y FQ(k)p FO(\027)5 b FQ(k)p FO(:)118 2790 y FP(Th)n(us)36 b FQ(k)p FP(\001\()p FO(B)t FP(\))p FQ(k)h(\024)g FO(C)6 b FQ(k)p FO(B)t FQ(k)14 b FO("K)1106 2759 y FL(1)p FM(=)p FL(2)1245 2790 y FP(\()p FO(K)42 b FP(=)37 b FO(K)6 b FP(\()p FO(")p FP(\)\).)62 b(Since)37 b(dim)14 b FO(\033)s FP(\()p FO(M)9 b FP(\))38 b FO(<)e FP(2,)118 2889 y FO(K)6 b FP(\()p FO(")p FP(\))23 b(=)g FO(o)p FP(\()p FO(")520 2859 y FN(\000)p FL(2)609 2889 y FP(\),)28 b(this)g(implies)g(\001\()p FO(B)t FP(\))c(=)e(0.)p 2514 2889 4 57 v 2518 2836 50 4 v 2518 2889 V 2567 2889 4 57 v 118 3054 a FC(R)l(emark)43 b FP(13)p FC(.)g FP(The)32 b(restriction)e(on)i(dim)14 b FO(\033)s FP(\()p FO(M)9 b FP(\))32 b(in)g(Theorem)f(8)g(can)h(b)r(e) g(re-)118 3153 y(mo)n(v)n(ed)g(\(see)h([263)n(]\),)i(but)e(the)g(pro)r (of)f(b)r(ecomes)h(m)n(uc)n(h)f(more)g(complicated,)118 3253 y(so)27 b(w)n(e)g(omit)h(this)g(generalization)e(here.)118 3429 y FR(5.)34 b FP(As)21 b(a)g(corollary)e(from)i(Theorem)f(8)h(w)n (e)g(obtain)g(the)h(follo)n(wing)e(Kleinec)n(k)n(e{)118 3529 y(Shirok)n(o)n(v)26 b(t)n(yp)r(e)h(theorems.)118 3689 y FR(Theorem)34 b(9.)43 b FC(If)33 b(two)g(p)l(olynomial)i (semiline)l(ar)e(r)l(elations)h(have)g(the)e(same)118 3788 y(gr)l(aph,)g(then)d(their)h(r)l(epr)l(esentations)g(c)l(oincide.) 118 3948 y(Example)40 b FP(6)p FC(.)j FP(By)31 b(Theorem)f(9,)h(the)g (semilinear)f(relations)g(\(ad)2161 3960 y FM(A;q)2267 3948 y FP(\))2299 3918 y FM(n)2345 3948 y FO(B)i FP(=)c(0)118 4048 y(and)36 b(ad)376 4060 y FM(A;q)496 4048 y FO(B)42 b FP(=)37 b FO(AB)29 b FQ(\000)24 b FO(q)s(B)t(A)38 b FP(=)g(0)e(ha)n(v)n(e)f(the)i(same)f(represen)n(tations)e(b)n(y)118 4147 y(the)25 b(b)r(ounded)g(op)r(erators)e FO(A)g FP(=)g FO(A)1194 4117 y FN(\003)1232 4147 y FP(,)j FO(B)t FP(,)f(since)g(the)g (c)n(haracteristic)e(functions)p eop %%Page: 55 59 55 58 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(55)118 333 y(\010\()p FO(t;)14 b(s)p FP(\))32 b(=)g FO(t)22 b FQ(\000)f FO(q)s(s)33 b FP(and)g(\010\()p FO(t;)14 b(s)p FP(\))32 b(=)g(\()p FO(t)22 b FQ(\000)f FO(q)s(s)p FP(\))1534 303 y FM(n)1580 333 y FP(,)34 b(corresp)r(onding) d(to)i(the)g(\014rst)118 432 y(and)40 b(the)h(second)f(relations)f (resp)r(ectiv)n(ely)-7 b(,)43 b(de\014ne)d(the)h(same)f(graph.)74 b(In)118 532 y(particular,)30 b(for)g FO(q)h FP(=)c(1)j(w)n(e)g(obtain) g(that)h(b)r(ounded)f(op)r(erators)f FO(A)f FP(=)f FO(A)2415 502 y FN(\003)2453 532 y FP(,)32 b FO(B)118 632 y FP(satisfy)23 b(the)h(relation)e(\(ad)932 644 y FM(A)986 632 y FP(\))1018 601 y FM(n)1064 632 y FO(B)27 b FP(=)c([)p FO(A;)14 b(:)g(:)g(:)g FP([)p FO(A;)g(B)t FP(])g FO(:)g(:)g(:)g FP(])23 b(=)f(0)h(if)h(and)g (only)f(if)g FO(A)p FP(,)118 731 y FO(B)k FP(comm)n(ute,)d(and)f(for)g FO(q)j FP(=)c FQ(\000)p FP(1,)i(a)e(pair)h(of)g(b)r(ounded)g(op)r (erators)e FO(A)i FP(=)g FO(A)2422 701 y FN(\003)2461 731 y FP(,)h FO(B)118 831 y FP(is)32 b(a)g(solution)g(of)h(the)f (equation)g(\(ad)1310 843 y FM(A;)p FN(\000)p FL(1)1469 831 y FP(\))1501 801 y FM(n)1546 831 y FO(B)j FP(=)c FQ(f)p FO(A;)14 b(:)g(:)g(:)f FQ(f)p FO(A;)h(B)t FQ(g)g FO(:)g(:)g(:)f FQ(g)31 b FP(=)f(0)118 930 y(i\013)e FO(AB)g FP(=)22 b FQ(\000)p FO(B)t(A)p FP(.)118 1055 y FR(6.)66 b FP(As)38 b(corollaries)d(from)i(Theorem)g(8)g(w)n(e)g(will)h(also)f (giv)n(e)f(some)h(results,)118 1155 y(where)d(the)i(form)e(of)h(the)g (c)n(haracteristic)e(binary)h(relation)g(corresp)r(onding)118 1254 y(to)27 b(the)h(semilinear)e(relation)h(\(1.10\))f(is)h(more)g (imp)r(ortan)n(t)g(then)g(the)h(form)f(of)118 1354 y(the)h(functions)g FO(f)660 1366 y FM(i)687 1354 y FP(,)g FO(g)778 1366 y FM(i)805 1354 y FP(.)118 1504 y FR(Theorem)d(10.)36 b FC(L)l(et)25 b FO(f)9 b FC(,)27 b FO(g)s FC(,)f(b)l(e)f(b)l(ounde)l (d)h(Bor)l(el)h(functions.)37 b(A)25 b(p)l(air)h(of)g(b)l(oun-)118 1604 y(de)l(d)31 b(op)l(er)l(ators)f FO(A)24 b FP(=)e FO(A)862 1574 y FN(\003)901 1604 y FC(,)30 b FO(B)k FC(satis\014es)c (the)f(e)l(quation)1051 1766 y FO(f)9 b FP(\()p FO(A)p FP(\))p FO(B)28 b FP(=)22 b FO(B)t(g)s FP(\()p FO(A)p FP(\))118 1929 y FC(if)31 b(and)f(only)g(if)h FP(supp)792 1949 y FM(A)846 1929 y FP(\()p FO(B)t FP(\))24 b FQ(\032)f FP(\000)p FC(,)30 b(wher)l(e)g FP(\000)23 b(=)g FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(j)g FO(f)9 b FP(\()p FO(t)p FP(\))23 b(=)g FO(g)s FP(\()p FO(s)p FP(\))p FQ(g)p FC(.)118 2079 y(Pr)l(o)l(of.)43 b FP(Set)32 b FO(M)38 b FP(=)29 b FO(f)9 b FP(\()p FO(A)p FP(\),)33 b FO(N)38 b FP(=)29 b FO(g)s FP(\()p FO(A)p FP(\).)49 b(Since)32 b FO(E)1692 2091 y FM(M)1766 2079 y FP(\()p FO(\013)p FP(\))e(=)f FO(E)2068 2091 y FM(A)2123 2079 y FP(\()p FO(f)2205 2049 y FN(\000)p FL(1)2294 2079 y FP(\()p FO(\013)p FP(\)\))j(for)118 2178 y(an)n(y)i(Borel)g(set)g FO(\013)p FP(,)j(the)e(condition)g(supp)1452 2199 y FM(A)1506 2178 y FP(\()p FO(B)t FP(\))h FQ(\032)e FP(\000)h(is)f(equiv)-5 b(alen)n(t)35 b(to)f(the)118 2278 y(follo)n(wing)27 b(one)803 2378 y(supp)974 2398 y FM(M)s(;N)1123 2378 y FP(\()p FO(B)t FP(\))c FQ(\032)g(f)p FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(j)g FO(t)g FP(=)g FO(s)p FQ(g)p FO(:)118 2515 y FP(Theorem)k(8)g(giv)n(es)f(the)i(required)f(statemen)n(t.)p 2514 2515 4 57 v 2518 2462 50 4 v 2518 2515 V 2567 2515 4 57 v 243 2677 a(In)f(a)g(similar)f(w)n(a)n(y)g(one)g(sho)n(ws)g(that) i(Theorem)e(8)h(implies)g(the)g(F)-7 b(uglede{)118 2777 y(Putnam{Rozen)n(blum)37 b(theorem)g(see,)j(for)e(example)f([241)o(])h (and)f(also)g(Sec-)118 2876 y(tion)24 b(1.4.3.)35 b(In)24 b(particular,)g(if)g(b)r(ounded)h(op)r(erators)d FO(A)h FP(=)g FO(A)2026 2846 y FN(\003)2064 2876 y FP(,)i FO(B)k FP(satisfy)23 b(the)118 2976 y(relation)28 b FO(AB)j FP(=)25 b FO(q)s(B)t(A)p FP(,)31 b(where)d FQ(j)p FO(q)s FQ(j)e FP(=)g(1,)j(then)h FO(AB)g FP(=)i(\026)-48 b FO(q)s(B)t(A)p FP(,)30 b(since)f(the)h(sets)118 3075 y FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(2)g FJ(R)i FQ(\002)18 b FJ(R)29 b FQ(j)23 b FO(t)g FP(=)g FO(q)s(s)p FQ(g)k FP(and)g FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(2)h FJ(R)g FQ(\002)18 b FJ(R)29 b FQ(j)23 b FO(t)g FP(=)p 1917 3030 41 4 v 23 w FO(q)s(s)p FQ(g)k FP(coincide.)118 3200 y FC(R)l(emark)43 b FP(14)p FC(.)h FP(Sets)33 b(of)g(the)g(form)f FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))32 b FQ(j)f FO(f)9 b FP(\()p FO(t)p FP(\))32 b(=)f FO(g)s FP(\()p FO(s)p FP(\))p FQ(g)p FP(,)j(where)e FO(f)9 b FP(,)34 b FO(g)h FP(are)118 3300 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FO(B)h FC(gives)e(a)f(r)l(epr)l(esentation)f(of)h(any)g(r)l(elation)g (whose)h(binary)118 4147 y(r)l(elation)31 b(c)l(ontains)e FO(F)12 b FC(.)p eop %%Page: 56 60 56 59 bop 118 100 a FP(56)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FC(Pr)l(o)l(of.)43 b FP(Let)28 b FO(F)35 b FQ(\032)23 b FP(\000)g(=)865 265 y Fz(\010)914 333 y FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(j)1153 270 y Fz(P)1241 291 y FM(n)1241 358 y(i)p FL(=1)1366 333 y FO(f)1407 345 y FM(i)1435 333 y FP(\()p FO(t)p FP(\))p FO(g)1569 345 y FM(i)1596 333 y FP(\()p FO(s)p FP(\))h(=)f(0)1853 265 y Fz(\011)1901 333 y FP(.)37 b(Since)293 502 y(supp)464 522 y FM(A)518 502 y FP(\()p FO(B)t FP(\))24 b FQ(\032)f(f)p FP(\()p FO(t;)14 b(s)p FP(\))22 b FQ(j)i FO(s)f FP(=)f FO(')p FP(\()p FO(t)p FP(\))p FQ(g)i(\032)e(f)p FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(j)g FO(g)1813 514 y FM(i)1841 502 y FP(\()p FO(s)p FP(\))g(=)g FO(g)2095 514 y FM(i)2122 502 y FP(\()p FO(')p FP(\()p FO(t)p FP(\)\))p FQ(g)p FO(;)118 671 y FP(w)n(e)h(ha)n(v)n(e)f FO(B)t(g)532 683 y FM(i)559 671 y FP(\()p FO(A)p FP(\))h(=)f FO(g)837 683 y FM(i)864 671 y FP(\()p FO(')p FP(\()p FO(A)p FP(\)\))p FO(B)t FP(,)k FO(i)c FP(=)f(1,)i FO(:)14 b(:)g(:)28 b FP(,)d FO(n)p FP(,)f(b)n(y)g(Theorem)g(10.)35 b(On)23 b(the)118 771 y(other)k(hand,)1014 829 y FM(n)975 853 y Fz(X)981 1030 y FM(i)p FL(=1)1109 932 y FO(f)1150 944 y FM(i)1177 932 y FP(\()p FO(t)p FP(\))14 b FO(g)1325 944 y FM(i)1352 932 y FP(\()p FO(')p FP(\()p FO(t)p FP(\)\))25 b(=)e(0)118 1155 y(for)k(an)n(y)g FO(t)c FQ(2)g FO(\033)s FP(\()p FO(A)p FP(\),)29 b(hence)992 1093 y Fz(P)1080 1114 y FM(n)1080 1180 y(i)p FL(=1)1205 1155 y FO(f)1246 1167 y FM(i)1274 1155 y FP(\()p FO(A)p FP(\))14 b FO(g)1454 1167 y FM(i)1481 1155 y FP(\()p FO(')p FP(\()p FO(A)p FP(\)\))25 b(=)e(0.)36 b(It)28 b(follo)n(ws)f(that)368 1292 y FM(n)329 1316 y Fz(X)335 1493 y FM(i)p FL(=1)463 1395 y FO(f)504 1407 y FM(i)531 1395 y FP(\()p FO(A)p FP(\))p FO(B)t(g)764 1407 y FM(i)792 1395 y FP(\()p FO(A)p FP(\))d(=)1069 1292 y FM(n)1030 1316 y Fz(X)1036 1493 y FM(i)p FL(=1)1163 1395 y FO(f)1204 1407 y FM(i)1232 1395 y FP(\()p FO(A)p FP(\)\()p FO(B)t(g)1497 1407 y FM(i)1525 1395 y FP(\()p FO(A)p FP(\))19 b FQ(\000)f FO(g)1793 1407 y FM(i)1821 1395 y FP(\()p FO(')p FP(\()p FO(A)p FP(\)\))p FO(B)t FP(\))25 b(=)e(0)p FO(:)p 2514 1395 4 57 v 2518 1343 50 4 v 2518 1395 V 2567 1395 4 57 v 118 1631 a FR(Corollary)42 b(1.)j FC(L)l(et)36 b FO(A)f FP(=)g FO(A)1082 1601 y FN(\003)1120 1631 y FC(,)k(and)e(let)f (ther)l(e)h(exist)f(a)h(de)l(c)l(omp)l(osition)h(of)118 1731 y FO(\033)s FP(\()p FO(A)p FP(\))30 b FC(into)f(Bor)l(el)h(sets)e FO(P)930 1743 y FM(i)958 1731 y FC(,)h FO(\033)s FP(\()p FO(A)p FP(\))24 b(=)f FQ([)1355 1743 y FM(i)1383 1731 y FO(P)1436 1743 y FM(i)1464 1731 y FC(,)29 b(such)g(that)g(e)l(ach)h FO(P)2114 1743 y FM(i)2158 1731 y FQ(\002)15 b FO(P)2291 1743 y FM(j)2343 1731 y FQ(\\)i FP(\000)p FC(,)29 b FO(i)p FC(,)118 1830 y FO(j)f FP(=)23 b(1)p FC(,)29 b FP(2)p FC(,)h FO(:)14 b(:)g(:)28 b FC(,)i(is)g(the)g(gr)l(aph)h(of)f(a)g (function.)39 b(Then)31 b FP(k)n(er)12 b(\001)2034 1842 y FM(A)2112 1830 y FP(=)22 b FA(M)2286 1842 y FM(A)2341 1830 y FP(\(\000\))p FC(.)118 1986 y(Pr)l(o)l(of.)43 b FP(Theorem)26 b(11)g(implies)h FO(E)1180 1998 y FM(A)1234 1986 y FP(\()p FO(P)1319 1998 y FM(i)1347 1986 y FP(\)\001\()p FO(B)t FP(\))p FO(E)1640 1998 y FM(A)1696 1986 y FP(\()p FO(P)1781 1998 y FM(j)1817 1986 y FP(\))c(=)f(0)27 b(for)f(an)n(y)g FO(i;)14 b(j)31 b FP(and)118 2085 y(hence)d(\001\()p FO(B)t FP(\))c(=)e(0.)p 2514 2085 V 2518 2033 50 4 v 2518 2085 V 2567 2085 4 57 v 118 2249 a FC(R)l(emark)49 b FP(15)p FC(.)e FP(Let)38 b(us)g(note)h(that)f(if)45 b(\000)38 b(is)g(suc)n(h)g(that)h(the)f(set)h FQ(f)p FO(\026)h FQ(2)h FJ(R)47 b FQ(j)118 2348 y FP(\()p FO(\025;)14 b(\026)p FP(\))29 b FQ(2)f FP(\000)p FQ(g)i FP(is)g(\014nite)h(or)f (coun)n(table)f(for)h(an)n(y)g FO(\025)e FQ(2)g FJ(R)p FP(,)37 b(then)31 b(the)g(condition)118 2448 y(of)d(Corollary)d(1)i(is) g(satis\014ed.)118 2620 y FR(7.)36 b FP(As)26 b(w)n(as)g(noticed,)h (the)f(con)n(v)n(erse)f(of)33 b(Theorem)25 b(7)h(is)h(false)f(in)g(the) h(general)118 2720 y(case.)35 b(Here)24 b(w)n(e)g(construct)f(an)h (example)g(whic)n(h)g(sim)n(ultaneously)g(solv)n(es)e(the)118 2819 y(F)-7 b(uglede{W)g(eiss)28 b(problem.)37 b(Namely)-7 b(,)29 b(w)n(e)e(sho)n(w)h(that)g(there)g(exist)g(a)g(pair)f(of)118 2919 y(b)r(ounded)j(op)r(erators)d FO(A)g FP(=)e FO(A)1069 2889 y FN(\003)1108 2919 y FP(,)k FO(B)34 b FP(and)29 b(con)n(tin)n(uous)g(functions)g FO(f)2238 2931 y FM(i)2266 2919 y FP(,)h FO(g)2359 2931 y FM(i)2415 2919 y FP(suc)n(h)118 3018 y(that)446 3130 y FM(m)415 3155 y Fz(X)421 3331 y FM(i)p FL(=1)549 3233 y FO(f)590 3245 y FM(i)617 3233 y FP(\()p FO(A)p FP(\))p FO(B)t(g)850 3245 y FM(i)878 3233 y FP(\()p FO(A)p FP(\))24 b(=)f(0)p FO(;)96 b FP(but)1529 3130 y FM(m)1499 3155 y Fz(X)1505 3331 y FM(i)p FL(=1)1650 3212 y FP(\026)1632 3233 y FO(f)1673 3245 y FM(i)1701 3233 y FP(\()p FO(A)p FP(\))p FO(B)21 b FP(\026)-44 b FO(g)1949 3245 y FM(i)1976 3233 y FP(\()p FO(A)p FP(\))24 b FQ(6)p FP(=)e(0)p FO(:)118 3480 y FC(Example)50 b FP(7)p FC(.)f FP(Let)42 b FO(D)r FP(\()p FJ(R)910 3450 y FM(n)961 3480 y FP(\))g(b)r(e)f(the)h(space)e(of)h(compactly)g(supp)r(orted)g (in-)118 3580 y(\014nitely)h(di\013eren)n(tiable)f(functions)h(on)f FJ(R)1478 3549 y FM(n)1529 3580 y FP(,)k FO(D)1668 3549 y FN(0)1692 3580 y FP(\()p FJ(R)1778 3549 y FM(n)1829 3580 y FP(\))d(b)r(e)g(its)f(dual)h(space,)118 3679 y FO(F)12 b(L)240 3649 y FL(1)277 3679 y FP(\()p FJ(R)363 3649 y FM(n)414 3679 y FP(\))31 b(the)g(F)-7 b(ourier)29 b(algebra,)g FO(P)12 b(M)d FP(\()p FJ(R)1471 3649 y FM(n)1522 3679 y FP(\))31 b(the)f(space)g(dual)g(to)g FO(F)12 b(L)2368 3649 y FL(1)2405 3679 y FP(\()p FJ(R)2491 3649 y FM(n)2543 3679 y FP(\))118 3779 y(\(the)30 b(space)e(of)h(pseudo-measures\),)f FO(F)41 b FP(the)29 b(F)-7 b(ourier)28 b(transform,)h FO(')19 b FQ(\003)g FO( )32 b FP(the)118 3878 y(con)n(v)n(olution)26 b(of)i(t)n(w)n(o)f(functions)g(in)h FO(D)r FP(\()p FJ(R)1427 3848 y FM(n)1478 3878 y FP(\),)41 b(~)-55 b FO(')p FP(\()p FO(x)p FP(\))24 b(=)p 1838 3806 231 4 v 23 w FO(')p FP(\()p FQ(\000)p FO(x)p FP(\))q(.)243 3978 y(Consider)i(the)i(follo)n(wing)f (p)r(olynomial)g(in)h(six)f(v)-5 b(ariables)345 4147 y FO(p)p FP(\()p FO(x)466 4159 y FL(1)504 4147 y FO(;)14 b(:)g(:)g(:)27 b(;)14 b(x)749 4159 y FL(6)787 4147 y FP(\))23 b(=)g FO(x)977 4113 y FL(2)977 4168 y(1)1033 4147 y FP(+)18 b FO(x)1163 4113 y FL(2)1163 4168 y(2)1219 4147 y FP(+)g FO(x)1349 4113 y FL(2)1349 4168 y(3)1405 4147 y FQ(\000)g FP(1)g(+)g FO(i)p FP(\()p FO(x)1739 4113 y FL(2)1739 4168 y(4)1795 4147 y FP(+)h FO(x)1926 4113 y FL(2)1926 4168 y(5)1982 4147 y FP(+)f FO(x)2112 4113 y FL(2)2112 4168 y(6)2168 4147 y FQ(\000)g FP(1\))p FO(:)p eop %%Page: 57 61 57 60 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(57)243 333 y(Let)27 b FO(s)430 345 y FM(i)458 333 y FP(,)h FO(r)546 345 y FM(i)574 333 y FP(,)g FO(i)22 b FP(=)h(1,)k FO(:)14 b(:)g(:)28 b FP(,)f FO(m)p FP(,)h(b)r(e)g(p)r(olynomials)f(satisfying)g(the)h(relation)892 564 y FO(p)p FP(\()p FO(x)19 b FQ(\000)f FO(y)s FP(\))23 b(=)1332 460 y FM(m)1302 485 y Fz(X)1308 662 y FM(i)p FL(=1)1435 564 y FO(s)1474 576 y FM(i)1502 564 y FP(\()p FO(x)p FP(\))14 b FO(r)1664 576 y FM(i)1693 564 y FP(\()p FO(y)s FP(\))118 816 y(for)31 b(an)n(y)f FO(x;)14 b(y)31 b FQ(2)e FJ(R)703 785 y FL(6)747 816 y FP(.)47 b(Let)31 b FO(u;)14 b(v)31 b FQ(2)e FO(D)r FP(\()p FJ(R)1366 785 y FM(n)1418 816 y FP(\))i(and)g FO(a)1690 828 y FM(i)1746 816 y FP(=)d FO(us)1926 828 y FM(i)1953 816 y FP(,)k FO(b)2044 828 y FM(i)2100 816 y FP(=)c FO(v)s(r)2273 828 y FM(i)2302 816 y FP(.)47 b(Ob)n(vi-)118 915 y(ously)-7 b(,)29 b FO(a)395 927 y FM(i)422 915 y FP(,)h FO(b)511 927 y FM(i)563 915 y FQ(2)c FO(D)r FP(\()p FJ(R)801 885 y FL(6)844 915 y FP(\).)41 b(Consider)28 b(the)i(op)r(erators)d FO(A)1864 927 y FM(i)1917 915 y FP(=)e FO(M)2088 927 y FM(a)2124 935 y Fw(i)2154 915 y FP(,)k FO(B)2269 927 y FM(i)2322 915 y FP(=)c FO(M)2493 927 y FM(b)2522 935 y Fw(i)2552 915 y FP(,)118 1015 y FO(i)46 b FP(=)f(1,)c FO(:)14 b(:)g(:)27 b FP(,)45 b FO(m)c FP(in)h(the)f(space)g FO(L)1275 985 y FL(2)1312 1015 y FP(\()p FJ(R)1398 985 y FL(6)1441 1015 y FP(\))h(\(here)f FO(M)1822 1027 y FM(f)1906 1015 y FP(is)g(the)g(op)r(erator)f(of)118 1114 y(m)n(ultiplication)31 b(b)n(y)f(the)h(function)g FO(f)9 b FP(\).)45 b(Since)30 b(the)h(F)-7 b(ourier)30 b(transform)f(of)h(a) 118 1214 y(pseudo-measure)23 b(\010)h(b)r(elongs)g(to)g FO(L)1250 1184 y FN(1)1320 1214 y FP(\()p FJ(R)1407 1184 y FL(6)1450 1214 y FP(\),)h(the)g(op)r(erator)e FO(T)34 b FP(=)23 b FO(F)2238 1184 y FN(\000)p FL(1)2327 1214 y FO(M)2408 1226 y FM(F)9 b FL(\010)2510 1214 y FO(F)118 1314 y FP(is)29 b(w)n(ell)f(de\014ned)h(in)h(the)f(space)f FO(L)1182 1284 y FL(2)1218 1314 y FP(\()p FJ(R)1305 1284 y FL(6)1348 1314 y FP(\).)41 b(F)-7 b(urthermore)27 b(a)i(direct)f (computa-)118 1413 y(tion)g(sho)n(ws)e(that)678 1536 y Fz(\020)758 1525 y FM(m)728 1549 y Fz(X)734 1726 y FM(i)p FL(=1)861 1628 y FO(M)942 1640 y FM(b)971 1648 y Fw(i)1002 1628 y FO(T)12 b(M)1144 1640 y FM(a)1180 1648 y Fw(i)1209 1628 y FO(';)i( )1357 1536 y Fz(\021)1430 1628 y FP(=)22 b FQ(h)p FO(p)14 b FP(\010)p FO(;)g(u')k FQ(\003)1891 1606 y Fz(f)1885 1628 y FP(\026)-45 b FO(v)s( )t FQ(i)348 b FP(\(1.16\))118 1880 y(for)30 b(an)n(y)f FO(u)p FP(,)i FO(v)s FP(,)g FO(')p FP(,)g FO( )g FQ(2)d FO(D)r FP(\()p FJ(R)1039 1850 y FL(6)1082 1880 y FP(\),)k(where)d FQ(h\001)p FO(;)14 b FQ(\001i)31 b FP(is)f(the)h(pairing)e(of)h(the)h (spaces)118 1980 y FO(D)r FP(\()p FJ(R)275 1950 y FL(6)319 1980 y FP(\),)43 b FO(D)488 1950 y FN(0)511 1980 y FP(\()p FJ(R)598 1950 y FL(6)641 1980 y FP(\))d(\(since)g FO(P)12 b(M)d FP(\()p FJ(R)1202 1950 y FL(6)1245 1980 y FP(\))44 b FQ(\032)f FO(D)r FP(\()p FJ(R)1587 1950 y FL(6)1630 1980 y FP(\))1662 1950 y FN(0)1686 1980 y FP(,)g FO(p)14 b FP(\010)43 b FQ(2)h FO(D)2081 1950 y FN(0)2104 1980 y FP(\()p FJ(R)2191 1950 y FL(6)2234 1980 y FP(\))c(for)g(an)n(y)118 2079 y(p)r(olynomial)27 b FO(p)h FP(in)f(six)h(v)-5 b(ariables\).)36 b(Since)27 b(the)h(set)723 2249 y FO(L)23 b FP(=)f FQ(f)p FO(u')c FQ(\003)1141 2227 y FP(~)1115 2249 y(\026)-45 b FO(v)s( )26 b FQ(j)d FO(';)14 b( )s(;)g(u;)g(v)26 b FQ(2)e FO(D)r FP(\()p FJ(R)1853 2214 y FL(6)1896 2249 y FP(\))p FQ(g)118 2418 y FP(is)35 b(dense)f(in)h FO(D)r FP(\()p FJ(R)703 2388 y FL(6)746 2418 y FP(\),)i(the)e(existence)f(of)h (a)f(pseudo-measure)f(\010)h(suc)n(h)h(that)118 2517 y FO(p)14 b FP(\010)23 b(=)f(0)h(and)30 b(\026)-49 b FO(p)14 b FP(\010)23 b FQ(6)p FP(=)f(0)h(w)n(ould)g(imply)g(the)h (existence)f(of)g(functions)g FO(u)p FP(,)h FO(v)i FP(suc)n(h)118 2617 y(that)382 2740 y Fz(\020)462 2728 y FM(m)432 2753 y Fz(X)438 2930 y FM(i)p FL(=1)565 2832 y FO(M)655 2798 y FN(\003)646 2853 y FM(b)675 2861 y Fw(i)706 2832 y FO(T)12 b(M)857 2798 y FN(\003)848 2853 y FM(a)884 2861 y Fw(i)913 2832 y FO(';)i( )1061 2740 y Fz(\021)1134 2832 y FP(=)22 b FQ(h)7 b FP(\026)-49 b FO(p)14 b FP(\010)p FO(;)19 b FP(\026)-47 b FO(u)o(')19 b FQ(\003)1595 2810 y Fz(f)1586 2832 y FO(v)s( )s FQ(i)24 b(6)p FP(=)e(0)p FO(;)1443 2977 y FP(for)27 b(some)g FO(')p FP(,)h FO( )e FQ(2)d FO(D)r FP(\()p FJ(R)2198 2943 y FL(6)2242 2977 y FP(\))p FO(;)382 3085 y Fz(\020)471 3073 y FM(n)432 3098 y Fz(X)438 3275 y FM(i)p FL(=1)565 3177 y FO(M)646 3189 y FM(b)675 3197 y Fw(i)706 3177 y FO(T)12 b(M)848 3189 y FM(a)884 3197 y Fw(i)913 3177 y FO(';)i( )1061 3085 y Fz(\021)1134 3177 y FP(=)22 b FQ(h)p FO(p)14 b FP(\010)p FO(;)g(u')k FQ(\003)1595 3155 y Fz(f)1589 3177 y FP(\026)-45 b FO(v)s( )s FQ(i)24 b FP(=)e(0)p FO(;)1443 3322 y FP(for)27 b(all)g FO(')p FP(,)h FO( )e FQ(2)d FO(D)r FP(\()p FJ(R)2106 3288 y FL(6)2149 3322 y FP(\))p FO(:)118 3491 y FP(Hence,)30 b(for)f(the)h(op)r(erator)e FO(A)1063 3503 y FM(i)1091 3491 y FP(,)i FO(B)1207 3503 y FM(i)1235 3491 y FP(,)g FO(T)40 b FP(constructed)29 b(relativ)n(ely)f(to)i(\010,)g FO(u)p FP(,)g FO(v)s FP(,)118 3591 y(w)n(e)d(obtain)669 3702 y FM(m)639 3727 y Fz(X)645 3904 y FM(i)p FL(=1)773 3806 y FO(B)836 3818 y FM(i)863 3806 y FO(T)12 b(A)986 3818 y FM(i)1036 3806 y FP(=)23 b(0)83 b(and)1510 3702 y FM(m)1479 3727 y Fz(X)1485 3904 y FM(i)p FL(=1)1613 3806 y FO(B)1680 3772 y FN(\003)1676 3826 y FM(i)1718 3806 y FO(T)12 b(A)1841 3772 y FN(\003)1841 3826 y FM(i)1902 3806 y FQ(6)p FP(=)22 b(0)p FO(:)309 b FP(\(1.17\))118 4048 y(Since)23 b(an)n(y)f(\014nite)h(comm)n(utativ)n (e)f(family)h(of)f(normal)g(op)r(erators)f(can)h(b)r(e)h(real-)118 4147 y(ized)k(as)f(a)g(family)g(of)h(con)n(tin)n(uous)e(functions)i(of) f(one)h(self-adjoin)n(t)f(op)r(erator,)p eop %%Page: 58 62 58 61 bop 118 100 a FP(58)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(one)k(can)g(\014nd)h(a)f(self-adjoin)n(t)g(op)r(erator)f FO(A)i FP(and)f(con)n(tin)n(uous)g(functions)g FO(f)2524 345 y FM(i)2552 333 y FP(,)118 432 y FO(g)158 444 y FM(i)218 432 y FP(suc)n(h)h(that)g FO(B)657 444 y FM(i)716 432 y FP(=)e FO(f)852 444 y FM(i)880 432 y FP(\()p FO(A)p FP(\),)k FO(A)1125 444 y FM(i)1184 432 y FP(=)c FO(g)1319 444 y FM(i)1347 432 y FP(\()p FO(A)p FP(\).)52 b(Then)32 b(\(1.17\))g(is)g(equiv)-5 b(alen)n(t)32 b(to)118 532 y(the)c(follo)n(wing)446 686 y FM(m)415 711 y Fz(X)421 888 y FM(i)p FL(=1)549 790 y FO(f)590 802 y FM(i)617 790 y FP(\()p FO(A)p FP(\))14 b FO(T)26 b(g)872 802 y FM(i)899 790 y FP(\()p FO(A)p FP(\))e(=)f(0)82 b(and)1522 686 y FM(m)1492 711 y Fz(X)1498 888 y FM(i)p FL(=1)1643 768 y FP(\026)1626 790 y FO(f)1667 802 y FM(i)1694 790 y FP(\()p FO(A)p FP(\))14 b FO(T)28 b FP(\026)-45 b FO(g)1948 802 y FM(i)1976 790 y FP(\()p FO(A)p FP(\))23 b FQ(6)p FP(=)g(0)p FO(:)118 1059 y FP(Th)n(us,)k(it)h(remains)e(to)i(pro)n(v)n (e)d(the)j(existence)f(of)g(a)g(pseudo-measure)f(\010)h(suc)n(h)118 1159 y(that)38 b FO(p)14 b FP(\010)41 b(=)f(0)d(and)45 b(\026)-49 b FO(p)14 b FP(\010)40 b FQ(6)p FP(=)g(0.)68 b(In)38 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3946 y FO(m)p FP(".)243 4048 y(It)c(w)n(as)e(sho)n(wn)h(b)n(y)g(Arv)n(eson)g(that)h(the)g (pseudo-in)n(tegral)d(op)r(erator)h FO(T)2512 4060 y FM(m)118 4147 y FP(is)k(supp)r(orted)g(b)n(y)g(an)n(y)f(pseudo-closed)g (set)h(on)g(whic)n(h)g(the)h(measure)e FO(m)h FP(is)p eop %%Page: 60 64 60 63 bop 118 100 a FP(60)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(concen)n(trated)h(\(the)j(set)e FO(E)j FQ(\032)26 b FJ(R)f FQ(\002)20 b FJ(R)35 b FP(is)30 b(called)f(pseudo-closed)f(if) i(its)g(com-)118 432 y(plemen)n(t)j(is)f(a)g(union)g(of)g(coun)n(tably) g(man)n(y)f(measurable)g(sets)h(of)h(the)f(form)118 532 y FO(X)25 b FQ(\002)18 b FO(Y)42 b FQ(\032)22 b FJ(R)j FQ(\002)18 b FJ(R)p FP(\).)243 632 y(Let)30 b(\000)g(b)r(e)g(the)g(c)n (haracteristic)e(binary)i(relation)f(corresp)r(onding)f(to)h(the)118 731 y(semilinear)19 b(relation)f(\(1.10\),)i(and)f FO(m)h FP(a)f(regular)e(measure)h(with)i(supp)14 b FO(m)23 b FQ(\032)g FP(\000.)118 831 y(Then,)28 b FO(A)p FP(,)g FO(T)520 843 y FM(m)610 831 y FP(is)g(a)f(represen)n(tation)f(of)h (relation)g(\(1.10\).)36 b(Indeed,)440 1003 y Fz(\000)517 967 y FM(n)478 992 y Fz(X)484 1168 y FM(i)p FL(=1)612 1070 y FO(f)653 1082 y FM(i)680 1070 y FP(\()p FO(A)p FP(\))14 b FO(T)869 1082 y FM(m)946 1070 y FO(g)986 1082 y FM(i)1013 1070 y FP(\()p FO(A)p FP(\))o FO(~)-41 b(x)q(;)13 b(~)-41 b(y)1268 978 y Fz(\021)1341 1070 y FP(=)1468 967 y FM(n)1428 992 y Fz(X)1435 1168 y FM(i)p FL(=1)1548 1003 y Fz(\000)1586 1070 y FO(T)1635 1082 y FM(m)1698 1070 y FO(g)1738 1082 y FM(i)1765 1070 y FP(\()p FO(A)p FP(\))o FO(~)g(x)q(;)p 1976 1003 69 4 v 14 w(f)2017 1082 y FM(i)2045 1070 y FP(\()p FO(A)p FP(\))o FO(~)g(y)2215 1003 y Fz(\001)629 1347 y FP(=)756 1243 y FM(n)716 1268 y Fz(X)723 1445 y FM(i)p FL(=1)850 1234 y Fz(Z)10 b(Z)989 1423 y FL(\000)1048 1347 y FO(g)1088 1359 y FM(i)1115 1347 y FP(\()p FO(s)p FP(\))k FO(f)1273 1359 y FM(i)1301 1347 y FP(\()p FO(t)p FP(\))g(\()o FO(~)-41 b(x)q FP(\()p FO(s)p FP(\))p FO(;)13 b(~)-41 b(y)s FP(\()p FO(t)p FP(\)\))14 b FO(dm)p FP(\()p FO(t;)g(s)p FP(\))629 1599 y(=)716 1486 y Fz(Z)c(Z)855 1675 y FL(\000)914 1599 y FP(\010\()p FO(t;)k(s)p FP(\))g(\()o FO(~)-41 b(x)q FP(\()p FO(s)p FP(\))p FO(;)13 b(~)-41 b(y)s FP(\()p FO(t)p FP(\)\))14 b FO(dm)p FP(\()p FO(t;)g(s)p FP(\))24 b(=)f(0)p FO(:)118 1856 y FR(1.3.4)94 b(Irreducible)32 b(represen)m(tations)f(of)h (semilinear)d(relations)118 2009 y FP(No)n(w)21 b(w)n(e)g(study)h (irreducible)f(represen)n(tations)f(of)h(semilinear)g(relations,)h (i.e.,)118 2108 y(an)27 b(irreducible)g(families)h(of)g(op)r(erators)d FO(A)e FP(=)g FO(A)1644 2078 y FN(\003)1683 2108 y FP(,)k FO(B)t FP(,)h FO(B)1918 2078 y FN(\003)1984 2108 y FP(satisfying)989 2240 y FM(n)950 2265 y Fz(X)956 2441 y FM(i)p FL(=1)1084 2343 y FO(f)1125 2355 y FM(i)1152 2343 y FP(\()p FO(A)p FP(\))14 b FO(B)19 b(g)1414 2355 y FM(i)1441 2343 y FP(\()p FO(A)p FP(\))24 b(=)e(0)p FO(:)118 2589 y FP(It)k(is)f(clear)f(that)i (if)g FO(f)776 2601 y FM(i)803 2589 y FP(,)g FO(g)892 2601 y FM(i)945 2589 y FP(are)e(p)r(olynomials,)h(then)h(suc)n(h)f(a)g (family)g(of)g(op)r(era-)118 2689 y(tors)i(de\014nes)g(an)h (irreducible)f(represen)n(tation)f(of)h(the)h FQ(\003)p FP(-algebra)d(generated)118 2788 y(b)n(y)35 b FO(a)g FP(=)g FO(a)464 2758 y FN(\003)502 2788 y FP(,)h FO(b)p FP(,)h FO(b)693 2758 y FN(\003)766 2788 y FP(and)d(the)i(relation)1397 2726 y Fz(P)1484 2747 y FM(n)1484 2813 y(i)p FL(=1)1610 2788 y FO(f)1651 2800 y FM(i)1678 2788 y FP(\()p FO(a)p FP(\))14 b FO(b)g(g)1890 2800 y FM(i)1917 2788 y FP(\()p FO(a)p FP(\))36 b(=)f(0.)58 b(In)35 b(what)118 2888 y(follo)n(ws)28 b(w)n(e)g(shall)h(mean)f(b)n(y)h(a)f(represen)n(tation)f(of)i(the)g (semilinear)f(relation)118 2988 y(a)e(triple)h(\()p FO(A)c FP(=)g FO(A)674 2957 y FN(\003)712 2988 y FO(;)14 b(B)t(;)g(B)920 2957 y FN(\003)958 2988 y FP(\))27 b(satisfying)f(the)h(semilinear)e (relation)h(instead)g(of)118 3087 y(the)i(pair)f(\()p FO(A)d FP(=)e FO(A)699 3057 y FN(\003)738 3087 y FO(;)14 b(B)t FP(\),)28 b(if)g(it)g(do)r(es)f(not)h(lead)f(to)g(an)n(y)g (confusion.)118 3233 y FR(1.)58 b FP(W)-7 b(e)36 b(b)r(egin)f(with)g (\014nite-dimensional)g(representations)e(and)i(establish)118 3332 y(a)e(connection)g(b)r(et)n(w)n(een)h(irreducible)f(represen)n (tations)e(of)40 b(\(1.10\))33 b(and)g(the)118 3432 y(corresp)r(onding) 26 b(graph.)118 3590 y FR(Prop)s(osition)f(20.)37 b FC(If)26 b(a)g(family)i FP(\()p FO(A;)14 b(B)t(;)g(B)1491 3560 y FN(\003)1530 3590 y FP(\))26 b FC(de\014nes)g(a)g (\014nite-dimensional)118 3690 y(irr)l(e)l(ducible)43 b(r)l(epr)l(esentation)f(of)61 b FP(\(1.10\))o FC(,)45 b(then)d(the)f(c)l(orr)l(esp)l(onding)j(gr)l(aph)118 3790 y FP(\000)23 b Fs(\026)228 3805 y FM(\033)r FL(\()p FM(A)p FL(\))404 3790 y FC(is)30 b(c)l(onne)l(cte)l(d.)39 b(F)-6 b(or)30 b(every)h(\014nite)f(c)l(onne)l(cte)l(d)f(sub)l(gr)l (aph)i FP(\()p FO(D)r(;)14 b FP(\000)23 b Fs(\026)2457 3802 y FM(D)2517 3790 y FP(\))p FC(,)118 3889 y(ther)l(e)29 b(exists)f(an)h(irr)l(e)l(ducible)h(r)l(epr)l(esentation)f FP(\()p FO(A;)14 b(B)t(;)g(B)1912 3859 y FN(\003)1950 3889 y FP(\))29 b FC(with)g FO(\033)s FP(\()p FO(A)p FP(\))c(=)d FO(D)r FC(.)118 4048 y(Pr)l(o)l(of.)43 b FP(In)24 b(fact,)g(if)g(\000)f Fs(\026)843 4063 y FM(\033)r FL(\()p FM(A)p FL(\))989 4048 y FP(=)f(\000)1128 4060 y FL(1)1175 4048 y FQ([)10 b FP(\000)1292 4060 y FL(2)1353 4048 y FP(is)23 b(the)g(union)g(of)h(t)n(w)n(o)e(connected)h(sub-)118 4147 y(graphs)i(then,)i(b)n(y)f(Theorem)f(6,)h(the)h(sp)r(ectral)e (subspaces)g FO(A)i FP(corresp)r(onding)p eop %%Page: 61 65 61 64 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(61)118 333 y(to)25 b(the)g(v)n(ertices)f(of)39 b(\000)813 345 y FL(1)875 333 y FP(and)24 b(\000)1085 345 y FL(2)1147 333 y FP(are)g(in)n(v)-5 b(arian)n(t)24 b(with)h(resp)r(ect)g(to)g FO(B)k FP(and)24 b FO(B)2513 303 y FN(\003)2552 333 y FP(.)118 432 y(This)k(sho)n(ws)e(that)i(\()p FO(A;)14 b(B)t(;)g(B)1028 402 y FN(\003)1067 432 y FP(\))28 b(is)f(reducible.)243 532 y(Let)38 b(\000)h Fs(\026)528 547 y FM(\033)r FL(\()p FM(A)p FL(\))712 532 y FP(b)r(e)g(connected.)67 b(The)38 b(family)g(of)f(op)r(erators)f FO(A)p FP(,)41 b FO(B)t FP(,)f FO(B)2536 502 y FN(\003)118 632 y FP(de\014ned)28 b(b)n(y)478 916 y FO(A)23 b FP(=)651 724 y Fz(0)651 870 y(B)651 923 y(@)723 788 y FO(\025)771 800 y FL(1)1107 788 y FP(0)897 885 y(.)929 909 y(.)961 935 y(.)745 1042 y(0)285 b FO(\025)1120 1054 y FM(m)1183 724 y Fz(1)1183 870 y(C)1183 923 y(A)1270 916 y FO(;)180 b FQ(f)p FO(\025)1563 928 y FL(1)1600 916 y FO(;)14 b(:)g(:)g(:)g(;)g(\025)1833 928 y FM(m)1896 916 y FQ(g)23 b FP(=)f FO(D)r(;)1536 1134 y(\025)1584 1146 y FM(i)1635 1134 y FQ(6)p FP(=)h FO(\025)1771 1146 y FM(j)1807 1134 y FO(;)179 b(i)23 b FQ(6)p FP(=)g FO(j;)473 1367 y(B)k FP(=)c(\()p FO(b)719 1379 y FM(ij)777 1367 y FP(\))809 1333 y FM(m)809 1387 y(i;j)s FL(=1)972 1367 y FO(;)180 b(b)1211 1379 y FM(ij)1292 1367 y FP(=)1379 1225 y Fz(\()1446 1310 y FP(0)p FO(;)83 b FP(\()p FO(\025)1674 1322 y FM(i)1702 1310 y FO(;)14 b(\025)1787 1322 y FM(j)1823 1310 y FP(\))32 b FO(=)-51 b FQ(2)23 b FP(\000)p FQ(j)2031 1322 y FM(D)2091 1310 y FO(;)1446 1430 y FP(1)p FO(;)83 b FP(\()p FO(\025)1674 1442 y FM(i)1702 1430 y FO(;)14 b(\025)1787 1442 y FM(j)1823 1430 y FP(\))23 b FQ(2)g FP(\000)p FQ(j)2031 1442 y FM(D)2091 1430 y FO(;)118 1603 y FP(is)41 b(irreducible.)76 b(Indeed,)44 b(the)d(relation)f([)p FO(C)q(;)14 b(A)p FP(])46 b(=)f(0)c(implies)g (that)g FO(C)47 b FP(is)118 1703 y(diagonal,)40 b FO(C)47 b FP(=)41 b(diag\()p FO(c)920 1715 y FL(1)957 1703 y FO(;)14 b(:)g(:)g(:)28 b(;)14 b(c)1192 1715 y FL(2)1229 1703 y FP(\).)69 b(F)-7 b(rom)38 b([)p FO(C)q(;)14 b(B)t FP(])42 b(=)f(0,)f(it)f(follo)n(ws)f(that)118 1803 y FO(c)154 1815 y FM(k)195 1803 y FO(b)231 1815 y FM(k)q(l)321 1803 y FP(=)28 b FO(b)450 1815 y FM(k)q(l)512 1803 y FO(c)548 1815 y FM(l)573 1803 y FP(.)46 b(Since)31 b(\000)d Fs(\026)977 1818 y FM(\033)r FL(\()p FM(A)p FL(\))1154 1803 y FP(is)j(connected,)g(w)n(e)f(ha)n(v)n(e)g(that)h(there)f(exists) 118 1902 y(a)36 b(p)r(erm)n(utation)g(\()p FO(l)737 1914 y FL(1)774 1902 y FO(;)14 b(:)g(:)g(:)28 b(;)14 b(l)998 1914 y FM(m)1061 1902 y FP(\))37 b FQ(2)h FO(S)1274 1914 y FM(m)1374 1902 y FP(suc)n(h)e(that)g(\()p FO(\025)1838 1914 y FM(l)1859 1923 y Fw(k)1901 1902 y FO(;)14 b(\025)1986 1914 y FM(l)2007 1923 y Fw(k)q Fy(+1)2118 1902 y FP(\))38 b FQ(2)g FP(\000)g Fs(\026)2406 1917 y FM(\033)r FL(\()p FM(A)p FL(\))2552 1902 y FP(,)118 2002 y(hence)32 b FO(b)389 2014 y FM(l)410 2023 y Fw(k)446 2014 y FM(;l)487 2023 y Fw(k)q Fy(+1)629 2002 y FP(=)e(1.)49 b(This)32 b(implies)h(that)f FO(c)1539 2014 y FL(1)1606 2002 y FP(=)e FO(:)14 b(:)g(:)31 b FP(=)f FO(c)1960 2014 y FM(m)2023 2002 y FP(,)j(and)f FO(C)37 b FP(=)30 b FO(c)2472 2014 y FL(1)2509 2002 y FO(I)7 b FP(.)118 2101 y(Moreo)n(v)n(er,)25 b(w)n(e)i(ha)n(v)n(e)g FO(\033)s FP(\()p FO(A)p FP(\))d(=)f FO(D)r FP(.)p 2514 2101 4 57 v 2518 2049 50 4 v 2518 2101 V 2567 2101 4 57 v 243 2263 a(W)-7 b(e)19 b(also)f(giv)n(e)g(a)g(reform)n(ulation)f (of)i(this)g(statemen)n(t)g(for)f(the)i(symmetrical)118 2363 y(case.)118 2512 y FR(Prop)s(osition)j(21.)35 b FC(If)43 b FO(A)23 b FP(=)f FO(A)1119 2482 y FN(\003)1158 2512 y FC(,)k FO(B)h FP(=)22 b FO(B)1453 2482 y FN(\003)1516 2512 y FC(is)i(an)h(irr)l(e)l(ducible)g(\014nite-dimen-)118 2612 y(sional)i(r)l(epr)l(esentation)e(of)44 b FP(\(1.10\))o FC(,)27 b(then)e(the)g(gr)l(aph)i FP(\000)1854 2624 y FM(s)1912 2612 y Fs(\026)1947 2627 y FM(\033)r FL(\()p FM(A)p FL(\))2119 2612 y FC(is)f(c)l(onne)l(cte)l(d.)118 2711 y(F)-6 b(or)41 b(every)h(\014nite)e(c)l(onne)l(cte)l(d)h(sub)l(gr) l(aph)g FP(\()p FO(D)r(;)14 b FP(\000)1668 2723 y FM(s)1747 2711 y Fs(\026)1782 2726 y FM(\033)r FL(\()p FM(A)p FL(\))1928 2711 y FP(\))p FC(,)44 b(ther)l(e)d(exists)g(an)118 2811 y(irr)l(e)l(ducible)c(p)l(air)g FO(A)e FP(=)e FO(A)961 2781 y FN(\003)1000 2811 y FC(,)38 b FO(B)g FP(=)c FO(B)1330 2781 y FN(\003)1404 2811 y FC(satisfying)44 b FP(\(1.10\))35 b FC(such)h(that)g FO(D)g FP(=)118 2911 y FO(\033)s FP(\()p FO(A)p FP(\))p FC(.)243 3060 y FP(The)22 b(pro)r(of)g(is)h(the)g(same)f (as)g(the)h(one)f(giv)n(en)g(ab)r(o)n(v)n(e,)h(but)g(with)g(\000)2272 3072 y FM(s)2330 3060 y FP(replac-)118 3160 y(ing)29 b(\000.)40 b(Note)29 b(that)g(the)h(constructed)e(op)r(erator)f FO(B)33 b FP(is)c(self-adjoin)n(t)f(b)r(ecause)118 3259 y(the)g(graph)f(\000)549 3271 y FM(s)612 3259 y FP(is)g(symmetrical.) 118 3400 y FR(2.)80 b FP(In)42 b(what)h(follo)n(ws,)h(w)n(e)e(in)n(v)n (estigate)f(irreducible)h(represen)n(tations)e(of)118 3500 y(\(1.10\))26 b(on)h(a)g(Hilb)r(ert)g(space)g FO(H)7 b FP(,)27 b(where)f FO(H)34 b FP(is)27 b(not)g(necessary)f (\014nite-dimen-)118 3599 y(sional.)35 b(First,)25 b(w)n(e)f(pro)n(v)n (e)f(an)h(analogue)f(of)h(the)h(theorem)f(on)h(connectedness)118 3699 y(of)i(the)h(graph)e(\(Prop)r(osition)f(20\).)37 b(F)-7 b(or)26 b(this)i(purp)r(ose,)e(w)n(e)h(recall)f(some)h(def-)118 3799 y(initions.)118 3948 y FR(De\014nition)38 b(4.)45 b FC(A)36 b(subset)f FO(F)45 b FQ(\032)34 b FJ(R)29 b FQ(\002)22 b FJ(R)42 b FC(is)36 b(c)l(al)t(le)l(d)h(mar)l(ginal)t(ly)h (nul)t(l)d(with)118 4048 y(r)l(esp)l(e)l(ct)i(to)h(a)g(me)l(asur)l(e)f FO(\026)g FP(\()p FC(or)h(a)g(class)g(of)h(me)l(asur)l(es)e(e)l (quivalent)h(to)f FO(\026)p FP(\))h FC(if)118 4147 y FO(F)d FQ(\032)23 b FP(\()p FO(\013)c FQ(\002)f FJ(R)p FP(\))25 b FQ([)18 b FP(\()p FJ(R)25 b FQ(\002)18 b FO(\014)t FP(\))p FC(,)31 b(wher)l(e)f FO(\026)p FP(\()p FO(\013)p FP(\))24 b(=)f FO(\026)p FP(\()p FO(\014)t FP(\))h(=)f(0)p FC(.)p eop %%Page: 62 66 62 65 bop 118 100 a FP(62)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)243 333 y FP(Let,)g(further,)771 512 y(\000\()p FO(M)9 b FP(\))23 b(=)g FQ(f)p FO(y)i FQ(j)e(9)p FO(x)h FQ(2)f FO(M)18 b FP(:)28 b(\()p FO(x;)14 b(y)s FP(\))23 b FQ(2)h FP(\000)p FQ(g)p FO(;)682 647 y FP(\000)734 613 y FN(\000)p FL(1)823 647 y FP(\()p FO(M)9 b FP(\))23 b(=)g FQ(f)p FO(x)g FQ(j)g(9)p FO(y)j FQ(2)d FO(M)18 b FP(:)28 b(\()p FO(x;)14 b(y)s FP(\))23 b FQ(2)h FP(\000)p FQ(g)p FO(;)118 827 y FP(and)35 b(let)g FO(M)504 796 y FM(c)572 827 y FP(b)r(e)h(the)f(complemen)n(t)g(of)g(the)g(set)g FO(M)9 b FP(.)58 b(In)35 b(what)g(follo)n(ws)f(w)n(e)118 926 y(shall)25 b(assume)g(that)g(\000\()p FO(M)9 b FP(\))25 b(and)h(\000)1215 896 y FN(\000)p FL(1)1304 926 y FP(\()p FO(M)9 b FP(\))25 b(are)f(Borel)h(for)f(an)n(y)h(Borel)f(set)h FO(M)9 b FP(.)243 1026 y(The)31 b(concepts)h(de\014ned)g(b)r(elo)n(w)f (generalize)f(those)i(of)f(quasi-in)n(v)-5 b(ariance)118 1125 y(and)32 b(ergo)r(dicit)n(y)g(of)g(a)g(measure.)51 b(F)-7 b(or)32 b(graphs)f(related)h(to)g(dynamical)g(sys-)118 1225 y(tems,)c(w)n(e)f(will)h(also)f(discuss)g(these)g(notions)g(in)h (Section)g(2.1.1.)118 1389 y FR(De\014nition)h(5.)40 b FC(We)28 b(c)l(al)t(l)i(a)f(set)f FO(M)k FQ(\032)22 b FJ(R)35 b FC(right-invariant)i FP(\()p FC(left-invariant)8 b FP(\))118 1488 y FC(with)25 b(r)l(esp)l(e)l(ct)f(to)h FP(\000)f FC(and)h(a)g(me)l(asur)l(e)f FO(\026)p FC(,)i(if)f(the)g(set) f FO(M)1791 1458 y FM(c)1832 1488 y FQ(\002)7 b FO(M)15 b FQ(\\)7 b FP(\000)24 b(\()p FC(r)l(esp)l(e)l(ctively)118 1588 y FO(M)15 b FQ(\002)6 b FO(M)375 1558 y FM(c)414 1588 y FQ(\\)g FP(\000\))25 b FC(is)g(mar)l(ginal)t(ly)g(nul)t(l)9 b FP(;)26 b FO(M)33 b FC(is)24 b(invariant)h(if)g(it)f(is)h(left-)f (and)g(right-)118 1688 y(invariant.)47 b(A)32 b(me)l(asur)l(e)h FO(\026)f FC(is)h(c)l(al)t(le)l(d)42 b FP(\()p FC(left-,)34 b(right-)p FP(\))f FC(quasi-invariant)g(with)118 1787 y(r)l(esp)l(e)l(ct)28 b(to)g FP(\000)g FC(if)h(the)f(set)g FP(\000\()p FO(M)9 b FP(\))15 b FQ([)h FP(\000)1252 1757 y FN(\000)p FL(1)1341 1787 y FP(\()p FO(M)9 b FP(\))28 b(\()p FC(r)l(esp)l(e)l(ctively)h FP(\000)2039 1757 y FN(\000)p FL(1)2128 1787 y FP(\()p FO(M)9 b FP(\))p FC(,)29 b FP(\000\()p FO(M)9 b FP(\)\))118 1887 y FC(is)33 b(of)h(non-zer)l(o)f (me)l(asur)l(e)f(for)i(every)g(Bor)l(el)g(set)e FO(M)42 b FC(such)33 b(that)g FO(\026)p FP(\()p FO(M)9 b FP(\))29 b FQ(6)p FP(=)f(0)p FC(,)118 1986 y FP(\000\()p FO(M)9 b FP(\))p FC(,)39 b FP(\000)440 1956 y FN(\000)p FL(1)529 1986 y FP(\()p FO(M)9 b FP(\))36 b FQ(2)g FA(B)p FP(\()p FJ(R)q FP(\))p FC(.)66 b(A)37 b(sp)l(e)l(ctr)l(al)g(typ)l(e)g FO(\026)f FC(is)i(c)l(al)t(le)l(d)46 b FP(\()p FC(left-,)40 b(right-)r FP(\))118 2086 y FC(er)l(go)l(dic)26 b(with)g(r)l(esp)l(e)l (ct)f(to)g FP(\000)f FC(if)i(any)33 b FP(\()p FC(left-,)27 b(right-)p FP(\))e FC(invariant)h(set)f(is)g(a)g FO(\026)p FC(-nul)t(l)118 2186 y(set)30 b(or)g(has)g(the)g FO(\026)p FC(-nul)t(l)f(c)l(omplement.)243 2349 y FP(Note)d(that)h(for)f(an)n(y)f (op)r(erator)g FO(A)e FP(=)g FO(A)1471 2319 y FN(\003)1536 2349 y FP(there)j(exists)g(the)g(trivial)g(repre-)118 2449 y(sen)n(tation)31 b(\()p FO(A;)14 b FP(0)p FO(;)g FP(0\))30 b(of)h(the)h(relation)e(\(1.10\).)46 b(Therefore,)31 b(it)h(is)f(natural)f(to)118 2548 y(consider)d(represen)n(tations)e (without)j(trivial)f(parts.)118 2712 y FR(De\014nition)32 b(6.)41 b FC(We)30 b(c)l(al)t(l)i(a)f(r)l(epr)l(esentation)f FO(A)p FC(,)h FO(B)t FC(,)h FO(B)1909 2682 y FN(\003)1977 2712 y FQ(\003)p FC(-c)l(omplete)f(if)g(for)118 2812 y(any)d(non-zer)l(o)g(sp)l(e)l(ctr)l(al)g(subsp)l(ac)l(e)h FO(W)39 b FC(of)29 b(the)f(op)l(er)l(ator)h FO(A)f FC(one)g(of)h(the)f (op)l(er-)118 2911 y(ators)i FO(B)k FC(and)c FO(B)654 2881 y FN(\003)722 2911 y FC(is)g(not)g(e)l(qual)g(to)g(zer)l(o)g(on)f FO(W)12 b FC(.)243 3075 y FP(It)19 b(is)h(easy)e(to)i(sho)n(w)e(that)i (an)n(y)f(irreducible)g(represen)n(tation)f(of)h(dimension)118 3174 y(greater)26 b(than)i(one)f(is)g FQ(\003)p FP(-complete.)118 3338 y FR(Prop)s(osition)22 b(22.)34 b FC(If)24 b(a)g(r)l(epr)l (esentation)f FP(\()p FO(A;)14 b(B)t(;)g(B)1766 3308 y FN(\003)1805 3338 y FP(\))23 b FC(of)42 b FP(\(1.10\))23 b FC(is)g FQ(\003)p FC(-c)l(omp-)118 3438 y(lete,)36 b(then)f(the)f(sp)l(e)l(ctr)l(al)h(me)l(asur)l(e)f(of)i(the)e(op)l(er)l (ator)i FO(A)f FC(is)g(quasi-invariant)118 3537 y(with)30 b(r)l(esp)l(e)l(ct)g(to)g FP(\000)p FC(.)118 3701 y(Pr)l(o)l(of.)43 b FP(If)37 b FO(E)531 3713 y FM(A)586 3701 y FP(\()p FO(M)9 b FP(\))38 b FQ(6)p FP(=)f(0)f(but)i FO(E)1181 3713 y FM(A)1235 3701 y FP(\(\000)1319 3671 y FN(\000)p FL(1)1408 3701 y FP(\()p FO(M)9 b FP(\)\))39 b(=)e(0)f(and)h FO(E)2045 3713 y FM(A)2099 3701 y FP(\(\000\()p FO(M)9 b FP(\)\))39 b(=)e(0,)118 3800 y(then)641 3900 y FO(B)t(E)769 3912 y FM(A)823 3900 y FP(\()p FO(M)9 b FP(\))24 b(=)e FO(E)1149 3912 y FM(A)1204 3900 y FP(\(\000)1288 3866 y FN(\000)p FL(1)1377 3900 y FP(\()p FO(M)9 b FP(\)\))p FO(B)t(E)1691 3912 y FM(A)1746 3900 y FP(\()p FO(M)g FP(\))23 b(=)g(0)118 4048 y(and)636 4147 y FO(B)703 4113 y FN(\003)741 4147 y FO(E)802 4159 y FM(A)856 4147 y FP(\()p FO(M)9 b FP(\))24 b(=)e FO(E)1182 4159 y FM(A)1237 4147 y FP(\(\000\()p FO(M)9 b FP(\)\))p FO(B)1574 4113 y FN(\003)1612 4147 y FO(E)1673 4159 y FM(A)1728 4147 y FP(\()p FO(M)g FP(\))23 b(=)g(0)p FO(;)p eop %%Page: 63 67 63 66 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(63)118 333 y(hence)30 b FO(B)i Fs(\026)27 b FO(E)569 345 y FM(A)624 333 y FP(\()p FO(M)9 b FP(\))p FO(H)34 b FP(=)28 b(0)i(and)g FO(B)1277 303 y FN(\003)1343 333 y Fs(\026)d FO(E)1466 345 y FM(A)1520 333 y FP(\()p FO(M)9 b FP(\))p FO(H)35 b FP(=)27 b(0.)45 b(Th)n(us,)30 b(the)h(repre-)118 432 y(sen)n(tation)c(\()p FO(A;)14 b(B)t(;)g(B)778 402 y FN(\003)817 432 y FP(\))28 b(is)f(not)h FQ(\003)p FP(-complete.)p 2514 432 4 57 v 2518 380 50 4 v 2518 432 V 2567 432 4 57 v 243 611 a(One)36 b(can)h(also)f(sho)n(w) h(that)g(if)h(the)f(sp)r(ectral)g(measure)f(of)h FO(A)h FP(is)f(quasi-)118 710 y(in)n(v)-5 b(arian)n(t,)43 b(then)e(there)g (exists)f(a)g(b)r(ounded)h(op)r(erator)e FO(B)45 b FP(suc)n(h)c(that)f (the)118 810 y(triple)c(\()p FO(A;)14 b(B)t(;)g(B)650 780 y FN(\003)688 810 y FP(\))36 b(is)g(a)f FQ(\003)p FP(-complete)f(represen)n(tation)g(of)42 b(\(1.10\))o(.)60 b(But)36 b(w)n(e)118 910 y(omit)28 b(the)g(pro)r(of)f(here.)118 1080 y FR(Prop)s(osition)40 b(23.)45 b FC(If)38 b(a)g(r)l(epr)l (esentation)g FP(\()p FO(A;)14 b(B)t(;)g(B)1838 1050 y FN(\003)1877 1080 y FP(\))37 b FC(of)56 b FP(\(1.10\))37 b FC(is)h(irr)l(e-)118 1180 y(ducible,)32 b(then)d(the)h(sp)l(e)l(ctr)l (al)g(me)l(asur)l(e)f(of)i FO(A)f FC(is)g(er)l(go)l(dic)h(with)g(r)l (esp)l(e)l(ct)e(to)h FP(\000)p FC(.)118 1351 y(Pr)l(o)l(of.)43 b FP(Let)22 b(\()p FO(A;)14 b(B)t(;)g(B)823 1321 y FN(\003)862 1351 y FP(\))21 b(b)r(e)h(an)f(irreducible)g(represen)n(tation)f(of)h (the)h(relation)118 1450 y(\(1.10\))o(.)35 b(If)22 b(the)g(sp)r(ectral) f(t)n(yp)r(e)g(of)g FO(A)h FP(is)f(not)h(ergo)r(dic,)f(then)h(there)g (exists)f FO(M)31 b FQ(\032)118 1550 y FJ(R)36 b FP(suc)n(h)30 b(that)g FO(E)641 1562 y FM(A)695 1550 y FP(\()p FO(M)9 b FP(\))28 b FQ(6)p FP(=)e(0,)k FO(E)1124 1562 y FM(A)1179 1550 y FP(\()p FO(M)9 b FP(\))27 b FQ(6)p FP(=)g FO(I)7 b FP(,)30 b(and)g(the)g(sets)g(\000)20 b FQ(\\)h FP(\()p FO(M)2292 1520 y FM(c)2345 1550 y FQ(\002)f FO(M)9 b FP(\),)118 1650 y(\000)23 b FQ(\\)h FP(\()p FO(M)33 b FQ(\002)23 b FO(M)596 1620 y FM(c)629 1650 y FP(\))36 b(are)e(marginally)f(n)n(ull)j(sets.)59 b(Hence,)37 b(\000)24 b FQ(\\)f FP(\()p FO(M)2207 1620 y FM(c)2265 1650 y FQ(\002)g FO(M)9 b FP(\))35 b FQ(\032)118 1749 y FP(\()p FO(M)231 1761 y FL(1)293 1749 y FQ(\002)24 b FJ(R)p FP(\))31 b FQ([)24 b FP(\()p FJ(R)31 b FQ(\002)24 b FO(M)865 1761 y FL(2)902 1749 y FP(\),)39 b(where)e FO(\026)p FP(\()p FO(M)1409 1761 y FL(1)1446 1749 y FP(\))h(=)g FO(\026)p FP(\()p FO(M)1782 1761 y FL(2)1819 1749 y FP(\))h(=)e(0.)64 b(F)-7 b(rom)36 b(this)h(it)118 1849 y(follo)n(ws)31 b(that)h FO(E)640 1861 y FM(A)694 1849 y FP(\()p FO(M)816 1819 y FM(c)850 1849 y FP(\))p FO(B)t(E)1010 1861 y FM(A)1065 1849 y FP(\()p FO(M)9 b FP(\))29 b(=)h FO(E)1404 1861 y FM(A)1458 1849 y FP(\()p FO(M)1580 1819 y FM(c)1635 1849 y FQ(n)21 b FO(M)1779 1861 y FL(1)1816 1849 y FP(\))p FO(B)t(E)1976 1861 y FM(A)2030 1849 y FP(\()p FO(M)30 b FQ(n)21 b FO(M)2317 1861 y FL(2)2354 1849 y FP(\))30 b(=)f(0,)118 1949 y(i.e.,)j(the)f(sp)r(ectral)f(subspace)g FO(E)1151 1961 y FM(A)1206 1949 y FP(\()p FO(M)9 b FP(\))p FO(H)38 b FP(is)30 b(in)n(v)-5 b(arian)n(t)30 b(with)h(resp)r(ect)g(to) f FO(A)p FP(,)118 2048 y FO(B)t FP(.)48 b(In)31 b(the)h(same)e(w)n(a)n (y)g(one)h(can)g(sho)n(w)f(that)i FO(E)1659 2060 y FM(A)1713 2048 y FP(\()p FO(M)9 b FP(\))p FO(H)38 b FP(is)31 b(in)n(v)-5 b(arian)n(t)30 b(with)118 2148 y(resp)r(ect)21 b(to)g FO(A)p FP(,)i FO(B)666 2118 y FN(\003)704 2148 y FP(.)35 b(This)21 b(con)n(tradicts)f(the)i(irreducibilit)n(y)e(of)h FO(A)p FP(,)i FO(B)t FP(,)g FO(B)2366 2118 y FN(\003)2404 2148 y FP(.)p 2514 2148 V 2518 2095 50 4 v 2518 2148 V 2567 2148 4 57 v 243 2326 a(Note)32 b(that)h(the)f(con)n(v)n(erse)f (statemen)n(t)h(is)g(also)f(true.)51 b(Namely)-7 b(,)34 b(for)e(an)n(y)118 2426 y(ergo)r(dic)g(measure)h FO(\026)g FP(there)g(exists)g(an)g(irreducible)g(represen)n(tation)e FO(A)p FP(,)36 b FO(B)t FP(,)118 2526 y FO(B)185 2496 y FN(\003)254 2526 y FP(of)30 b(relation)g(\(1.10\))f(suc)n(h)h(that)h FO(\026)f FP(is)g(a)g(sp)r(ectral)g(scalar)f(measure)g(of)h FO(A)p FP(.)118 2625 y(Ho)n(w)n(ev)n(er)c(w)n(e)h(will)h(not)f(discuss) h(the)f(pro)r(of)g(here.)118 2779 y FR(3.)36 b FP(Let)28 b(\()p FO(A;)14 b(B)t FP(\))29 b(satisfy)e(the)h(follo)n(wing)e (semilinear)h(relation:)1079 2965 y FO(AB)g FP(=)c FO(B)t(F)12 b FP(\()p FO(A)p FP(\))p FO(;)763 b FP(\(1.19\))118 3151 y(where)29 b FO(F)12 b FP(\()p FQ(\001)p FP(\))31 b(is)e(a)h(b)r (ounded)g(Borel)e(mapping)i(de\014ned)g(on)f FJ(R)p FP(.)50 b(It)30 b(is)f(easy)g(to)118 3251 y(sho)n(w)24 b(that)h(in)g(this)g (case)f(the)h(in)n(v)-5 b(ariance)24 b(of)g(a)h(set)f(\001)g FQ(2)f FA(B)p FP(\()p FJ(R)q FP(\))31 b(with)25 b(resp)r(ect)118 3350 y(to)h(\000)g(and)g(a)f(measure)g FO(\026)h FP(means)g(its)g(in)n (v)-5 b(ariance)25 b(with)h(resp)r(ect)g(to)g(the)g(map-)118 3450 y(ping)d FO(F)12 b FP(\()p FQ(\001)p FP(\),)25 b(i.e.,)f FO(\026)p FP(\()p FO(F)12 b FP(\(\001\)\))24 b(=)f FO(\026)p FP(\(\001\))g(and)g FO(\026)p FP(\()p FO(F)1585 3420 y FN(\000)p FL(1)1675 3450 y FP(\(\001\)\))h(=)e FO(\026)p FP(\(\001\).)37 b(A)23 b(measure)118 3550 y FO(\026)35 b FP(is)f(quasi-in)n(v)-5 b(arian)n(t)32 b(with)j(resp)r(ect)f(to)g (\000)g(if)h(the)g(measures)e FO(\026)p FP(\()p FO(F)12 b FP(\()p FQ(\001)p FP(\)\))35 b(and)118 3649 y FO(\026)p FP(\()p FO(F)265 3619 y FN(\000)p FL(1)354 3649 y FP(\()p FQ(\001)p FP(\)\))27 b(are)d(absolutely)g(con)n(tin)n(uous)g(with)i (resp)r(ect)f(to)g FO(\026)p FP(.)36 b(The)25 b(ergo)r(dic-)118 3749 y(it)n(y)f(of)g(the)g(sp)r(ectral)f(measure)g(with)h(resp)r(ect)g (to)g(\000)g(means)f(ergo)r(dicit)n(y)g(of)g(the)118 3848 y(measure)f FO(E)498 3860 y FM(A)552 3848 y FP(\()p FQ(\001)p FP(\))i(with)f(resp)r(ect)g(to)f FO(F)12 b FP(\()p FQ(\001)p FP(\))24 b(\(i.e.,)g(for)e(an)n(y)g FO(F)12 b FP(\()p FQ(\001)p FP(\)-in)n(v)-5 b(arian)n(t)22 b(Borel)118 3948 y(set)28 b(\001)d FQ(\032)e FJ(R)p FP(,)35 b(either)28 b FO(E)839 3960 y FM(A)893 3948 y FP(\(\001\))d(=)f(0)j(or) h FO(E)1373 3960 y FM(A)1427 3948 y FP(\(\001\))d(=)e FO(I)7 b FP(\).)40 b(Therefore,)27 b(b)n(y)h(Prop)r(o-)118 4048 y(sition)d(23,)g(w)n(e)g(ha)n(v)n(e)f(that,)i(if)g(\()p FO(A;)14 b(B)t(;)g(B)1360 4018 y FN(\003)1399 4048 y FP(\))26 b(is)f(an)g(irreducible)g(represen)n(tation)118 4147 y(of)34 b(\(1.19\))o(,)28 b(then)g FO(E)732 4159 y FM(A)786 4147 y FP(\()p FQ(\001)p FP(\))h(is)e(ergo)r(dic)g(with)h (resp)r(ect)f(to)h FO(F)12 b FP(\()p FQ(\001)p FP(\).)p eop %%Page: 64 68 64 67 bop 118 100 a FP(64)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FR(4.)34 b FP(In)20 b(the)h(analysis)e(of)h(the)g(represen)n (tations)e(of)27 b(\(1.19\))o(,)22 b(the)f(b)r(eha)n(vior)d(of)i(the) 118 432 y(dynamical)28 b(system)h FO(F)21 b FP(:)29 b FJ(R)i FQ(\000)-49 b(!)26 b FJ(R)35 b FP(pla)n(ys)28 b(a)g(cen)n(tral)g(role.)40 b(The)29 b(structure)g(of)118 532 y(represen)n(tations)19 b(of)26 b(\(1.19\))20 b(dep)r(ends)h(on)f (the)h(structure)f(of)g(the)h(orbits)f(of)g(the)118 632 y(corresp)r(onding)i(dynamical)g(system)i(\(see)f(Section)h(2.1.1)e (for)h(details\).)35 b(Here)118 731 y(w)n(e)j(giv)n(e)f(the)i(corresp)r (onding)d(results)h(for)h(general)f(semilinear)g(relations.)118 831 y(W)-7 b(e)28 b(b)r(egin)g(with)g(the)g(follo)n(wing)e (de\014nition.)118 999 y FR(De\014nition)k(7.)40 b FC(A)28 b(set)g FO(E)34 b FC(is)29 b(c)l(al)t(le)l(d)39 b FP(\000)p FC(-invariant)29 b(if)h FO(E)22 b FQ(\002)15 b FO(E)2065 969 y FM(c)2116 999 y FQ(\\)h FP(\000)23 b(=)g FJ(?)28 b FC(and)118 1099 y FO(E)184 1068 y FM(c)237 1099 y FQ(\002)19 b FO(E)24 b FQ(\\)19 b FP(\000)25 b(=)f FJ(?)p FC(.)40 b(A)30 b(minimal)i FP(\000)24 b Fs(\026)1307 1111 y FM(M)1381 1099 y FC(-invariant)31 b(set)f FO(O)1964 1111 y FM(M)2038 1099 y FP(\()p FO(E)5 b FP(\))31 b FC(c)l(ontaining)118 1198 y(the)23 b(set)f FO(E)27 b FC(is)c(said)g(to)g(b)l(e)f(a)h(tr)l (aje)l(ctory)30 b FP(\()p FC(semi-tr)l(aje)l(ctory)7 b FP(\))25 b FC(of)e(the)f(set)g FO(E)29 b FQ(\032)22 b FO(M)118 1298 y FC(with)30 b(r)l(esp)l(e)l(ct)g(to)g FP(\000)23 b Fs(\026)779 1310 y FM(M)852 1298 y FC(.)243 1466 y FP(The)32 b(concept)g(of)g(a)g(tra)5 b(jectory)31 b(generalizes)g(that)h(of)g(an)g(orbit)g(for)g(dy-)118 1566 y(namical)25 b(systems.)35 b(The)25 b(follo)n(wing)g(result)f(is)h (an)g(analogue)f(of)h(the)g(theorem)118 1665 y(on)31 b(the)h(connectedness)e(of)i(the)f(graph)f(supp)r(orting)h(an)g (irreducible)g(\014nite-)118 1765 y(dimensional)c(represen)n(tation.) 118 1933 y FR(Theorem)k(12.)40 b FC(L)l(et)29 b FO(M)39 b FC(b)l(e)29 b(a)i(c)l(omp)l(act)f(set.)243 2033 y FP(\()p FC(a)6 b FP(\))32 b FC(If)f(ther)l(e)g(is)h FO(x)25 b FQ(2)h FO(M)39 b FC(such)31 b(that)g(the)g(tr)l(aje)l(ctory)h FO(O)1985 2045 y FM(M)2059 2033 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))f FC(is)g(dense)118 2133 y(in)f FO(M)9 b FC(,)31 b(then)e FO(M)39 b FC(is)31 b(the)f(sp)l(e)l(ctrum)f(of)i(an)f (irr)l(e)l(ducible)i(r)l(epr)l(esentation,)f(i.e.,)118 2233 y(ther)l(e)22 b(exists)f(an)g(irr)l(e)l(ducible)i(r)l(epr)l (esentation)f FP(\()p FO(A;)14 b(B)t(;)g(B)1876 2203 y FN(\003)1914 2233 y FP(\))22 b FC(such)g(that)f FO(\033)s FP(\()p FO(A)p FP(\))j(=)118 2332 y FO(M)9 b FC(.)243 2433 y FP(\()p FC(b)c FP(\))32 b FC(If)f(the)g(set)g FO(M)39 b FC(is)32 b(the)f(sp)l(e)l(ctrum)f(of)i(an)f(irr)l(e)l (ducible)h(r)l(epr)l(esentation)118 2532 y(of)56 b FP(\(1.10\))o FC(,)40 b(then)d(the)h(tr)l(aje)l(ctory)g FO(O)1300 2544 y FM(M)1374 2532 y FP(\()p FO(G)25 b FQ(\\)f FO(M)9 b FP(\))38 b FC(is)f(dense)h(in)g FO(M)46 b FC(for)38 b(any)118 2632 y(op)l(en)30 b(set)g FO(G)p FC(,)g(wher)l(e)h FO(G)18 b FQ(\\)h FO(M)32 b FQ(6)p FP(=)22 b FJ(?)p FC(.)118 2800 y(Pr)l(o)l(of.)43 b FP(Let)i FO(O)607 2812 y FM(M)681 2800 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))g(b)r(e)g(dense)f(in)h FO(M)9 b FP(.)88 b(Then)44 b(there)h(is)f(a)g(sequence)118 2900 y FQ(f)p FO(\025)208 2912 y FM(n)253 2900 y FQ(g)295 2870 y FN(1)295 2920 y FM(n)p FL(=1)462 2900 y FQ(\032)37 b FO(O)627 2912 y FM(M)701 2900 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))f(whic)n(h)g(is)h(dense)f(in)g FO(M)9 b FP(.)63 b(No)n(w)36 b(let)h FO(\026)f FP(b)r(e)h(a)f(mea-)118 2999 y(sure)25 b(concen)n(trated)g(on)g FQ(f)p FO(\025)985 3011 y FM(n)1030 2999 y FQ(g)1072 2969 y FN(1)1072 3020 y FM(n)p FL(=1)1201 2999 y FP(,)h(and)f FO(\026)p FP(\()p FO(\025)1539 3011 y FM(n)1585 2999 y FP(\))f FQ(6)p FP(=)e(0)k(for)f (an)n(y)g FO(\025)2124 3011 y FM(n)2169 2999 y FP(.)36 b(Then)26 b FO(\026)g FP(is)118 3099 y(ergo)r(dic.)36 b(In)27 b(fact,)h(let)f FO(S)32 b FP(b)r(e)c(in)n(v)-5 b(arian)n(t)26 b(with)i(resp)r(ect)f(to)g(\000,)g(and)g FO(\026)p FP(\()p FO(S)5 b FP(\))24 b FQ(6)p FP(=)e(0,)118 3199 y FO(\026)p FP(\()p FO(S)256 3169 y FM(c)290 3199 y FP(\))45 b FQ(6)p FP(=)f(0.)76 b(Then)40 b FO(S)32 b FQ(\\)c(f)p FO(\025)1102 3211 y FM(n)1147 3199 y FQ(g)1189 3169 y FN(1)1189 3219 y FM(n)p FL(=1)1362 3199 y FQ(6)p FP(=)45 b FJ(?)p FP(.)75 b(F)-7 b(rom)40 b(this)h(it)g(follo)n(ws)f (that)118 3298 y FO(S)25 b FQ(\\)20 b FO(O)332 3310 y FM(M)406 3298 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))27 b FQ(6)p FP(=)g FJ(?)p FP(.)43 b(If)31 b FO(x)c FQ(2)h FO(S)5 b FP(,)30 b(then)h FO(O)1458 3310 y FM(M)1532 3298 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))c FQ(\032)f FO(S)5 b FP(,)31 b(hence)f FO(\026)p FP(\()p FO(S)2326 3268 y FM(c)2359 3298 y FP(\))e(=)e(0,)118 3398 y(whic)n(h)20 b(con)n(tradicts)e(the)i(assumption.)33 b(Assume)20 b(that)g FO(x)j FQ(62)h FO(S)5 b FP(,)21 b(hence)e FO(x)24 b FQ(2)f FO(S)2518 3368 y FM(c)2552 3398 y FP(.)118 3498 y(Since)h FO(S)k FP(is)23 b(in)n(v)-5 b(arian)n(t)23 b(with)h(resp)r(ect)f(to)g (\000,)i(so)e(is)g(the)h(complemen)n(t)f(of)h FO(S)k FP(and)118 3597 y FO(O)181 3609 y FM(M)255 3597 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))9 b FQ(\\)g FO(S)579 3567 y FM(c)637 3597 y FQ(6)p FP(=)22 b FJ(?)p FP(.)35 b(F)-7 b(rom)23 b(the)g(condition)g FO(x)h FQ(2)f FO(O)1771 3609 y FM(M)1845 3597 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))9 b FQ(\\)g FO(S)2169 3567 y FM(c)2227 3597 y FP(w)n(e)22 b(obtain)118 3697 y FO(O)181 3709 y FM(M)255 3697 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))h FQ(\\)f FO(S)606 3667 y FM(c)673 3697 y FP(=)32 b FO(O)833 3709 y FM(M)907 3697 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\).)55 b(Hence)33 b FO(O)1495 3709 y FM(M)1569 3697 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))g FQ(\032)g FO(S)1951 3667 y FM(c)1984 3697 y FP(,)i(and)f FO(\026)p FP(\()p FO(S)5 b FP(\))33 b(=)f(0,)118 3796 y(whic)n(h)c(giv)n(es)e(a)h(con)n(tradiction.)118 3948 y FR(5.)36 b FP(Let)26 b FO(G)h FP(b)r(e)g(an)f(op)r(en)h(set,)f FO(G)17 b FQ(\\)f FO(M)32 b FQ(6)p FP(=)22 b FJ(?)p FP(,)p 1513 3876 449 4 v 27 w FO(O)1576 3960 y FM(M)1650 3948 y FP(\()p FO(G)d FQ(\\)g FO(M)9 b FP(\))23 b FQ(6)p FP(=)f FO(M)9 b FP(,)27 b(and)f FO(E)2433 3960 y FM(A)2487 3948 y FP(\()p FQ(\001)p FP(\))118 4048 y(b)r(e)j(the)g(sp)r(ectral)f (measure)f(of)i(the)g(op)r(erator)e FO(A)i FP(with)g(sp)r(ectrum)f FO(M)9 b FP(.)40 b(Then)118 4147 y FO(E)179 4159 y FM(A)233 4147 y FP(\()p FO(G)p FP(\))25 b FQ(6)p FP(=)e(0)k(and)h(there)f (exists)h(a)f(compact)g(set)h FO(F)36 b FQ(\032)23 b FO(G)18 b FQ(\\)h FO(M)9 b FP(,)28 b FO(E)2215 4159 y FM(A)2269 4147 y FP(\()p FO(F)12 b FP(\))24 b FQ(6)p FP(=)f(0.)p eop %%Page: 65 69 65 68 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(65)118 333 y(The)24 b(set)g FO(O)474 345 y FM(M)548 333 y FP(\()p FO(F)12 b FP(\))24 b(is)g(a)g(Borel)e (set.)36 b(Since)p 1437 260 267 4 v 24 w FO(O)1500 345 y FM(M)1574 333 y FP(\()p FO(F)12 b FP(\))24 b FQ(6)p FP(=)e FO(M)9 b FP(,)25 b FO(E)2013 345 y FM(A)2067 333 y FP(\()p FO(O)2162 345 y FM(M)2236 333 y FP(\()p FO(F)12 b FP(\)\))24 b FQ(6)p FP(=)f FO(I)7 b FP(.)118 432 y(Th)n(us,)28 b FO(E)413 444 y FM(A)468 432 y FP(\()p FQ(\001)p FP(\))g(is)g(not)g (ergo)r(dic)g(with)g(resp)r(ect)g(to)g(\000)c Fs(\026)1792 444 y FM(M)1865 432 y FP(,)29 b(b)r(ecause)e FO(O)2287 444 y FM(M)2361 432 y FP(\()p FO(F)12 b FP(\))29 b(is)118 532 y(an)e(in)n(v)-5 b(arian)n(t)27 b(set.)p 2514 532 4 57 v 2518 479 50 4 v 2518 532 V 2567 532 4 57 v 118 798 a FR(De\014nition)35 b(8.)42 b FC(A)33 b(set)f FO(\034)43 b FC(is)33 b(said)h(to)f(b)l(e)f(a)i(me)l(asur)l(able)f(se)l(ction)g (of)g FP(\()p FO(D)r(;)14 b FP(\000\))118 897 y FC(if)29 b FO(\034)k FQ(\032)23 b FO(D)30 b FC(is)f(a)g(Bor)l(el)g(set)f(and)h (every)h(tr)l(aje)l(ctory)f FO(O)1773 909 y FM(D)1833 897 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))g FC(with)g(r)l(esp)l(e)l(ct)f (to)118 997 y FP(\000)23 b Fs(\026)228 1009 y FM(D)318 997 y FC(interse)l(cts)29 b FO(\034)39 b FC(at)30 b(exactly)g(one)g(p)l (oint.)118 1200 y FR(Prop)s(osition)k(24.)43 b FC(L)l(et)33 b FP(\()p FO(D)r(;)14 b FP(\000\))33 b FC(have)h(a)g(me)l(asur)l(able)f (se)l(ction.)49 b(Then,)35 b(for)118 1300 y(any)30 b(irr)l(e)l(ducible) h(r)l(epr)l(esentation)e(of)48 b FP(\(1.10\))29 b FC(with)h FO(\033)s FP(\()p FO(A)p FP(\))24 b FQ(\032)f FO(D)r FC(,)30 b(ther)l(e)f(exists)118 1400 y(a)h(unique)g(tr)l(aje)l(ctory)g FO(O)894 1412 y FM(D)955 1400 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))f FC(for)i(which)g FO(E)1606 1412 y FM(A)1661 1400 y FP(\()p FO(O)1756 1412 y FM(D)1816 1400 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\)\))24 b(=)e FO(I)7 b FC(.)118 1603 y(Pr)l(o)l(of.)43 b FP(Let)34 b FO(\034)44 b FP(b)r(e)34 b(a)f(measurable)g(section)g(of)h(\()p FO(D)r(;)14 b FP(\000\).)56 b(Assume)34 b(that)g FO(\026)g FP(is)118 1703 y(not)28 b(concen)n(trated)f(on)g(an)n(y)g(orbit)h FO(O)1297 1715 y FM(D)1357 1703 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\).)38 b(W)-7 b(e)28 b(pro)n(v)n(e)e(that)j(there)e(exists) 118 1802 y(a)34 b(partition)g(of)g FO(\034)45 b FP(in)n(to)34 b(t)n(w)n(o)f(sets)h FO(\034)44 b FP(=)34 b FO(\034)1451 1814 y FL(1)1512 1802 y FQ([)23 b FO(\034)1626 1814 y FL(2)1664 1802 y FP(,)36 b FO(\034)1759 1814 y FL(1)1820 1802 y FQ(\\)23 b FO(\034)1934 1814 y FL(2)2006 1802 y FP(=)34 b FJ(?)p FP(,)i(suc)n(h)e(that)118 1902 y FO(\026)p FP(\()p FO(O)263 1914 y FM(D)324 1902 y FP(\()p FO(\034)392 1914 y FL(1)430 1902 y FP(\)\))45 b FO(>)f FP(0,)f FO(\026)p FP(\()p FO(O)901 1914 y FM(D)962 1902 y FP(\()p FO(\034)1030 1914 y FL(2)1067 1902 y FP(\)\))i FO(>)f FP(0.)75 b(Consider)39 b(the)i(tra)5 b(jectories)39 b FO(T)2439 1914 y FM(i)2510 1902 y FP(=)118 2002 y FO(O)183 1971 y FM(i)181 2024 y(D)242 2002 y FP(\()p FO(\034)310 2014 y FM(i)338 2002 y FP(\),)c FO(i)d FP(=)f(1,)k(2,)f(for)e(an)n(y)h(t)n(w)n(o)f(sets)h FO(\034)1445 2014 y FM(i)1506 2002 y FP(satisfying)f(the)i(conditions)e FO(\034)42 b FP(=)118 2101 y FO(\034)154 2113 y FL(1)209 2101 y FQ([)18 b FO(\034)318 2113 y FL(2)356 2101 y FP(,)27 b FO(\034)442 2113 y FL(1)497 2101 y FQ(\\)18 b FO(\034)606 2113 y FL(2)667 2101 y FP(=)k FJ(?)p FP(.)37 b(By)26 b(the)i(de\014nition)g(of)f(a)f(measurable)g(section,)h FO(T)2465 2113 y FL(1)2519 2101 y FQ(\\)118 2201 y FO(T)167 2213 y FL(2)227 2201 y FP(=)c FJ(?)f FP(and)h FO(T)608 2213 y FL(1)655 2201 y FQ([)9 b FO(T)768 2213 y FL(2)828 2201 y FP(=)23 b FO(D)r FP(.)35 b(Assuming)23 b(that)h(for)e(an)n(y)h (suc)n(h)f(decomp)r(osition)118 2300 y FO(D)28 b FP(=)e FO(T)355 2312 y FL(1)412 2300 y FQ([)20 b FO(T)536 2312 y FL(2)602 2300 y FP(one)29 b(of)h(the)f(v)-5 b(alues)29 b FO(\026)p FP(\()p FO(T)1377 2312 y FL(1)1415 2300 y FP(\))g(and)h FO(\026)p FP(\()p FO(T)1771 2312 y FL(2)1808 2300 y FP(\))g(is)f(equal)g(to)g(zero,)g(w)n(e)118 2400 y(can)23 b(\014nd)h(a)f(decreasing)e(sequence)i FQ(f)p FO(T)1333 2370 y FM(k)1372 2400 y FQ(g)g FP(suc)n(h)g(that)h FO(E)1857 2412 y FM(A)1911 2400 y FP(\()p FO(T)2004 2370 y FM(k)2044 2400 y FP(\))g(=)e FO(I)31 b FP(for)22 b(an)n(y)h FO(k)118 2500 y FP(and)280 2437 y Fz(T)363 2500 y FO(T)424 2470 y FM(k)488 2500 y FP(=)h FO(O)640 2512 y FM(D)700 2500 y FP(\()p FQ(f)p FO(x)p FQ(g)p FP(\))29 b(for)f(some)f FO(x)e FQ(2)f FO(\034)9 b FP(.)40 b(F)-7 b(rom)28 b(the)g(latter)g (argumen)n(t)f(w)n(e)118 2599 y(can)33 b(conclude)h(that)f FO(\026)h FP(is)g(concen)n(trated)e(on)h(an)g(orbit)h(whic)n(h)f(con)n (tradicts)118 2699 y(the)28 b(assumption.)243 2811 y(Let)20 b FO(\034)33 b FP(=)22 b FO(\034)576 2823 y FL(1)617 2811 y FQ([)s FO(\034)711 2823 y FL(2)769 2811 y FP(b)r(e)f(the)f (required)f(decomp)r(osition,)i(i.e.)35 b FO(\026)p FP(\()p FO(O)2178 2823 y FM(D)2239 2811 y FP(\()p FO(\034)2307 2823 y FM(i)2335 2811 y FP(\)\))24 b FO(>)e FP(0,)118 2911 y FO(i)28 b FP(=)g(1)p FO(;)14 b FP(2.)46 b(Since)31 b(b)r(oth)g(sets)f FO(O)1105 2923 y FM(D)1166 2911 y FP(\()p FO(\034)1234 2923 y FL(1)1272 2911 y FP(\))h(and)f FO(O)1562 2923 y FM(D)1623 2911 y FP(\()p FO(\034)1691 2923 y FL(2)1729 2911 y FP(\))h(are)f(in)n(v)-5 b(arian)n(t)29 b(with)j(re-)118 3010 y(sp)r(ect)24 b(to)f(\000)p FQ(j)501 3022 y FM(D)562 3010 y FP(,)h(the)g(existence)f(of)h(the)g(decomp)r (osition)f(implies)g(a)h(con)n(tradic-)118 3110 y(tion)29 b(to)f(ergo)r(dicit)n(y)f(of)h(the)h(measure)f FO(\026)p FP(.)39 b(Th)n(us,)28 b FO(\026)h FP(is)f(concen)n(trated)f(on)h(the) 118 3209 y(tra)5 b(jectory)26 b(of)i(some)f(p)r(oin)n(t)g FO(x)p FP(.)p 2514 3209 V 2518 3157 50 4 v 2518 3209 V 2567 3209 4 57 v 243 3475 a(If)40 b(there)f(is)g(no)g(measurable)f (section)h(for)g(the)h(graph)f(\()p FO(D)r FP(,)j(\000\))e(of)f(the)118 3575 y(semilinear)24 b(relation)g(\(1.10\))o(,)h(then)g(the)g (structure)f(of)h(represen)n(tations)d(with)118 3674 y(b)r(ounded)f(op)r(erators)d(\()p FO(A;)c(B)t(;)g(B)1112 3644 y FN(\003)1151 3674 y FP(\))21 b(is)f(more)g(complicated:)33 b(there)20 b(migh)n(t)g(exist)118 3774 y(irreducible)j(represen)n (tations)e(suc)n(h)i(that)h(the)g(sp)r(ectrum)g(of)f(the)h(op)r(erator) e FO(A)118 3874 y FP(is)28 b(not)f(discrete.)118 4031 y FC(R)l(emark)33 b FP(18)p FC(.)j FP(The)22 b(same)f(theorems)g(are)f (v)-5 b(alid)22 b(for)f(the)h(represen)n(tation)e(with)118 4131 y FO(B)27 b FP(=)c FO(B)363 4101 y FN(\003)401 4131 y FP(,)28 b(but)g(with)g(\000)845 4143 y FM(s)908 4131 y FP(instead)g(of)f(\000.)p eop %%Page: 66 70 66 69 bop 118 100 a FP(66)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FR(1.3.5)94 b(Represen)m(tations)31 b(of)g(semilinear)e FO(F)1727 345 y FL(4)1765 333 y FR(-relations)118 486 y FP(The)e(problem)g(of)g(describing)g(all)g(irreducible)f(represen)n (tations)g(of)h(semilin-)118 586 y(ear)h(relations)g(up)h(to)g(unitary) f(equiv)-5 b(alence)28 b(migh)n(t)h(b)r(e)h(v)n(ery)d(di\016cult.)42 b(The)118 685 y(complexit)n(y)26 b(of)h(the)g(description)f(dep)r(ends) h(on)g(the)g(structure)f(of)h(the)g(corre-)118 785 y(sp)r(onding)g (graph.)118 951 y FR(Theorem)37 b(13.)44 b FP(1)p FC(.)55 b(If)35 b(al)t(l)h(c)l(onne)l(cte)l(d)f(c)l(omp)l(onents)g(of)h(the)f (gr)l(aph)i FP(\000)e FC(c)l(or-)118 1051 y(r)l(esp)l(onding)c(to)f(a)g (semiline)l(ar)h(r)l(elation)f(ar)l(e)h(of)f(the)g(form:)2060 1033 y Fo(r)2102 1051 y FC(,)2199 1033 y Fo(r)p 2199 1035 117 4 v 8 w Fu(-)8 b Fo(r)2365 1051 y FC(,)30 b(then)118 1162 y(any)37 b(irr)l(e)l(ducible)g(r)l(epr)l(esentation)g FP(\()p FO(A;)14 b(B)t(;)g(B)1543 1131 y FN(\003)1581 1162 y FP(\))37 b FC(of)g(the)f(r)l(elation)h(is)f(one-)g(or)118 1261 y(two-dimensional)9 b FP(:)186 1428 y(\()p FC(i)f FP(\))43 b FO(A)23 b FP(=)g FO(\025)p FC(,)30 b FO(B)d FP(=)c(0)p FC(,)30 b FO(\025)23 b FQ(2)h FJ(R)p FP(;)160 1594 y(\()p FC(ii)8 b FP(\))44 b FO(A)34 b FP(=)521 1527 y Fz(\000)573 1559 y FM(\025)612 1567 y Fy(1)688 1559 y FL(0)593 1620 y(0)43 b FM(\025)708 1628 y Fy(2)754 1527 y Fz(\001)792 1594 y FC(,)38 b FO(B)h FP(=)1056 1527 y Fz(\000)1108 1565 y FL(0)23 b(0)1110 1618 y FM(b)i FL(0)1211 1527 y Fz(\001)1249 1594 y FC(,)38 b(wher)l(e)f FP(\()p FO(\025)1633 1606 y FL(1)1671 1594 y FO(;)14 b(\025)1756 1606 y FL(2)1794 1594 y FO(;)g(b)p FP(\))35 b FC(b)l(elongs)i(to)f(the)g(set)326 1707 y FO(K)397 1719 y FL(1)456 1707 y FP(=)544 1640 y Fz(\010)593 1707 y FP(\()p FO(\025)673 1719 y FL(1)711 1707 y FO(;)14 b(\025)796 1719 y FL(2)833 1707 y FO(;)g(b)p FP(\))23 b FQ(2)g FJ(R)1093 1677 y FL(3)1160 1707 y FQ(j)g FP(\010\()p FO(\025)1346 1719 y FL(1)1384 1707 y FO(;)14 b(\025)1469 1719 y FL(2)1506 1707 y FP(\))24 b(=)e(0;)27 b FO(\025)1789 1719 y FL(1)1850 1707 y FQ(6)p FP(=)c FO(\025)1986 1719 y FL(2)2024 1707 y FO(;)k(b)c(>)g FP(0)2263 1640 y Fz(\011)2311 1707 y FC(.)243 1873 y FP(2)p FC(.)53 b(If)36 b(al)t(l)f(c)l(onne)l (cte)l(d)g(c)l(omp)l(onents)g(of)h(the)f(gr)l(aph)h FP(\000)1942 1885 y FM(s)2012 1873 y FC(ar)l(e)f(of)h(the)f(form)160 1989 y Fo(r)201 2007 y FC(,)306 1989 y Fo(r)p 306 1991 4 4 v 302 1986 V 297 1981 V 293 1976 V 289 1972 V 286 1967 V 283 1963 V 281 1959 V 279 1955 V 277 1951 V 275 1947 V 274 1943 V 274 1940 V 273 1936 V 273 1933 V 274 1930 V 275 1927 V 276 1924 V 277 1921 V 279 1918 V 282 1916 V 282 1916 V 284 1913 V 286 1911 V 289 1909 V 291 1908 V 294 1907 V 296 1905 V 299 1905 V 301 1904 V 304 1904 V 306 1903 V 309 1904 V 311 1904 V 314 1905 V 316 1905 V 319 1907 V 321 1908 V 324 1909 V 326 1911 V 329 1913 V 331 1916 V 306 1991 V 311 1986 V 316 1981 V 320 1976 V 323 1972 V 327 1967 V 330 1963 V 332 1959 V 334 1955 V 336 1951 V 338 1947 V 339 1943 V 339 1940 V 340 1936 V 340 1933 V 339 1930 V 338 1927 V 337 1924 V 336 1921 V 334 1918 V 331 1916 V 348 2007 a FC(,)453 1989 y Fo(r)p 453 1991 125 4 v 100 w(r)619 2007 y FC(,)k(then)e(any)g(irr)l (e)l(ducible)h(r)l(epr)l(esentation)e FP(\()p FO(A;)14 b(B)41 b FP(=)35 b FO(B)2400 1977 y FN(\003)2438 2007 y FP(\))i FC(of)118 2106 y(the)30 b(r)l(elation)g(is)h(one-)e(or)h (two-dimensional)9 b FP(:)186 2273 y(\()p FC(i)f FP(\))43 b FO(A)23 b FP(=)g FO(\025)p FC(,)30 b FO(B)d FP(=)c(0)p FC(,)30 b FO(\025)23 b FQ(2)h FJ(R)p FP(;)160 2439 y(\()p FC(ii)8 b FP(\))44 b FO(A)30 b FP(=)f FO(\025)p FC(,)35 b FO(B)f FP(=)c FO(b)p FC(,)k(wher)l(e)g FP(\()p FO(\025;)14 b(b)p FP(\))34 b FC(b)l(elongs)g(to)g(the)f(set)h FO(K)2102 2451 y FL(2)2168 2439 y FP(=)c FQ(f)p FP(\()p FO(\025;)14 b(b)p FP(\))29 b FQ(2)326 2539 y FJ(R)380 2509 y FL(2)446 2539 y FQ(j)23 b FP(\010\()p FO(\025;)14 b(\025)p FP(\))24 b(=)f(0)p FQ(g)p FP(;)135 2705 y(\()p FC(iii)8 b FP(\))44 b FO(A)g FP(=)540 2638 y Fz(\000)592 2670 y FM(\025)631 2678 y Fy(1)707 2670 y FL(0)611 2731 y(0)f FM(\025)726 2739 y Fy(2)773 2638 y Fz(\001)811 2705 y FC(,)i FO(B)j FP(=)1057 2638 y Fz(\000)1108 2678 y FL(0)26 b FM(b)1110 2731 y(b)g FL(0)1212 2638 y Fz(\001)1250 2705 y FC(,)44 b(wher)l(e)e FP(\()p FO(\025)1645 2717 y FL(1)1683 2705 y FO(;)14 b(\025)1768 2717 y FL(2)1806 2705 y FO(;)g(b)p FP(\))41 b FC(b)l(elongs)g(to)h FO(K)2430 2717 y FL(3)2510 2705 y FP(=)326 2751 y Fz(\010)374 2818 y FP(\()p FO(\025)454 2830 y FL(1)492 2818 y FO(;)14 b(\025)577 2830 y FL(2)615 2818 y FO(;)g(b)p FP(\))23 b FQ(2)g FJ(R)875 2788 y FL(3)941 2818 y FQ(j)g FP(\010\()p FO(\025)1127 2830 y FL(1)1165 2818 y FO(;)14 b(\025)1250 2830 y FL(2)1288 2818 y FP(\))23 b(=)g(\010\()p FO(\025)1571 2830 y FL(2)1609 2818 y FO(;)14 b(\025)1694 2830 y FL(1)1731 2818 y FP(\))24 b(=)e(0;)28 b FO(\025)2015 2830 y FL(1)2075 2818 y FQ(6)p FP(=)23 b FO(\025)2211 2830 y FL(2)2249 2818 y FO(;)k(b)c(>)g FP(0)2488 2751 y Fz(\011)2536 2818 y FC(.)243 2984 y FP(Note)40 b(that)i(if)f FO(f)780 2996 y FM(i)807 2984 y FP(,)j FO(g)914 2996 y FM(i)982 2984 y FP(are)c(p)r(olynomials,)k (then)d(the)g(algebra)e FJ(C)15 b FQ(h)p FO(a;)f(b)51 b FQ(j)118 3022 y Fz(P)206 3042 y FM(n)206 3109 y(i)p FL(=1)331 3084 y FO(f)372 3096 y FM(i)400 3084 y FP(\()p FO(a)p FP(\))14 b FO(b)g(g)612 3096 y FM(i)639 3084 y FP(\()p FO(a)p FP(\))23 b(=)g(0)p FQ(i)k FP(is)h(an)f FO(F)1211 3096 y FL(4)1249 3084 y FP(-algebra.)118 3251 y FC(Pr)l(o)l(of.)43 b FP(Under)37 b(the)g(ab)r(o)n(v)n(e)e(condition)i (w)n(e)f(ha)n(v)n(e)g(that)h(\000)g(is)f(the)h(graph)f(of)118 3350 y(a)g(bijectiv)n(e)g(mapping)g FO(\036)9 b FP(:)31 b FO(N)1063 3362 y FL(1)1137 3350 y FQ(!)37 b FO(N)1324 3362 y FL(2)1361 3350 y FP(,)h(where)e FO(N)1738 3362 y FL(1)1799 3350 y FQ(\\)24 b FO(N)1945 3362 y FL(2)2019 3350 y FP(=)37 b FJ(?)f FP(\(i.e.,)i(\000)f(=)118 3450 y FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))35 b FQ(2)h FO(N)523 3462 y FL(2)583 3450 y FQ(\002)23 b FO(N)738 3462 y FL(1)810 3450 y FQ(j)35 b FO(t)g FP(=)g FO(\036)p FP(\()p FO(s)p FP(\))p FQ(g)p FP(\),)j(and)c(\000)1540 3462 y FM(s)1611 3450 y FP(is)g(the)i(graph)d(of)i(a)g(bijectiv)n(e)118 3550 y(mapping)30 b FO(\036)9 b FP(:)29 b FO(N)37 b FQ(!)27 b FO(N)39 b FP(\(i.e.,)32 b(\000)27 b(=)h FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))27 b FQ(2)h FO(N)1655 3519 y FL(2)1719 3550 y FQ(j)g FO(t)g FP(=)f FO(\036)p FP(\()p FO(s)p FP(\))p FQ(g)p FP(\).)46 b(Moreo)n(v)n(er,)118 3649 y FO(\036)32 b FP(is)f(measurable)f(under)h(the)g(assumption)g(that)g (\000\()p FO(M)9 b FP(\))31 b(and)g(\000)2189 3619 y FN(\000)p FL(1)2278 3649 y FP(\()p FO(M)9 b FP(\))32 b(are)118 3749 y(Borel)f(sets)g(for)g(an)n(y)g(Borel)f FO(M)9 b FP(.)49 b(Therefore,)31 b(b)n(y)g(Corollary)f(8,)i(if)g(\()p FO(A;)14 b(B)t FP(\))30 b FQ(2)118 3848 y FO(L)p FP(\()p FO(H)7 b FP(\))22 b(is)g(a)g(represen)n(tation)e(of)i(a)g(relation)f (with)h(suc)n(h)g(a)g(graph,)g(then)h(\()p FO(A)p FP(,)g FO(B)t FP(\))118 3948 y(satisfy)28 b(the)g(relation)f FO(AB)h FP(=)23 b FO(B)t(\036)p FP(\()p FO(A)p FP(\),)29 b(and)f(the)g(sp)r(ectrum)h(of)e(the)i(op)r(erator)118 4048 y FO(A)g FP(b)r(elongs)e(to)h FO(N)676 4060 y FL(1)732 4048 y FQ([)19 b FO(N)873 4060 y FL(2)938 4048 y FP(in)28 b(the)h(\014rst)f(case,)f(and)h(to)g FO(N)37 b FP(in)28 b(the)h(second)e(one.)118 4147 y(Clearly)-7 b(,)27 b(the)h(op)r (erators)d FO(A)p FP(,)j FO(B)t(B)1180 4117 y FN(\003)1246 4147 y FP(comm)n(ute.)p eop %%Page: 67 71 67 70 bop 118 100 a FK(1.3.)36 b(Lie)28 b(algebras)d(and)j(semilinear)e (relations)880 b FP(67)243 333 y(Let)33 b(us)g(consider)f(the)h (\014rst)g(case.)52 b(W)-7 b(e)33 b(ha)n(v)n(e)f FO(B)1799 303 y FL(2)1868 333 y FP(=)g(0)h(and,)h(therefore,)118 432 y(k)n(er)13 b FO(B)39 b FQ(6)p FP(=)c(0.)57 b(Denote)35 b(the)g(subspace)f(k)n(er)13 b FO(B)28 b FQ(\\)23 b FP(k)n(er)13 b FO(B)1849 402 y FN(\003)1922 432 y FP(b)n(y)34 b FO(H)2113 444 y FL(1)2151 432 y FP(.)58 b(It)35 b(is)g(easy)118 532 y(to)42 b(sho)n(w)e(that)i FO(H)716 544 y FL(1)795 532 y FP(is)g(in)n(v)-5 b(arian)n(t)40 b(with)i(resp)r(ect)g(to)f FO(A)p FP(,)46 b FO(B)t FP(,)f FO(B)2205 502 y FN(\003)2243 532 y FP(,)h(and)41 b(all)118 632 y(irreducible)35 b(represen)n (tations)e(de\014ned)i(on)g FO(H)1605 644 y FL(1)1678 632 y FP(are)f(one-dimensional)g(and)118 731 y(giv)n(en)39 b(b)n(y)h(\()p FO(i)p FP(\).)75 b(No)n(w)39 b(assume)h(that)g(k)n(er)13 b FO(B)31 b FQ(\\)c FP(k)n(er)13 b FO(B)1852 701 y FN(\003)1934 731 y FP(=)44 b FJ(?)p FP(.)74 b(Set)40 b FO(H)2429 743 y FL(0)2510 731 y FP(=)118 831 y(k)n(er)13 b FO(B)27 b FQ(\\)d FP(\(k)n(er)13 b FO(B)636 801 y FN(\003)675 831 y FP(\))707 801 y FN(?)763 831 y FP(.)59 b(Since)35 b FO(H)1138 843 y FL(0)1211 831 y FQ(\032)g FO(E)1372 843 y FM(A)1427 831 y FP(\()p FO(\036)p FP(\()p FO(N)1607 843 y FL(1)1645 831 y FP(\)\))p FO(H)7 b FP(,)37 b FO(B)1912 801 y FN(\003)1950 831 y FO(H)2019 843 y FL(0)2092 831 y FQ(\032)e FO(E)2253 843 y FM(A)2307 831 y FP(\()p FO(N)2406 843 y FL(1)2444 831 y FP(\))p FO(H)7 b FP(,)118 930 y(and)22 b FO(N)341 942 y FL(1)384 930 y FQ(\\)7 b FO(\036)p FP(\()p FO(N)594 942 y FL(1)632 930 y FP(\))23 b(=)g FJ(?)p FP(,)f(the)g (subspaces)f FO(H)1465 942 y FL(0)1502 930 y FP(,)i FO(B)1615 900 y FN(\003)1654 930 y FO(H)1723 942 y FL(0)1782 930 y FP(are)d(orthogonal.)33 b(More-)118 1030 y(o)n(v)n(er,)28 b FO(H)390 1042 y FL(0)427 1030 y FP(,)i FO(B)547 1000 y FN(\003)585 1030 y FO(H)654 1042 y FL(0)721 1030 y FP(are)e(in)n(v)-5 b(arian)n(t)28 b(with)h(resp)r(ect)g(to)g FO(A)p FP(,)h FO(B)t(B)2039 1000 y FN(\003)2078 1030 y FP(,)f(whic)n(h)h(imply)118 1130 y(that)24 b(giv)n(en)e(\001)h FQ(2)h FA(B)p FP(\()p FJ(R)836 1100 y FL(2)880 1130 y FP(\),)g(the)g(subspace)e FO(E)5 b FP(\(\001\))p FO(H)1708 1142 y FL(0)1756 1130 y FQ(\010)10 b FO(B)1898 1100 y FN(\003)1935 1130 y FO(E)5 b FP(\(\001\))p FO(H)2203 1142 y FL(0)2265 1130 y FP(is)23 b(in)n(v)-5 b(ari-)118 1229 y(an)n(t)21 b(with)h(resp)r(ect)f(to)g FO(A)p FP(,)i FO(B)t FP(,)f FO(B)1100 1199 y FN(\003)1139 1229 y FP(,)g(where)f FO(E)5 b FP(\()p FQ(\001)p FP(\))22 b(is)f(the)h(join)n(t)f(resolution) f(of)h(the)118 1329 y(iden)n(tit)n(y)34 b(for)e(the)i(comm)n(uting)f (pair)g(of)g(op)r(erators)e FO(A)p FP(,)36 b FO(B)t(B)2054 1299 y FN(\003)2125 1329 y FP(restricted)d(to)118 1429 y FO(H)187 1441 y FL(0)224 1429 y FP(.)j(F)-7 b(rom)24 b(this)h(it)g(follo)n(ws)e(that)i(\001)f(is)h(concen)n(trated)e(in)i (one)f(p)r(oin)n(t)g(if)h FO(A)p FP(,)h FO(B)t FP(,)118 1528 y FO(B)185 1498 y FN(\003)246 1528 y FP(is)c(an)g(irreducible)g (family)-7 b(.)35 b(Th)n(us)22 b(for)g(suc)n(h)g(a)g(family)g(of)h(op)r (erators)d(there)118 1628 y(exists)34 b(a)g(join)n(t)g(eigen)n(v)n (ector)e FO(e)i FQ(2)h FO(H)1300 1640 y FL(0)1371 1628 y FP(for)f FO(A)p FP(,)i FO(B)t(B)1760 1598 y FN(\003)1799 1628 y FP(,)g(and)e FQ(f)p FO(e;)14 b(B)2211 1598 y FN(\003)2248 1628 y FO(e)p FQ(g)34 b FP(de\014ne)118 1727 y(an)29 b(orthogonal)d(basis)i(of)h(the)g(represen)n(tation)e(space.)39 b(The)29 b(corresp)r(onding)118 1827 y(irreducible)e(represen)n(tation) f(is)h(giv)n(en)g(b)n(y)h(\()p FO(ii)p FP(\).)243 1933 y(In)j(the)g(case)f FO(A)f FP(=)f FO(A)921 1903 y FN(\003)960 1933 y FP(,)k FO(B)j FP(w)n(e)30 b(ha)n(v)n(e)g(that)h(the)h(op)r (erators)d FO(A\036)p FP(\()p FO(A)p FP(\),)k FO(A)21 b FP(+)118 2033 y FO(\036)p FP(\()p FO(A)p FP(\))30 b(comm)n(ute)f (with)g FO(A)p FP(,)g FO(B)t FP(,)g(and)g(hence)g(due)g(to)f(the)h (irreducibilit)n(y)-7 b(,)29 b(they)118 2132 y(are)c(m)n(ultiples)i(of) f(the)g(iden)n(tit)n(y)-7 b(,)27 b(i.e.,)f FO(A\036)p FP(\()p FO(A)p FP(\))f(=)e FO(a)1717 2144 y FL(1)1754 2132 y FO(I)33 b FP(and)26 b FO(A)16 b FP(+)f FO(\036)p FP(\()p FO(A)p FP(\))24 b(=)f FO(a)2472 2144 y FL(2)2509 2132 y FO(I)7 b FP(.)118 2232 y(Then)38 b FO(A)407 2202 y FL(2)469 2232 y FQ(\000)25 b FO(a)603 2244 y FL(2)640 2232 y FO(A)g FP(+)g FO(a)861 2244 y FL(1)898 2232 y FO(I)46 b FP(=)39 b(0,)g(and)e(so)g(the)h(sp)r(ectrum)f(of)h FO(A)f FP(is)h FO(\033)s FP(\()p FO(A)p FP(\))i(=)118 2332 y FQ(f)p FO(\025)208 2344 y FL(1)245 2332 y FO(;)14 b(\025)330 2344 y FL(2)368 2332 y FQ(g)p FP(,)22 b(where)e FO(\025)736 2344 y FL(1)774 2332 y FP(,)i FO(\025)867 2344 y FL(2)926 2332 y FP(are)e(the)h(ro)r(ots)f(of)h(the)g(equation)f FO(\025)2002 2302 y FL(2)2045 2332 y FQ(\000)5 b FO(a)2159 2344 y FL(2)2195 2332 y FO(\025)g FP(+)g FO(a)2362 2344 y FL(1)2423 2332 y FP(=)22 b(0.)118 2431 y(Hence)k(the)g(sp)r(ectrum)f (of)h FO(A)g FP(is)f(discrete)g(as)g(so)r(on)g(as)f(\()p FO(A)p FP(,)j FO(B)t FP(\))f(is)f(irreducible.)118 2531 y(In)c(addition,)h FO(B)627 2501 y FL(2)685 2531 y FP(comm)n(utes)e (with)h FO(A)p FP(,)h FO(B)j FP(and)20 b(is)h(a)f(m)n(ultiple)h(of)g (the)g(iden)n(tit)n(y)-7 b(,)118 2631 y FO(B)185 2600 y FL(2)253 2631 y FP(=)29 b FO(b)383 2600 y FL(2)420 2631 y FO(I)7 b FP(.)50 b(If)32 b FO(b)e FQ(6)p FP(=)g(0)h(and)h FO(e)1062 2643 y FM(\025)1101 2651 y Fy(1)1169 2631 y FP(is)g(an)f(eigen)n(v)n(ector)f(of)i FO(A)p FP(,)h(then)f FO(e)2261 2643 y FM(\025)2300 2651 y Fy(1)2337 2631 y FP(,)h FO(B)t(e)2499 2643 y FM(\025)2538 2651 y Fy(1)118 2730 y FP(de\014ne)d(an)g(in)n(v)-5 b(arian)n(t)29 b(subspace;)i(moreo) n(v)n(er)d FO(B)t(e)1670 2742 y FM(\025)1709 2750 y Fy(1)1775 2730 y FP(is)i(an)g(eigen)n(v)n(ector)e(with)118 2830 y(the)40 b(eigen)n(v)-5 b(alue)39 b FO(\036)p FP(\()p FO(\025)811 2842 y FL(1)849 2830 y FP(\))44 b(=:)f FO(\025)1104 2842 y FL(2)1141 2830 y FP(.)73 b(Therefore,)42 b(b)n(y)d(the)h (irreducibilit)n(y)-7 b(,)42 b(the)118 2929 y(op)r(erators)25 b FO(A)p FP(,)j FO(B)k FP(can)26 b(b)r(e)i(at)f(most)g(t)n(w)n (o-dimensional.)35 b(If)28 b FO(\036)p FP(\()p FO(\025)2119 2941 y FL(1)2157 2929 y FP(\))c FQ(6)p FP(=)e FO(\025)2348 2941 y FL(1)2413 2929 y FP(\(i.e.,)118 3029 y(\010\()p FO(\025)258 3041 y FL(1)296 3029 y FO(;)14 b(\025)381 3041 y FL(1)419 3029 y FP(\))25 b FQ(6)p FP(=)g(0\),)k(then)h (normalizing)d(the)j(orthogonal)c(basis)j FO(e)2142 3041 y FM(\025)2181 3049 y Fy(1)2217 3029 y FP(,)h FO(B)t(e)2376 3041 y FM(\025)2415 3049 y Fy(1)2480 3029 y FP(w)n(e)118 3129 y(get)22 b(an)g(orthogonal)d(basis)j(in)g(whic)n(h)g(op)r(erators) e FO(A)p FP(,)j FO(B)j FP(are)21 b(of)h(the)h(form)e(\()p FO(iii)p FP(\).)118 3228 y(F)-7 b(or)21 b FO(\036)p FP(\()p FO(\025)p FP(\))j(=)f FO(\025)e FP(one)g(has)g(that)g FO(A)p FP(,)i FO(B)i FP(comm)n(ute,)e(and)e(hence)g(w)n(e)g(can)f(c)n (ho)r(ose)g(a)118 3328 y(join)n(t)25 b(eigen)n(v)n(ector)e FO(e)777 3340 y FM(\025;b)869 3328 y FP(,)i(whic)n(h)g(de\014ne)g(an)f (in)n(v)-5 b(arian)n(t)24 b(subspace.)35 b(F)-7 b(rom)25 b(this)118 3428 y(one)31 b(can)h(conclude)f(that)h(the)g(corresp)r (onding)e(irreducible)h(represen)n(tation)118 3527 y(is)d(one)f (dimensional)g(and)g(giv)n(en)g(b)n(y)g(\()p FO(i)p FP(\))h(or)f(\()p FO(ii)p FP(\).)p 2514 3527 4 57 v 2518 3474 50 4 v 2518 3527 V 2567 3527 4 57 v 243 3745 a(If)38 b(the)g(graph)f(\000)g (corresp)r(onding)f(to)i(a)f(semilinear)g(relation)g(con)n(tains)118 3879 y(the)g(subgraphs:)756 3861 y Fo(r)p 756 3862 4 4 v 751 3858 V 747 3853 V 743 3848 V 739 3844 V 736 3839 V 733 3835 V 730 3831 V 728 3827 V 726 3823 V 725 3819 V 724 3815 V 723 3812 V 723 3808 V 723 3805 V 723 3802 V 724 3799 V 725 3796 V 727 3793 V 729 3790 V 731 3788 V 731 3788 V 734 3785 V 736 3783 V 739 3781 V 741 3780 V 743 3778 V 746 3777 V 748 3776 V 751 3776 V 753 3775 V 756 3775 V 758 3775 V 761 3776 V 763 3776 V 766 3777 V 768 3778 V 771 3780 V 773 3781 V 776 3783 V 778 3785 V 781 3788 V 756 3862 V 761 3858 V 765 3853 V 769 3848 V 773 3844 V 776 3839 V 779 3835 V 782 3831 V 784 3827 V 786 3823 V 787 3819 V 788 3815 V 789 3812 V 789 3808 V 789 3805 V 789 3802 V 788 3799 V 787 3796 V 785 3793 V 783 3790 V 781 3788 V 834 3879 a FP(or)987 3861 y Fo(r)p 987 3862 117 4 v 1020 3860 a Fu(-)1111 3861 y Fo(r)p 1111 3862 V 1145 3860 a Fu(-)1236 3861 y Fo(r)1331 3879 y FP(\(and)g(with)g(an)n(y)g(other)f(orien)n(tation\),)118 4036 y(and)26 b(the)h(graph)e(\000)706 4048 y FM(s)767 4036 y FP(con)n(tains)g(the)i(subgraphs:)1699 4019 y Fo(r)p 1699 4020 4 4 v 1694 4015 V 1690 4011 V 1686 4006 V 1682 4001 V 1679 3997 V 1676 3993 V 1674 3988 V 1671 3984 V 1670 3980 V 1668 3977 V 1667 3973 V 1666 3969 V 1666 3966 V 1666 3963 V 1667 3960 V 1667 3956 V 1669 3954 V 1670 3951 V 1672 3948 V 1674 3946 V 1674 3946 V 1677 3943 V 1679 3941 V 1682 3939 V 1684 3938 V 1687 3936 V 1689 3935 V 1692 3934 V 1694 3934 V 1697 3933 V 1699 3933 V 1702 3933 V 1704 3934 V 1707 3934 V 1709 3935 V 1712 3936 V 1714 3938 V 1717 3939 V 1719 3941 V 1722 3943 V 1724 3946 V 1699 4020 V 1704 4015 V 1708 4011 V 1712 4006 V 1716 4001 V 1719 3997 V 1722 3993 V 1725 3988 V 1727 3984 V 1729 3980 V 1730 3977 V 1731 3973 V 1732 3969 V 1732 3966 V 1732 3963 V 1732 3960 V 1731 3956 V 1730 3954 V 1728 3951 V 1726 3948 V 1724 3946 V 1699 4020 125 4 v 100 w(r)1882 4036 y FP(,)f(or)2073 4019 y Fo(r)p 2073 4020 V 100 w(r)p 2198 4020 V 99 w(r)2364 4036 y FP(,)g(then)118 4147 y(the)32 b(problem)f(of)h(describing)f(all) g(irreducible)g(represen)n(tations)f(\()p FO(A;)14 b(B)t(;)g(B)2504 4117 y FN(\003)2543 4147 y FP(\))p eop %%Page: 68 72 68 71 bop 118 100 a FP(68)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(and)i(\()p FO(A;)14 b(B)29 b FP(=)24 b FO(B)660 303 y FN(\003)699 333 y FP(\))29 b(resp)r(ectiv)n(ely)e(b)r(ecomes)i(v) n(ery)e(complicated)h(\(the)i(corre-)118 432 y(sp)r(onding)20 b FQ(\003)p FP(-algebra)e(is)j(wild\).)35 b(W)-7 b(e)21 b(refer)e(the)i(reader)e(to)i(Sections)f(3.1.1)f(and)118 532 y(3.1.2)27 b(for)h(a)f(precise)h(de\014nition)g(of)g FQ(\003)p FP(-wild)g(algebras,)e(and)i(to)g(Section)g(3.1.4)118 632 y(for)f(the)h(pro)r(of)f(of)h(the)g(ab)r(o)n(v)n(e)e(fact.)118 870 y FH(1.4)112 b(Represen)m(tations)37 b(of)h Fr(q)t FH(-relations)118 1052 y FR(1.4.1)94 b(Finite-dimensional)27 b(represen)m(tations)32 b(of)g FO(q)s FR(-relations)118 1205 y(1.)k FP(When)28 b(studying)g(represen)n(tations)e(of)h FO(q)s FP(-relations,)262 1406 y(\()p FO(V)19 b(I)397 1418 y FL(0)435 1406 y FP(\))p FO(;)14 b FP(\()p FO(V)19 b(I)7 b(I)682 1418 y FL(0)720 1406 y FP(\))845 1350 y(1)p 845 1387 42 4 v 851 1463 a FO(i)896 1406 y FP([)p FO(A;)14 b(B)t FP(])24 b(=)f FO(\013)p FP(\()p FO(A)1367 1372 y FL(2)1423 1406 y FQ(\006)18 b FO(B)1573 1372 y FL(2)1611 1406 y FP(\))p FO(;)180 b(\013)23 b(>)g FP(0)p FO(;)262 1606 y FP(\()p FO(V)c(I)397 1618 y FL(1)435 1606 y FP(\))p FO(;)14 b FP(\()p FO(V)19 b(I)7 b(I)682 1618 y FL(1)720 1606 y FP(\))845 1550 y(1)p 845 1587 V 851 1663 a FO(i)896 1606 y FP([)p FO(A;)14 b(B)t FP(])24 b(=)f FO(\013)p FP(\()p FO(A)1367 1572 y FL(2)1423 1606 y FQ(\006)18 b FO(B)1573 1572 y FL(2)1611 1606 y FP(\))g(+)g FO(I)7 b(;)180 b(\013)24 b FQ(2)f FJ(R)i FQ(n)18 b(f)p FP(0)p FQ(g)p FO(;)118 1795 y FP(it)24 b(w)n(ould)e(b)r(e)h(nice)h(to)e(ha)n (v)n(e)g(a)h(kind)g(of)g(theorem)f(of)h(the)h(Kleinec)n(k)n(e{Shirok)n (o)n(v)118 1895 y(t)n(yp)r(e)k(for)f(general)f(relations)h(of)g(the)h (form)923 2040 y(1)p 923 2077 V 929 2153 a FO(i)975 2096 y FP([)p FO(A;)14 b(B)t FP(])23 b(=)g FO(f)9 b FP(\()p FO(A)p FP(\))19 b(+)f FO(\036)p FP(\()p FO(B)t FP(\))p FO(;)118 2295 y FP(where)i FO(f)9 b FP(\()p FQ(\001)p FP(\),)22 b FO(\036)p FP(\()p FQ(\001)p FP(\))g(are)d(real)h(functions) g(of)h FO(t)i FQ(2)g FJ(R)1600 2265 y FL(1)1643 2295 y FP(.)35 b(In)20 b(the)h(\014nite-dimensional)118 2395 y(case)27 b(\(dim)14 b FO(H)30 b(<)23 b FQ(1)p FP(\))28 b(w)n(e)f(ha)n(v)n(e)f(the)i(follo)n(wing)f(statemen)n(t.)118 2548 y FR(Prop)s(osition)j(25.)41 b FC(If)48 b FO(A)30 b FC(and)g FO(B)k FC(ar)l(e)d(self-adjoint)g(op)l(er)l(ators)g(in)f(a)h (\014nite-)118 2648 y(dimensional)h(sp)l(ac)l(e)e FO(H)7 b FC(,)30 b(and)1067 2814 y FP([)p FO(A;)14 b(B)t FP(])24 b(=)e FO(T)30 b FP(+)18 b FO(S;)118 2980 y FC(wher)l(e)998 3147 y FP([)p FO(A;)c(T)e FP(])22 b(=)h([)p FO(B)t(;)14 b(S)5 b FP(])23 b(=)f(0)p FO(;)118 3313 y FC(then)30 b FP([)p FO(A;)14 b(B)t FP(])23 b(=)g(0)p FC(.)118 3466 y(Pr)l(o)l(of.)43 b FP(Indeed,)28 b(since)330 3632 y([)p FO(A;)14 b(B)t FP(])542 3598 y FL(2)603 3632 y FP(=)23 b(\()p FO(T)30 b FP(+)18 b FO(S)5 b FP(\)[)p FO(A;)14 b(B)t FP(])23 b(=)g FO(T)12 b FP(\()p FO(AB)22 b FQ(\000)c FO(B)t(A)p FP(\))h(+)f FO(S)5 b FP(\()p FO(AB)23 b FQ(\000)18 b FO(B)t(A)p FP(\))603 3757 y(=)23 b FO(A)p FP(\()p FO(T)12 b(B)t FP(\))18 b FQ(\000)g FP(\()p FO(T)12 b(B)t FP(\))p FO(A)19 b FP(+)f(\()p FO(S)5 b(A)p FP(\))p FO(B)23 b FQ(\000)18 b FO(B)t FP(\()p FO(S)5 b(A)p FP(\))603 3881 y(=)23 b([)p FO(A;)14 b(T)e(B)t FP(])18 b(+)g([)p FO(S)5 b(A;)14 b(B)t FP(])p FO(;)118 4048 y FP(w)n(e)27 b(conclude)h(that)f(T) -7 b(r[)p FO(A;)14 b(B)t FP(])1059 4018 y FL(2)1120 4048 y FP(=)22 b(0.)37 b(But)28 b([)p FO(A;)14 b(B)t FP(])28 b(is)f(a)g(sk)n(ew-adjoin)n(t)f(op)r(era-)118 4147 y(tor,)h(therefore,) g(T)-7 b(r)o([)p FO(A;)14 b(B)t FP(])944 4117 y FL(2)1005 4147 y FP(=)23 b(0)k(implies)h([)p FO(A;)14 b(B)t FP(])23 b(=)g(0.)p 2514 4147 4 57 v 2518 4095 50 4 v 2518 4147 V 2567 4147 4 57 v eop %%Page: 69 73 69 72 bop 118 100 a FK(1.4.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-relations)1130 b FP(69)118 333 y FR(Corollary)32 b(2.)40 b FC(Irr)l(e)l(ducible)29 b(\014nite-dimensional)h(r)l(epr)l (esentations)f(of)g(r)l(ela-)118 432 y(tions)36 b FP(\()p FO(V)19 b(I)465 444 y FL(0)503 432 y FP(\))p FC({)p FP(\()p FO(V)h(I)7 b(I)756 444 y FL(1)794 432 y FP(\))36 b FC(ar)l(e)h (one-dimensional,)j FO(A)35 b FP(=)f FO(\025)p FC(,)39 b FO(B)f FP(=)d FO(\026)p FC(,)j FP(\()p FO(\025;)14 b(\026)p FP(\))35 b FQ(2)118 532 y FO(M)199 547 y FL(\()p FN(\001)p FL(\))274 532 y FP(\()p FO(\013)p FP(\))p FC(,)d(wher)l(e)242 732 y FO(M)323 747 y FL(\()p FM(V)14 b(I)431 755 y Fy(0)463 747 y FL(\))494 732 y FP(\()p FO(\013)p FP(\))24 b(=)e FQ(f)p FP(\()p FO(\025;)14 b(\026)p FP(\))24 b FQ(2)f FJ(R)1119 698 y FL(2)1185 732 y FQ(j)h FO(\025)f FP(=)g(0)p FO(;)14 b(\026)22 b FP(=)h(0)p FQ(g)p FO(;)182 b FC(for)31 b(al)t(l)g FO(\013)23 b(>)g FP(0)p FC(,)242 857 y FO(M)323 872 y FL(\()p FM(V)14 b(I)431 880 y Fy(1)463 872 y FL(\))494 857 y FP(\()p FO(\013)p FP(\))24 b(=)e FJ(?)p FO(;)183 b(\013)24 b(>)f FP(0)p FO(;)242 1030 y(M)323 1045 y FL(\()p FM(V)14 b(I)431 1053 y Fy(1)463 1045 y FL(\))494 1030 y FP(\()p FO(\013)p FP(\))24 b(=)e FQ(f)p FP(\()p FO(\025;)14 b(\026)p FP(\))24 b FQ(2)f FJ(R)1119 995 y FL(2)1185 1030 y FQ(j)h FO(\025)1280 995 y FL(2)1336 1030 y FP(+)18 b FO(\026)1469 995 y FL(2)1529 1030 y FP(=)23 b FQ(\000)1697 973 y FP(1)p 1692 1010 54 4 v 1692 1087 a FO(\013)1755 1030 y FQ(g)p FO(;)183 b(\013)23 b(<)g FP(0)p FO(;)208 1191 y(M)289 1206 y FL(\()p FM(V)14 b(I)5 b(I)431 1214 y Fy(0)463 1206 y FL(\))494 1191 y FP(\()p FO(\013)p FP(\))24 b(=)e FQ(f)p FP(\()p FO(\025;)14 b(\026)p FP(\))24 b FQ(2)f FJ(R)1119 1157 y FL(2)1185 1191 y FQ(j)h FO(\025)1280 1157 y FL(2)1340 1191 y FP(=)f FO(\026)1478 1157 y FL(2)1515 1191 y FQ(g)p FO(;)183 b FC(for)31 b(al)t(l)g FO(\013)23 b(>)g FP(0)p FC(,)208 1364 y FO(M)289 1379 y FL(\()p FM(V)14 b(I)5 b(I)431 1387 y Fy(1)463 1379 y FL(\))494 1364 y FP(\()p FO(\013)p FP(\))24 b(=)e FQ(f)p FP(\()p FO(\025;)14 b(\026)p FP(\))24 b FQ(2)f FJ(R)1119 1330 y FL(2)1185 1364 y FQ(j)h FO(\025)1280 1330 y FL(2)1336 1364 y FQ(\000)18 b FO(\026)1469 1330 y FL(2)1529 1364 y FP(=)23 b FQ(\000)1697 1308 y FP(1)p 1692 1345 V 1692 1421 a FO(\013)1755 1364 y FQ(g)p FO(;)183 b FC(for)30 b(al)t(l)h FO(\013)24 b FQ(6)p FP(=)e(0)p FC(.)118 1582 y(R)l(emark)44 b FP(19)p FC(.)h FP(In)34 b(the)g(\014nite-dimensional)f (case,)h(the)g(corresp)r(onding)e(gen-)118 1681 y(eralization)j(of)i (the)g(Jacobson)e(theorem)i(is)f(not)h(true.)65 b(Indeed,)39 b(ev)n(en)d(for)118 1781 y(dim)14 b FO(H)32 b FP(=)24 b(3,)29 b(there)f(exist)h(matrices)f FO(A)p FP(,)h FO(B)t FP(,)g FO(T)12 b FP(,)28 b FO(S)5 b FP(,)29 b(suc)n(h)f(that)h(the)g (relations)118 1881 y([)p FO(P)r(;)14 b(T)e FP(])23 b(=)f([)p FO(Q;)14 b(S)5 b FP(])23 b(=)g(0)k(hold,)g(but)h(the)g(matrix)g([)p FO(P)r(;)14 b(Q)p FP(])27 b(is)h(not)g(nilp)r(oten)n(t.)243 1984 y(The)c(follo)n(wing)f(example)h(is)g(due)g(to)g(V.S.)h(Guba)f (and)g(is)g(giv)n(en)f(in)49 b([244)o(].)118 2125 y FC(Example)37 b FP(8)p FC(.)42 b FP(Let)578 2410 y FO(A)24 b FP(=)751 2243 y Fz(0)751 2393 y(@)824 2310 y FP(2)83 b(0)f(0)824 2409 y(0)h(1)f(0)824 2509 y(0)h(0)f(0)1115 2243 y Fz(1)1115 2393 y(A)1201 2410 y FO(;)97 b(B)27 b FP(=)1499 2243 y Fz(0)1499 2393 y(@)1604 2310 y FP(2)147 b(4)115 b(4)1604 2409 y(6)g FQ(\000)p FP(4)82 b(4)1571 2509 y FQ(\000)p FP(3)115 b(6)g(2)1991 2243 y Fz(1)1991 2393 y(A)2078 2410 y FO(;)422 2742 y(T)34 b FP(=)593 2575 y Fz(0)593 2725 y(@)666 2642 y FP(4)103 b(0)g(0)666 2742 y(0)82 b(16)g(0)666 2841 y(0)103 b(0)g(4)998 2575 y Fz(1)998 2725 y(A)1084 2742 y FO(;)97 b(S)28 b FP(=)1371 2575 y Fz(0)1371 2725 y(@)1476 2642 y FP(4)147 b(4)114 b(8)1443 2742 y FQ(\000)p FP(6)94 b(16)f(4)1476 2841 y(6)114 b FQ(\000)p FP(6)82 b(4)1863 2575 y Fz(1)1863 2725 y(A)1959 2742 y FP(=)22 b FO(B)2113 2708 y FL(2)2151 2742 y FO(=)p FP(4;)118 3032 y(then)28 b([)p FO(A;)14 b(B)t FP(])24 b(=)e FO(T)30 b FP(+)18 b FO(S)5 b FP(,)27 b([)p FO(T)7 b(;)14 b(A)p FP(])23 b(=)g([)p FO(S;)14 b(B)t FP(])23 b(=)g(0,)k(but)h FO(\033)s FP(\([)p FO(A;)14 b(B)t FP(]\))25 b FQ(6)p FP(=)d FQ(f)p FP(0)p FQ(g)p FP(.)118 3173 y FR(2.)34 b FP(A)23 b(similar)e(fact)h(to)g(Prop)r(osition)e(25)h(holds) h(for)f(op)r(erators)f(on)i(an)f(in\014nite-)118 3273 y(dimensional)27 b FO(H)7 b FP(,)28 b(but)g(under)g(the)g(additional)f (assumption)h(that)g(the)g(op)r(er-)118 3372 y(ators)e FO(T)39 b FP(and)28 b FO(S)k FP(are)26 b(compact.)118 3550 y FR(Prop)s(osition)j(26.)39 b FC(If)30 b FO(A)23 b FP(=)f FO(A)1116 3520 y FN(\003)1155 3550 y FC(,)29 b FO(B)e FP(=)c FO(B)1454 3520 y FN(\003)1515 3550 y FQ(2)h FO(L)p FP(\()p FO(H)7 b FP(\))p FC(,)29 b(and)38 b FP([)p FO(A;)14 b(B)t FP(])23 b(=)g FO(T)k FP(+)16 b FO(S)5 b FC(,)118 3650 y(wher)l(e)29 b FO(T)12 b FC(,)27 b FO(S)5 b FC(,)29 b(ar)l(e)f(c)l(omp)l(act)g(op)l(er)l(ators)h(on)f FO(H)35 b FC(such)27 b(that)h FP([)p FO(A;)14 b(T)e FP(])23 b(=)f([)p FO(B)t(;)14 b(S)5 b FP(])23 b(=)118 3749 y(0)p FC(,)30 b(then)f FP([)p FO(A;)14 b(B)t FP(])24 b(=)e(0)p FC(.)118 3927 y(Pr)l(o)l(of.)43 b FP(Since)28 b([)p FO(A;)14 b(B)t FP(])28 b(is)g(sk)n(ew-adjoin)n(t,)996 4117 y([)p FO(A;)14 b(B)t FP(])24 b(=)f(\()p FO(T)1401 4129 y FL(1)1456 4117 y FP(+)18 b FO(S)1590 4129 y FL(1)1627 4117 y FP(\))p FO(;)681 b FP(\(1.20\))p eop %%Page: 70 74 70 73 bop 118 100 a FP(70)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(where)d(the)h(compact)g(op)r(erators)d FO(T)1238 345 y FL(1)1298 333 y FP(=)h(\()p FO(T)h FQ(\000)13 b FO(T)1630 303 y FN(\003)1666 333 y FP(\))p FO(=)p FP(2,)25 b(and)g FO(S)2040 345 y FL(1)2100 333 y FP(=)e(\()p FO(S)17 b FQ(\000)c FO(S)2422 303 y FN(\003)2459 333 y FP(\))p FO(=)p FP(2)118 432 y(are)26 b(sk)n(ew-adjoin)n(t.)35 b(By)27 b(the)g(F)-7 b(uglede{Putnam{Rosen)n(blum)26 b(theorem)g(\(for)118 532 y(its)i(form)n(ulation)f(and)g(pro)r(of)g (see)g(Section)h(1.4.3)e(b)r(elo)n(w\),)i(w)n(e)f(ha)n(v)n(e)969 714 y([)p FO(T)1041 726 y FL(1)1078 714 y FO(;)14 b(A)p FP(])23 b(=)g([)p FO(S)1385 726 y FL(1)1422 714 y FO(;)14 b(B)t FP(])23 b(=)g(0)p FO(:)118 895 y FP(F)-7 b(urther,)20 b(let)f FQ(k)p FO(T)637 907 y FL(1)673 895 y FQ(k)k(\025)g(k)p FO(S)919 907 y FL(1)955 895 y FQ(k)g FO(>)g FP(0.)33 b(Cho)r(ose)18 b(an)g(eigen)n(v)-5 b(alue)18 b FO(\026)g FP(of)h(the)g(op)r(erator)118 995 y FO(T)167 1007 y FL(1)233 995 y FP(suc)n(h)29 b(that)g FQ(j)p FO(\026)p FQ(j)d FP(=)f FQ(k)p FO(T)906 1007 y FL(1)942 995 y FQ(k)p FP(,)k(and)g(write) g FO(H)1482 1007 y FM(\026)1552 995 y FP(=)c FQ(f)p FO(f)33 b FQ(2)26 b FO(H)16 b FP(:)28 b FO(T)2024 1007 y FL(1)2061 995 y FO(f)34 b FP(=)25 b FO(\026f)9 b FQ(g)p FP(.)41 b(The)118 1095 y(space)34 b FO(H)416 1107 y FM(\026)494 1095 y FP(is)g(\014nite-dimensional)g(and)g(in)n(v)-5 b(arian)n(t)33 b(with)i(resp)r(ect)f(to)g FO(A)p FP(.)56 b(If)118 1194 y FO(e)23 b FQ(2)g FO(H)327 1206 y FM(\026)372 1194 y FP(,)28 b FQ(k)p FO(e)p FQ(k)21 b FP(=)i(1,)k(and)h FO(Ae)23 b FP(=)f FO(\025e)p FP(,)28 b FO(\025)23 b FQ(2)h FJ(R)p FP(,)34 b(then,)28 b(due)g(to)f(\(1.20\))o(,)h(w)n(e)f(ha)n(v)n (e)908 1376 y(\()p FO(A)19 b FQ(\000)f FO(\025I)7 b FP(\))p FO(B)t(e)24 b FP(=)f FO(\026e)18 b FP(+)g FO(S)1686 1388 y FL(1)1723 1376 y FO(e;)118 1558 y FP(whic)n(h)33 b(implies)g(\()p FO(S)731 1570 y FL(1)768 1558 y FO(e;)14 b(e)p FP(\))31 b(=)g FQ(\000)p FO(\026)p FP(,)j(and)f(therefore,)g FQ(k)p FO(S)1852 1570 y FL(1)1888 1558 y FQ(k)e(\025)h(j)p FO(\026)p FQ(j)p FP(.)52 b(But)33 b(since)118 1657 y FQ(k)p FO(S)211 1669 y FL(1)248 1657 y FQ(k)g(\024)f(k)p FO(T)511 1669 y FL(1)548 1657 y FQ(k)g FP(=)h FQ(j)p FO(\026)p FQ(j)p FP(,)j(w)n(e)d(ha)n(v)n(e)g FQ(k)p FO(S)1294 1669 y FL(1)1330 1657 y FQ(k)g FP(=)g FQ(j)p FO(\026)p FQ(j)p FP(,)i(whic)n(h)f(yields)g FO(S)2192 1669 y FL(1)2229 1657 y FO(e)f FP(=)f FQ(\000)p FO(\026e)p FP(.)118 1757 y(Then)37 b FO(S)395 1769 y FL(1)472 1757 y FP(=)h FQ(\000)p FO(T)689 1769 y FL(1)763 1757 y FP(on)f(the)g(whole)g(subspace)g FO(H)1711 1769 y FM(\026)1755 1757 y FP(.)66 b(Since)37 b(the)h(subspaces)118 1856 y FO(H)187 1868 y FM(\026)263 1856 y FP(and)31 b FQ(f)p FO(f)37 b FQ(2)29 b FO(H)36 b FQ(j)29 b FO(S)840 1868 y FL(1)877 1856 y FO(f)38 b FP(=)28 b FQ(\000)p FO(\026f)9 b FQ(g)30 b FP(coincide,)i(the)f(op)r(erator)f FO(B)35 b FP(also)30 b(maps)118 1956 y FO(H)187 1968 y FM(\026)268 1956 y FP(in)n(to)36 b FO(H)514 1968 y FM(\026)558 1956 y FP(.)63 b(Then)36 b FO(T)918 1968 y FL(1)991 1956 y FP(is)g(compact,)i(and)e(w)n(e)g(ha)n(v)n(e)f FO(S)2001 1968 y FL(1)2075 1956 y FP(=)i FQ(\000)p FO(T)2291 1968 y FL(1)2363 1956 y FP(on)f(all)118 2056 y(eigenspaces)19 b(of)h(the)h(op)r(erator)d FO(T)1149 2068 y FL(1)1186 2056 y FP(,)k(and)e(therefore,)h(on)f(the)h(whole)e FO(H)7 b FP(.)35 b(Hence)118 2155 y([)p FO(A;)14 b(B)t FP(])24 b(=)e(0.)p 2514 2155 4 57 v 2518 2103 50 4 v 2518 2155 V 2567 2155 4 57 v 243 2321 a(In)d(Section)g(1.4.2,)h(when)f(in)n(v)n (estigating)e(the)j(relation)e(\()p FO(V)h FP(1)2093 2333 y FL(1)2130 2321 y FP(\),)i(w)n(e)e(will)g(see)118 2421 y(that)36 b(the)g(corresp)r(onding)d(statemen)n(t)j(for)f (arbitrary)e(b)r(ounded)j(op)r(erators)118 2520 y(do)r(es)27 b(not)h(hold.)118 2736 y FR(1.4.2)94 b(Hermitian)30 b FO(q)s FR(-plane)h(and)h FO(q)s FR(-CCR)118 2889 y FP(W)-7 b(e)19 b(will)g(consecutiv)n(ely)f(study)h(represen)n(tations)d(of)j (relations)f(\()p FO(V)h(I)2233 2901 y FL(0)2271 2889 y FP(\),)i(\()p FO(V)e(I)2482 2901 y FL(1)2519 2889 y FP(\),)118 2989 y(\()p FO(V)g(I)7 b(I)296 3001 y FL(0)334 2989 y FP(\),)28 b(\()p FO(V)19 b(I)7 b(I)595 3001 y FL(1)633 2989 y FP(\))28 b(b)n(y)f(b)r(ounded)h(self-adjoin)n(t)f(op)r (erators.)118 3138 y FR(1.)42 b FP(\()p FO(V)19 b(I)370 3150 y FL(0)408 3138 y FP(\).)43 b(Consider)28 b(the)i(pairs)f(of)g(b)r (ounded)h(self-adjoin)n(t)f(op)r(erators)f(sat-)118 3238 y(isfying)f(the)h(relation)743 3419 y([)p FO(A;)14 b(B)t FP(])24 b(=)e FO(i\013)p FP(\()p FO(A)1242 3385 y FL(2)1298 3419 y FP(+)d FO(B)1449 3385 y FL(2)1486 3419 y FP(\))p FO(;)180 b(\013)24 b(>)e FP(0)p FO(:)118 3601 y FR(Prop)s(osition)34 b(27.)43 b FC(If)33 b(a)g(p)l(air)h(of)g(b)l(ounde)l(d)f(self-adjoint)i (op)l(er)l(ators)f FO(A)p FC(,)g FO(B)t FC(,)118 3701 y(satis\014es)c FP(\()p FO(V)19 b(I)565 3713 y FL(0)603 3701 y FP(\))p FC(,)30 b(then)g FO(A)23 b FP(=)g FO(B)k FP(=)c(0)p FC(.)118 3866 y(Pr)l(o)l(of.)43 b FP(In)n(tro)r(duce)32 b(the)h(op)r(erators)e FO(X)37 b FP(=)31 b FO(A)22 b FP(+)f FO(iB)t FP(,)34 b FO(X)1881 3836 y FN(\003)1949 3866 y FP(=)d FO(A)22 b FQ(\000)f FO(iB)t FP(.)52 b(Then)118 3966 y(the)28 b(op)r(erators)e FO(X)34 b FP(and)27 b FO(X)969 3935 y FN(\003)1034 3966 y FP(satisfy)g(the)h(Hermitian)g FO(q)s FP(-plane)f(relation:)825 4147 y(\(1)18 b FQ(\000)h FO(\013)p FP(\))14 b FO(X)7 b(X)1252 4113 y FN(\003)1312 4147 y FP(=)22 b(\(1)d(+)f FO(\013)p FP(\))p FO(X)1736 4113 y FN(\003)1774 4147 y FO(X)r(;)p eop %%Page: 71 75 71 74 bop 118 100 a FK(1.4.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-relations)1130 b FP(71)118 333 y(but)28 b(since)g FO(\013)23 b(>)g FP(0,)k(putting)h FO(q)e FP(=)d(\(1)18 b FQ(\000)g FO(\013)q FP(\))p FO(=)p FP(\(1)g(+)g FO(\013)p FP(\),)28 b(w)n(e)f(get)1071 493 y FO(X)1147 459 y FN(\003)1184 493 y FO(X)j FP(=)22 b FO(q)s(X)7 b(X)1562 459 y FN(\003)1599 493 y FO(:)741 b FP(\(1.21\))118 653 y(F)-7 b(or)35 b FO(q)k FQ(\024)d FP(0)e(\()p FO(\013)j FQ(\025)f FP(1\),)h(this)f (equation)e(p)r(ossesses)g(only)h(the)h(zero)e(solution)118 753 y FO(X)29 b FP(=)23 b FO(X)380 723 y FN(\003)441 753 y FP(=)f(0,)i(since)e(for)g FO(q)k FQ(\024)d FP(0,)g(the)g (non-negativ)n(e)e(op)r(erator)g FO(X)2204 723 y FN(\003)2241 753 y FO(X)29 b FP(should)118 853 y(b)r(e)f(equal)f(to)h(the)g(non-p)r (ositiv)n(e)e(one)h FO(q)s(X)7 b(X)1507 823 y FN(\003)1544 853 y FP(.)37 b(But)28 b(then)g FO(A)c FP(=)e FO(B)27 b FP(=)c(0.)243 952 y(W)-7 b(e)32 b(will)f(carry)f(out)i(a)f(more)g (detailed)g(in)n(v)n(estigation)f(of)i(the)f(case)g(1)e FO(>)118 1052 y(q)d(>)d FP(0)k(\(0)c FO(<)g(\013)g(<)g FP(1\).)118 1201 y FR(Lemma)46 b(7.)j FC(If)43 b FO(X)7 b FC(,)46 b FO(X)920 1171 y FN(\003)1004 1201 y FQ(2)h FO(L)p FP(\()p FO(H)7 b FP(\))p FC(,)46 b(and)52 b FP(\(1.21\))42 b FC(holds,)48 b(then)43 b FP(k)n(er)12 b FO(X)53 b FP(=)118 1301 y(k)n(er)13 b FO(X)319 1271 y FN(\003)356 1301 y FC(.)118 1463 y(Pr)l(o)l(of.)43 b FP(Indeed,)28 b(w)n(e)f(ha)n(v)n(e)g (k)n(er)13 b FO(X)29 b FP(=)23 b(k)n(er)12 b FO(X)1496 1432 y FN(\003)1534 1463 y FO(X)29 b FP(=)23 b(k)n(er)13 b FO(X)7 b(X)1997 1432 y FN(\003)2056 1463 y FP(=)23 b(k)n(er)13 b FO(X)2345 1432 y FN(\003)2382 1463 y FP(.)p 2514 1463 4 57 v 2518 1410 50 4 v 2518 1463 V 2567 1463 4 57 v 243 1624 a(The)26 b(represen)n(tation)e(space)h(of)g(the)i (relation)e(\(1.21\))g(no)n(w)g(has)g(the)h(form)118 1724 y FO(H)k FP(=)23 b FO(H)374 1736 y FL(0)417 1724 y FQ(\010)7 b FO(H)558 1736 y FL(1)595 1724 y FP(,)23 b(where)e FO(H)944 1736 y FL(0)1003 1724 y FP(and)g FO(H)1227 1736 y FL(1)1286 1724 y FP(are)g(subspaces)f(in)n(v)-5 b(arian)n(t)21 b(with)h(resp)r(ect)118 1823 y(to)37 b FO(X)7 b FP(,)38 b FO(X)442 1793 y FN(\003)480 1823 y FP(.)64 b(On)37 b FO(H)784 1835 y FL(0)859 1823 y FP(=)h(k)n(er)13 b FO(X)44 b FP(=)38 b(k)n(er)13 b FO(X)1504 1793 y FN(\003)1578 1823 y FP(w)n(e)37 b(ha)n(v)n(e)e FO(X)45 b FP(=)38 b FO(X)2203 1793 y FN(\003)2278 1823 y FP(=)g(0;)j(on)118 1923 y FO(H)187 1935 y FL(1)247 1923 y FP(=)23 b FO(H)411 1893 y FN(?)404 1944 y FL(0)495 1923 y FP(these)k(op)r(erators)f(are)g (non-degenerate.)243 2023 y(Consider)j(the)h(represen)n(tations)e(of)36 b(\(1.21\))29 b(on)h FO(H)1847 2035 y FL(1)1884 2023 y FP(.)44 b(F)-7 b(or)29 b(the)h(p)r(olar)f(de-)118 2122 y(comp)r(osition)j(of)g(the)h(op)r(erator)d FO(X)38 b FP(=)30 b FO(U)9 b(C)d FP(,)34 b(with)e(unitary)g FO(U)41 b FP(and)33 b FO(C)k(>)30 b FP(0,)118 2222 y(w)n(e)d(ha)n(v)n(e)426 2382 y FO(C)491 2348 y FL(2)528 2382 y FO(U)32 b FP(=)23 b FO(U)9 b FP(\()p FO(q)s(C)908 2348 y FL(2)945 2382 y FP(\))p FO(;)98 b FP(and)82 b FO(C)1379 2348 y FL(2)1417 2382 y FO(U)1483 2348 y FN(\003)1544 2382 y FP(=)23 b FO(U)1698 2348 y FN(\003)1735 2382 y FP(\()p FO(q)1807 2348 y FN(\000)p FL(1)1897 2382 y FO(C)1962 2348 y FL(2)1999 2382 y FP(\))p FO(:)309 b FP(\(1.22\))118 2543 y(But)24 b(then)g(if)g FO(\025)f(>)g FP(0)g(b)r(elongs)g(to)g(the)h(sp)r(ectrum) f FO(\033)s FP(\()p FO(C)1795 2512 y FL(2)1834 2543 y FP(\))g(of)h(the)g(op)r(erator)d FO(C)2514 2512 y FL(2)2552 2543 y FP(,)118 2642 y(then)35 b FO(\033)s FP(\()p FO(C)461 2612 y FL(2)499 2642 y FP(\))f FQ(\033)664 2580 y Fz(S)733 2667 y FM(k)q FN(2)p Fv(Z)871 2642 y FO(q)911 2612 y FM(k)952 2642 y FO(\025)p FP(.)57 b(F)-7 b(or)34 b FO(\025)h FP(whic)n(h)f(is)g(an)g(eigen)n(v)-5 b(alue)33 b(of)h FO(C)2344 2612 y FL(2)2382 2642 y FP(,)i(this)118 2742 y(fact)24 b(directly)f(follo)n(ws)f(from)h(\(1.22\))o(.)35 b(F)-7 b(or)23 b(the)h(case)e(where)h FO(\025)h FP(b)r(elongs)e(to)h (the)118 2841 y(con)n(tin)n(uous)k(sp)r(ectrum)g(of)h(the)g(op)r (erator)e FO(C)1534 2811 y FL(2)1571 2841 y FP(,)i(see)f(Section)h (2.1.1)e(b)r(elo)n(w.)243 2941 y(Since)18 b(the)h(set)705 2879 y Fz(S)774 2966 y FM(k)q FN(2)p Fv(Z)912 2941 y FO(q)952 2911 y FM(k)993 2941 y FO(\025)g FP(is)f(un)n(b)r(ounded,)j (for)d(b)r(ounded)h(represen)n(tations)118 3041 y(of)28 b(\()p FO(V)19 b(I)348 3053 y FL(0)386 3041 y FP(\))28 b(w)n(e)f(ha)n(v)n(e)f FO(A)d FP(=)g FO(B)k FP(=)c(0.)p 2514 3041 V 2518 2988 50 4 v 2518 3041 V 2567 3041 4 57 v 118 3202 a FC(R)l(emark)36 b FP(20)p FC(.)i FP(The)25 b(argumen)n(ts)e(ab)r(o)n(v)n(e)g(enable)h(us)g(to)h(write)f(an)g (explicit)h(for-)118 3302 y(m)n(ula)h(for)f(a)g(family)h(of)g (irreducible)f FC(unb)l(ounde)l(d)34 b FP(represen)n(tations)24 b(of)i(\()p FO(V)19 b(I)2482 3314 y FL(0)2519 3302 y FP(\),)253 3811 y FO(X)29 b FP(=)439 3395 y Fz(0)439 3541 y(B)439 3591 y(B)439 3641 y(B)439 3691 y(B)439 3740 y(B)439 3790 y(B)439 3840 y(B)439 3890 y(B)439 3940 y(B)439 3990 y(B)439 4043 y(@)516 3462 y FP(.)548 3487 y(.)581 3512 y(.)516 3617 y(.)548 3642 y(.)581 3667 y(.)801 3675 y(0)703 b Fp(0)691 3711 y Fz(p)p 774 3711 178 4 v 73 x FO(q)814 3760 y FN(\000)p FL(1)903 3784 y FO(\025)122 b FP(0)1035 3821 y FQ(p)p 1104 3821 49 4 v 70 x FO(\025)167 b FP(0)1236 3928 y Fz(p)p 1319 3928 126 4 v 73 x FO(q)1359 3977 y FL(2)1396 4001 y FO(\025)111 b FP(0)792 4156 y Fp(0)1532 4098 y FP(.)1564 4123 y(.)1596 4148 y(.)1712 4098 y(.)1744 4123 y(.)1776 4148 y(.)1804 3395 y Fz(1)1804 3541 y(C)1804 3591 y(C)1804 3641 y(C)1804 3691 y(C)1804 3740 y(C)1804 3790 y(C)1804 3840 y(C)1804 3890 y(C)1804 3940 y(C)1804 3990 y(C)1804 4043 y(A)1890 3811 y FO(;)180 b(\025)24 b FQ(2)f FP([1)p FO(;)14 b(q)s FP(\))p FO(;)p eop %%Page: 72 76 72 75 bop 118 100 a FP(72)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(if)f(w)n(e)g(de\014ne)g(what)g(is)f(a)h(correct)e(meaning)h (of)h(the)g(relation)f(for)h(un)n(b)r(ounded)118 432 y(op)r(erators)g(\(see)55 b([196)n(])28 b(etc.\).)243 532 y(Note)40 b(that)h(the)g(closure)e(of)h(the)h(symmetric)f(matrix)g FO(X)34 b FP(+)26 b FO(X)2328 502 y FN(\003)2406 532 y FP(from)118 632 y(the)k(set)f(of)g(\014nite)h(v)n(ectors,)e(is)h(not) g(a)g(self-adjoin)n(t)f(op)r(erator;)h(its)g(de\014ciency)118 731 y(indices)f(are)e(\(1,1\))i(\(see)55 b([24)o(]\).)118 856 y FR(2.)34 b FP(\()p FO(V)19 b(I)362 868 y FL(1)400 856 y FP(\).)35 b(Consider)21 b(pairs)g(of)g(b)r(ounded)h(self-adjoin)n (t)f(op)r(erators)f(satisfying)118 955 y(the)28 b(relation)507 1117 y FQ(\000)p FO(i)14 b FP([)p FO(A;)g(B)t FP(])22 b(=)h FO(\013)p FP(\()p FO(A)1084 1083 y FL(2)1141 1117 y FP(+)18 b FO(B)1291 1083 y FL(2)1328 1117 y FP(\))h(+)f FO(I)7 b(;)180 b(\013)23 b FQ(2)h FJ(R)1917 1083 y FL(1)1978 1117 y FQ(n)18 b(f)p FP(0)p FQ(g)p FO(:)118 1279 y FP(Making)25 b(a)g(c)n(hange)g(of)g(v)-5 b(ariables)25 b FO(X)k FP(=)23 b FO(A)15 b FP(+)f FO(iB)t FP(,)26 b FO(X)1754 1249 y FN(\003)1814 1279 y FP(=)d FO(A)15 b FQ(\000)f FO(iB)t FP(,)26 b(w)n(e)f(get)g(the)118 1379 y(follo)n(wing)723 1541 y(\(1)18 b FQ(\000)g FO(\013)p FP(\))c FO(X)7 b(X)1149 1506 y FN(\003)1210 1541 y FP(=)22 b(\(1)d(+)f FO(\013)p FP(\))c FO(X)1648 1506 y FN(\003)1686 1541 y FO(X)24 b FP(+)18 b(2)p FO(I)7 b(:)393 b FP(\(1.23\))243 1702 y(F)-7 b(or)31 b FO(\013)g FQ(\025)f FP(1)i(this)g(equation)g(do)r(es)g (not)g(ha)n(v)n(e)f(solutions,)h(since)g(the)h(left-)118 1802 y(hand)27 b(side)f(con)n(tains)g(a)g(non-p)r(ositiv)n(e)g(op)r (erator,)f(and)i(the)g(op)r(erator)e(on)h(the)118 1902 y(righ)n(t-hand)h(side)g(is)h(strictly)f(p)r(ositiv)n(e.)243 2001 y(Assuming)g FO(\013)d(<)e FP(1,)27 b(w)n(e)h(ha)n(v)n(e)323 2198 y FO(X)7 b(X)475 2163 y FN(\003)535 2198 y FP(=)633 2142 y(1)18 b(+)g FO(\013)p 633 2179 197 4 v 633 2255 a FP(1)g FQ(\000)g FO(\013)839 2198 y(X)915 2163 y FN(\003)953 2198 y FO(X)24 b FP(+)1217 2142 y(2)p 1139 2179 V 1139 2255 a(1)18 b FQ(\000)g FO(\013)1346 2198 y(I)30 b FP(=)23 b FO(q)s(X)1616 2163 y FN(\003)1653 2198 y FO(X)i FP(+)18 b(\()p FO(q)k FP(+)c(1\))p FO(I)7 b(;)219 b FP(\(1.24\))118 2394 y(where)30 b FO(q)g FP(=)d(\(1)19 b(+)f FO(\013)p FP(\))p FO(=)p FP(\(1)g FQ(\000)g FO(\013)q FP(\))27 b FO(>)g FQ(\000)p FP(1.)44 b(T)-7 b(o)30 b(rewrite)g(the)g(latter)g (in)h(the)f(form)118 2494 y(used)j(in)h(the)f(literature,)h(in)n(tro)r (duce)f(the)h(op)r(erator)d FO(a)i FP(=)1997 2433 y FQ(p)p 2066 2433 184 4 v 61 x FO(q)21 b FP(+)d(1)c FO(X)7 b FP(,)34 b FO(a)2440 2464 y FN(\003)2510 2494 y FP(=)118 2533 y FQ(p)p 187 2533 V 61 x FO(q)22 b FP(+)c(1)13 b FO(X)460 2564 y FN(\003)498 2594 y FP(.)37 b(Then)27 b(w)n(e)h(ha)n(v)n(e)1061 2756 y FO(aa)1149 2721 y FN(\003)1210 2756 y FP(=)23 b FO(q)s(a)1382 2721 y FN(\003)1420 2756 y FO(a)18 b FP(+)g FO(I)7 b(:)732 b FP(\(1.25\))118 2917 y FC(R)l(emark)49 b FP(21)p FC(.)f FP(F)-7 b(or)38 b FO(q)44 b FP(=)d FQ(\000)p FP(1,)g(equation)d(\(1.25\))o(,)j FQ(f)p FO(a;)14 b(a)1913 2887 y FN(\003)1951 2917 y FQ(g)41 b FP(=)g FO(I)7 b FP(,)41 b(p)r(ossesses)118 3017 y(irreducible)27 b(one-dimensional)f(and)i(t)n(w)n(o-dimensional)e(solutions.)243 3142 y(F)-7 b(urther,)26 b(in)g(the)h(in)n(terv)-5 b(al)25 b FO(q)h(>)d FQ(\000)p FP(1)i(select)h(the)h(p)r(oin)n(ts)f FO(q)g FP(=)c(0)k(\()p FO(\013)e FP(=)e FQ(\000)p FP(1\),)118 3241 y(and)32 b FO(q)j FP(=)c(1)h(\()p FO(\013)g FP(=)e(0\),)k(corresp) r(onding)d(to)h(the)h(co-isometry)d FO(aa)2200 3211 y FN(\003)2270 3241 y FP(=)g FO(I)40 b FP(and)118 3341 y(CCR)28 b([)p FO(a;)14 b(a)475 3311 y FN(\003)513 3341 y FP(])23 b(=)f FO(I)7 b FP(:)p 188 3423 2317 4 v 186 3523 4 100 v 245 3493 a FO(q)p 340 3523 V 192 w FP(1)23 b FO(<)f(q)p 797 3523 V 192 w(q)k FP(=)d(1)p 1102 3523 V 145 w(0)f FO(<)h(q)j(<)d FP(1)p 1624 3523 V 177 w FO(q)j FP(=)d(0)p 1994 3523 V 138 w FQ(\000)p FP(1)f FO(<)h(q)j(<)c FP(0)p 2503 3523 V 188 3526 2317 4 v 186 3626 4 100 v 238 3596 a FO(\013)p 340 3626 V 100 w FP(0)h FO(<)f(\013)i(<)f FP(1)p 797 3626 V 99 w FO(\013)g FP(=)g(0)p 1102 3626 V 99 w FQ(\000)p FP(1)f FO(<)h(\013)g(<)g FP(0)p 1624 3626 V 99 w FO(\013)g FP(=)g FQ(\000)p FP(1)p 1994 3626 V 168 w FO(\013)h(<)e FQ(\000)p FP(1)p 2503 3626 V 188 3629 2317 4 v 118 3786 a FR(3.)38 b FP(F)-7 b(or)27 b FO(q)g(>)c FP(1,)28 b(the)h(op)r(erator)d FO(a)1148 3756 y FN(\003)1214 3786 y FP(is)i(non-degenerate.)37 b(Let)28 b FO(a)c FP(=)f FO(U)9 b(C)34 b FP(b)r(e)29 b(its)118 3886 y(p)r(olar)20 b(decomp)r(osition)g(suc)n(h)g(that)g(k)n(er)13 b FO(U)32 b FP(=)23 b(k)n(er)12 b FO(C)6 b FP(,)23 b(and)d FO(U)29 b FP(is)21 b(a)f(co-isometry;)118 3985 y FO(C)29 b FQ(\025)23 b FP(0.)37 b(Then)702 4147 y FO(U)9 b(C)833 4113 y FL(2)871 4147 y FO(U)937 4113 y FN(\003)997 4147 y FP(=)23 b FO(q)s(C)6 b(U)1256 4113 y FN(\003)1294 4147 y FO(U)j(C)25 b FP(+)18 b FO(I)30 b FP(=)23 b FO(q)s(C)1786 4113 y FL(2)1842 4147 y FP(+)18 b FO(I)7 b(;)p eop %%Page: 73 77 73 76 bop 118 100 a FK(1.4.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-relations)1130 b FP(73)118 333 y(whic)n(h)28 b(giv)n(es)715 499 y FO(C)780 465 y FL(2)817 499 y FO(U)883 465 y FN(\003)944 499 y FP(=)23 b FO(U)1098 465 y FN(\003)1136 499 y FP(\()p FO(q)s(C)1273 465 y FL(2)1329 499 y FP(+)18 b FO(I)7 b FP(\))24 b(=)e FO(U)1664 465 y FN(\003)1702 499 y FO(f)9 b FP(\()p FO(C)1849 465 y FL(2)1886 499 y FP(\))p FO(;)753 634 y(C)818 600 y FL(2)856 634 y FO(U)31 b FP(=)23 b FO(U)9 b(q)1138 600 y FN(\000)p FL(1)1227 634 y FP(\()p FO(C)1324 600 y FL(2)1380 634 y FQ(\000)18 b FO(I)7 b FP(\))24 b(=)e FO(f)1699 600 y FN(\000)p FL(1)1788 634 y FP(\()p FO(C)1885 600 y FL(2)1923 634 y FP(\))p FO(:)385 b FP(\(1.26\))118 800 y(If)28 b FO(\025)23 b FQ(2)h FO(\033)s FP(\()p FO(C)498 770 y FL(2)536 800 y FP(\),)k(then)g(the)g(p)r(oin)n(ts)f FO(f)9 b FP(\()p FO(\025)p FP(\))24 b(=)f FO(q)s(\025)18 b FP(+)g(1,)27 b FO(f)9 b FP(\()p FO(f)g FP(\()p FO(\025)p FP(\)\))24 b(=)f FO(q)2183 770 y FL(2)2220 800 y FO(\025)18 b FP(+)g FO(q)j FP(+)d(1,)118 900 y FO(:)c(:)g(:)28 b FP(,)g(also)f(b)r(elong)g (to)h FO(\033)s FP(\()p FO(C)975 870 y FL(2)1013 900 y FP(\).)39 b(But)28 b(this)g(set)g(of)g(p)r(oin)n(ts)g(is)g(un)n(b)r (ounded,)h(i.e.,)118 1000 y FC(for)i FO(q)26 b(>)c FP(1)30 b FC(ther)l(e)g(ar)l(e)g(no)f(r)l(epr)l(esentations)h(by)h(b)l(ounde)l (d)f(op)l(er)l(ators)p FP(.)118 1126 y FC(R)l(emark)k FP(22)p FC(.)i FP(F)-7 b(or)22 b FO(q)k(>)c FP(1)g(one)g(can)g (consider)f(a)h(formal)g(un)n(b)r(ounded)g(solution)118 1226 y(of)34 b(\(1.25\))27 b(giv)n(en)g(b)n(y)g(the)h(follo)n(wing)f (Jacobi)f(matrix)547 1750 y FO(a)d FP(=)702 1334 y Fz(0)702 1480 y(B)702 1529 y(B)702 1579 y(B)702 1629 y(B)702 1679 y(B)702 1729 y(B)702 1779 y(B)702 1828 y(B)702 1878 y(B)702 1928 y(B)702 1981 y(@)775 1391 y FP(0)188 b(0)775 1491 y(1)g(0)731 b Fp(0)899 1527 y Fz(q)p 982 1527 170 4 v 992 1586 a FL(1)p FN(\000)p FM(q)1109 1569 y Fy(2)p 992 1604 150 4 v 1008 1651 a FL(1)p FN(\000)p FM(q)1263 1623 y FP(0)1240 1740 y(.)1272 1765 y(.)1304 1790 y(.)1502 1740 y(.)1534 1765 y(.)1566 1790 y(.)1415 1835 y Fz(q)p 1498 1835 179 4 v 1508 1890 a FL(1)p FN(\000)p FM(q)1625 1874 y Fw(n)p 1508 1908 159 4 v 1528 1956 a FL(1)p FN(\000)p FM(q)1787 1927 y FP(0)996 2106 y Fp(0)1764 2048 y FP(.)1796 2073 y(.)1828 2098 y(.)1944 2048 y(.)1976 2073 y(.)2008 2098 y(.)2036 1334 y Fz(1)2036 1480 y(C)2036 1529 y(C)2036 1579 y(C)2036 1629 y(C)2036 1679 y(C)2036 1729 y(C)2036 1779 y(C)2036 1828 y(C)2036 1878 y(C)2036 1928 y(C)2036 1981 y(A)2122 1750 y FO(:)118 2269 y FP(In)32 b(this)h(case,)f(the)h (problem)e(with)i(in)n(tro)r(ducing)e(un)n(b)r(ounded)i(op)r(erators)d (is)118 2368 y(not)k(quite)h(trivial.)57 b(F)-7 b(or)33 b(example,)j(the)f(closure)e(of)h(the)h(follo)n(wing)e(Jacobi)118 2468 y(matrix)314 3037 y FO(a)18 b FP(+)g FO(a)503 3003 y FN(\003)564 3037 y FP(=)652 2546 y Fz(0)652 2693 y(B)652 2742 y(B)652 2792 y(B)652 2842 y(B)652 2892 y(B)652 2942 y(B)652 2991 y(B)652 3041 y(B)652 3091 y(B)652 3141 y(B)652 3191 y(B)652 3241 y(B)652 3290 y(B)652 3344 y(@)725 2616 y FP(0)188 b(1)725 2748 y(1)g(0)1185 2652 y Fz(q)p 1268 2652 170 4 v 1278 2711 a FL(1)p FN(\000)p FM(q)1395 2694 y Fy(2)p 1278 2729 150 4 v 1294 2776 a FL(1)p FN(\000)p FM(q)1967 2748 y Fp(0)849 2828 y Fz(q)p 932 2828 170 4 v 942 2887 a FL(1)p FN(\000)p FM(q)1059 2870 y Fy(2)p 942 2904 150 4 v 958 2952 a FL(1)p FN(\000)p FM(q)1291 2924 y FP(0)1608 2865 y(.)1640 2890 y(.)1672 2916 y(.)1268 3041 y(.)1300 3066 y(.)1332 3091 y(.)1608 3041 y(.)1640 3066 y(.)1672 3091 y(.)1865 3007 y Fz(q)p 1948 3007 179 4 v 1958 3062 a FL(1)p FN(\000)p FM(q)2075 3046 y Fw(n)p 1958 3080 159 4 v 1979 3128 a FL(1)p FN(\000)p FM(q)1521 3186 y Fz(q)p 1604 3186 179 4 v 1614 3241 a FL(1)p FN(\000)p FM(q)1731 3224 y Fw(n)p 1614 3259 159 4 v 1634 3307 a FL(1)p FN(\000)p FM(q)1975 3278 y FP(0)2214 3220 y(.)2246 3245 y(.)2279 3270 y(.)946 3457 y Fp(0)1952 3399 y FP(.)1984 3424 y(.)2017 3449 y(.)2214 3399 y(.)2246 3424 y(.)2279 3449 y(.)2306 2546 y Fz(1)2306 2693 y(C)2306 2742 y(C)2306 2792 y(C)2306 2842 y(C)2306 2892 y(C)2306 2942 y(C)2306 2991 y(C)2306 3041 y(C)2306 3091 y(C)2306 3141 y(C)2306 3191 y(C)2306 3241 y(C)2306 3290 y(C)2306 3344 y(A)118 3622 y FP(de\014ned)31 b(on)f(the)g(set)g(of)h(\014nite)f(v)n(ectors,)g (is)g(not)g(self-adjoin)n(t,)g(but)h(has)f(de\014-)118 3722 y(ciency)d(indices)h(\(1,1\))f(see)55 b([24)o(,)28 b(53)o(].)118 3848 y FR(4.)34 b FP(F)-7 b(or)21 b(0)h FO(<)h(q)j(<)d FP(1,)f(the)g(op)r(erator)d FO(a)1269 3818 y FN(\003)1328 3848 y FP(is)j(also)e(non-degenerate,)h(in)g(the)h (p)r(olar)118 3948 y(decomp)r(osition)d FO(a)k FP(=)g FO(U)9 b(C)d FP(,)22 b FO(U)28 b FP(is)20 b(a)g(co-isometry)-7 b(,)20 b(and)g(equalit)n(y)g(\(1.26\))f(holds.)243 4048 y(If)h FO(\025)k FQ(2)f FO(\033)s FP(\()p FO(C)615 4018 y FL(2)654 4048 y FP(\),)f(then)f(all)f(p)r(oin)n(ts)g FO(f)9 b FP(\()p FO(\025)p FP(\),)23 b FO(f)9 b FP(\()p FO(f)g FP(\()p FO(\025)p FP(\)\),)21 b FO(:)14 b(:)g(:)34 b FP(should)20 b(also)f(b)r(elong)118 4147 y(to)29 b(the)h(sp)r(ectrum) f FO(\033)s FP(\()p FO(C)877 4117 y FL(2)915 4147 y FP(\).)42 b(T)-7 b(o)29 b(the)g(p)r(oin)n(t)g FO(\025)e FP(=)e(\(1)19 b FQ(\000)g FO(q)s FP(\))1912 4117 y FN(\000)p FL(1)2031 4147 y FP(\(the)29 b(stationary)p eop %%Page: 74 78 74 77 bop 118 100 a FP(74)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(p)r(oin)n(t)f(of)f(this)g(mapping\))h(there)f(corresp)r(onds)e (a)i(circle)g(of)g(one-dimensional)118 432 y(op)r(erators)30 b FO(a)g FP(=)g(\(1)21 b FQ(\000)f FO(q)s FP(\))911 402 y FN(\000)p FL(1)p FM(=)p FL(2)1068 432 y FO(e)1107 402 y FM(i\036)1174 432 y FP(,)33 b FO(\036)e FQ(2)f FO(S)1451 402 y FL(1)1488 432 y FP(.)50 b(If)32 b FO(\025)f(>)e FP(\(1)21 b FQ(\000)g FO(q)s FP(\))2074 402 y FN(\000)p FL(1)2164 432 y FP(,)33 b(the)f(corre-)118 532 y(sp)r(onding)27 b(solutions)g(are)g(un)n(b)r(ounded.)118 664 y FC(R)l(emark)48 b FP(23)p FC(.)e FP(There)37 b(exists)g(a)g(series)f(of)h(suc)n(h)g (inequiv)-5 b(alen)n(t)38 b(un)n(b)r(ounded)118 763 y(represen)n (tations)20 b(of)29 b(\(1.25\))21 b(dep)r(ending)i(on)f(a)f(parameter)g FO(\025)j FQ(2)f FP(\()p FO(\025)2207 775 y FL(0)2245 763 y FO(;)14 b(q)s(\025)2370 775 y FL(0)2415 763 y FP(+)8 b(1],)118 863 y FO(\025)166 875 y FL(0)227 863 y FO(>)22 b FP(\(1)d FQ(\000)f FO(q)s FP(\))562 833 y FN(\000)p FL(1)651 863 y FP(.)243 995 y(If)33 b(\(1)23 b FQ(\000)e FO(q)s FP(\))586 964 y FN(\000)p FL(1)708 995 y FO(>)33 b(\025)g(>)f FP(0,)j(then)e FO(\025)g FQ(2)g FO(\033)s FP(\()p FO(C)1594 964 y FL(2)1633 995 y FP(\))g(implies)h(that)g FO(\033)s FP(\()p FO(C)2319 964 y FL(2)2357 995 y FP(\))g(con-)118 1094 y(tains)27 b(all)h(p)r(oin)n(ts)f FO(f)9 b FP(\()p FO(\025)p FP(\),)29 b FO(f)9 b FP(\()p FO(f)g FP(\()p FO(\025)p FP(\)\),)28 b FO(:)14 b(:)g(:)28 b FP(,)f(tending)h(to)f(\(1) 19 b FQ(\000)f FO(q)s FP(\))2051 1064 y FN(\000)p FL(1)2140 1094 y FP(;)28 b(also)e FO(\033)s FP(\()p FO(C)2504 1064 y FL(2)2543 1094 y FP(\))118 1194 y(con)n(tains)34 b(all)g(p)r(oin)n (ts)g FO(f)879 1164 y FN(\000)p FL(1)968 1194 y FP(\()p FO(\025)p FP(\),)k FO(f)1191 1164 y FN(\000)p FL(1)1279 1194 y FP(\()p FO(f)1361 1164 y FN(\000)p FL(1)1450 1194 y FP(\()p FO(\025)p FP(\)\),)e FO(:)14 b(:)g(:)28 b FP(,)36 b(unless)f(this)f(sequence)118 1293 y(con)n(tains)24 b(the)i(zero)e(p)r(oin)n(t.)36 b(Otherwise)24 b(k)n(er)13 b FO(U)32 b FP(=)23 b(k)n(er)12 b FO(C)30 b FP(=)22 b(0,)k(whic)n(h)f (implies)118 1393 y FO(f)168 1363 y FL(\()p FN(\000)p FM(n)p FL(\))317 1393 y FP(\()p FO(\025)p FP(\))35 b FQ(2)g FO(\033)s FP(\()p FO(C)701 1363 y FL(2)739 1393 y FP(\))g(for)f(an)n(y)g FO(n)p FP(.)57 b(But)35 b FO(f)1456 1363 y FL(\()p FN(\000)p FM(m)p FL(\))1623 1393 y FP(\()p FO(\025)p FP(\))g FO(<)f FP(0)g(for)g(some)g FO(m)h FQ(2)g FJ(N)t FP(.)118 1493 y(This)28 b(means)g(that)h(equation)e(\(1.25\))h (has)f(the)i FC(unique)f FP(irreducible)f(in\014nite-)118 1592 y(dimensional)g(represen)n(tation)f(b)n(y)h(b)r(ounded)h(op)r (erators,)547 2125 y FO(a)23 b FP(=)702 1709 y Fz(0)702 1855 y(B)702 1905 y(B)702 1955 y(B)702 2005 y(B)702 2054 y(B)702 2104 y(B)702 2154 y(B)702 2204 y(B)702 2254 y(B)702 2304 y(B)702 2357 y(@)775 1767 y FP(0)188 b(0)775 1866 y(1)g(0)731 b Fp(0)899 1903 y Fz(q)p 982 1903 170 4 v 992 1961 a FL(1)p FN(\000)p FM(q)1109 1945 y Fy(2)p 992 1979 150 4 v 1008 2027 a FL(1)p FN(\000)p FM(q)1263 1998 y FP(0)1240 2116 y(.)1272 2141 y(.)1304 2166 y(.)1502 2116 y(.)1534 2141 y(.)1566 2166 y(.)1415 2211 y Fz(q)p 1498 2211 179 4 v 1508 2266 a FL(1)p FN(\000)p FM(q)1625 2249 y Fw(n)p 1508 2284 159 4 v 1528 2332 a FL(1)p FN(\000)p FM(q)1787 2303 y FP(0)996 2482 y Fp(0)1764 2424 y FP(.)1796 2448 y(.)1828 2474 y(.)1944 2424 y(.)1976 2448 y(.)2008 2474 y(.)2036 1709 y Fz(1)2036 1855 y(C)2036 1905 y(C)2036 1955 y(C)2036 2005 y(C)2036 2054 y(C)2036 2104 y(C)2036 2154 y(C)2036 2204 y(C)2036 2254 y(C)2036 2304 y(C)2036 2357 y(A)2122 2125 y FO(;)118 2662 y FP(with)21 b(a)f(v)-5 b(acuum)21 b(v)n(ector)e FO(e)943 2674 y FL(0)1000 2662 y FP(suc)n(h)i(that)f FO(ae)1436 2674 y FL(0)1496 2662 y FP(=)j(0)d(\(the)h(F)-7 b(o)r(c)n(k)20 b(represen)n(tation\).)243 2762 y(Notice)27 b(that)h(the)g(sp)r(ectrum)g(of)f(the)h(b)r(ounded)g (self-adjoin)n(t)g(op)r(erator)658 3179 y FO(a)19 b FP(+)f FO(a)848 3145 y FN(\003)909 3179 y FP(=)996 2887 y Fz(0)996 3034 y(B)996 3083 y(B)996 3133 y(B)996 3183 y(B)996 3233 y(B)996 3286 y(@)1069 2936 y FP(0)188 b(1)1069 3068 y(1)g(0)1529 2973 y Fz(q)p 1612 2973 170 4 v 1622 3031 a FL(1)p FN(\000)p FM(q)1739 3015 y Fy(2)p 1622 3049 150 4 v 1639 3097 a FL(1)p FN(\000)p FM(q)1193 3149 y Fz(q)p 1276 3149 170 4 v 1286 3207 a FL(1)p FN(\000)p FM(q)1403 3190 y Fy(2)p 1286 3225 150 4 v 1303 3273 a FL(1)p FN(\000)p FM(q)1635 3244 y FP(0)1870 3186 y(.)1902 3211 y(.)1934 3236 y(.)1612 3362 y(.)1644 3387 y(.)1677 3412 y(.)1870 3362 y(.)1902 3387 y(.)1934 3412 y(.)1962 2887 y Fz(1)1962 3034 y(C)1962 3083 y(C)1962 3133 y(C)1962 3183 y(C)1962 3233 y(C)1962 3286 y(A)118 3600 y FP(is)28 b(concen)n(trated)e(on)h(the)h(in)n(terv) -5 b(al)27 b([)p FQ(\000)p FP(2)p FO(=)p FP(\(1)17 b FQ(\000)h FO(q)s FP(\))p FO(;)c FP(2)p FO(=)p FP(\(1)k FQ(\000)g FO(q)s FP(\)].)118 3749 y FR(5.)45 b FP(F)-7 b(or)30 b FO(q)h FP(=)d(0,)j(w)n(e)f(ha)n(v)n(e)g FO(aa)1055 3719 y FN(\003)1121 3749 y FP(=)d FO(I)7 b FP(,)32 b(i.e.,)f FO(a)g FP(is)f(a)g(co-isometry)-7 b(.)45 b(Irreducible)118 3848 y(represen)n(tations)21 b(are)g(the)i(follo)n(wing:)33 b(a)22 b(circle)g(of)g(one-dimensional)f(unitary)118 3948 y(op)r(erators)37 b FO(a)43 b FP(=)f FO(e)730 3918 y FM(i\036)797 3948 y FP(,)g FO(\036)h FQ(2)g FO(S)1108 3918 y FL(1)1145 3948 y FP(,)f(and)d(a)g(single)g(op)r(erator)e(adjoin) n(t)i(to)g(the)118 4048 y(unilateral)29 b(shift)i(op)r(erator)e(in)h FO(l)1146 4060 y FL(2)1183 4048 y FP(\()p FJ(Z)1277 4060 y FL(+)1326 4048 y FP(\).)45 b(In)30 b(this)g(case,)g(an)n(y)f (isometry)h(in)g FO(H)118 4147 y FP(giv)n(es)22 b(rise)h(to)g(a)g (unique)g(decomp)r(osition)g(of)g FO(H)30 b FP(in)n(to)23 b(in)n(v)-5 b(arian)n(t)22 b(with)i(resp)r(ect)p eop %%Page: 75 79 75 78 bop 118 100 a FK(1.4.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-relations)1130 b FP(75)118 333 y(to)19 b FO(a)g FP(and)f FO(a)470 303 y FN(\003)528 333 y FP(subspaces,)h FO(H)30 b FP(=)23 b FO(H)1179 345 y FL(uni)1276 333 y FQ(\010)q FO(H)1411 345 y FL(shift)1562 333 y FP(suc)n(h)c(that)g FO(a)k Fs(\026)2014 345 y FM(H)2068 353 y Fy(uni)2174 333 y FP(is)c(a)f(unitary)118 432 y(op)r(erator,)38 b(and)e FO(a)j Fs(\026)776 444 y FM(H)830 453 y Fy(shift)987 432 y FP(is)d(unitarily)h(equiv)-5 b(alen)n(t)37 b(to)g(a)f(m)n (ultiple)i(of)e(the)118 532 y(op)r(erator)31 b(adjoin)n(t)i(to)g(the)g (unilateral)f(shift)i(\(H.)f(W)-7 b(old's)33 b(decomp)r(osition\).)118 632 y(One)27 b(can)h(easily)e(obtain)i(this)g(decomp)r(osition)f (putting)825 884 y FO(H)894 896 y FL(shift)1049 884 y FP(=)1164 780 y FN(1)1151 805 y Fz([)1137 984 y FM(k)q FL(=0)1258 884 y FP(\()p FO(a)1334 850 y FN(\003)1372 884 y FP(\))1404 850 y FM(k)1445 884 y FP(\()p FO(H)f FQ(\011)18 b FO(a)1699 850 y FN(\003)1737 884 y FO(H)7 b FP(\))p FO(:)118 1154 y FR(6.)36 b FP(F)-7 b(or)27 b FQ(\000)p FP(1)22 b FO(<)h(q)j(<)d FP(0,)k(w)n(e)g(ha)n(v)n(e)g(the)h (follo)n(wing:)243 1257 y(a\))d(a)g(circle)g(of)g(one-dimensional)f (represen)n(tations)f FO(a)g FP(=)g(\(1)13 b FQ(\000)h FO(q)s FP(\))2289 1227 y FN(\000)p FL(1)p FM(=)p FL(2)2446 1257 y FO(e)2485 1227 y FM(i\036)2552 1257 y FP(,)118 1357 y FO(\036)24 b FQ(2)f FO(S)325 1327 y FL(1)362 1357 y FP(;)243 1461 y(b\))28 b(the)g(F)-7 b(o)r(c)n(k)27 b(represen)n(tation)f(b)n(y)h(b)r(ounded)h(op)r(erators)547 2009 y FO(a)23 b FP(=)702 1593 y Fz(0)702 1739 y(B)702 1789 y(B)702 1839 y(B)702 1888 y(B)702 1938 y(B)702 1988 y(B)702 2038 y(B)702 2088 y(B)702 2137 y(B)702 2187 y(B)702 2240 y(@)775 1650 y FP(0)188 b(0)775 1750 y(1)g(0)740 b(0)899 1787 y Fz(q)p 982 1787 170 4 v 992 1845 a FL(1)p FN(\000)p FM(q)1109 1828 y Fy(2)p 992 1863 150 4 v 1008 1911 a FL(1)p FN(\000)p FM(q)1263 1882 y FP(0)1240 2000 y(.)1272 2025 y(.)1304 2050 y(.)1502 2000 y(.)1534 2025 y(.)1566 2050 y(.)1005 2187 y(0)1415 2094 y Fz(q)p 1498 2094 179 4 v 1508 2150 a FL(1)p FN(\000)p FM(q)1625 2133 y Fw(n)p 1508 2168 159 4 v 1528 2215 a FL(1)p FN(\000)p FM(q)1787 2187 y FP(0)1764 2307 y(.)1796 2332 y(.)1828 2357 y(.)1944 2307 y(.)1976 2332 y(.)2008 2357 y(.)2036 1593 y Fz(1)2036 1739 y(C)2036 1789 y(C)2036 1839 y(C)2036 1888 y(C)2036 1938 y(C)2036 1988 y(C)2036 2038 y(C)2036 2088 y(C)2036 2137 y(C)2036 2187 y(C)2036 2240 y(A)2122 2009 y FO(:)118 2552 y FP(There)23 b(are)f(no)h(represen)n(tations)f(b) n(y)h(un)n(b)r(ounded)g(op)r(erators,)g(since)g(for)g(eac)n(h)118 2652 y FQ(\000)p FO(q)223 2622 y FN(\000)p FL(1)335 2652 y FO(<)g(\025)g FQ(2)g FO(\033)s FP(\()p FO(C)719 2622 y FL(2)758 2652 y FP(\))28 b(the)g(p)r(oin)n(t)f FO(f)1227 2622 y FN(\000)p FL(1)1316 2652 y FP(\()p FO(\025)p FP(\))d FO(<)f FP(0)k(also)g(b)r(elongs)f(to)i FO(\033)s FP(\()p FO(C)2322 2622 y FL(2)2360 2652 y FP(\).)118 2889 y FR(1.4.3)94 b(Real)31 b(quan)m(tum)h(plane)f(and)i(real)f(quan)m(tum)f(h)m(yp)s (erb)s(oloid)118 3050 y(1.)37 b FP(Consider,)27 b(\014nally)-7 b(,)28 b(pairs)f(of)h(self-adjoin)n(t)f(op)r(erators)f(whic)n(h)i (satisfy)f(the)118 3150 y(relations)g(\()p FO(V)19 b(I)7 b(I)634 3162 y FL(0)671 3150 y FP(\))28 b(and)g(\()p FO(V)19 b(I)7 b(I)1071 3162 y FL(1)1109 3150 y FP(\),)550 3341 y([)p FO(A;)14 b(B)t FP(])24 b(=)f FO(i\013)p FP(\()p FO(A)1050 3306 y FL(2)1106 3341 y FQ(\000)18 b FO(B)1256 3306 y FL(2)1293 3341 y FP(\))p FO(;)181 b(\013)23 b(>)g FP(0)p FO(;)550 3475 y FP([)p FO(A;)14 b(B)t FP(])24 b(=)f FO(i\013)p FP(\()p FO(A)1050 3441 y FL(2)1106 3475 y FQ(\000)18 b FO(B)1256 3441 y FL(2)1293 3475 y FP(\))h(+)f FO(iI)7 b(;)180 b(\013)23 b FQ(2)g FJ(R)i FQ(n)18 b(f)p FP(0)p FQ(g)p FO(:)243 3670 y FP(Represen)n(tations)24 b(of)h(relations)f(\()p FO(V)c(I)7 b(I)1451 3682 y FL(0)1488 3670 y FP(\))26 b(and)f(\()p FO(V)19 b(I)7 b(I)1883 3682 y FL(1)1921 3670 y FP(\))26 b(b)n(y)f(b)r(ounded)h(self-)118 3770 y(adjoin)n(t)h(op)r(erators)f(can)h(b)r(e)h(obtained)g(from)f(the) h(follo)n(wing)e(theorem.)118 3948 y FR(Theorem)k(14.)41 b FP(\(B.)27 b(F)-7 b(uglede,)28 b(C.R.)f(Putnam,)h(M.)f(Rosen)n (blum\))p FC(.)39 b(L)l(et)29 b FO(M)9 b FC(,)118 4048 y FO(N)g FC(,)25 b FO(T)34 b FQ(2)23 b FO(L)p FP(\()p FO(H)7 b FP(\))p FC(,)25 b(and)f(the)g(op)l(er)l(ators)g FO(M)32 b FC(and)24 b FO(N)32 b FC(b)l(e)23 b(normal.)37 b(If)24 b FO(M)9 b(T)34 b FP(=)23 b FO(T)12 b(N)d FC(,)118 4147 y(then)30 b FO(M)393 4117 y FN(\003)430 4147 y FO(T)35 b FP(=)22 b FO(T)12 b(N)738 4117 y FN(\003)805 4147 y FC(as)30 b(wel)t(l.)p eop %%Page: 76 80 76 79 bop 118 100 a FP(76)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FC(Pr)l(o)l(of.)43 b FP(The)37 b(pro)r(of)g(is)g(based)f(on)h (the)g(unitarit)n(y)g(of)g(the)g(op)r(erator-v)-5 b(alued)118 432 y(functions)22 b(exp\()p FO(z)t(M)762 402 y FN(\003)806 432 y FQ(\000)12 b FP(\026)-47 b FO(z)s(M)9 b FP(\))22 b(and)f(exp\()p FQ(\000)p FO(z)t(N)1562 402 y FN(\003)1606 432 y FP(+)12 b(\026)-47 b FO(z)s(N)9 b FP(\).)35 b(Since)22 b FO(M)2187 402 y FM(k)2227 432 y FO(T)35 b FP(=)22 b FO(T)12 b(N)2535 402 y FM(k)118 532 y FP(for)38 b(all)h FO(k)45 b FP(=)c(1,)g(2,)e FO(:)14 b(:)g(:)27 b FP(,)42 b(and)d(exp\()5 b(\026)-47 b FO(z)s(M)9 b FP(\))p FO(T)53 b FP(=)41 b FO(T)25 b FP(exp\()5 b(\026)-47 b FO(z)s(N)9 b FP(\),)42 b FQ(8)p FO(z)i FQ(2)f FJ(C)14 b FP(,)48 b(w)n(e)118 632 y(conclude)27 b(that)h(for)f(the)h(b)r(ounded)g(en)n (tire)g(op)r(erator-v)-5 b(alued)25 b(function)j FO(F)12 b FP(\()p FQ(\001)p FP(\))378 816 y FO(F)g FP(\()p FO(z)t FP(\))22 b(=)h(exp\()p FO(z)t(M)952 782 y FN(\003)1008 816 y FQ(\000)g FP(\026)-47 b FO(z)s(M)9 b FP(\))p FO(T)25 b FP(exp)o(\()5 b(\026)-47 b FO(z)t(N)27 b FQ(\000)18 b FO(z)t(N)1826 782 y FN(\003)1863 816 y FP(\))572 940 y(=)23 b(exp\()p FO(z)t(M)952 906 y FN(\003)989 940 y FP(\)\(exp)q(\()p FQ(\000)5 b FP(\026)-47 b FO(z)s(M)9 b FP(\)\))p FO(T)25 b FP(exp\()5 b(\026)-47 b FO(z)s(N)9 b FP(\)\))14 b(exp\()p FQ(\000)p FO(z)t(N)2246 906 y FN(\003)2283 940 y FP(\))572 1065 y(=)23 b FO(F)12 b FP(\(0\))23 b(=)g FO(T)12 b FP(;)118 1249 y(therefore,)33 b(exp\()p FO(z)t(M)788 1219 y FN(\003)826 1249 y FP(\))p FO(T)43 b FP(=)31 b FO(T)25 b FP(exp\()p FO(z)t(N)1398 1219 y FN(\003)1435 1249 y FP(\))34 b(for)e(all)h FO(z)i FQ(2)d FJ(C)15 b FP(.)59 b(Th)n(us,)34 b FO(M)2381 1219 y FN(\003)2418 1249 y FO(T)43 b FP(=)118 1349 y FO(T)12 b(N)255 1319 y FN(\003)292 1349 y FP(.)p 2514 1349 4 57 v 2518 1296 50 4 v 2518 1349 V 2567 1349 4 57 v 118 1522 a FR(2.)36 b FP(Relation)28 b(\()p FO(V)19 b(I)7 b(I)741 1534 y FL(0)778 1522 y FP(\),)743 1706 y([)p FO(A;)14 b(B)t FP(])24 b(=)e FO(i\013)p FP(\()p FO(A)1242 1672 y FL(2)1298 1706 y FQ(\000)d FO(B)1449 1672 y FL(2)1486 1706 y FP(\))p FO(;)180 b(\013)24 b(>)e FP(0)p FO(;)118 1891 y FP(can)27 b(b)r(e)h(rewritten)g(in)f(the)h(form)577 2095 y FO(X)7 b(Y)41 b FP(=)23 b FO(q)s(Y)18 b(X)r(;)180 b(q)26 b FP(=)1371 2039 y(1)18 b FQ(\000)g FO(i\013)p 1371 2076 226 4 v 1371 2152 a FP(1)g(+)g FO(i\013)1629 2095 y FQ(6)p FP(=)k(1)p FO(;)97 b FQ(j)p FO(q)s FQ(j)23 b FP(=)g(1)118 2319 y(\(real)38 b(quan)n(tum)h(sphere)f FJ(R)1009 2289 y FL(2)1009 2340 y FM(q)1052 2319 y FP(\);)44 b(here,)d FO(X)48 b FP(=)41 b FO(A)26 b FP(+)f FO(B)t FP(,)42 b FO(Y)60 b FP(=)41 b FO(A)26 b FQ(\000)f FO(B)t FP(.)70 b(F)-7 b(or)118 2419 y(b)r(ounded)39 b(self-adjoin)n(t)f(op)r (erators,)h(w)n(e)f(also)f(ha)n(v)n(e)g(due)i(to)f(the)g(F)-7 b(uglede{)118 2519 y(Putnam{Rosen)n(blum)27 b(theorem:)1113 2703 y FO(X)7 b(Y)41 b FP(=)29 b(\026)-48 b FO(q)17 b(Y)h(X)r(;)118 2887 y FP(whic)n(h)28 b(implies,)g(since)g FO(q)f FQ(6)p FP(=)j(\026)-49 b FO(q)s FP(,)29 b(that)f FO(X)7 b(Y)42 b FP(=)23 b FO(Y)c(X)30 b FP(=)23 b(0,)28 b(i.e.,)g([)p FO(A;)14 b(B)t FP(])24 b(=)f(0)28 b(and)118 2987 y FO(A)180 2957 y FL(2)241 2987 y FP(=)22 b FO(B)395 2957 y FL(2)433 2987 y FP(.)243 3087 y(Therefore,)32 b(all)g(irreducible)f(represen)n (tations)g(of)h(\()p FO(V)19 b(I)7 b(I)2041 3099 y FL(0)2079 3087 y FP(\))33 b(b)n(y)e(b)r(ounded)118 3187 y(op)r(erators)26 b(are)g(one-dimensional.)118 3339 y FR(3.)36 b FP(No)n(w)27 b(consider)g(b)r(ounded)h(pairs)f(satisfying)f(\()p FO(V)20 b(I)7 b(I)1833 3351 y FL(1)1870 3339 y FP(\),)532 3524 y FQ(\000)p FO(i)p FP([)p FO(A;)14 b(B)t FP(])23 b(=)g FO(\013)p FP(\()p FO(A)1096 3489 y FL(2)1152 3524 y FQ(\000)18 b FO(B)1302 3489 y FL(2)1340 3524 y FP(\))h(+)f FO(I)7 b(;)180 b(\013)23 b FQ(2)g FJ(R)i FQ(n)18 b(f)p FP(0)p FQ(g)p FO(:)118 3708 y FP(F)-7 b(or)27 b(the)h(self-adjoin)n(t)f(op)r (erators)f FO(X)j FP(=)23 b FO(A)18 b FP(+)h FO(B)t FP(,)27 b FO(Y)42 b FP(=)23 b FO(A)18 b FQ(\000)g FO(B)t FP(,)28 b(w)n(e)g(ha)n(v)n(e)247 3928 y FO(X)7 b(Y)41 b FP(=)22 b FO(q)s(Y)d(X)25 b FQ(\000)18 b FO(i)p FP(\()p FO(q)j FP(+)d(1\))p FO(I)7 b(;)180 b(q)26 b FP(=)1466 3872 y(1)18 b FQ(\000)g FO(\013i)p 1466 3909 V 1466 3985 a FP(1)g(+)g FO(\013i)1724 3928 y FQ(6)p FP(=)k(1)p FO(;)97 b FQ(j)p FO(q)s FQ(j)23 b FP(=)g(1)p FO(:)128 b FP(\(1.27\))118 4147 y(A)20 b FQ(\003)p FP(-algebra)d(generated)h(b)n(y)j(\(1.27\))e (is)g(called)g(a)g(real)g(quan)n(tum)h(h)n(yp)r(erb)r(oloid.)p eop %%Page: 77 81 77 80 bop 118 100 a FK(Commen)n(ts)27 b(to)h(Chapter)f(1)1494 b FP(77)243 333 y(F)-7 b(or)27 b(b)r(ounded)h(op)r(erators,)d(\(1.27\)) i(implies)686 519 y FO(X)7 b FP(\()p FO(X)g(Y)36 b FP(+)18 b FO(\013)1090 484 y FN(\000)p FL(1)1180 519 y FO(I)7 b FP(\))23 b(=)g FO(q)17 b FP(\()p FO(X)7 b(Y)36 b FP(+)18 b FO(\013)1748 484 y FN(\000)p FL(1)1838 519 y FO(I)7 b FP(\))p FO(X)r(;)118 705 y FP(and,)27 b(due)f(to)h(the)g(F)-7 b(uglede{Putnam{Rosen)n(blum)24 b(theorem,)j(w)n(e)f(also)f(ha)n(v)n(e) 686 891 y FO(X)7 b FP(\()p FO(X)g(Y)36 b FP(+)18 b FO(\013)1090 856 y FN(\000)p FL(1)1180 891 y FO(I)7 b FP(\))23 b(=)29 b(\026)-48 b FO(q)17 b FP(\()p FO(X)7 b(Y)36 b FP(+)18 b FO(\013)1748 856 y FN(\000)p FL(1)1838 891 y FO(I)7 b FP(\))p FO(X)r(;)118 1077 y FP(whic)n(h)28 b(is)f(p)r(ossible)g(only) h(if)887 1263 y FO(X)7 b(Y)18 b(X)29 b FP(=)23 b FO(X)1291 1228 y FL(2)1328 1263 y FO(Y)41 b FP(=)23 b FQ(\000)p FO(\013)1623 1228 y FN(\000)p FL(1)1712 1263 y FO(X)r(:)557 b FP(\(1.28\))118 1449 y(By)30 b(\(1.28\))o(,)g FO(H)585 1461 y FL(0)649 1449 y FP(=)25 b(k)n(er)13 b FO(X)35 b FP(is)30 b(in)n(v)-5 b(arian)n(t)28 b(under)h FO(A)p FP(,)h FO(B)t FP(;)h(but)f(then)f(b)n(y)i(\(1.27\))o(,)118 1548 y(w)n(e)23 b(get)g FO(H)439 1560 y FL(0)500 1548 y FP(=)f FQ(f)p FP(0)p FQ(g)p FP(.)34 b(On)23 b(the)h(subspace)f FO(H)1462 1518 y FN(?)1455 1569 y FL(0)1518 1548 y FP(,)h(the)g(op)r (erator)d FO(X)30 b FP(is)24 b(in)n(v)n(ertible,)118 1648 y(and)j FO(X)7 b(Y)41 b FP(=)23 b FO(Y)c(X)7 b FP(.)36 b(Th)n(us)27 b(w)n(e)g(ha)n(v)n(e)f([)p FO(A;)14 b(B)t FP(])24 b(=)e(0,)27 b(and)g FO(\013)p FP(\()p FO(A)1980 1618 y FL(2)2037 1648 y FQ(\000)17 b FO(B)2186 1618 y FL(2)2224 1648 y FP(\))h(+)g FO(I)30 b FP(=)22 b(0.)243 1749 y(Therefore,)27 b(irreducible)h(represen)n(tations)e(of)j(\()p FO(V)19 b(I)7 b(I)1905 1761 y FL(1)1943 1749 y FP(\))28 b(b)n(y)h(b)r(ounded)f(op-)118 1849 y(erators)e(are)g(one-dimensional.) 118 1985 y FC(R)l(emark)49 b FP(24)p FC(.)f FP(F)-7 b(or)38 b(a)g(p)r(ossible)g(de\014nition)h(of)g(in)n(tegrable)e(pairs)h(of)g (un)n(b)r(o-)118 2085 y(unded)33 b(pairs)e(of)h(self-adjoin)n(t)g(op)r (erators)e(satisfying)h(\()p FO(V)19 b(I)7 b(I)2035 2097 y FL(0)2073 2085 y FP(\))33 b(and)f(\()p FO(V)19 b(I)7 b(I)2482 2097 y FL(1)2519 2085 y FP(\),)118 2184 y(and)28 b(the)g(structure)f(of)g(suc)n(h)g(pairs,)g(see)g([250)o(,)h(251)n(].) 118 2435 y FH(Commen)m(ts)36 b(to)h(Chapter)h(1)118 2620 y FR(Section)32 b(1.1.)243 2721 y FP(1.1.1.)54 b(W)-7 b(e)34 b(giv)n(e)f(basic)g(de\014nitions)h(and)g(prop)r(erties)f(of)h FQ(\003)p FP(-represen)n(ta-)118 2820 y(tions)29 b(of)f FQ(\003)p FP(-algebras)e(b)n(y)i(b)r(ounded)h(op)r(erators,)e(i.e.,)i FQ(\003)p FP(-homomorphisms)e(of)118 2920 y FQ(\003)p FP(-algebras)e(in)n(to)i(the)h FQ(\003)p FP(-algebra)d(of)j(op)r (erators)d(on)j(a)f(Hilb)r(ert)h(space)f FO(H)7 b FP(.)243 3021 y(Notice)36 b(that)g(in)h(the)f(b)r(o)r(ok)g(w)n(e)g(do)g(not)g (consider)f FQ(\003)p FP(-homomorphisms)118 3121 y(in)n(to)26 b(the)h FQ(\003)p FP(-algebra)c(of)k(op)r(erators)d(on)i(spaces)f(with) i(inde\014nite)g(metric.)36 b(F)-7 b(or)118 3221 y(suc)n(h)23 b(represen)n(tations,)f(see,)h(e.g.,)h([97)o(,)f(14)o(,)g(98)o(,)g(256) n(,)g(257)o(,)g(172)n(,)g(136)o(],)h(as)e(w)n(ell)118 3320 y(as)27 b(the)h(bibliograph)n(y)e(cited)i(there.)243 3448 y(1.1.2.)80 b(Since)43 b(in)g(this)f(b)r(o)r(ok,)k(our)c(main)h (concern)e(is)h(the)h(represen-)118 3548 y(tation)35 b(theory)f(of)h FQ(\003)p FP(-algebras,)f(the)h(problems)g(of)f FO(C)1845 3518 y FN(\003)1884 3548 y FP(-represen)n(tabilit)n(y)f(of) 118 3647 y FQ(\003)p FP(-algebras)38 b(is)j(v)n(ery)f(imp)r(ortan)n(t)g (in)i(what)f(follo)n(ws.)75 b(But)42 b(it)f(seems)g(that)118 3747 y(there)23 b(has)g(not)g(b)r(een)h(m)n(uc)n(h)f(w)n(ork)f(on)h (the)g(dev)n(elopmen)n(t)g(of)g(these)h(problems)118 3847 y(\(see)k([161)n(])g(and)f(the)h(bibliograph)n(y)e(therein\).)243 3948 y(The)38 b(fact)h(that)g(in)h(Prop)r(osition)d(5)h(the)h (implications)g(\()p FO(i)p FP(\))j FQ(\))f FP(\()p FO(ii)p FP(\))h FQ(\))118 4048 y FP(\()p FO(iii)p FP(\))23 b FQ(\))g FP(\()p FO(iv)s FP(\))e(hold)f(is)g(easy)-7 b(.)34 b(Coun)n(terexamples)19 b(to)h(the)h(implications)f(\()p FO(ii)p FP(\))j FQ(\))118 4147 y FP(\()p FO(i)p FP(\))34 b(and)g(\()p FO(iii)p FP(\))f FQ(\))g FP(\()p FO(ii)p FP(\))h(ha)n(v)n(e)f(b)r(een)h(constructed)f(b)n(y)h(S.)g(P)n(op)r(o)n (vyc)n(h.)53 b(F)-7 b(or)33 b(a)p eop %%Page: 78 82 78 81 bop 118 100 a FP(78)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(coun)n(terexample)32 b(to)i(\()p FO(iv)s FP(\))g FQ(\))f FP(\()p FO(iii)p FP(\),)h(see)g([299)n(,)g(72)o(])g(and)g(the)g (bibliograph)n(y)118 432 y(therein.)243 534 y(The)21 b(exp)r(osition)g(of)g(questions)g(ab)r(out)g(residual)g(family)g(of)h (\014nite-dimen-)118 634 y(sional)27 b(represen)n(tations)e(of)j(a)f (group)g FO(C)1394 604 y FN(\003)1432 634 y FP(-algebra)f(follo)n(ws)g ([57)o(].)243 736 y(F)-7 b(or)33 b(Remark)g(3,)i(see)f(the)g(related)f (bibliograph)n(y:)48 b([56)o(,)34 b(219)n(,)g(107)o(])g(and)118 835 y(the)28 b(bibliograph)n(y)e(giv)n(en)h(there.)243 964 y(1.1.3.)34 b(In)24 b(some)f(cases,)g(the)h(represen)n(tation)e (theory)h(of)g(a)g FQ(\003)p FP(-algebra)e(can)118 1064 y(b)r(e)29 b(reduced)f(to)h(that)f(of)h(one)f(of)g(its)h(en)n(v)n (eloping)e FQ(\003)p FP(-algebra,)f FO(\033)s FP(-)p FO(C)2222 1034 y FN(\003)2261 1064 y FP(-algebra,)118 1163 y(or)35 b FO(C)293 1133 y FN(\003)331 1163 y FP(-algebra)e(\(and)j (vice)f(v)n(ersa\).)59 b(F)-7 b(or)34 b(a)h(\014nitely)h(generated)e FQ(\003)p FP(-algebra)118 1263 y FA(A)p FP(,)k(its)e(en)n(v)n(eloping)e FO(C)844 1233 y FN(\003)883 1263 y FP(-algebra)f(exists)j(and)g(is)f (unique)h(if)h(and)e(only)h(if)g FA(A)118 1363 y FP(is)j FQ(\003)p FP(-b)r(ounded)f(\(see,)j(e.g.,)g([69)o(,)e(133)o(]\).)70 b(The)39 b FQ(\003)p FP(-algebra)d(generated)h(b)n(y)h(a)118 1462 y(pair)29 b(of)g(orthogonal)e(pro)5 b(jections,)28 b(whic)n(h)i(w)n(as)e(considered)g(in)i(Example)e(3,)118 1562 y(is)h FQ(\003)p FP(-b)r(ounded.)39 b(F)-7 b(or)28 b(an)g(in)n(v)n(estigation)f(of)i(its)g(en)n(v)n(eloping)e FO(C)2087 1532 y FN(\003)2125 1562 y FP(-algebra,)g(see)118 1661 y([286)o(,)g(234)o(])h(and)f(others.)243 1763 y(If)k(a)g (\014nitely)g(generated)f FQ(\003)p FP(-algebra)e(is)j(not)g FQ(\003)p FP(-b)r(ounded,)g(then)h(one)e(can)118 1863 y(construct)25 b(an)h(en)n(v)n(eloping)e(pro-)p FO(C)1215 1833 y FN(\003)1252 1863 y FP(-algebra)g(\(see)h([8,)h(10)o(,)f(208)o (])h(and)f(the)h(ref-)118 1963 y(erences)33 b(therein\))i(whic)n(h)f(p) r(ossesses)f(man)n(y)h(useful)h(prop)r(erties)e(of)h(the)h(en-)118 2062 y(v)n(eloping)27 b FO(C)511 2032 y FN(\003)549 2062 y FP(-algebra.)35 b(The)27 b(exp)r(osition)h(of)f(Theorem)g(1)g(follo)n (ws)g([213)n(].)243 2191 y(1.1.4.)33 b(F)-7 b(or)21 b(\014nitely)h (generated)e FQ(\003)p FP(-algebras,)g(w)n(e)h(set)g(a)g(natural)g (language)118 2291 y(of)28 b(represen)n(tation)e(of)h(generators)e(and) j(relations.)243 2393 y(The)c(sp)r(ectral)g(theorem)g(for)h(a)f(single) g(self-adjoin)n(t)g(op)r(erator)f FO(A)g FP(=)g FO(A)2449 2362 y FN(\003)2510 2393 y FP(=)118 2440 y Fz(R)173 2461 y FN(k)p FM(A)p FN(k)157 2536 y(\000k)p FM(A)p FN(k)345 2507 y FO(\025)14 b(dE)5 b FP(\()p FO(\025)p FP(\))42 b(giv)n(es)d(a)h(decomp)r(osition)g(of)g(the)h(op)r(erator)d FO(A)j FP(in)n(to)f(irre-)118 2617 y(ducible)31 b(op)r(erators)e(of)i (m)n(ultiplication)g(b)n(y)f FO(\025)f FQ(2)g FJ(R)p FP(.)52 b(It)31 b(is)g(a)g(standard)e(part)118 2717 y(of)36 b(a)g(univ)n(ersit)n(y)f(course)g(and)h(is)h(exp)r(ounded)f(in)g (detail)h(in)f(textb)r(o)r(oks)g(on)118 2816 y(sp)r(ectral)27 b(theory)g([4,)g(241)o(,)h(235)n(,)g(37)o(,)g(29)o(],)f(etc.)243 2918 y(P)n(airs)19 b(of)i(self-adjoin)n(t)f(op)r(erators)f FO(A)j FP(and)f FO(B)k FP(ha)n(v)n(e)20 b(a)g(m)n(uc)n(h)h(more)f (compli-)118 3018 y(cated)27 b(structure.)36 b(W)-7 b(e)27 b(discuss)f(the)h(complexit)n(y)f(of)h(their)g(unitary)f(descrip-)118 3117 y(tion)h(in)g(Section)g(3.1.2.)35 b(F)-7 b(or)27 b(a)f(unitary)g(reduction)h(of)g(a)f(pair)g(of)h(Hermitian)118 3217 y(matrices,)g(see)g([254)o(,)h(259)n(])g(and)f(the)h(bibliograph)n (y)e(therein.)243 3319 y(The)f(sp)r(ectral)g(theorem)g(for)f(a)h(pair)g (of)g FC(c)l(ommuting)32 b FP(self-adjoin)n(t)25 b(op)r(era-)118 3418 y(tors)g(\(see,)h(e.g.,)g([37)o(,)g(29)o(],)g(etc.\))37 b(giv)n(es)24 b(a)h(decomp)r(osition)g(of)h(a)f(pair)g(of)h(op)r(er-) 118 3518 y(ators,)31 b FO(A)416 3530 y FL(1)483 3518 y FP(=)576 3451 y Fz(R)616 3548 y Fv(R)663 3531 y Fy(2)708 3518 y FO(\025)756 3530 y FL(1)807 3518 y FO(dE)911 3533 y FL(\()p FM(A)987 3541 y Fy(1)1020 3533 y FM(;A)1090 3541 y Fy(2)1122 3533 y FL(\))1152 3518 y FP(\()p FO(\025)1232 3530 y FL(1)1270 3518 y FO(;)14 b(\025)1355 3530 y FL(2)1393 3518 y FP(\),)33 b FO(A)1543 3530 y FL(2)1609 3518 y FP(=)1703 3451 y Fz(R)1742 3548 y Fv(R)1789 3531 y Fy(2)1834 3518 y FO(\025)1882 3530 y FL(2)1934 3518 y FO(dE)2038 3533 y FL(\()p FM(A)2114 3541 y Fy(1)2147 3533 y FM(;A)2217 3541 y Fy(2)2249 3533 y FL(\))2279 3518 y FP(\()p FO(\025)2359 3530 y FL(1)2397 3518 y FO(;)14 b(\025)2482 3530 y FL(2)2519 3518 y FP(\),)118 3618 y(in)n(to)29 b(irreducible)f(pairs)f(of)i(op)r (erators)e(of)h(m)n(ultiplication)h(b)n(y)f FO(\025)2162 3630 y FL(1)2229 3618 y FP(and)g FO(\025)2439 3630 y FL(2)2506 3618 y FP(in)118 3717 y(a)f(one-dimensional)g(space.)243 3819 y(The)k(pro)r(of)f(that)h(there)g(are)f(no)h(b)r(ounded)g (self-adjoin)n(t)g(op)r(erators)e(sat-)118 3919 y(isfying)e(CCR)h (follo)n(ws)f([300)n(])h(\(see)g(also)e([104)o(,)h(235)o(],)h(etc.\).) 243 4048 y(1.1.5.)34 b(F)-7 b(or)22 b(a)g(reduction)g(of)h(algebras)e (\(without)i(an)f(in)n(v)n(olution\))h(de\014ned)118 4147 y(b)n(y)34 b(a)f(pair)g(of)h(generators)d(that)j(are)f(related)g (b)n(y)h(a)f(quadratic)g(relation)g(to)p eop %%Page: 79 83 79 82 bop 118 100 a FK(Commen)n(ts)27 b(to)h(Chapter)f(1)1494 b FP(79)118 333 y(a)31 b(canonical)f(form,)j(see,)f(e.g.,)g([292)n(].) 49 b(Canonical)30 b(forms)h(for)g(pairs)g(of)g(self-)118 432 y(adjoin)n(t)g(op)r(erators)d(\()p FQ(\003)p FP(-algebras)g(with)k (a)e(pair)g(of)h(self-adjoin)n(t)f(generators\))118 532 y(whic)n(h)j(satisfy)f(a)g(quadratic)g(relation)f(\(\\non-comm)n (utativ)n(e)g(conics")h(on)g(a)118 632 y(real)f(plane\))h(can)g(b)r(e)g (found)g(in,)h(e.g.,)g([196)o(,)f(200)n(],)h(etc.)50 b(Newton's)32 b(classi\014-)118 731 y(cation)f([183)n(])h(\(see)f(also) f([271)n(])i(etc.\))48 b(of)31 b(third-degree)f(curv)n(es)g (discouraged)118 831 y(the)d(authors)f(to)h(in)n(v)n(estigate)e(the)i (problem)g(of)g(classi\014cation)e(of)i FQ(\003)p FP(-algebras)118 930 y(with)h(t)n(w)n(o)f(generators)e(and)j(one)f(cubic)h(relation.)243 1081 y(The)h(remaining)g(part)g(of)h(this)g(section)f(is)g(dev)n(oted)g (to)h(represen)n(tations)118 1180 y(of)d(\\non-comm)n(utativ)n(e)d (curv)n(es)i(of)g(the)h(second)f(degree)g(on)g(the)h(real)f(plane")118 1280 y(b)n(y)32 b(b)r(ounded)h(op)r(erators,)e(and)i(represen)n (tations)d(of)i FQ(\003)p FP(-algebras)e(whic)n(h)i(are)118 1380 y(more)27 b(general)f(than)i(these)f(\\curv)n(es".)118 1530 y FR(Section)32 b(1.2)243 1630 y FP(1.2.1.)i(W)-7 b(e)26 b(exp)r(ose)e(some)g(kno)n(wn)g(facts)h(ab)r(out)g(algebras)d (satisfying)i(the)118 1730 y(standard)h(p)r(olynomial)g(iden)n(tit)n(y) h(\(see,)h(e.g.,)e([113)o(,)h(209)o(])g(and)f(the)i(bibliogra-)118 1829 y(ph)n(y)h(therein\),)f(and)h(their)f(represen)n(tations)f(\(see)i ([153)n(])g(and)f(others\).)243 1929 y(The)g(exp)r(osition)g(of)h (Theorem)f(2)g(follo)n(ws)g([229)n(,)h(230)n(].)243 2054 y(1.2.2.)36 b(F)-7 b(or)28 b(a)f(n)n(um)n(b)r(er)h(of)g(examples)f(of)h (algebras)e(and)i FQ(\003)p FP(-algebras)d(gen-)118 2154 y(erated)h(b)n(y)g(idemp)r(oten)n(ts)g(and)g(orthogonal)e(pro)5 b(jections,)26 b(their)g(represen)n(ta-)118 2254 y(tions)i(are)e (studied.)243 2354 y(Represen)n(tations)31 b(of)h(the)h(algebra)d FO(Q)1460 2366 y FL(2)1530 2354 y FP(\(without)j(an)f(in)n(v)n (olution\))g(gen-)118 2453 y(erated)39 b(b)n(y)h(a)f(pair)g(of)h(idemp) r(oten)n(ts)g(w)n(ere)f(studied,)k(e.g.,)g(in)d([297)o(].)73 b(All)118 2553 y(irreducible)25 b(represen)n(tations)e(of)i FO(Q)1258 2565 y FL(2)1320 2553 y FP(are)f(either)h(one-)f(or)h(t)n(w)n (o-dimensional.)118 2653 y(The)37 b(description)f(of)h(indecomp)r (osable)f(represen)n(tations)f(can)h(b)r(e)i(deriv)n(ed)118 2752 y(from)e([181)o(,)g(94)o(].)64 b(The)36 b(problem)g(of)h(the)g (description)e(of)i(represen)n(tations)118 2852 y(of)31 b(the)f FQ(\003)p FP(-algebra)e(generated)h(b)n(y)i(a)f(pair)f(of)i (orthogonal)d(pro)5 b(jections)29 b(on)h(a)118 2951 y (\014nite-dimensional)h(space)g(is)g(reduced)g(to)g(a)g(description)g (of)g(Jordan's)e(an-)118 3051 y(gles)20 b(b)r(et)n(w)n(een)h(subspaces) e([122)o(];)k(in)e(the)g(case)f(of)g(a)g(separable)f(Hilb)r(ert)j (space,)118 3151 y(see)35 b([205)n(,)g(105)o(],)i(etc.)e(\(see,)i (e.g.,)g(a)d(detailed)h(bibliograph)n(y)f(in)h([40)o(]\).)59 b(F)-7 b(or)118 3250 y(\014nite-dimensional)32 b(represen)n(tations)e (of)j(the)f FQ(\003)p FP(-algebra)e(generated)h(b)n(y)h(an)118 3350 y(idemp)r(oten)n(t)20 b(and)f(its)g(adjoin)n(t,)i(see)d([70)o(,)i (115)n(],)h(etc.;)h(for)d(in\014nite-dimensional)118 3450 y(\(and)28 b(ev)n(en)f(un)n(b)r(ounded\))h(represen)n(tations,)e (see)h([213)o(].)243 3550 y(The)h(algebra)e FO(Q)771 3562 y FL(2)836 3550 y FP(generated)i(b)n(y)f(a)h(pair)g(of)g(idemp)r (oten)n(ts)g(is)g(the)h(group)118 3649 y(algebra)35 b(of)h(the)h (simplest)f(in\014nite)h(group)e FJ(Z)1601 3661 y FL(2)1656 3649 y FQ(\003)24 b FJ(Z)1783 3661 y FL(2)1852 3649 y FP(=)38 b FJ(Z)17 b(o)24 b(Z)2184 3661 y FL(2)2216 3649 y FP(.)63 b(F)-7 b(or)35 b(the)118 3749 y(group)30 b FO(G)f FP(=)g FJ(Z)606 3719 y FM(k)662 3749 y FJ(o)20 b FO(G)812 3761 y FM(f)856 3749 y FP(,)32 b FO(k)g(>)c FP(1,)k(where)f FO(G)1485 3761 y FM(f)1559 3749 y FP(is)g(a)g(\014nite) h(group,)f FJ(C)15 b FP([)p FO(G)p FP(])37 b(is)31 b(an)118 3848 y FO(F)171 3863 y FL(2)p FN(j)p FM(G)276 3872 y Fw(f)313 3863 y FN(j)337 3848 y FP(-algebra,)24 b(and)h(its)h (irreducible)e FQ(\003)p FP(-represen)n(tations)f(can)h(b)r(e)i (obtained)118 3948 y(using)20 b(Mac)n(k)n(ey's)f(formalism)h(of)h (induced)g(represen)n(tations)d([166)o(].)35 b(Ho)n(w)n(ev)n(er,)118 4048 y(the)29 b(description)f(of)h(all)f(indecomp)r(osable)g(represen)n (tations)f(of)h FO(G)h FP(is)g(a)f(v)n(ery)118 4147 y(complicated)38 b(problem)g([39)o(].)68 b(Examples)38 b(of)g FO(F)1691 4159 y FM(n)1736 4147 y FP(-algebras)e(generated)h(b)n(y)p eop %%Page: 80 84 80 83 bop 118 100 a FP(80)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)118 333 y FP(idemp)r(oten)n(ts,)c(whic)n(h)e(are)e(not)i(group)f(algebras,) g(and)h(their)g(represen)n(tations)118 432 y(are)27 b(discussed)g(in)h ([239)n(,)g(83)o(,)g(155)n(].)243 533 y(The)18 b(exp)r(osition)g(of)h (Section)f(1.2.2)f(and)i(the)g(pro)r(of)f(of)g(Theorem)g(3)g(follo)n (ws)118 633 y([229)o(,)27 b(230)o(].)243 761 y(1.2.3.)73 b(Non-comm)n(utativ)n(e)40 b(\\circle",)i(\\h)n(yp)r(erb)r(ola",)f (\\pair)e(of)i(in)n(ter-)118 860 y(secting)35 b(lines",)h(are)e(also)g (natural)g(examples)g(of)h FO(F)1804 872 y FL(4)1842 860 y FP(-algebras.)57 b(F)-7 b(or)34 b(their)118 960 y(represen)n(tations,)26 b(see,)h(e.g.,)g([200)o(,)h(196)n(].)243 1061 y(F)-7 b(or)27 b(algebras)f(that)i(are)f(generated)g(b)n(y)h (idemp)r(oten)n(ts)g(satisfying)g(linear)118 1161 y(relations,)h(see)g ([32)o(].)43 b(The)30 b(fact)g(that)f FO(Q)1398 1173 y FL(4)p FM(;)p FL(2)1518 1161 y FP(is)g(an)h FO(F)1774 1173 y FL(4)1811 1161 y FP(-algebra)e(follo)n(ws)g(from)118 1260 y(the)37 b(description)f(of)h(its)g FQ(\003)p FP(-represen)n (tations)d([90)o(])j(\(see)g(also)e(Section)i(2.2.1)118 1360 y(b)r(elo)n(w\).)243 1461 y(Giv)n(en)32 b(in)g(Section)g(1.2.3,)g (items)g(4)g(and)g(5,)g(are)g(examples)f(of)h(algebras)118 1561 y(generated)27 b(b)n(y)g(idemp)r(oten)n(ts)h(whic)n(h)f(are)g(not) h FO(F)1660 1573 y FM(n)1705 1561 y FP(-algebras)d(for)j(an)n(y)e FO(n)d FQ(2)h FJ(N)t FP(.)118 1660 y(This)34 b(follo)n(ws)e(from)h(the) h(structure)f(of)h(their)f(irreducible)g(represen)n(tations)118 1760 y([32)o(,)28 b(90)o(,)g(91)o(])f(\(see)h(also)e(Section)56 b(2.2.1\).)118 1914 y FR(Section)32 b(1.3)243 2015 y FP(1.3.1.)60 b(Bounded)36 b(represen)n(tations)e(of)i(t)n(w)n (o-dimensional)e(real)h(Lie)h(al-)118 2115 y(gebras)31 b(and)i(their)f(nonlinear)g(transformations)f(can)h(easily)g(b)r(e)h (describ)r(ed)118 2214 y(using)20 b(the)g(Kleinec)n(k)n(e{Shirok)n(o)n (v)d(theorem)i([262)o(,)h(137)o(])g(\(see)g(also)f([104)n(],)j(etc.\).) 243 2342 y(1.3.2{1.3.5.)62 b(In)37 b(these)g(sections,)i(a)d(more)g (general)g(class)g(of)h(semilin-)118 2441 y(ear)32 b(relations)h(is)g (studied.)54 b(The)34 b(pro)r(of)e(of)i(the)f(Kleinec)n(k)n(e{Shirok)n (o)n(v)d(t)n(yp)r(e)118 2541 y(theorems)36 b(and)h(the)h(study)f(of)g (their)g(irreducible)g(represen)n(tations)e(follo)n(w)118 2641 y([35)o(,)28 b(248)n(].)118 2795 y FR(Section)k(1.4)243 2896 y FP(1.4.1.)81 b(Irreducible)42 b(represen)n(tations)f(of)i(the)h FO(q)s FP(-relations)d(b)n(y)i(\014nite-)118 2995 y(dimensional)30 b(and)f(compact)h(op)r(erators)e(are)h(one-dimensional.)42 b(Their)30 b(de-)118 3095 y(scription)d(using)g(Prop)r(ositions)f(25)h (and)g(26)g(follo)n(ws)f([244)o(].)243 3223 y(1.4.2.)56 b(It)34 b(w)n(as)g(noticed)g(in)h([292)n(],)h(etc.,)h(that)d(the)h (Hermitian)f FO(q)s FP(-plane)118 3322 y(do)r(es)24 b(not)h(ha)n(v)n(e) e(non-trivial)g(represen)n(tations)g(b)n(y)h(b)r(ounded)h(op)r (erators.)33 b(W)-7 b(e)118 3422 y(follo)n(w)38 b([196)n(].)69 b(Bounded)38 b(represen)n(tations)f(of)h(one-dimensional)f FO(q)s FP(-CCR,)118 3522 y FO(q)45 b FQ(2)d FJ(R)p FP(,)48 b(w)n(ere)37 b(studied)j(in)f(n)n(umerous)e(pap)r(ers)i(\(see,)i(e.g.,) g([36)o(,)e(164)o(];)44 b(for)118 3621 y(detailed)19 b(references,)h(see,)g(e.g.,)h([140)n(]\).)35 b(The)19 b(exp)r(osition)f(follo)n(ws)g([194)o(,)h(196)o(].)118 3721 y(F)-7 b(or)31 b(un)n(b)r(ounded)g(represen)n(tations)e(of)i FO(q)s FP(-CCR,)g FO(q)h FQ(2)d FJ(R)p FP(,)38 b(see)30 b([142)o(,)h(196)o(,)g(53)o(,)118 3820 y(111)o(,)c(200)o(],)h(etc.)243 3948 y(1.4.3.)34 b(Real)24 b(quan)n(tum)h(plane)f(and)g(real)g(quan)n (tum)g(h)n(yp)r(erb)r(oloid)g(do)g(not)118 4048 y(ha)n(v)n(e)c (non-trivial)g(represen)n(tations)g(b)n(y)h(b)r(ounded)h(op)r(erators.) 33 b(In)21 b([194)o(,)g(196)o(],)118 4147 y(this)41 b(is)f(pro)n(v)n (ed)f(b)n(y)h(using)g(the)h(F)-7 b(uglede{Putnam{Rosen)n(blum)38 b(theorem)p eop %%Page: 81 85 81 84 bop 118 100 a FK(Commen)n(ts)27 b(to)h(Chapter)f(1)1494 b FP(81)118 333 y(\(see,)42 b(e.g.,)f([104)o(,)d(241)o(]\).)71 b(F)-7 b(or)38 b(un)n(b)r(ounded)h(represen)n(tations)e(of)i(the)g (real)118 432 y(quan)n(tum)28 b(h)n(yp)r(erb)r(oloid,)f(see)g([250)n(,) h(251)o(].)p eop %%Page: 82 86 82 85 bop 118 100 a FP(82)875 b FK(Chapter)27 b(1.)37 b(P)n(airs)25 b(of)j(self-adjoin)n(t)f(op)r(erators)p eop %%Page: 83 87 83 86 bop 118 900 a FS(Chapter)46 b(2)118 1218 y(Represen)l(tations)h (of)f(dynamical)f Fm(\003)p FS(-algebras)118 1669 y FH(2.1)112 b(Op)s(erator)54 b(relations)d(and)j(one-dimensional)e(dy-)373 1785 y(namical)36 b(systems)118 1972 y FR(2.1.1)94 b(Op)s(erator)38 b(relations)g(connected)h(with)f(one-dimensional)410 2072 y(dynamical)31 b(systems)118 2230 y(1.)37 b FP(Consider)27 b(an)g(op)r(erator)f FO(X)34 b FP(satisfying,)28 b(together)e(with)j (its)f(adjoin)n(t)f FO(X)2514 2200 y FN(\003)2552 2230 y FP(,)118 2330 y(an)g(algebraic)f(relation)h(of)g(the)h(form)1019 2518 y FO(X)7 b(X)1171 2483 y FN(\003)1231 2518 y FP(=)23 b FO(F)12 b FP(\()p FO(X)1492 2483 y FN(\003)1529 2518 y FO(X)7 b FP(\))p FO(;)744 b FP(\(2.1\))118 2706 y(where)34 b FO(F)12 b FP(\()p FQ(\001)p FP(\))d(:)31 b FJ(R)42 b FQ(\000)-49 b(!)35 b FJ(R)41 b FP(is)35 b(a)g(mapping)f(measurable)g (with)h(resp)r(ect)g(to)f(the)118 2805 y(Borel)27 b FO(\033)s FP(-algebra.)243 2908 y(If)33 b FO(F)12 b FP(\()p FQ(\001)p FP(\))32 b(=)f FO(P)664 2920 y FM(n)709 2908 y FP(\()p FQ(\001)p FP(\))j(is)e(a)g(real)g(p)r(olynomial,)h(then)g(the)g(pair)f (of)h(op)r(erators)118 3007 y FO(X)7 b FP(,)31 b FO(X)324 2977 y FN(\003)392 3007 y FP(satisfying)f(relation)f(\(2.1\))i(is)f(a)g (represen)n(tation)f(of)i(a)f FQ(\003)p FP(-algebra)e FB(A)118 3107 y FP(generated)h(b)n(y)h(elemen)n(ts)g FO(x)p FP(,)i FO(x)1108 3077 y FN(\003)1177 3107 y FP(satisfying)e(the) g(relation)g FO(xx)2097 3077 y FN(\003)2163 3107 y FP(=)e FO(P)2309 3119 y FM(n)2354 3107 y FP(\()p FO(x)2433 3077 y FN(\003)2472 3107 y FO(x)p FP(\).)118 3207 y(In)e(some)f(non-p)r (olynomial)f(cases,)h(pairs)g(of)h(op)r(erators)d FO(X)7 b FP(,)26 b FO(X)2105 3176 y FN(\003)2168 3207 y FP(form)f(repre-)118 3306 y(sen)n(tations)30 b(of)g FO(C)674 3276 y FN(\003)712 3306 y FP(-algebras)e(or)i(other)g(top)r(ological)f(algebras,)g(but)i (w)n(e)f(will)118 3406 y(not)35 b(concen)n(trate)e(our)h(atten)n(tion)h (on)f(the)h(underlying)f(algebraic)f(ob)5 b(jects,)118 3506 y(restricting)27 b(ourselv)n(es)e(to)j(the)g(study)g(of)f (represen)n(tations)f(of)h(the)h(relation.)243 3608 y(W)-7 b(e)35 b(ha)n(v)n(e)f(already)g(considered)g(examples)g(of)h(represen)n (tations)e(of)i(the)118 3707 y(form)25 b(\(2.1\))g(ab)r(o)n(v)n(e.)35 b(F)-7 b(or)24 b(the)i(Hermitian)f FO(q)s FP(-plane,)g FO(xx)1858 3677 y FN(\003)1920 3707 y FP(=)e FO(q)s(x)2095 3677 y FN(\003)2134 3707 y FO(x)p FP(,)j FO(q)g FQ(2)d FJ(R)p FP(,)32 b(w)n(e)118 3807 y(ha)n(v)n(e)g FO(F)12 b FP(\()p FO(\025)p FP(\))32 b(=)f FO(q)s(\025)p FP(,)j(and)f(for)f FO(q)s FP(-CCR,)g FO(xx)1462 3777 y FN(\003)1533 3807 y FP(=)f FO(q)s(x)1716 3777 y FN(\003)1755 3807 y FO(x)22 b FP(+)f(1,)34 b FO(q)g FQ(2)e FJ(R)p FP(,)40 b(w)n(e)32 b(ha)n(v)n(e)118 3907 y FO(F)12 b FP(\()p FO(\025)p FP(\))24 b(=)f FO(q)s(\025)c FP(+)f(1.)243 4009 y(Belo)n(w)26 b(w)n(e)h(consider)g(other)g(examples)g(of)h(relations)e(of)h(the)h (form)g(\(2.1\))o(.)118 4147 y FC(Example)36 b FP(9)p FC(.)41 b FP(\(Second)26 b(degree)g(mappings\).)36 b(Consider)26 b(the)h(follo)n(wing)e(rela-)1305 4357 y(83)p eop %%Page: 84 88 84 87 bop 118 100 a FP(84)485 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(tion)749 516 y FO(xx)843 482 y FN(\003)905 516 y FP(=)c FO(\013x)1093 482 y FN(\003)1132 516 y FO(x)p FP(\()p FO(I)j FQ(\000)18 b FO(x)1403 482 y FN(\003)1442 516 y FO(x)p FP(\))p FO(;)181 b(a)22 b(>)h FP(0)p FO(:)460 b FP(\(2.2\))118 700 y(The)35 b(corresp)r(onding)d(p)r (olynomial)i(is)h FO(F)12 b FP(\()p FO(\025)p FP(\))35 b(=)f FO(\013\025)p FP(\(1)24 b FQ(\000)f FO(\025)p FP(\).)58 b(An)n(y)35 b(second-)118 799 y(degree)f(mapping)g(can)h(b)r(e)g (reduced)g(to)f(suc)n(h)h(a)f(form)h(b)n(y)f(using)h(an)f(a\016ne)118 899 y(c)n(hange)e(of)h(v)-5 b(ariables;)34 b(ho)n(w)n(ev)n(er,)e(p)r (ositivit)n(y)h(considerations)e(that)i(will)g(b)r(e)118 998 y(discussed)23 b(b)r(elo)n(w)g(mak)n(e)f(the)i(represen)n(tation)d (problem)i(di\013eren)n(t)g(for)g(di\013er-)118 1098 y(en)n(t)30 b(forms)g(of)g(the)h(p)r(olynomial.)44 b(Here,)31 b(w)n(e)f(consider)f(only)h(relations)f(\(2.2\))118 1198 y(and)1015 1381 y FO(xx)1109 1347 y FN(\003)1171 1381 y FP(=)23 b(\()p FO(x)1338 1347 y FN(\003)1377 1381 y FO(x)18 b FQ(\000)g FO(q)s(I)7 b FP(\))1640 1347 y FL(2)2404 1381 y FP(\(2.3\))118 1580 y(whic)n(h)41 b(corresp)r(onds)f(to)h(the)g (p)r(olynomial)1572 1559 y(~)1553 1580 y FO(F)12 b FP(\()p FO(t)p FP(\))46 b(=)g(\()p FO(t)28 b FQ(\000)f FO(q)s FP(\))2123 1549 y FL(2)2160 1580 y FP(.)78 b(The)42 b(t)n(w)n(o)118 1679 y(latter)30 b(p)r(olynomials)g(can)g(b)r(e)h(transformed)e(to)h (eac)n(h)g(other)g(b)n(y)g(the)h(c)n(hange)118 1779 y(of)38 b(v)-5 b(ariables)36 b FO(t)j FP(=)g FQ(\000)p FO(\013\025)26 b FP(+)e FO(b)p FP(,)40 b(with)e FO(\013)i FP(=)f(1)25 b(+)1682 1718 y FQ(p)p 1752 1718 225 4 v 1752 1779 a FP(4)p FO(q)c FP(+)d(1)o(.)67 b(The)37 b(structure)118 1878 y(of)27 b(b)r(ounded)h(represen)n(tations)d(of)j(suc)n(h)f (relations)f(greatly)g(dep)r(ends)h(on)g(the)118 1978 y(v)-5 b(alue)28 b(of)f FO(\013)h FP(\(see)g(b)r(elo)n(w\).)118 2112 y FC(Example)38 b FP(10)p FC(.)j FP(\(Con)n(tin)n(ued)28 b(fractions\).)38 b(Another)28 b(in)n(teresting)f(class)h(of)g(re-)118 2211 y(lations)e(related)f(to)h(con)n(tin)n(uous)f(fractions)g(is)h (giv)n(en)f(b)n(y)h(the)g(follo)n(wing)f(rela-)118 2311 y(tions)783 2494 y FO(xx)877 2460 y FN(\003)939 2494 y FP(=)e(\()p FO(ax)1150 2460 y FN(\003)1189 2494 y FO(x)c FP(+)f FO(c)p FP(\)\()p FO(bx)1521 2460 y FN(\003)1559 2494 y FO(x)h FP(+)f FO(d)p FP(\))1783 2460 y FN(\000)p FL(1)1873 2494 y FO(;)700 2619 y(a;)c(b;)g(c;)g(d)22 b FQ(2)h FJ(R)p FO(;)103 b(b)23 b FQ(6)p FP(=)g(0)p FO(;)96 b(ad)19 b FQ(\000)f FO(bc)k FQ(6)p FP(=)h(0)p FO(:)118 2802 y FP(Belo)n(w,)31 b(w)n(e)g(will)g(sho)n(w)g(ho)n(w)f(represen)n (tations)f(of)j(the)f(relation)f(dep)r(end)i(on)118 2902 y(the)c(v)-5 b(alues)27 b(of)h(the)g(parameters.)243 3002 y(In)23 b(particular,)g(the)g(t)n(w)n(o-parameter)d(unit)k(quan)n (tum)f(disk)g(algebra)e([138)o(])118 3102 y(is)28 b(generated)e(b)n(y)h (the)h(relation)582 3285 y FO(q)s(z)t(z)708 3251 y FN(\003)763 3285 y FQ(\000)18 b FO(z)889 3251 y FN(\003)926 3285 y FO(z)27 b FP(=)22 b FO(q)g FQ(\000)c FP(1)g(+)g FO(\026)p FP(\(1)g FQ(\000)g FO(z)t(z)1675 3251 y FN(\003)1712 3285 y FP(\)\(1)g FQ(\000)g FO(z)1962 3251 y FN(\003)2000 3285 y FO(z)t FP(\))p FO(;)615 3410 y FP(0)23 b FQ(\024)f FO(\026)i FQ(\024)e FP(1)p FO(;)97 b FP(0)22 b FQ(\024)h FO(q)j FQ(\024)d FP(1)p FO(;)96 b FP(\()p FO(\026;)14 b(q)s FP(\))24 b FQ(6)p FP(=)e(\(0)p FO(;)14 b FP(1\))p FO(;)118 3593 y FP(whic)n(h)28 b(can)f(b)r(e)h(rewritten)f(in)h(the)g (form)f(\(2.1\))h(with)810 3824 y FO(F)12 b FP(\()p FO(\025)p FP(\))24 b(=)1109 3768 y(\()p FO(q)e FP(+)c FO(\026)p FP(\))p FO(\025)h FP(+)f(1)g FQ(\000)g FO(q)j FQ(\000)d FO(\026)p 1109 3805 741 4 v 1283 3881 a(\026\025)h FP(+)f(1)g FQ(\000)g FO(\026)1859 3824 y(:)118 4048 y FC(R)l(emark)37 b FP(25)p FC(.)i FP(As)26 b(w)n(e)f(will)h(see,)g(in)g(man)n(y)f (examples)g(un)n(b)r(ounded)h(op)r(erators)118 4147 y FO(X)32 b FP(naturally)25 b(arise)f(as)h(represen)n(tations)f(of)h(suc) n(h)g(a)g(relation.)36 b(Therefore,)24 b(it)p eop %%Page: 85 89 85 88 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f (systems)896 b FP(85)118 333 y(is)28 b(necessary)e(to)h(accurately)f (de\014ne)i(the)g(meaning)f(of)h(the)g(relation)f(for)g(un-)118 432 y(b)r(ounded)k(op)r(erators.)44 b(Occasionally)-7 b(,)29 b(deriv)n(ed)h(form)n(ulae)g(will)h(mak)n(e)e(sense)118 532 y(for)38 b(un)n(b)r(ounded)g(op)r(erators,)h(to)r(o,)i(or)c(will)i (yield)f(un)n(b)r(ounded)h(op)r(erators;)118 632 y(ho)n(w)n(ev)n(er,)28 b(w)n(e)h(will)h(not)g(discuss)f(problems)g(related)g(to)g(un)n(b)r (ounded)h(op)r(era-)118 731 y(tors)d(here.)118 893 y FR(2.)35 b FP(Let)24 b FO(X)30 b FP(b)r(e)24 b(b)r(ounded.)36 b(F)-7 b(or)23 b(a)h(p)r(olar)e(decomp)r(osition)i(of)f(the)h(op)r (erator)e FO(X)7 b FP(,)118 992 y(w)n(e)29 b(ha)n(v)n(e)f FO(X)k FP(=)25 b FO(U)9 b(C)d FP(,)29 b(where)g FO(C)j FP(=)25 b FO(C)1297 962 y FN(\003)1361 992 y FP(=)g(\()p FO(X)1559 962 y FN(\003)1596 992 y FO(X)7 b FP(\))1704 962 y FL(1)p FM(=)p FL(2)1808 992 y FP(,)30 b FO(U)38 b FP(is)29 b(a)f(partial)h(isom-)118 1092 y(etry)-7 b(,)34 b(and)f(k)n(er)13 b FO(U)40 b FP(=)32 b(k)n(er)12 b FO(C)38 b FP(=)32 b(k)n(er)13 b FO(X)7 b FP(,)33 b FO(U)1441 1062 y FN(\003)1479 1092 y FO(U)42 b FP(is)33 b(an)f(orthogonal)f(pro)5 b(jection)118 1192 y(on)n(to)27 b(\(k)n(er)13 b FO(C)6 b FP(\))559 1162 y FN(?)616 1192 y FP(.)37 b(T)-7 b(aking)26 b(in)n(to)i(accoun)n(t)e(relation)h(\(2.1\),)g(w)n(e)h(get)1021 1348 y FO(U)9 b(C)1152 1314 y FL(2)1189 1348 y FO(U)1255 1314 y FN(\003)1316 1348 y FP(=)22 b FO(F)12 b FP(\()p FO(C)1565 1314 y FL(2)1603 1348 y FP(\))p FO(;)746 b FP(\(2.4\))118 1505 y(whic)n(h)28 b(implies)f(that)665 1661 y FO(U)9 b(C)796 1627 y FL(2)856 1661 y FP(=)23 b FO(F)12 b FP(\()p FO(C)1106 1627 y FL(2)1143 1661 y FP(\))p FO(U;)97 b(C)1417 1627 y FL(2)1455 1661 y FO(U)1521 1627 y FN(\003)1582 1661 y FP(=)22 b FO(U)1735 1627 y FN(\003)1773 1661 y FO(F)12 b FP(\()p FO(C)1935 1627 y FL(2)1973 1661 y FP(\))p FO(:)376 b FP(\(2.5\))118 1818 y(Notice)28 b(that)g(\(2.4\))f(implies)h(that)g(k)n(er)12 b FO(U)1409 1788 y FN(\003)1470 1818 y FP(=)23 b(k)n(er)13 b FO(X)1759 1788 y FN(\003)1819 1818 y FP(=)23 b(k)n(er)13 b FO(F)f FP(\()p FO(C)2194 1788 y FL(2)2231 1818 y FP(\))28 b(as)f(w)n(ell.)118 1957 y FR(3.)51 b FP(No)n(w)32 b(w)n(e)g(establish) g(some)g(relations)f(whic)n(h)h(follo)n(w)g(from)g(\(2.1\).)51 b(First)118 2056 y(notice)35 b(that)g FO(X)7 b FP(\()p FO(X)741 2026 y FN(\003)778 2056 y FO(X)g FP(\))35 b(=)g FO(F)12 b FP(\()p FO(X)1194 2026 y FN(\003)1231 2056 y FO(X)7 b FP(\))p FO(X)41 b FP(and)35 b(\()p FO(X)1726 2026 y FN(\003)1764 2056 y FO(X)7 b FP(\))p FO(X)1948 2026 y FN(\003)2020 2056 y FP(=)35 b FO(X)2196 2026 y FN(\003)2233 2056 y FO(F)12 b FP(\()p FO(X)2406 2026 y FN(\003)2444 2056 y FO(X)7 b FP(\).)118 2156 y(F)-7 b(or)29 b(an)n(y)f FO(k)g FP(=)d(1,)k(2,)g FO(:)14 b(:)g(:)28 b 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FN(\003)596 496 y FO(X)g FP(\))27 b(=)f FO(\036)p FP(\()p FO(F)12 b FP(\()p FO(X)1076 466 y FN(\003)1115 496 y FO(X)7 b FP(\)\))p FO(X)g FC(,)32 b(for)g(any)g(me)l(asur)l(able)h(function)f FO(\036)p FP(\()p FQ(\001)p FP(\))326 595 y FC(b)l(ounde)l(d)e(on)g (the)g(sp)l(e)l(ctrum)f(of)h FO(X)1421 565 y FN(\003)1459 595 y FO(X)7 b FP(;)209 758 y FC(c)e FP(\))42 b FO(\036)p FP(\()p FO(C)472 728 y FL(2)510 758 y FP(\))p FO(U)608 728 y FN(\003)697 758 y FP(=)50 b FO(U)878 728 y FN(\003)915 758 y FO(\036)p FP(\()p FO(F)12 b FP(\()p FO(C)1158 728 y FL(2)1197 758 y FP(\)\))p FC(,)50 b(for)45 b(any)g(me)l(asur)l(able)h (function)f FO(\036)p FP(\()p FQ(\001)p FP(\))326 858 y FC(b)l(ounde)l(d)30 b(on)g(the)g(sp)l(e)l(ctrum)f(of)h FO(X)1421 828 y FN(\003)1459 858 y FO(X)7 b FP(;)201 1021 y FC(d)i FP(\))42 b FO(E)387 1036 y FM(C)439 1019 y Fy(2)475 1021 y FP(\(\001\))p FO(U)674 991 y FN(\003)736 1021 y FP(=)23 b FO(U)890 991 y FN(\003)927 1021 y FO(E)988 1036 y FM(C)1040 1019 y Fy(2)1077 1021 y FP(\()p FO(F)1174 991 y FN(\000)p FL(1)1263 1021 y FP(\(\001\)\))31 b FC(for)g(al)t(l)f (me)l(asur)l(able)h FP(\001)23 b FQ(\032)g FJ(R)p FP(.)118 1197 y FC(Her)l(e)34 b FO(E)382 1212 y FM(C)434 1195 y Fy(2)471 1197 y FP(\()p FQ(\001)p FP(\))h FC(is)g(a)f(r)l(esolution)h (of)g(the)g(identity)g(of)g(the)f(op)l(er)l(ator)i FO(C)2345 1166 y FL(2)2382 1197 y FC(,)g(and)118 1296 y FO(F)183 1266 y FN(\000)p FL(1)272 1296 y FP(\(\001\))42 b FC(denotes)g(the)g (ful)t(l)g(pr)l(e-image)h(of)f FP(\001)g(\()p FC(mor)l(e)g(pr)l(e)l (cisely,)k(the)c(ful)t(l)118 1396 y(pr)l(e-image)31 b(of)g FP(\001)18 b FQ(\\)h FO(F)12 b FP(\()p FJ(R)p FP(\))q(\))p FC(.)118 1556 y(Pr)l(o)l(of.)43 b FP(It)29 b(w)n(as)e(already)g(sho)n (wn)g(that)h(\(2.1\))g(is)g(equiv)-5 b(alen)n(t)28 b(to)g(a\),)h(and)f (that)118 1656 y(\(2.1\))e(implies)g(b\).)37 b(T)-7 b(o)26 b(sho)n(w)f(that)i(b\))g(implies)f(\(2.1\),)g(tak)n(e)g FO(\036)p FP(\()p FO(\025)p FP(\))e(=)f FO(\025)p FP(.)37 b(Mul-)118 1755 y(tiplying)28 b(b)n(y)f FO(X)616 1725 y FN(\003)681 1755 y FP(from)g(the)h(righ)n(t,)e(w)n(e)h(ha)n(v)n(e)g FO(X)7 b(X)1709 1725 y FN(\003)1745 1755 y FO(X)g(X)1897 1725 y FN(\003)1957 1755 y FP(=)23 b FO(F)12 b FP(\()p FO(X)2218 1725 y FN(\003)2255 1755 y FO(X)7 b FP(\))p FO(X)g(X)2515 1725 y FN(\003)2552 1755 y FP(,)118 1855 y(whic)n(h)44 b(implies)h(that)g(relation)e(\(2.1\))h(holds)g(on)g(v)n (ectors)f(orthogonal)f(to)118 1954 y(k)n(er)13 b FO(X)7 b(X)395 1924 y FN(\003)432 1954 y FP(.)37 b(On)27 b(the)h(other)f (hand,)h(k)n(er)12 b FO(X)1421 1924 y FN(\003)1482 1954 y FP(=)22 b(k)n(er)13 b FO(U)1760 1924 y FN(\003)1821 1954 y FP(=)23 b(k)n(er)13 b FO(F)f FP(\()p FO(X)2207 1924 y FN(\003)2244 1954 y FO(X)7 b FP(\).)243 2054 y(The)35 b(pro)r(of)g(that)h(c\))g(is)f(equiv)-5 b(alen)n(t)36 b(to)f(a\))g(can)h(b)r(e)f(done)h(in)g(the)f(same)118 2154 y(w)n(a)n(y)e(as)g(b\))h(w)n(as)f(deriv)n(ed.)55 b(Relation)33 b(d\))i(is)e(a)h(particular)e(case)h(of)h(c\))g(with)118 2253 y FO(\036)p FP(\()p FO(\025)p FP(\))43 b(=)e FO(\037)480 2265 y FL(\001)539 2253 y FP(\()p FO(\025)p FP(\))f(\(notice)f(that)f FO(E)1230 2268 y FM(F)9 b FL(\()p FM(C)1359 2252 y Fy(2)1392 2268 y FL(\))1422 2253 y FP(\(\001\))42 b(=)f FO(E)1764 2268 y FM(C)1816 2252 y Fy(2)1853 2253 y FP(\()p FO(F)1950 2223 y FN(\000)p FL(1)2039 2253 y FP(\(\001\)\)\);)46 b(using)38 b(a)118 2353 y(sp)r(ectral)26 b(represen)n(tation)f(for)h (measurable)f(functions)i(of)g(the)g(p)r(ositiv)n(e)f(self-)118 2453 y(adjoin)n(t)38 b(op)r(erator)f FO(C)821 2423 y FL(2)859 2453 y FP(,)k FO(\036)p FP(\()p FO(C)1069 2423 y FL(2)1107 2453 y FP(\))h(=)1287 2386 y Fz(R)1356 2453 y FO(\036)p FP(\()p FO(\025)p FP(\))14 b FO(dE)1635 2468 y FM(C)1687 2451 y Fy(2)1725 2453 y FP(\()p FO(\025)p FP(\),)42 b(one)c(can)h(easily)e(see)118 2552 y(that)28 b(c\))g(follo)n(ws)e(from)i(d\))g(as)f(w)n(ell.)p 2514 2552 4 57 v 2518 2500 50 4 v 2518 2552 V 2567 2552 4 57 v 118 2717 a FR(4.)34 b FP(Op)r(erators)18 b(of)h(the)i(form)e (\(2.1\))h(form)f(a)g(sub)r(class)h(in)g(the)g(class)f(of)g(cen)n (tered)118 2816 y(op)r(erators.)72 b(Recall)40 b(that)g(a)g(b)r(ounded) h(op)r(erator)d FO(X)46 b FP(is)40 b(cen)n(tered,)j(if)d(the)118 2916 y(op)r(erators)34 b FO(X)570 2886 y FM(l)595 2916 y FP(\()p FO(X)703 2886 y FN(\003)740 2916 y FP(\))772 2886 y FM(l)798 2916 y FP(,)k(\()p FO(X)967 2886 y FN(\003)1005 2916 y FP(\))1037 2886 y FM(k)1078 2916 y FO(X)1154 2886 y FM(k)1194 2916 y FP(,)g FO(k)s(;)14 b(l)37 b FP(=)g(1,)g(2,)e FO(:)14 b(:)g(:)28 b FP(,)38 b(form)d(a)g(comm)n(uting)118 3016 y(family)-7 b(.)118 3176 y FR(Prop)s(osition)24 b(29.)36 b FC(A)24 b(b)l(ounde)l(d)i(op)l(er)l(ator)f FO(X)31 b FC(satisfying)j FP(\(2.1\))24 b FC(is)i(c)l(enter)l(e)l(d.) 118 3276 y(If)f(a)f(p)l(air)h(of)g(op)l(er)l(ators,)i(a)d(self-adjoint) j FO(C)j FC(and)24 b(a)h(p)l(artial)g(isometry)g FO(U)9 b FC(,)26 b(such)118 3375 y(that)k FP(k)n(er)13 b FO(U)32 b FP(=)22 b(k)n(er)13 b FO(C)779 3345 y FL(2)817 3375 y FC(,)30 b FP(k)n(er)13 b FO(U)1063 3345 y FN(\003)1124 3375 y FP(=)22 b(k)n(er)13 b FO(F)f FP(\()p FO(C)1498 3345 y FN(\003)1537 3375 y FP(\))p FC(,)30 b(satisfy)h(r)l(elation)37 b FP(\(2.5\))p FC(,)30 b(then)118 3475 y FO(U)39 b FC(is)30 b(a)g(c)l(enter)l(e)l(d)f(p)l(artial)i(isometry.)118 3635 y(Pr)l(o)l(of.)43 b FP(Let)28 b(us)g(sho)n(w)e(that)i FO(X)34 b FP(is)28 b(cen)n(tered.)36 b(W)-7 b(rite)28 b FO(A)1861 3647 y FM(k)1925 3635 y FP(=)23 b FO(X)2089 3605 y FM(k)2129 3635 y FP(\()p FO(X)2237 3605 y FN(\003)2275 3635 y FP(\))2307 3605 y FM(k)2348 3635 y FP(,)28 b FO(B)2462 3647 y FM(l)2510 3635 y FP(=)118 3735 y(\()p FO(X)226 3704 y FN(\003)264 3735 y FP(\))296 3704 y FM(l)322 3735 y FO(X)398 3704 y FM(l)422 3735 y FP(,)k FO(k)s(;)14 b(l)30 b FQ(\025)d FP(1.)46 b(First)31 b(consider)f(the)h(op)r(erators) e FO(A)1930 3747 y FM(k)1971 3735 y FP(.)46 b(Applying)32 b(\(2.7\))118 3834 y(w)n(e)27 b(ha)n(v)n(e)334 4010 y FO(A)396 4022 y FM(k)460 4010 y FP(=)c FO(X)624 3975 y FM(k)664 4010 y FP(\()p FO(X)772 3975 y FN(\003)810 4010 y FP(\))842 3975 y FM(k)906 4010 y FP(=)g FO(X)1070 3975 y FM(k)q FN(\000)p FL(1)1195 4010 y FP(\()p FO(X)7 b(X)1379 3975 y FN(\003)1416 4010 y FP(\)\()p FO(X)1556 3975 y FN(\003)1594 4010 y FP(\))1626 3975 y FM(k)q FN(\000)p FL(1)460 4147 y FP(=)23 b FO(X)624 4113 y FM(k)q FN(\000)p FL(1)749 4147 y FO(F)12 b FP(\()p FO(X)922 4113 y FN(\003)960 4147 y FO(X)7 b FP(\)\()p FO(X)1176 4113 y FN(\003)1213 4147 y FP(\))1245 4113 y FM(k)q FN(\000)p FL(1)1394 4147 y FP(=)23 b FO(F)1547 4113 y FM(n)1592 4147 y FP(\()p FO(X)1700 4113 y FN(\003)1738 4147 y FO(X)7 b FP(\))p FO(X)1922 4113 y FM(n)p FN(\000)p FL(1)2051 4147 y FP(\()p FO(X)2159 4113 y FN(\003)2196 4147 y FP(\))2228 4113 y FM(n)p FN(\000)p FL(1)p eop %%Page: 87 91 87 90 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f (systems)896 b FP(87)460 333 y(=)23 b FO(F)613 298 y FM(n)658 333 y FP(\()p FO(X)766 298 y FN(\003)804 333 y FO(X)7 b FP(\))p FO(X)988 298 y FM(n)p FN(\000)p FL(2)1117 333 y FO(F)12 b FP(\()p FO(X)1290 298 y FN(\003)1327 333 y FO(X)7 b FP(\)\()p FO(X)1543 298 y FN(\003)1581 333 y FP(\))1613 298 y FM(n)p FN(\000)p FL(2)460 467 y FP(=)23 b FO(F)613 433 y FM(n)658 467 y FP(\()p FO(X)766 433 y FN(\003)804 467 y FO(X)7 b FP(\))p FO(F)977 433 y FM(n)p FN(\000)p FL(1)1106 467 y FP(\()p FO(X)1214 433 y FN(\003)1252 467 y FO(X)g FP(\))14 b FO(:)g(:)g(:)f(F)f FP(\()p FO(X)1657 433 y FN(\003)1695 467 y FO(X)7 b FP(\))p FO(:)118 637 y FP(Since)22 b(all)g(the)g(op)r(erators)e FO(A)1000 649 y FM(k)1041 637 y FP(,)j FO(k)j FQ(\025)d FP(1,)f(are)f(functions)h(of)g(the)g(single)g(op)r(erator)118 737 y FO(X)194 706 y FN(\003)232 737 y FO(X)7 b FP(,)27 b(they)g(comm)n(ute)h(with)g(eac)n(h)f(other.)243 836 y(No)n(w)k(consider)f(a)h(pair)g FO(A)1073 848 y FM(k)1114 836 y FP(,)i FO(B)1233 848 y FM(l)1259 836 y FP(.)48 b(Again,)33 b(applying)f(\(2.7\))f(and)h(the)f(ob-)118 936 y(tained)d(represen)n(tation)e(for)h FO(A)1104 948 y FM(k)1145 936 y FP(,)h(w)n(e)f(get)258 1105 y FO(B)321 1117 y FM(l)346 1105 y FO(A)408 1117 y FM(k)472 1105 y FP(=)c(\()p FO(X)668 1071 y FN(\003)706 1105 y FP(\))738 1071 y FM(l)763 1105 y FO(X)839 1071 y FM(l)864 1105 y FO(X)940 1071 y FM(k)980 1105 y FP(\()p FO(X)1088 1071 y FN(\003)1126 1105 y FP(\))1158 1071 y FM(k)472 1243 y FP(=)g(\()p FO(X)668 1209 y FN(\003)706 1243 y FP(\))738 1209 y FM(l)763 1243 y FO(X)839 1209 y FM(l)864 1243 y FO(F)929 1209 y FM(k)970 1243 y FP(\()p FO(X)1078 1209 y FN(\003)1116 1243 y FO(X)7 b FP(\))p FO(F)1289 1209 y FM(k)q FN(\000)p FL(1)1414 1243 y FP(\()p FO(X)1522 1209 y FN(\003)1560 1243 y FO(X)g FP(\))14 b FO(:)g(:)g(:)f(F)f FP(\()p FO(X)1965 1209 y FN(\003)2002 1243 y FO(X)7 b FP(\))472 1381 y(=)23 b(\()p FO(X)668 1347 y FN(\003)706 1381 y FP(\))738 1347 y FM(l)763 1381 y FO(F)828 1347 y FM(k)q FL(+)p FM(l)941 1381 y FP(\()p FO(X)1049 1347 y FN(\003)1087 1381 y FO(X)7 b FP(\))p FO(F)1260 1347 y FM(k)q FL(+)p FM(l)p FN(\000)p FL(1)1458 1381 y FP(\()p FO(X)1566 1347 y FN(\003)1604 1381 y FO(X)g FP(\))14 b FO(:)g(:)g(:)f(F)1901 1347 y FM(l)p FL(+1)2010 1381 y FP(\()p FO(X)2118 1347 y FN(\003)2156 1381 y FO(X)7 b FP(\))p FO(X)2340 1347 y FM(l)472 1519 y FP(=)23 b FO(F)625 1484 y FM(k)666 1519 y FP(\()p FO(X)774 1484 y FN(\003)811 1519 y FO(X)7 b FP(\))p FO(F)984 1484 y FM(k)q FN(\000)p FL(1)1110 1519 y FP(\()p FO(X)1218 1484 y FN(\003)1255 1519 y FO(X)g FP(\))14 b FO(:)g(:)g(:)g(F)e FP(\()p FO(X)1661 1484 y FN(\003)1698 1519 y FO(X)7 b FP(\)\()p FO(X)1914 1484 y FN(\003)1952 1519 y FP(\))1984 1484 y FM(l)2009 1519 y FO(X)2085 1484 y FM(l)2133 1519 y FP(=)23 b FO(A)2283 1531 y FM(k)2324 1519 y FO(B)2387 1531 y FM(l)2412 1519 y FO(;)118 1688 y FP(and)j(therefore,)f(the)i(op) r(erators)c FO(A)1217 1700 y FM(k)1285 1688 y FP(and)i FO(B)1507 1700 y FM(l)1533 1688 y FP(,)h FO(k)s(;)14 b(l)24 b FQ(\025)f FP(1,)j(also)f(comm)n(ute)g(with)118 1788 y(eac)n(h)i(other.)243 1887 y(No)n(w)32 b(w)n(e)g(sho)n(w)g(that)h (the)g(op)r(erators)e FO(B)1544 1899 y FM(l)1569 1887 y FP(,)j FO(l)f FQ(\025)f FP(1,)h(also)f(comm)n(ute)g(with)118 1987 y(eac)n(h)27 b(other.)36 b(Indeed,)28 b(for)f FO(k)f(>)d(l)r FP(,)k(w)n(e)g(ha)n(v)n(e)275 2156 y FO(B)338 2168 y FM(k)379 2156 y FO(B)442 2168 y FM(l)490 2156 y FP(=)c(\()p FO(X)686 2122 y FN(\003)724 2156 y FP(\))756 2122 y FM(k)797 2156 y FO(X)873 2122 y FM(k)913 2156 y FP(\()p FO(X)1021 2122 y FN(\003)1059 2156 y FP(\))1091 2122 y FM(l)1116 2156 y FO(X)1192 2122 y FM(l)1240 2156 y FP(=)g(\()p FO(X)1436 2122 y FN(\003)1474 2156 y FP(\))1506 2122 y FM(l)1531 2156 y FP(\()p FO(X)1639 2122 y FN(\003)1677 2156 y FP(\))1709 2122 y FM(k)q FN(\000)p FM(l)1824 2156 y FO(X)1900 2122 y FM(k)q FN(\000)p FM(l)2013 2156 y FO(X)2089 2122 y FM(l)2114 2156 y FP(\()p FO(X)2222 2122 y FN(\003)2259 2156 y FP(\))2291 2122 y FM(l)2317 2156 y FO(X)2393 2122 y FM(l)490 2294 y FP(=)g(\()p FO(X)686 2260 y FN(\003)724 2294 y FP(\))756 2260 y FM(l)781 2294 y FO(B)844 2306 y FM(k)q FN(\000)p FM(l)958 2294 y FO(A)1020 2306 y FM(l)1046 2294 y FO(X)1122 2260 y FM(l)1170 2294 y FP(=)g(\()p FO(X)1366 2260 y FN(\003)1403 2294 y FP(\))1435 2260 y FM(l)1461 2294 y FO(A)1523 2306 y FM(l)1549 2294 y FO(B)1612 2306 y FM(k)q FN(\000)p FM(l)1726 2294 y FO(X)1802 2260 y FM(l)490 2432 y FP(=)g(\()p FO(X)686 2398 y FN(\003)724 2432 y FP(\))756 2398 y FM(l)781 2432 y FO(X)857 2398 y FM(l)882 2432 y FP(\()p FO(X)990 2398 y FN(\003)1028 2432 y FP(\))1060 2398 y FM(l)1086 2432 y FP(\()p FO(X)1194 2398 y FN(\003)1231 2432 y FP(\))1263 2398 y FM(k)q FN(\000)p FM(l)1378 2432 y FO(X)1454 2398 y FM(k)q FN(\000)p FM(l)1567 2432 y FO(X)1643 2398 y FM(l)1691 2432 y FP(=)g FO(B)1842 2444 y FM(l)1867 2432 y FO(B)1930 2444 y FM(k)1971 2432 y FO(:)118 2601 y FP(Th)n(us,)k(the)h (op)r(erator)e FO(X)34 b FP(is)28 b(cen)n(tered.)243 2701 y(It)e(remains)f(to)g(pro)n(v)n(e)f(the)j(second)e(statemen)n(t)h (of)f(the)h(prop)r(osition.)36 b(Let)118 2801 y(\001)23 b(=)g FO(\033)s FP(\()p FO(C)445 2771 y FL(2)483 2801 y FP(\))15 b FQ(n)e(f)p FP(0)p FQ(g)p FP(.)35 b(Then)26 b FO(E)1045 2816 y FM(C)1097 2799 y Fy(2)1133 2801 y FP(\(\001\))h(is)e(a)g(pro)5 b(jection)25 b(on)n(to)f(the)i(co-k)n (ernel)e(of)118 2900 y FO(C)183 2870 y FL(2)221 2900 y FP(,)31 b(and,)g(since)f(k)n(er)13 b FO(U)36 b FP(=)28 b(k)n(er)12 b FO(C)6 b FP(,)32 b(w)n(e)e(ha)n(v)n(e)f FO(E)1604 2915 y FM(C)1656 2898 y Fy(2)1693 2900 y FP(\(\001\))f(=)g FO(U)2013 2870 y FN(\003)2050 2900 y FO(U)9 b FP(.)46 b(Then)30 b(\(2.5\))118 3000 y(implies)e(that)g FO(U)9 b(U)712 2970 y FN(\003)773 3000 y FP(=)22 b FO(U)9 b(E)c FP(\(\001\))p FO(U)1191 2970 y FN(\003)1253 3000 y FP(=)22 b FO(E)1401 3015 y FM(C)1453 2998 y Fy(2)1490 3000 y FP(\()p FO(F)1587 2970 y FN(\000)p FL(1)1676 3000 y FP(\(\001\)\).)38 b(Similarly)-7 b(,)337 3169 y FO(U)403 3135 y FM(k)444 3169 y FP(\()p FO(U)542 3135 y FN(\003)580 3169 y FP(\))612 3135 y FM(k)676 3169 y FP(=)23 b FO(U)830 3135 y FM(k)q FN(\000)p FL(1)956 3169 y FO(U)9 b(U)1088 3135 y FN(\003)1125 3169 y FP(\()p FO(U)1223 3135 y FN(\003)1261 3169 y FP(\))1293 3135 y FM(k)q FN(\000)p FL(1)1443 3169 y FP(=)22 b FO(U)1596 3135 y FM(k)q FN(\000)p FL(1)1722 3169 y FO(U)9 b(U)1854 3135 y FN(\003)1891 3169 y FO(U)g(U)2023 3135 y FN(\003)2061 3169 y FP(\()p FO(U)2159 3135 y FN(\003)2197 3169 y FP(\))2229 3135 y FM(k)q FN(\000)p FL(1)676 3307 y FP(=)23 b FO(U)830 3273 y FM(k)871 3307 y FO(E)932 3322 y FM(C)984 3305 y Fy(2)1020 3307 y FP(\(\001\)\()p FO(U)1251 3273 y FN(\003)1290 3307 y FP(\))1322 3273 y FM(k)1386 3307 y FP(=)g FO(E)1535 3322 y FM(C)1587 3305 y Fy(2)1623 3307 y FP(\()p FO(F)1720 3273 y FN(\000)p FM(k)1813 3307 y FP(\(\001\)\))p FO(:)118 3486 y FP(Therefore,)f(the)h(pro)5 b(jections)20 b FO(U)1138 3456 y FM(k)1179 3486 y FP(\()p FO(U)1277 3456 y FN(\003)1315 3486 y FP(\))1347 3456 y FM(k)1388 3486 y FP(,)k FO(k)i FQ(\025)c FP(1,)h(comm)n(ute)f(b)r(oth)h(with)f FO(U)2471 3456 y FN(\003)2509 3486 y FO(U)118 3585 y FP(and)g(eac)n(h)f(other.)34 b(The)22 b(rest)g(of)g(the)g(comm)n(uting)g(relations)e(can)i(b)r(e)g (obtained)118 3685 y(in)30 b(the)h(same)e(w)n(a)n(y)g(as)g(w)n(as)g (done)h(for)f FO(X)37 b FP(in)30 b(the)g(pro)r(of)f(of)h(the)h(\014rst) e(part)h(of)118 3785 y(the)e(prop)r(osition.)p 2514 3785 4 57 v 2518 3732 50 4 v 2518 3785 V 2567 3785 4 57 v 118 3948 a FR(5.)34 b FP(No)n(w)19 b(w)n(e)g(will)g(sho)n(w)g(that)h (prop)r(erties)e(of)i(represen)n(tations)d(of)j(the)g(relation)118 4048 y(\(2.1\))33 b(dep)r(end)h(on)f(prop)r(erties)g(of)g(the)h (corresp)r(onding)d(dynamical)i(system)118 4147 y FO(\025)24 b FQ(7!)f FO(F)12 b FP(\()p FO(\025)p FP(\),)28 b(and)g(study)g(this)f (dep)r(endency)h(in)g(detail.)p eop %%Page: 88 92 88 91 bop 118 100 a FP(88)485 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FR(Prop)s(osition)33 b(30.)41 b FC(L)l(et)31 b(the)h(op)l(er)l(ators)g FO(C)38 b FC(and)32 b FO(U)40 b FC(satisfy)33 b(r)l(elation)39 b FP(\(2.5\))o FC(.)118 432 y(If)25 b FO(e)239 444 y FM(\025)307 432 y FC(is)g(an)f(eigenve)l(ctor)i(of)f(the)g(op)l(er)l(ator)h FO(C)1543 402 y FL(2)1580 432 y FC(,)g(c)l(orr)l(esp)l(onding)g(to)f (an)f(eigen-)118 532 y(value)33 b FO(\025)p FC(,)i(then)d FO(U)696 502 y FN(\003)734 532 y FO(e)773 544 y FM(\025)849 532 y FC(is)h(either)h(zer)l(o)f(or)g(is)h(again)g(an)e(eigenve)l(ctor) i(of)g FO(C)2512 502 y FL(2)2549 532 y FC(,)118 632 y(c)l(orr)l(esp)l (onding)d(to)f(the)g(eigenvalue)h FO(F)12 b FP(\()p FO(\025)p FP(\))p FC(.)118 795 y(Pr)l(o)l(of.)43 b FP(Indeed,)f(from)d(\(2.5\),)j (w)n(e)c(see)h(that)g FO(C)1664 765 y FL(2)1702 795 y FO(U)1768 765 y FN(\003)1805 795 y FO(e)1844 807 y FM(\025)1930 795 y FP(=)i FO(U)2102 765 y FN(\003)2140 795 y FO(F)12 b FP(\()p FO(C)2302 765 y FL(2)2340 795 y FP(\))i FO(e)2425 807 y FM(\025)2510 795 y FP(=)118 895 y FO(F)e FP(\()p FO(\025)p FP(\))i FO(U)375 865 y FN(\003)414 895 y FO(e)453 907 y FM(\025)496 895 y FP(.)p 2514 895 4 57 v 2518 842 50 4 v 2518 895 V 2567 895 4 57 v 243 1060 a(In)25 b(particular,)g(if)h (the)g(op)r(erator)e FO(U)35 b FP(is)25 b(unitary)-7 b(,)26 b(then)g FO(U)2031 1030 y FN(\003)2069 1060 y FO(e)2108 1072 y FM(\025)2177 1060 y FP(is)f(not)h(zero,)118 1160 y(whic)n(h)i(implies)h(that,)g(in)f(this)h(case,)e FO(F)12 b FP(\()p FQ(\001)p FP(\))29 b(maps)f(the)h(sp)r(ectrum)f(of)g FO(C)2368 1130 y FL(2)2434 1160 y FP(in)n(to)118 1259 y(itself.)243 1359 y(A)c(similar)e(fact)i(holds)f(for)g(all)h(p)r(oin)n (ts)f(of)h(the)f(sp)r(ectrum)h(of)g(the)g(op)r(erator)118 1459 y FO(C)183 1428 y FL(2)221 1459 y FP(.)118 1622 y FR(Prop)s(osition)29 b(31.)39 b FC(If)28 b(the)h(op)l(er)l(ator)g FO(X)35 b FC(in)f FP(\(2.1\))28 b FC(is)h(invertible)36 b FP(\()p FC(i.e.,)31 b FO(U)37 b FC(in)118 1722 y FP(\(2.5\))29 b FC(is)i(unitary)7 b FP(\))p FC(,)30 b(then)g FO(F)12 b FP(\()p FQ(\001)p FP(\))30 b FC(maps)h(the)f(sp)l(e)l(ctrum)f FO(\033)s FP(\()p FO(C)1982 1692 y FL(2)2020 1722 y FP(\))h FC(into)g(itself.)118 1885 y(Pr)l(o)l(of.)43 b FP(Let)30 b FO(\025)c FQ(2)g FO(\033)s FP(\()p FO(C)831 1855 y FL(2)870 1885 y FP(\).)42 b(Then)29 b(there)h(exists)e(a)h(sequence)g (of)g(unit)h(v)n(ectors)118 1985 y FO(e)157 1997 y FM(n)225 1985 y FQ(2)23 b FO(H)7 b FP(,)28 b FO(n)23 b FQ(\025)f FP(1,)28 b(suc)n(h)f(that)h FQ(k)p FO(C)1157 1955 y FL(2)1194 1985 y FO(e)1233 1997 y FM(n)1296 1985 y FQ(\000)18 b FO(\025e)1466 1997 y FM(n)1511 1985 y FQ(k)23 b(\000)-48 b(!)23 b(1)p FP(,)k FO(n)c FQ(\000)-48 b(!)23 b(1)p FP(.)37 b(Then)406 2164 y FQ(k)p FO(C)513 2130 y FL(2)550 2164 y FO(U)616 2130 y FN(\003)654 2164 y FO(e)693 2176 y FM(n)756 2164 y FQ(\000)18 b FO(F)12 b FP(\()p FO(\025)p FP(\))p FO(U)1082 2130 y FN(\003)1121 2164 y FO(e)1160 2176 y FM(n)1205 2164 y FQ(k)22 b FP(=)h FQ(k)p FO(U)1465 2130 y FN(\003)1502 2164 y FP(\()p FO(F)12 b FP(\()p FO(C)1696 2130 y FL(2)1734 2164 y FP(\))p FO(e)1805 2176 y FM(n)1869 2164 y FQ(\000)18 b FO(F)12 b FP(\()p FO(\025)p FP(\))p FO(e)2168 2176 y FM(n)2214 2164 y FP(\))p FQ(k)569 2299 y FP(=)22 b FQ(k)p FO(F)12 b FP(\()p FO(C)860 2265 y FL(2)898 2299 y FP(\))p FO(e)969 2311 y FM(n)1032 2299 y FQ(\000)18 b FO(F)12 b FP(\()p FO(\025)p FP(\))p FO(e)1331 2311 y FM(n)1377 2299 y FQ(k)23 b(\000)-49 b(!)23 b FP(0)p FO(;)180 b(n)23 b FQ(\000)-49 b(!)23 b(1)p FO(;)118 2479 y FP(and)28 b FQ(k)p FO(U)388 2448 y FN(\003)425 2479 y FO(e)464 2491 y FM(n)509 2479 y FQ(k)22 b FP(=)h(1,)k FO(n)c FQ(\025)g FP(1.)36 b(This)28 b(implies)g(that)g FO(F)12 b FP(\()p FO(\025)p FP(\))24 b FQ(2)f FO(\033)s FP(\()p FO(C)2093 2448 y FL(2)2131 2479 y FP(\).)p 2514 2479 V 2518 2426 50 4 v 2518 2479 V 2567 2479 4 57 v 243 2644 a(Prop)r(osition)34 b(30)h(pro)n(vides)f(a)i(w)n(a)n(y)e(for)i (constructing)f(represen)n(tations)118 2744 y(of)28 b(relation)g (\(2.1\))o(.)39 b(Indeed,)29 b(c)n(ho)r(ose)e(a)h(sequence)g(of)g(p)r (ositiv)n(e)g(n)n(um)n(b)r(ers)f FO(\025)2510 2756 y FM(k)2552 2744 y FP(,)118 2843 y FO(k)f FQ(2)d FJ(Z)p FP(,)d(\(if)27 b(it)g(exists\))e(suc)n(h)h(that)g FO(F)12 b FP(\()p FO(\025)1327 2855 y FM(k)1369 2843 y FP(\))23 b(=)g FO(\025)1560 2855 y FM(k)q FL(+1)1711 2843 y FP(for)j(all)g FO(k)g FQ(2)d FJ(Z)o FP(,)e(and)26 b(de\014ne)118 2943 y(the)i(op)r(erators)e FO(C)6 b FP(,)28 b FO(U)36 b FP(in)28 b FO(l)960 2955 y FL(2)997 2943 y FP(\()p FJ(Z)p FP(\))22 b(as)27 b(follo)n(ws:)671 3122 y FO(C)736 3088 y FL(2)773 3122 y FO(e)812 3134 y FM(k)876 3122 y FP(=)c FO(\025)1012 3134 y FM(k)1053 3122 y FO(;)97 b(U)9 b(e)1278 3134 y FM(k)1341 3122 y FP(=)23 b FO(e)1468 3134 y FM(k)q FN(\000)p FL(1)1593 3122 y FO(;)180 b(k)26 b FQ(2)e FJ(Z)o FO(:)376 b FP(\(2.8\))118 3302 y(Then)30 b(the)h(op)r(erators)d FO(U)9 b FP(,)30 b FO(C)6 b FP(,)31 b(satisfy)g(\(2.5\),)g(and)f (therefore,)f(giv)n(e)h(a)f(repre-)118 3401 y(sen)n(tation)k(of)41 b(\(2.1\))o(.)57 b(Belo)n(w,)34 b(w)n(e)g(will)g(study)g(whether)g (other)g(irreducible)118 3501 y(represen)n(tations)29 b(exist)i(and)f(what)h(is)g(their)f(structure.)46 b(In)31 b(particular,)f(for)118 3601 y(\\simple")24 b(mappings)g FO(F)12 b FP(\()p FQ(\001)p FP(\),)26 b(all)f(irreducible)f(represen)n (tations)f(with)i(unitary)118 3700 y FO(U)k FP(ha)n(v)n(e)19 b(suc)n(h)h(a)g(form,)h(and)f(can)g(b)r(e)h(classi\014ed)e(up)i(to)f(a) g(unitary)f(equiv)-5 b(alence.)118 3848 y FR(6.)81 b FP(Let)42 b(a)g(Borel)g(set)g(\001)h(b)r(e)f(suc)n(h)h(that)f(\001)48 b FQ(\032)f FO(F)12 b FP(\()p FJ(R)q FP(\))48 b(and)43 b FO(F)2240 3818 y FN(\000)p FL(1)2329 3848 y FP(\(\001\))48 b(=)118 3948 y(\001.)66 b(Prop)r(osition)36 b(28)h(implies)g(that)h FO(E)1396 3963 y FM(C)1448 3946 y Fy(2)1484 3948 y FP(\(\001\))p FO(U)1683 3918 y FN(\003)1761 3948 y FP(=)h FO(U)1931 3918 y FN(\003)1969 3948 y FO(E)2030 3963 y FM(C)2082 3946 y Fy(2)2119 3948 y FP(\()p FO(F)2216 3918 y FN(\000)p FL(1)2305 3948 y FP(\(\001\)\))h(=)118 4048 y FO(U)184 4018 y FN(\003)222 4048 y FO(E)283 4063 y FM(C)335 4046 y Fy(2)372 4048 y FP(\(\001\),)33 b(i.e.,)f FO(E)783 4063 y FM(C)835 4046 y Fy(2)871 4048 y FP(\(\001\))g(is)f(a)g(pro)5 b(jection)30 b(on)n(to)h(an)g(in)n(v)-5 b(arian)n(t)30 b(subspace)118 4147 y(in)36 b FO(H)7 b FP(.)62 b(Therefore,)37 b(to)e(study)h(irreducible)g(represen)n(tations,)g(w)n(e)f(need)h(to)p eop %%Page: 89 93 89 92 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f (systems)896 b FP(89)118 333 y(study)24 b(the)f(\\smallest")f(in)n(v)-5 b(arian)n(t)23 b(subsets)g(\(in)h(the)g(sense)e(indicated)i(ab)r(o)n(v) n(e\))118 432 y(that)c(con)n(tains)e(the)i(sp)r(ectrum)f(of)g FO(C)1247 402 y FL(2)1285 432 y FP(.)34 b(Belo)n(w)18 b(w)n(e)h(will)h(use)f(some)g(facts)g(ab)r(out)118 532 y(dynamical)30 b(systems)f(and)h(their)g(in)n(v)-5 b(arian)n(t)29 b(sets,)i(as)e(w)n(ell)h(as)g(prop)r(erties)f(of)118 632 y(the)f(corresp)r(onding)e(sp)r(ectral)h(measures.)118 783 y FR(7.)68 b FP(Recall)38 b(some)f(basic)h(notions)g(and)g(facts)g (ab)r(out)g(discrete)g(dynamical)118 882 y(systems.)45 b(A)31 b(discrete)f(time)h(one-dimensional)e(dynamical)h(system)g(is)g (just)118 982 y(a)d(\(con)n(tin)n(uous)g(or)g(measurable\))f(mapping)i FJ(R)1599 952 y FL(1)1665 982 y FQ(3)23 b FO(\025)h FQ(7!)f FO(F)12 b FP(\()p FO(\025)p FP(\))24 b FQ(2)f FJ(R)2254 952 y FL(1)2298 982 y FP(.)243 1082 y(First)36 b(of)g(all,)i(w)n(e)d (recall)h(the)g(notion)g(of)g(a)g(tra)5 b(jectory)34 b(or)h(an)h(orbit)g(of)118 1182 y(a)e(dynamical)f(system.)56 b(T)-7 b(raditionally)g(,)35 b(a)f(tra)5 b(jectory)32 b(or)h(an)h(orbit)f(of)h(the)118 1281 y(dynamical)27 b(system)g FO(F)12 b FP(\()p FQ(\001)p FP(\))24 b(:)f FJ(R)1068 1251 y FL(1)1134 1281 y FQ(7!)h FJ(R)1295 1251 y FL(1)1365 1281 y FP(is)k(the)g(set)542 1532 y(Orb)o(\()p FO(\025)p FP(\))c(=)f FQ(f)p FO(\025;)14 b(F)e FP(\()p FO(\025)p FP(\))p FO(;)i(F)1315 1497 y FL(2)1353 1532 y FP(\()p FO(\025)p FP(\))p FO(;)g(:)g(:)g(:)g FQ(g)23 b FP(=)1795 1428 y FN(1)1782 1453 y Fz([)1766 1629 y FM(n)p FL(=0)1905 1532 y FO(F)1970 1497 y FM(n)2015 1532 y FP(\()p FO(\025)p FP(\))q FO(:)118 1797 y FP(Here)33 b FO(F)385 1767 y FM(n)430 1797 y FP(\()p FQ(\001)p FP(\))f(=)f FO(F)12 b FP(\()p FO(F)807 1767 y FM(n)p FN(\000)p FL(1)937 1797 y FP(\()p FQ(\001)p FP(\)\),)35 b FO(n)d FP(=)f(1,)j(2,)e FO(:)14 b(:)g(:)28 b FP(,)34 b(and)e FO(F)1901 1767 y FL(0)1939 1797 y FP(\()p FQ(\001)p FP(\))h(is)g(the)g(iden)n(tit)n(y) 118 1897 y(transformation.)h(Since)25 b(w)n(e)f(are)g(in)n(terested)g (in)h(the)g(action)f(of)g FO(F)2198 1866 y FN(\000)p FL(1)2312 1897 y FP(as)g(w)n(ell,)118 1996 y(will)31 b(sa)n(y)f(that)h(t)n(w)n(o)f(p)r(oin)n(ts,)i FO(\025)1093 2008 y FL(1)1162 1996 y FP(and)f FO(\025)1375 2008 y FL(2)1412 1996 y FP(,)h(b)r(elong)f(to)g(the)g(same)f(tra)5 b(jectory)-7 b(,)118 2096 y(if)33 b FO(F)264 2066 y FM(k)305 2096 y FP(\()p FO(\025)385 2108 y FL(1)423 2096 y FP(\))e(=)g FO(\025)630 2108 y FL(2)700 2096 y FP(or)h FO(F)872 2066 y FM(k)913 2096 y FP(\()p FO(\025)993 2108 y FL(2)1031 2096 y FP(\))f(=)g FO(\025)1238 2108 y FL(1)1308 2096 y FP(for)h(some)g FO(k)i FQ(\025)d FP(0.)51 b(W)-7 b(e)33 b(also)e(need)h(the)118 2195 y(notion)e(of)h(the)g(tra)5 b(jectory)29 b(decomp)r(osition:)42 b(w)n(e)30 b(sa)n(y)f(that)i(t)n(w) n(o)f(p)r(oin)n(ts,)h FO(\025)2537 2207 y FL(1)118 2295 y FP(and)24 b FO(\025)324 2307 y FL(2)362 2295 y FP(,)h(b)r(elong)f(to) g(the)g(same)g(elemen)n(t)g(of)g(the)h(tra)5 b(jectory)22 b(decomp)r(osition,)118 2395 y(if)37 b(these)f(p)r(oin)n(ts)g(\\meet)g (in)g(the)h(future",)h(i.e.,)g(if)f FO(F)1831 2365 y FM(k)1872 2395 y FP(\()p FO(\025)1952 2407 y FL(1)1990 2395 y FP(\))g(=)g FO(F)2226 2365 y FM(m)2289 2395 y FP(\()p FO(\025)2369 2407 y FL(2)2407 2395 y FP(\))f(for)118 2494 y(some)h FO(k)s FP(,)k FO(m)f(>)g FP(0.)67 b(A)39 b(p)r(oin)n(t)f FO(\025)1172 2506 y FL(0)1250 2494 y FQ(2)i FJ(R)1399 2464 y FL(1)1443 2494 y FP(,)g(suc)n(h)e(that)g FO(F)1959 2464 y FM(m)2022 2494 y FP(\()p FO(\025)2102 2506 y FL(0)2140 2494 y FP(\))j(=)e FO(\025)2365 2506 y FL(0)2441 2494 y FP(and)118 2594 y FO(F)183 2564 y FM(n)228 2594 y FP(\()p FO(\025)308 2606 y FL(0)346 2594 y FP(\))e FQ(6)p FP(=)g FO(\025)565 2606 y FL(0)639 2594 y FP(for)e(0)h FO(<)h(n)g(<)f(m)p FP(,)i(is)e(called)g(a)f(p)r(erio)r (dic)h(p)r(oin)n(t)g(of)g(p)r(erio)r(d)118 2694 y FO(m)p FP(.)55 b(The)33 b(p)r(erio)r(dic)g(p)r(oin)n(ts)h FO(\025)1073 2706 y FL(0)1110 2694 y FP(,)i FO(F)12 b FP(\()p FO(\025)1314 2706 y FL(0)1351 2694 y FP(\),)34 b FO(:)14 b(:)g(:)28 b FP(,)35 b FO(F)1688 2663 y FM(m)p FN(\000)p FL(1)1836 2694 y FP(\()p FO(\025)1916 2706 y FL(0)1954 2694 y FP(\))f(form)f(a)g (cycle)g(of)118 2793 y(p)r(erio)r(d)55 b FO(m)p FP(.)243 2893 y(Bearing)39 b(in)j(mind)f(the)h(represen)n(tations)d(of)i (relation)g(\(2.5\))f(and)h(the)118 2993 y(fact)34 b(that)h(these)f(in) n(v)-5 b(arian)n(t)33 b(sets)h(should)g(carry)f(the)h(sp)r(ectral)g (measure)f(of)118 3093 y(the)k(op)r(erator)e FO(C)679 3062 y FL(2)717 3093 y FP(,)k(w)n(e)e(need)g(to)f(consider)g(a)g (measurable)g(mapping)h(of)f(a)118 3192 y(measurable)h(space.)68 b(In)38 b(this)h(case,)h(the)e(mapping)g FO(F)12 b FP(\()p FQ(\001)p FP(\))39 b(giv)n(es)e(rise)g(to)h(a)118 3292 y(mapping)27 b(of)h(Borel)e(measures)h(on)g(the)h(line)g(b)n(y)f(the)h (form)n(ula)962 3476 y FO(d\032)p FP(\()p FO(\025)p FP(\))c FQ(7!)f FO(d\032)p FP(\()p FO(F)1473 3441 y FN(\000)p FL(1)1563 3476 y FP(\()p FO(\025)p FP(\)\))p FO(:)243 3660 y FP(A)40 b(Borel)f(measure)f FO(\032)p FP(\()p FQ(\001)p FP(\))j(is)f(called)f(quasi-in)n(v)-5 b(arian)n(t)38 b(with)i(resp)r(ect)g(to)118 3760 y FO(F)12 b FP(\()p FQ(\001)p FP(\),)28 b(if)h FO(\032)p FP(\()p FO(F)538 3729 y FN(\000)p FL(1)627 3760 y FP(\()p FQ(\001)p FP(\)\))f(is)g (absolutely)f(con)n(tin)n(uous)f(with)i(resp)r(ect)g(to)f FO(\032)p FP(\()p FQ(\001)p FP(\).)118 3927 y FR(Prop)s(osition)32 b(32.)41 b FC(If)31 b(the)g(op)l(er)l(ator)g FO(U)40 b FC(in)d FP(\(2.5\))30 b FC(is)i(unitary,)f(and)g FO(F)12 b FP(\()p FQ(\001)p FP(\))31 b FC(is)118 4027 y(one-to-one,)i(then)e (the)h(sp)l(e)l(ctr)l(al)f(me)l(asur)l(e)h(of)50 b FO(C)1689 3997 y FL(2)1758 4027 y FC(is)31 b(quasi-invariant)i(with)118 4127 y(r)l(esp)l(e)l(ct)d(to)f FO(F)12 b FP(\()p FQ(\001)p FP(\))31 b FC(and)f FO(F)897 4096 y FN(\000)p FL(1)986 4127 y FP(\()p FQ(\001)p FP(\))p FC(.)p eop %%Page: 90 94 90 93 bop 118 100 a FP(90)485 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(Pr)l(o)l(of.)43 b FP(Indeed,)33 b(w)n(e)f(can)f(tak)n(e)h(the)g(sp)r(ectral)f(measure)g(to)h(b)r(e)g (in)g(the)g(form)118 432 y FO(\032)p FP(\()p FQ(\001)p FP(\))g(=)f(\()p FO(E)469 447 y FM(C)521 430 y Fy(2)558 432 y FP(\()p FQ(\001)p FP(\))p FO(!)s(;)14 b(!)s FP(\),)35 b(where)d FO(!)j FP(is)e(a)f(v)n(ector)g(of)g(maximal)g(sp)r(ectral)h (t)n(yp)r(e.)118 532 y(Then)28 b(the)g(statemen)n(t)f(follo)n(ws)g (from)g(the)h(equalit)n(y)296 721 y FO(\032)p FP(\()p FO(F)436 686 y FN(\000)p FL(1)525 721 y FP(\()p FQ(\001)p FP(\)\))c(=)f(\()p FO(E)849 736 y FM(C)901 719 y Fy(2)938 721 y FP(\()p FO(F)1035 686 y FN(\000)p FL(1)1124 721 y FP(\()p FQ(\001)p FP(\)\))p FO(!)s(;)14 b(!)s FP(\))668 856 y(=)23 b(\()p FO(U)854 821 y FN(\003)892 856 y FO(E)953 871 y FM(C)1005 854 y Fy(2)1042 856 y FP(\()p FO(F)1139 821 y FN(\000)p FL(1)1228 856 y FP(\()p FQ(\001)p FP(\)\))p FO(!)s(;)14 b(U)1505 821 y FN(\003)1543 856 y FO(!)s FP(\))23 b(=)g(\()p FO(E)1834 871 y FM(C)1886 854 y Fy(2)1923 856 y FP(\()p FQ(\001)p FP(\))p FO(U)2076 821 y FN(\003)2114 856 y FO(!)s(;)14 b(U)2272 821 y FN(\003)2310 856 y FO(!)s FP(\))118 1044 y(and)28 b(a)f(similar)g(relation)f(for)h FO(U)9 b FP(.)p 2514 1044 4 57 v 2518 992 50 4 v 2518 1044 V 2567 1044 4 57 v 243 1235 a(W)-7 b(e)23 b(also)e(need)i(the)g (notion)f(of)g(ergo)r(dicit)n(y)g(of)g(a)g(measure.)34 b(A)23 b(Borel)f(mea-)118 1335 y(sure)28 b FO(\032)p FP(\()p FQ(\001)p FP(\))i(is)e(called)g(ergo)r(dic)g(with)h(resp)r(ect) g(to)f(the)h(action)g(of)f(a)h(dynamical)118 1435 y(system)k FO(F)12 b FP(\()p FQ(\001)p FP(\),)36 b(if)d(for)g(an)n(y)g(measurable) f FO(F)12 b FP(\()p FQ(\001)p FP(\)-in)n(v)-5 b(arian)n(t)32 b(set)i(\001)e FQ(\032)h FJ(R)p FP(,)40 b(\(i.e.,)118 1534 y(the)29 b(set)f(\001)g(suc)n(h)g(that)g FO(F)922 1504 y FN(\000)p FL(1)1011 1534 y FP(\(\001\))d(=)f(\001\),)29 b(w)n(e)e(ha)n(v)n(e)g(that)i(either)f FO(\032)p FP(\(\001\))c(=)g(0)k (or)118 1634 y FO(\032)p FP(\()p FJ(R)d FQ(n)18 b FP(\001\))23 b(=)g(0.)118 1809 y FR(Prop)s(osition)37 b(33.)44 b FC(If)36 b(a)f(p)l(air)h FO(X)7 b FC(,)37 b FO(X)1368 1779 y FN(\003)1405 1809 y FC(,)g(satisfying)44 b FP(\(2.1\))p FC(,)36 b(is)g(irr)l(e)l (ducible,)118 1909 y(then)c(the)g(sp)l(e)l(ctr)l(al)g(me)l(asur)l(e)f (of)i(the)f(op)l(er)l(ator)41 b FO(C)1708 1879 y FL(2)1777 1909 y FC(is)33 b(er)l(go)l(dic)g(with)f(r)l(esp)l(e)l(ct)118 2008 y(to)e FO(F)12 b FP(\()p FQ(\001)p FP(\))p FC(.)118 2184 y(Pr)l(o)l(of.)43 b FP(Indeed,)g(otherwise)38 b(there)h(w)n(ould)g (exist)g(a)g(non-trivial)f(in)n(v)-5 b(arian)n(t)118 2283 y(subset)35 b(\001)g FQ(\032)g FO(\033)s FP(\()p FO(C)731 2253 y FL(2)769 2283 y FP(\).)59 b(As)35 b(w)n(as)f(noticed)g (ab)r(o)n(v)n(e,)i(in)f(this)g(case)f FO(E)2262 2298 y FM(C)2314 2282 y Fy(2)2350 2283 y FP(\(\001\))i(is)118 2383 y(a)d(pro)5 b(jection)32 b(on)n(to)h(an)f(in)n(v)-5 b(arian)n(t)33 b(subspace)f(whic)n(h)h(is)g(non-trivial)f(if)i(and)118 2483 y(only)27 b(if)h(the)g(sp)r(ectral)f(measure)g(of)h(\001)f(is)h (neither)f(zero)g(nor)g(one.)p 2514 2483 V 2518 2430 50 4 v 2518 2483 V 2567 2483 4 57 v 243 2674 a(The)c(simplest)h(in)n(v) -5 b(arian)n(t)23 b(sets)g(are)g(elemen)n(ts)h(of)f(the)h(tra)5 b(jectory)23 b(decom-)118 2773 y(p)r(osition)33 b(of)g(the)g(dynamical) f(system)h(\(in)g(the)g(one-to-one)e(case,)j(they)f(are)118 2873 y(just)e(orbits\).)45 b(The)30 b(simplest)h(class)e(of)i(quasi-in) n(v)-5 b(arian)n(t)28 b(ergo)r(dic)h(measures)118 2973 y(is)37 b(formed)f(b)n(y)h(atomic)f(measures)g(concen)n(trated)g(on)g (an)h(elemen)n(t)g(of)g(tra-)118 3072 y(jectory)24 b(decomp)r(osition;) h(ho)n(w)n(ev)n(er,)d(an)i(atomic)g(measure)f(concen)n(trated)g(on)118 3172 y(an)28 b(orbit)g(is)g(also)f(quasi-in)n(v)-5 b(arian)n(t)26 b(and)i(ergo)r(dic.)38 b(Belo)n(w,)27 b(w)n(e)h(will)h(see)f(that)118 3271 y(only)c(suc)n(h)g(measures)g(corresp)r(onding)e(to)i(an)h(orbit)f (giv)n(e)f(rise)h(to)h(irreducible)118 3371 y(represen)n(tations)h(of)h (the)h(relation.)243 3474 y(The)22 b(existence)g(of)g(non-atomic)f (quasi-in)n(v)-5 b(arian)n(t)19 b(ergo)r(dic)i(measures)g(de-)118 3573 y(p)r(ends)40 b(on)f(top)r(ological)e(prop)r(erties)h(of)i(the)f (dynamical)g(system.)71 b(In)39 b(the)118 3673 y(one-to-one)26 b(case,)h(w)n(e)g(ha)n(v)n(e)f(the)i(follo)n(wing)f(fact.)118 3848 y FR(Prop)s(osition)45 b(34.)i FC(If)42 b(a)g(dynamic)l(al)g (system)f FO(\025)k FQ(7!)e FO(F)12 b FP(\()p FO(\025)p FP(\))42 b FC(with)g(one-to-)118 3948 y(one)33 b FO(F)12 b FP(\()p FQ(\001)p FP(\))33 b FC(p)l(ossesses)g(a)f(me)l(asur)l(able)h (se)l(ction,)h(i.e.,)h(a)e(me)l(asur)l(able)g(set)f(that)118 4048 y(interse)l(cts)k(any)i(orbit)f(in)g(a)h(single)f(p)l(oint,)j (then)c(any)i(er)l(go)l(dic)g(me)l(asur)l(e)e(is)118 4147 y(c)l(onc)l(entr)l(ate)l(d)29 b(on)h(a)g(single)h(orbit)f(of)h (the)f(dynamic)l(al)h(system.)p eop %%Page: 91 95 91 94 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f (systems)896 b FP(91)243 333 y(In)37 b(the)h(non-bijectiv)n(e)f(case,)i (the)f(condition)f(of)h(existence)f(of)g(a)h(mea-)118 432 y(surable)26 b(section)h(is)g(replaced)f(b)n(y)g(the)i(follo)n (wing)e(condition)g(of)h(existence)g(of)118 532 y FO(M)9 b FP(-partition:)243 632 y FC(the)29 b(dynamic)l(al)i(system)e(p)l (ossesses)h(an)f FO(M)9 b FC(-p)l(artition,)30 b(i.e.,)h(ther)l(e)e (exists)118 731 y(a)h(p)l(artition)h FJ(R)e FP(=)701 669 y Fz(S)771 756 y FM(n)p FN(2)p Fv(N)916 731 y FP(\001)985 743 y FM(n)1031 731 y FC(,)h FP(\001)1155 743 y FM(k)1219 731 y FQ(2)23 b FA(B)p FP(\()p FJ(R)q FP(\))p FC(,)36 b(such)30 b(that)243 831 y FP(1\))36 b FC(for)i(any)g FO(k)h FC(ther)l(e)e(exists)g FO(j)42 b FC(such)37 b(that)g FO(F)12 b FP(\(\001)1806 843 y FM(k)1847 831 y FP(\))37 b(=)e(\001)2085 843 y FM(j)2157 831 y FC(and)j FO(F)12 b FP(\()p FQ(\001)p FP(\))37 b FC(is)118 930 y(one-to-one)30 b(on)g FP(\001)720 942 y FM(k)761 930 y FP(;)243 1030 y(2\))e FC(if)i(for)g(some)g FO(n)p FP(,)f FO(k)s FC(,)g(the)h(mapping) g FO(F)1508 1000 y FM(n)1553 1030 y FP(\()p FQ(\001)p FP(\))g FC(maps)g FP(\001)1955 1042 y FM(k)2025 1030 y FC(into)f(itself)p FP(,)h FC(then)118 1130 y FO(F)183 1100 y FM(n)228 1130 y FP(\()p FQ(\001)p FP(\))h FC(is)f(the)g (identity)g(on)g FP(\001)1064 1142 y FM(k)1105 1130 y FC(.)243 1229 y FP(W)-7 b(e)28 b(also)e(men)n(tion)i(the)g(follo)n (wing)e(statemen)n(t)i(from)f([289)o(].)118 1394 y FR(Theorem)33 b(15.)42 b FC(L)l(et)31 b FO(I)39 b FC(b)l(e)32 b(some)g(\014nite)g (interval,)h(and)g FO(F)12 b FP(\()p FQ(\001)p FP(\))d(:)29 b FO(I)34 b FQ(\000)-46 b(!)27 b FO(I)38 b FC(b)l(e)32 b(a)118 1494 y(c)l(ontinuous)k(p)l(artial)t(ly)j(monotone)e(mapping)46 b FP(\()p FC(i.e.,)41 b FO(I)i FC(de)l(c)l(omp)l(oses)38 b(into)f(a)118 1594 y(\014nite)29 b(union)g(of)h(sub-intervals,)f(on)h (which)g FO(F)12 b FP(\()p FQ(\001)p FP(\))30 b FC(is)g(monotone)6 b FP(\))p FC(.)39 b(Then)30 b(the)118 1693 y(fol)t(lowing)i(c)l (onditions)f(ar)l(e)f(e)l(quivalent)8 b FP(:)243 1793 y FC(i)g FP(\))33 b FC(ther)l(e)e(exists)h(an)f FO(M)9 b FC(-p)l(artition)32 b(of)h FO(I)7 b FC(,)32 b(e)l(ach)h(element)f(of) g(which)i(is)e(an)118 1892 y(interval)39 b FP(\()p FC(p)l(ossibly,)32 b(a)e(single)h(p)l(oint)8 b FP(\);)243 1992 y FC(ii)g FP(\))29 b FC(any)f(quasi-invariant)h(er)l(go)l(dic)h(me)l(asur)l(e)d (is)h(c)l(onc)l(entr)l(ate)l(d)g(on)g(a)g(sin-)118 2092 y(gle)i(element)g(of)h(the)f(tr)l(aje)l(ctory)g(de)l(c)l(omp)l(osition) 6 b FP(;)243 2191 y FC(iii)i FP(\))31 b FC(the)f(set)g(of)g(p)l(erio)l (dic)i(p)l(oints,)f FP(P)n(er)12 b FO(F)g FC(,)30 b(is)g(close)l(d)9 b FP(;)243 2303 y FC(iv)g FP(\))34 b FC(for)g(some)f FO(m)d FQ(\025)f FP(0)p FC(,)34 b(the)f(r)l(elation)h FP(Fix\()p FO(F)1697 2273 y FL(2)1730 2248 y Fw(m)p Fy(+1)1861 2303 y FP(\))29 b(=)g(Fix\()p FO(F)2234 2273 y FL(2)2267 2248 y Fw(m)2327 2303 y FP(\))k FC(holds)118 2403 y FP(\(Fix\()p FO(F)12 b FP(\))31 b FC(denotes)f(the)g(set)f(of)i(\014xe)l(d)e(p)l (oints)h(of)h FO(F)12 b FP(\))p FC(.)243 2568 y FP(Belo)n(w)26 b(w)n(e)h(study)h(the)g(corresp)r(ondence)e(b)r(et)n(w)n(een)h(the)h (orbits)f(and)g(irre-)118 2667 y(ducible)h(represen)n(tations)e(of)h (the)h(relation.)118 2883 y FR(2.1.2)94 b(Finite-dimensional)27 b(represen)m(tations)118 3036 y FP(If)g(the)g(sequence)f FO(\025)733 3048 y FM(k)801 3036 y FP(in)h(\(2.8\))f(is)g(p)r(erio)r (dic,)h(the)g(corresp)r(onding)e(irreducible)118 3136 y(represen)n(tation)18 b(is)i(\014nite-dimensional.)34 b(W)-7 b(e)20 b(will)g(sho)n(w)f(here)h(that)g(all)g(\014nite-)118 3236 y(dimensional)28 b(represen)n(tations)e(of)i(relation)f(\(2.1\))h (are)f(related)g(to)h(cycles)g(of)118 3335 y(the)22 b(corresp)r(onding) d(dynamical)h(system.)35 b(Then)21 b(w)n(e)f(apply)h(the)h(Shark)n(o)n (vsky)118 3435 y(theorem)30 b(on)h(existence)f(of)h(cycles)f(to)h(to)f (study)h(irreducible)g(\014nite-dimen-)118 3534 y(sional)c(represen)n (tations;)e(these)j(results)f(are)g(illustrated)g(with)h(examples.)118 3683 y FR(1.)49 b FP(Let)33 b(us)f(classify)f(irreducible)g(pairs)g FO(X)7 b FP(,)33 b FO(X)1629 3653 y FN(\003)1698 3683 y FP(of)f(op)r(erators)e(on)i(a)g(\014nite-)118 3783 y(dimensional)h(space,)i(ob)r(eying)e(relation)g(\(2.1\))g (\(irreducible)h(\014nite-dimen-)118 3883 y(sional)27 b(represen)n(tations)e(of)34 b(\(2.1\)\))28 b(up)g(to)f(unitary)h (equiv)-5 b(alence.)118 4048 y FR(Theorem)30 b(16.)41 b FC(A)n(ny)29 b(cycle)i FO(O)1132 4060 y FM(\025)1199 4048 y FP(=)22 b FQ(f)p FO(\025;)14 b(F)e FP(\()p FO(\025)p FP(\))p FO(;)i(:)g(:)g(:)h(;)f(F)1841 4018 y FM(n)p FN(\000)p FL(1)1971 4048 y FP(\()p FO(\025)p FP(\))p FQ(g)30 b FC(of)g(p)l(erio)l(d)i FO(n)p FC(,)118 4147 y(such)38 b(that)f FO(\025)h FQ(\025)f FP(0)g FC(and)h FO(t)f(>)f FP(0)h FC(for)i(al)t(l)f(other)h(p)l(oints)e FO(t)h FQ(2)f FO(O)2140 4159 y FM(\025)2184 4147 y FC(,)j(de\014nes)d(a)p eop %%Page: 92 96 92 95 bop 118 100 a FP(92)485 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(family)32 b(of)e FO(n)p FC(-dimensional)h (irr)l(e)l(ducible)g(r)l(epr)l(esentations)f(of)49 b FP(\(2.1\))o(:)572 670 y FO(X)30 b FP(=)758 429 y Fz(0)758 575 y(B)758 625 y(B)758 674 y(B)758 728 y(@)887 492 y FP(0)1069 421 y Fz(p)p 1152 421 178 4 v 71 x FO(F)12 b FP(\()p FO(\025)p FP(\))452 b(0)895 581 y FC(.)895 614 y(.)895 647 y(.)1152 589 y(.)1186 614 y(.)1221 639 y(.)1420 589 y(.)1455 614 y(.)1490 639 y(.)887 754 y FP(0)518 b(0)1606 683 y Fz(p)p 1689 683 309 4 v 71 x FO(F)1754 730 y FM(n)p FN(\000)p FL(1)1884 754 y FP(\()p FO(\025)p FP(\))831 855 y FO(e)870 825 y FM(i\036)937 855 y FO(\025)193 b FP(0)f FO(:)14 b(:)g(:)272 b FP(0)1997 429 y Fz(1)1997 575 y(C)1997 625 y(C)1997 674 y(C)1997 728 y(A)2084 670 y FO(;)297 b FP(\(2.9\))118 1008 y FC(wher)l(e)31 b FO(\036)23 b FQ(2)g FP([0)p FO(;)14 b FP(2)p FO(\031)s FP(\))p FC(,)30 b(if)h FO(\025)23 b(>)g FP(0)p FC(,)30 b(or)g(a)g(single)g(r)l(epr)l(esentation,)h(if)f FO(\025)24 b FP(=)f(0)p FC(.)243 1108 y(These)30 b(ar)l(e)g(al)t(l,)i(up)d(to)h (unitary)g(e)l(quivalenc)l(e,)h(distinct)f(irr)l(e)l(ducible)h(r)l(ep-) 118 1207 y(r)l(esentations)f(of)48 b FP(\(2.1\))29 b FC(in)h(a)g(\014nite-dimensional)h(sp)l(ac)l(e.)118 1349 y(Pr)l(o)l(of.)43 b FP(Indeed,)28 b(b)n(y)g(a)f(direct)g(calculation)g (one)g(can)g(c)n(hec)n(k)g(that)h(the)g(repre-)118 1449 y(sen)n(tation)c(\(2.9\))g(satis\014es)g(the)g(necessary)f(relation.)35 b(Let)25 b(us)f(sho)n(w)f(that)i(it)g(is)118 1548 y(irreducible.)44 b(An)n(y)30 b(b)r(ounded)h(self-adjoin)n(t)f(op)r(erator)e FO(T)12 b FP(,)30 b(whic)n(h)g(comm)n(utes)118 1648 y(with)25 b FO(X)30 b FP(and)24 b FO(X)637 1618 y FN(\003)675 1648 y FP(,)g(comm)n(utes)g(with)h FO(C)1360 1618 y FL(2)1421 1648 y FP(and)f FO(U)33 b FP(in)n(tro)r(duced)24 b(ab)r(o)n(v)n(e)f(as) g(w)n(ell.)118 1748 y(But,)38 b(in)d(the)h(selected)f(basis)g FO(C)1165 1717 y FL(2)1237 1748 y FP(is)h(diagonal)d(with)j(distinct)g (eigen)n(v)-5 b(alues)118 1847 y FO(\025)p FP(,)29 b FO(:)14 b(:)g(:)27 b FP(,)i FO(F)459 1817 y FM(n)p FN(\000)p FL(1)589 1847 y FP(\()p FO(\025)p FP(\).)39 b(Then)29 b FO(T)39 b FP(is)28 b(diagonal,)e(and)i(the)h(comm)n(utation)e(with)i FO(U)118 1947 y FP(implies)f(that)g FO(T)34 b FP(=)23 b FO(cI)7 b FP(.)243 2046 y(W)-7 b(e)19 b(sho)n(w)f(that)h(these)g(are) f(all)g(the)i(irreducible)e(represen)n(tations)f(of)25 b(\(2.1\).)243 2146 y(Assume)33 b(that)g FO(U)42 b FP(is)33 b(unitary)-7 b(.)53 b(Since)33 b(dim)14 b FO(H)39 b(<)32 b FQ(1)p FP(,)j(the)e(op)r(erator)e FO(C)2537 2116 y FL(2)118 2246 y FP(can)41 b(b)r(e)i(diagonalized,)g(and)f(has)f (non-negativ)n(e)f(eigen)n(v)-5 b(alues.)78 b(T)-7 b(ak)n(e)41 b(an)118 2345 y(eigen)n(v)-5 b(alue)37 b FO(t)i FP(and)f(a)f(unit)i (eigen)n(v)n(ector)d FO(e)1512 2357 y FM(t)1579 2345 y FP(of)i FO(C)1749 2315 y FL(2)1787 2345 y FP(,)j FO(C)1916 2315 y FL(2)1953 2345 y FO(e)1992 2357 y FM(t)2062 2345 y FP(=)f FO(te)2236 2357 y FM(t)2265 2345 y FP(.)69 b(It)38 b(fol-)118 2445 y(lo)n(ws)28 b(from)g(\(2.5\))h(that)g FO(U)946 2415 y FN(\003)984 2445 y FO(e)1023 2457 y FM(t)1077 2445 y FP(=)c FO(e)1206 2460 y FM(F)9 b FL(\()p FM(t)p FL(\))1366 2445 y FP(is)29 b(again)f(a)g(unit)i(eigen)n(v)n(ector)c(of) j FO(C)2514 2415 y FL(2)2552 2445 y FP(,)118 2553 y FO(C)183 2523 y FL(2)221 2553 y FO(U)287 2523 y FN(\003)325 2553 y FO(e)364 2565 y FM(t)415 2553 y FP(=)23 b FO(F)12 b FP(\()p FO(t)p FP(\))p FO(e)701 2568 y FM(F)d FL(\()p FM(t)p FL(\))833 2553 y FP(.)37 b(Consider)26 b(the)h(sequence)g(of)g (unit)g(v)n(ectors)f(\()p FO(U)2373 2523 y FN(\003)2411 2553 y FP(\))2443 2523 y FM(k)2484 2553 y FO(e)2523 2565 y FM(t)2552 2553 y FP(,)118 2653 y FO(k)31 b FQ(\025)d FP(0.)45 b(Since)31 b(there)g(is)f(only)h(a)f(\014nite)h(n)n(um)n(b)r (er)f(of)h(eigen)n(v)-5 b(alues)30 b(of)g FO(C)2388 2623 y FL(2)2426 2653 y FP(,)h(w)n(e)118 2752 y(conclude)e(that,)g(for)g (some)f FO(n)d FQ(\025)g FP(1)j(and)h(some)f(eigen)n(v)-5 b(alue)28 b FO(t)2038 2764 y FL(0)2075 2752 y FP(,)i FO(F)2193 2722 y FM(n)2238 2752 y FP(\()p FO(t)2300 2764 y FL(0)2337 2752 y FP(\))c(=)f FO(t)2515 2764 y FL(0)2552 2752 y FP(.)118 2852 y(The)i(linear)g(span)g FO(H)782 2864 y FL(0)846 2852 y FP(of)h(eigenspaces)d(of)j FO(C)1539 2822 y FL(2)1576 2852 y FP(,)g(corresp)r(onding)d(to)i(eigen)n(v)-5 b(al-)118 2952 y(ues)28 b FO(t)292 2964 y FL(0)329 2952 y FP(,)g FO(:)14 b(:)g(:)27 b FP(,)h FO(F)620 2922 y FM(n)p FN(\000)p FL(1)750 2952 y FP(\()p FO(t)812 2964 y FL(0)849 2952 y FP(\),)g(is)g(in)n(v)-5 b(arian)n(t)26 b(with)i(resp)r(ect)g(to)f FO(C)2004 2922 y FL(2)2069 2952 y FP(and)h FO(U)2297 2922 y FN(\003)2335 2952 y FP(.)243 3051 y(W)-7 b(e)31 b(sho)n(w)f(that)i(the)f(space)g FO(H)1223 3063 y FL(0)1291 3051 y FP(is)g(also)f(in)n(v)-5 b(arian)n(t)30 b(with)i(resp)r(ect)f(to)g FO(U)9 b FP(.)118 3151 y(Assume)21 b(that)g FO(U)9 b(e)697 3163 y FM(t)722 3171 y Fy(0)779 3151 y FP(do)r(es)20 b(not)h(b)r(elong)f(to)h FO(H)1522 3163 y FL(0)1559 3151 y FP(.)35 b(Denote)21 b(b)n(y)f FO(u)h FP(the)g(pro)5 b(jection)118 3251 y(of)39 b FO(U)9 b(e)329 3263 y FM(t)354 3271 y Fy(0)428 3251 y FP(on)39 b(the)g(orthogonal)d(complemen)n(t)j(of)f FO(H)1784 3263 y FL(0)1822 3251 y FP(.)70 b(F)-7 b(rom)38 b(the)h(relation)118 3350 y FO(E)179 3365 y FM(C)231 3348 y Fy(2)268 3350 y FP(\(\001\))p FO(U)467 3320 y FN(\003)532 3350 y FP(=)26 b FO(U)689 3320 y FN(\003)727 3350 y FO(E)788 3365 y FM(C)840 3348 y Fy(2)876 3350 y FP(\()p FO(F)973 3320 y FN(\000)p FL(1)1063 3350 y FP(\(\001\)\))k(for)f(an)n(y)g(measurable)f(\001,)j(w)n(e)e(ha)n(v)n(e) g(that)118 3450 y FO(u)g FP(b)r(elongs)h(to)f(the)h(space)f FO(H)1037 3462 y FL(1)1104 3450 y FP(generated)g(b)n(y)g(eigen)n(v)n (ectors)f(corresp)r(onding)118 3550 y(to)g(all)f(eigen)n(v)-5 b(alues)26 b FO(\025)e FQ(2)f FP(\003)g(=)g FQ(f)p FO(F)12 b FP(\()p FO(\025)p FP(\))23 b(=)g FO(t)1443 3562 y FL(0)1480 3550 y FO(;)14 b(\025)23 b FQ(6)p FP(=)g FO(F)1741 3519 y FM(n)p FN(\000)p FL(1)1871 3550 y FP(\()p FO(t)1933 3562 y FL(0)1971 3550 y FP(\))p FQ(g)p FP(.)243 3649 y(Since)30 b(an)n(y)g(p)r(oin)n(t)g(has)g(a)g(single)g(image,)g(for)g (all)g FO(k)g(>)e FP(0,)i(the)h(sets)f(\003)2442 3661 y FM(k)2510 3649 y FP(=)118 3749 y FO(F)183 3719 y FN(\000)p FM(k)276 3749 y FP(\(\003\))20 b(are)f(disjoin)n(t.)34 b(W)-7 b(e)20 b(ha)n(v)n(e)f(that)h FO(U)1431 3719 y FM(k)1472 3749 y FO(H)1541 3761 y FL(1)1601 3749 y FQ(\032)j FO(E)1750 3764 y FM(C)1802 3747 y Fy(2)1838 3749 y FP(\()p FO(F)1935 3719 y FN(\000)p FM(k)2028 3749 y FP(\(\003\)\))p FO(H)2251 3761 y FL(1)2289 3749 y FP(,)e(and)f(all)118 3848 y(these)29 b(spaces)f(are)g(non-zero)f(and)i(orthogonal)d(to)j (eac)n(h)f(other.)41 b(But)29 b(this)g(is)118 3948 y(imp)r(ossible,)f (since)f FO(H)34 b FP(is)28 b(\014nite-dimensional.)36 b(Therefore,)27 b FO(H)2106 3960 y FL(0)2171 3948 y FP(is)g(in)n(v)-5 b(arian)n(t)118 4048 y(with)31 b(resp)r(ect)f(to)h FO(X)37 b FP(and)30 b FO(X)1048 4018 y FN(\003)1085 4048 y FP(,)i(and,)f(due)g (to)f(the)h(irreducibilit)n(y)-7 b(,)30 b(coincides)118 4147 y(with)e(the)g(whole)f(of)h FO(H)7 b FP(.)p eop %%Page: 93 97 93 96 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f (systems)896 b FP(93)243 333 y(The)30 b(op)r(erator)f FO(U)820 303 y FM(n)896 333 y FP(comm)n(utes)h(with)h FO(C)1546 303 y FL(2)1584 333 y FP(,)g(since)f FO(F)1909 303 y FM(n)1954 333 y FP(\()p FO(t)2016 345 y FL(0)2054 333 y FP(\))e(=)g FO(t)2237 345 y FL(0)2274 333 y FP(.)46 b(It)31 b(also)118 432 y(ob)n(viously)j(comm)n(utes)i(with)g FO(U)44 b FP(and)36 b FO(U)1426 402 y FN(\003)1464 432 y FP(.)61 b(Therefore,)36 b(the)g(irreducibilit)n(y)118 532 y(implies)20 b FO(U)458 502 y FM(n)526 532 y FP(=)j FO(\013I)7 b FP(,)22 b FQ(j)p FO(\013)p FQ(j)h FP(=)g(1.)34 b(In)20 b(the)g(basis)f FO(e)1530 544 y FM(t)1559 532 y FP(,)h FO(:)14 b(:)g(:)27 b FP(,)22 b FO(e)1810 548 y FM(F)1861 531 y Fw(n)p Fx(\000)p Fy(1)1974 548 y FL(\()p FM(t)p FL(\))2055 532 y FP(,)g(the)e(op)r(erators)118 632 y(act)27 b(as)g(needed.)243 740 y(Consider)e(the)i(case)f(of)g (non-unitary)f FO(U)9 b FP(.)37 b(No)n(w,)26 b(k)n(er)13 b FO(C)1978 710 y FL(2)2038 740 y FP(=)23 b(k)n(er)13 b FO(U)32 b FQ(6)p FP(=)22 b FQ(f)p FP(0)p FQ(g)p FP(,)118 839 y(and)28 b(there)g(exists)g(a)f(unit)i(v)n(ector)e FO(e)1257 851 y FL(0)1322 839 y FP(suc)n(h)g(that)i FO(U)9 b(e)1795 851 y FL(0)1855 839 y FP(=)24 b FO(C)2009 809 y FL(2)2046 839 y FO(e)2085 851 y FL(0)2146 839 y FP(=)f(0.)38 b(Again,)118 939 y(consider)31 b(the)h(v)n(ectors)e(\()p FO(U)977 909 y FN(\003)1015 939 y FP(\))1047 909 y FM(k)1088 939 y FO(e)1127 951 y FL(0)1164 939 y FP(,)j FO(k)g FP(=)c(0,)k(1,)e FO(:)14 b(:)g(:)28 b FP(.)49 b(Relation)31 b(\(2.5\))h(implies)118 1039 y(that)38 b FO(e)347 1051 y FM(k)426 1039 y FP(is)g(either)g(an)g (eigen)n(v)n(ector)e(of)i FO(C)1504 1008 y FL(2)1580 1039 y FP(with)g(the)h(eigen)n(v)-5 b(alue)37 b FO(F)2405 1008 y FM(k)2446 1039 y FP(\(0\),)118 1138 y(or)d(the)h(zero)f(v)n (ector.)57 b(Consider)34 b(t)n(w)n(o)g(p)r(ossibilities:)51 b(there)35 b(exists)f FO(n)h FP(suc)n(h)118 1238 y(that)f FO(e)343 1250 y FM(k)417 1238 y FP(=)e(0,)j(or)e(else)g(there)h(are)e (n)n(um)n(b)r(ers)h FO(k)k FP(and)c FO(n)p FP(,)j FO(k)f(<)e(n)p FP(,)i(suc)n(h)f(that)118 1338 y FO(F)183 1307 y FM(n)228 1338 y FP(\(0\))23 b(=)g FO(F)510 1307 y FM(k)551 1338 y FP(\(0\).)243 1446 y(W)-7 b(e)26 b(sho)n(w)f(that)i(the)f(second)g (alternativ)n(e)f(is)h(imp)r(ossible.)36 b(Indeed,)27 b(since)118 1545 y(k)n(er)13 b FO(U)309 1515 y FN(\003)374 1545 y FP(=)27 b(k)n(er)12 b FO(F)g FP(\()p FO(C)752 1515 y FL(2)790 1545 y FP(\),)31 b(w)n(e)f(conclude)g(that)g(the)h(m)n (ultiplicit)n(y)f(of)g(the)h(eigen-)118 1645 y(v)-5 b(alue)31 b FO(F)401 1615 y FM(k)442 1645 y FP(\(0\))h(is)f(the)h(same)e(as)h (that)h(of)f FO(F)1478 1615 y FM(n)1523 1645 y FP(\(0\).)49 b(On)31 b(the)h(other)e(hand,)j(t)n(w)n(o)118 1745 y(orthogonal)f (eigenspaces,)i FO(H)1079 1757 y FM(k)q FN(\000)p FL(1)1239 1745 y FP(and)f FO(H)1475 1757 y FM(n)1555 1745 y FP(that)h(corresp)r (ond)e(to)i FO(F)2343 1714 y FM(k)q FN(\000)p FL(1)2469 1745 y FP(\(0\))118 1844 y(and)e FO(F)349 1814 y FM(n)394 1844 y FP(\(0\),)i(are)e(mapp)r(ed)g(b)n(y)g FO(U)1207 1814 y FN(\003)1278 1844 y FP(in)n(to)g(the)g(eigenspace)g FO(H)2079 1856 y FM(k)2152 1844 y FP(corresp)r(ond-)118 1944 y(ing)k(to)g FO(F)440 1914 y FM(k)481 1944 y FP(\(0\).)62 b(Since)37 b(the)f(space)g(is)g(\014nite-dimensional,)i(there)e(exists) f(a)118 2043 y(non-zero)25 b(v)n(ector)g FO(w)k FP(in)e(the)g(direct)f (sum)h FO(H)1506 2055 y FM(k)q FN(\000)p FL(1)1648 2043 y FQ(\010)15 b FO(H)1797 2055 y FM(n)1869 2043 y FP(suc)n(h)26 b(that)h FO(U)2300 2013 y FN(\003)2338 2043 y FO(w)f FP(=)c(0,)118 2143 y(whic)n(h)28 b(con)n(tradicts)e(the)i(condition)f (k)n(er)13 b FO(U)1481 2113 y FN(\003)1542 2143 y FP(=)23 b(k)n(er)12 b FO(F)g FP(\()p FO(C)1916 2113 y FL(2)1954 2143 y FP(\).)243 2251 y(Then,)45 b(there)d(exists)g FO(n)f FP(suc)n(h)h(that)g FO(e)1497 2263 y FM(k)1585 2251 y FP(=)k(0,)f FO(k)50 b FQ(\025)d FO(n)p FP(,)e(i.e.,)h FO(e)2343 2263 y FM(n)p FN(\000)p FL(1)2519 2251 y FQ(2)118 2351 y FP(k)n(er)13 b FO(U)309 2321 y FN(\003)382 2351 y FP(=)35 b(k)n(er)13 b FO(F)f FP(\()p FO(C)769 2321 y FL(2)807 2351 y FP(\),)37 b(and)e FO(F)1133 2321 y FM(n)1178 2351 y FP(\(0\))h(=)f(0;)j(th)n(us)d(0)g(is)g(a)g(p)r(erio)r (dic)g(p)r(oin)n(t)g(of)118 2451 y(p)r(erio)r(d)28 b FO(n)p FP(,)f(and)h(the)g(form)n(ula)e(follo)n(ws.)p 2514 2451 4 57 v 2518 2398 50 4 v 2518 2451 V 2567 2451 4 57 v 118 2685 a FR(2.)56 b FP(The)34 b(presen)n(ted)f(theorem)h (reduces)f(the)i(problem)e(of)h(classi\014cation)f(of)118 2785 y(\014nite-dimensional)24 b(irreducible)f(represen)n(tations)e(\() p FO(C)1859 2755 y FL(2)1897 2785 y FO(;)14 b(U)9 b FP(\))24 b(to)g(the)g(descrip-)118 2920 y(tion)34 b(of)g(cycles)f(of)g(the)h (dynamical)g(system)f FJ(R)1630 2890 y FL(1)1706 2866 y FM(F)9 b FL(\()p FN(\001)p FL(\))1718 2920 y FQ(\000)-48 b(!)44 b FJ(R)1916 2890 y FL(1)1960 2920 y FP(.)55 b(So,)35 b(let)f(us)g(lo)r(ok)118 3020 y(ho)n(w)27 b(the)h(facts)g(ab)r(out)f (cycles)g(of)h(dynamical)f(systems)g(can)g(b)r(e)h(used)g(in)f(the)118 3119 y(con)n(text)g(of)h(represen)n(tation)e(theory)-7 b(.)243 3228 y(Shark)n(o)n(vsky's)31 b(theorem)i(establishes)g(the)h (follo)n(wing)f(order)g(in)h(the)g(set)118 3327 y(of)28 b(natural)e(n)n(um)n(b)r(ers.)118 3519 y FR(Theorem)31 b(17.)40 b FC(L)l(et)29 b FO(F)21 b FP(:)28 b FO(I)i FQ(7!)23 b FO(I)37 b FC(b)l(e)30 b(a)g(c)l(ontinuous)f(mapping)i(of)g (the)f(close)l(d)118 3619 y(interval)37 b FO(I)43 b FC(into)36 b(itself.)58 b(If)36 b(the)g(dynamic)l(al)i(system)e(p)l(ossesses)h(a)f (cycle)h(of)118 3718 y(p)l(erio)l(d)j FO(m)p FC(,)h(then)e(for)g(any)g FO(m)1089 3688 y FN(0)1137 3718 y FO(/)25 b(m)p FC(,)41 b(ther)l(e)e(exists)f(a)h(cycle)h(of)f(p)l(erio)l(d)h FO(m)2526 3688 y FN(0)2549 3718 y FC(,)118 3818 y(wher)l(e)27 b FO(/)e FC(denotes)i(the)f(fol)t(lowing)i(or)l(der)f(on)f(the)g(set)f FJ(N)36 b FC(of)27 b(natur)l(al)f(numb)l(ers)7 b FP(:)221 4018 y(1)18 b FO(/)g FP(2)g FO(/)g FQ(\001)c(\001)g(\001)k FO(/)g FP(2)678 3983 y FM(n)741 4018 y FO(/)g FQ(\001)c(\001)g(\001)19 b FO(/)f FP(2)1019 3983 y FL(2)1074 4018 y FQ(\001)g FP(5)g FO(/)g FP(2)1277 3983 y FL(2)1333 4018 y FQ(\001)g FP(3)g FO(/)g FQ(\001)c(\001)g(\001)k FO(/)g FP(2)g FQ(\001)h FP(5)f FO(/)g FP(2)g FQ(\001)g FP(3)g FO(/)g FQ(\001)c(\001)g(\001)k FO(/)g FP(5)g FO(/)g FP(3)p FO(:)2363 4117 y FP(\(2.10\))p eop %%Page: 94 98 94 97 bop 118 100 a FP(94)485 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(F)-6 b(or)34 b(any)g FO(m)p FC(,)h(ther)l(e)f(exists)f(a)h(c)l(ontinuous)f(mapping)1871 312 y FP(~)1852 333 y FO(F)22 b FP(:)29 b FO(I)37 b FQ(7!)30 b FO(I)41 b FC(such)34 b(that)118 432 y(the)39 b(dynamic)l(al)h(system) e(has)h(a)f(cycle)i(of)f(p)l(erio)l(d)g FO(m)p FC(,)i(and)e(do)l(es)g (not)f(have)118 532 y(cycles)31 b(of)f(p)l(erio)l(ds)i FO(m)810 502 y FN(0)862 532 y FC(for)f FO(m)18 b(/)g(m)1219 502 y FN(0)1243 532 y FC(.)243 706 y FP(This)k(theorem)g(giv)n(es)f (the)i(follo)n(wing)f(statemen)n(t)g(ab)r(out)g(the)h(dimensions)118 805 y(of)28 b(irreducible)f(represen)n(tations)e(of)34 b(\(2.1\).)118 979 y FR(Prop)s(osition)43 b(35.)k FC(L)l(et)40 b FO(F)21 b FP(:)31 b FO(I)49 b FQ(7!)41 b FO(I)47 b FC(b)l(e)40 b(a)g(c)l(ontinuous)f(mapping)j(of)f(the)118 1079 y(interval)31 b FO(I)37 b FC(into)31 b(itself.)41 b(If)31 b(ther)l(e)f(exist)g(irr)l(e)l(ducible)i FO(m)p FC(-dimensional)g(r)l(epr)l(e-)118 1178 y(sentations)e(of)49 b FP(\(2.1\))29 b FC(such)i(that)f FO(\033)s FP(\()p FO(C)1336 1148 y FL(2)1374 1178 y FP(\))24 b FQ(\032)f FO(I)7 b FC(,)31 b(then)f(for)h(any)f FO(m)2167 1148 y FN(0)2209 1178 y FO(/)18 b(m)p FC(,)31 b(ther)l(e)118 1278 y(exist)f FO(m)385 1248 y FN(0)408 1278 y FC(-dimensional)h(irr)l (e)l(ducible)g(r)l(epr)l(esentations)f(of)48 b FP(\(2.1\))p FC(.)243 1380 y(F)-6 b(or)23 b(any)g FO(m)p FC(,)h(ther)l(e)f(exists)g (a)g(c)l(ontinuous)f(mapping)1909 1359 y FP(~)1890 1380 y FO(F)f FP(:)28 b FO(I)i FQ(7!)23 b FO(I)30 b FC(such)23 b(that)118 1480 y(the)39 b(r)l(elation)45 b FP(\(2.1\))38 b FC(has)h(an)g FO(m)p FC(-dimensional)h(irr)l(e)l(ducible)f(r)l(epr)l (esentation)118 1579 y(and)f(do)l(es)h(not)e(have)i(irr)l(e)l(ducible)g (r)l(epr)l(esentations)f(of)h(dimension)g FO(m)2411 1549 y FN(0)2472 1579 y FC(for)118 1679 y FO(m)18 b(/)g(m)342 1649 y FN(0)366 1679 y FC(.)243 1853 y FP(F)-7 b(or)27 b(con)n(tin)n(uous)f(mappings)h FO(F)12 b FP(\()p FQ(\001)p FP(\),)29 b(the)f(follo)n(wing)e(corollaries)f(hold.)118 2027 y FR(Corollary)35 b(3.)42 b FC(If)32 b(r)l(elation)38 b FP(\(2.1\))31 b FC(with)h(c)l(ontinuous)e FO(F)12 b FP(\()p FQ(\001)p FP(\))32 b FC(has)g(irr)l(e)l(ducible)118 2126 y(r)l(epr)l(esentations)37 b(with)g(a)f(dimension)i(which)g(is)f (not)f(e)l(qual)g(to)h(a)g(p)l(ower)g(of)118 2226 y FP(2)p FC(,)45 b(then)e(ther)l(e)f(ar)l(e)g(in\014nitely)h(many)f(dimensions)i (for)f(which)50 b FP(\(2.1\))42 b FC(has)118 2326 y(irr)l(e)l(ducible) 31 b(r)l(epr)l(esentations.)118 2499 y FR(Corollary)36 b(4.)42 b FC(The)32 b(existenc)l(e)g(of)g(a)g(thr)l(e)l(e-dimensional)i (irr)l(e)l(ducible)f(r)l(ep-)118 2599 y(r)l(esentation)24 b(of)42 b FP(\(2.1\))23 b FC(implies)j(that)31 b FP(\(2.1\))24 b FC(has)g(irr)l(e)l(ducible)i(r)l(epr)l(esentations)118 2699 y(of)31 b(any)f(dimension)h FO(n)23 b FQ(2)g FJ(N)t FC(.)118 2872 y(Example)35 b FP(11)p FC(.)j FP(\(Second-degree)24 b(mapping.)36 b(Finite-dimensional)24 b(represen-)118 2972 y(tations\).)37 b(Consider)27 b(the)g(follo)n(wing)g(relation)1025 3160 y FO(xx)1119 3126 y FN(\003)1181 3160 y FP(=)c(\()p FO(x)1348 3126 y FN(\003)1387 3160 y FO(x)18 b FQ(\000)g FO(q)s FP(\))1607 3126 y FL(2)1645 3160 y FO(:)118 3348 y FP(The)k(corresp)r(onding)f(dynamical)g(system)h(is)g(generated)f(b)n (y)h(the)h(p)r(olynomial)118 3447 y FO(P)171 3459 y FM(q)208 3447 y FP(\()p FO(\025)p FP(\))44 b(=)e(\()p FO(\025)27 b FQ(\000)e FO(q)s FP(\))740 3417 y FL(2)778 3447 y FP(.)72 b(According)38 b(to)h(the)h(argumen)n(ts)e(ab)r(o)n(v)n(e,)j(all)e (\014nite-)118 3547 y(dimensional)30 b(represen)n(tations)f(are)h (describ)r(ed)g(in)h(terms)g(of)g(cycles)f(of)h(this)118 3647 y(mapping.)44 b(Let)30 b(us)f(lo)r(ok)h(at)f(ho)n(w)h(the)g(v)-5 b(alue)30 b(of)f FO(q)k FP(a\013ects)d(the)g(existence)g(of)118 3746 y(cycles)d(and)h(their)f(order)f(\(see,)i(e.g.,)f([260)o(]\))243 3848 y(F)-7 b(or)20 b FO(q)26 b(<)c FQ(\000)p FP(1)p FO(=)p FP(4,)f(there)f(are)f(no)i(stationary)e(p)r(oin)n(ts,)j(and)e (therefore,)h(there)118 3948 y(are)28 b(no)h(cycles)g(at)g(all.)42 b(F)-7 b(or)29 b FO(q)g FP(=)c FQ(\000)p FP(1)p FO(=)p FP(4,)j(there)h(exists)g(a)g(unique)g(stationary)118 4048 y(p)r(oin)n(t,)24 b FO(\025)f FP(=)g(1)p FO(=)p FP(4,)f(and)g(no)g(other)g(cycles.)34 b(T)-7 b(o)22 b(this)h(p)r(oin)n (t,)h(there)e(corresp)r(onds)118 4147 y(a)27 b(circle)g(of)h (one-dimensional)e(represen)n(tations,)g FO(X)j FP(=)23 b FO(e)1932 4117 y FM(i\036)1999 4147 y FO(=)p FP(2,)k FO(\036)c FQ(2)h FP([0)p FO(;)14 b FP(2)p FO(\031)s FP(\).)p eop %%Page: 95 99 95 98 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f (systems)896 b FP(95)243 333 y(F)-7 b(or)36 b FQ(\000)p FP(1)p FO(=)p FP(4)i FO(<)h(q)k(<)c FP(3)p FO(=)p FP(4,)f(there)f(are)g (t)n(w)n(o)g(stationary)e(p)r(oin)n(ts,)40 b FO(\025)2380 345 y FL(0)p FM(;)p FL(1)2510 333 y FP(=)118 432 y(\(2)p FO(q)28 b FP(+)e(1)f FQ(\006)505 372 y(p)p 574 372 225 4 v 60 x FP(4)p FO(q)c FP(+)d(1)o(\))p FO(=)p FP(2,)41 b(whic)n(h)d(giv)n(e)f(t)n(w)n(o)h(one-dimensional)f(families)h(of)118 532 y(irreducible)29 b(represen)n(tations.)41 b(There)29 b(are)g(no)g(other)h(cycles)f(and)g(no)g(other)118 632 y(irreducible)e(\014nite-dimensional)g(represen)n(tations.)243 734 y(As)32 b FO(q)k FP(increases)31 b(from)i(3/4)e(to)h FO(q)1300 704 y FN(\003)1370 734 y FQ(\031)f FP(1)p FO(:)p FP(4,)i(cycles)f(of)h(order)e(2,)j(2)2335 704 y FL(2)2371 734 y FP(,)f FO(:)14 b(:)g(:)28 b FP(,)118 833 y(2)160 803 y FM(n)205 833 y FP(,)k(and)f(the)h(corresp)r(onding)d(families)j (of)f(irreducible)g(represen)n(tations)e(of)118 933 y(the)f(corresp)r (onding)e(dimensions)h(arise.)243 1035 y(F)-7 b(or)35 b FO(q)40 b FP(=)d FO(q)619 1005 y FN(\003)658 1035 y FP(,)h(there)e(exist)g(cycles)f(of)h(an)n(y)g(order)f(2)1928 1005 y FM(k)1968 1035 y FP(,)j FO(k)i FQ(\025)d FP(1,)h(and)e(no)118 1135 y(other)j(cycles;)45 b(an)n(y)39 b(irreducible)g (\014nite-dimensional)g(represen)n(tation)f(has)118 1235 y(dimension)28 b(2)553 1204 y FM(k)621 1235 y FP(for)f(some)g FO(k)f FQ(\025)c FP(0.)243 1337 y(As)29 b FO(q)g(>)c(q)563 1307 y FN(\003)630 1337 y FP(increases,)j(other)h(cycles)f(arise)g(in)i (the)f(order)f(describ)r(ed)h(b)n(y)118 1436 y(the)37 b(Shark)n(o)n(vsky)d(theorem.)62 b(Starting)36 b(from)g(some)g FO(q)41 b FQ(\031)c FP(1)p FO(:)p FP(75,)h(there)e(are)118 1536 y(cycles)23 b(of)h(order)f(3,)h(and)f(therefore,)h(cycles)f(of)h (all)g(other)f(orders.)34 b(Therefore,)118 1636 y(for)c(suc)n(h)f FO(q)k FP(there)d(are)f(irreducible)h(represen)n(tations)e(of)i(an)n(y) f(\014nite)i(dimen-)118 1735 y(sion.)59 b(Notice)36 b(that)f(for)g (some)g(v)-5 b(alues)35 b(of)g FO(q)s FP(,)i(zero)d(ma)n(y)h(b)r(ecome) g(p)r(erio)r(dic)118 1835 y(p)r(oin)n(t)24 b(\(e.g.,)g(for)f FO(q)k FP(=)22 b(1,)i(zero)f(is)g(a)h(p)r(erio)r(dic)f(p)r(oin)n(t)h (of)g(the)g(second)f(order\);)h(in)118 1935 y(this)g(case,)g(the)g (corresp)r(onding)e(one-parameter)f(family)j(of)g(represen)n(tations) 118 2034 y(degenerates)i(in)n(to)i(a)f(single)g(irreducible)g(represen) n(tation.)243 2137 y(Relation)g(\(2.2\))960 2324 y FO(xx)1054 2290 y FN(\003)1116 2324 y FP(=)c FO(\013x)1304 2290 y FN(\003)1343 2324 y FO(x)p FP(\()p FO(I)j FQ(\000)18 b FO(x)1614 2290 y FN(\003)1653 2324 y FO(x)p FP(\))118 2512 y(has)k(a)g(similar)f(set)i(of)f(\014nite-dimensional)g(represen)n (tations,)f(but)i(the)g(family)118 2612 y(corresp)r(onding)28 b(to)i(the)h(\014xed)f(p)r(oin)n(t)g FO(\025)e FP(=)f(0)j(degenerates)e (in)n(to)i(the)h(unique)118 2712 y(trivial)19 b(represen)n(tation,)h (and)g(there)f(is)h(no)g(degeneration)e(of)i(represen)n(tations)118 2811 y(corresp)r(onding)h(to)h(other)g(cycles.)35 b(The)22 b(corresp)r(onding)f(critical)h(v)-5 b(alues)22 b(of)g FO(\013)118 2911 y FP(are:)39 b FO(\013)26 b FP(=)f(3)k(\(t)n(w)n (o-dimensional)f(represen)n(tations)f(arise\),)i FO(\013)d FP(=)f FO(\013)2231 2881 y FN(\003)2295 2911 y FQ(\031)h FP(3)p FO(:)p FP(569)118 3011 y(\(there)35 b(are)e(represen)n(tations)f (with)j(dimensions)f(of)g(an)n(y)g(p)r(o)n(w)n(er)f(of)h(2,)i(and)118 3110 y(no)24 b(others\),)g FO(\013)f FQ(\031)g FP(3)p FO(:)p FP(8)g(\(there)h(is)g(a)f(three-dimensional)g(represen)n (tation,)g(and)118 3210 y(th)n(us)28 b(represen)n(tations)d(with)j(an)n (y)f(dimensions\).)118 3348 y FC(Example)40 b FP(12)p FC(.)j FP(\(Con)n(tin)n(uous)30 b(fractions.)45 b(Finite-dimensional)31 b(represen)n(ta-)118 3448 y(tions\).)77 b(Consider)40 b(op)r(erator)f(relations)h(whic)n(h)h(arise)e(from)i(the)g(M\177)-42 b(obius)118 3547 y(mapping)783 3735 y FO(xx)877 3701 y FN(\003)939 3735 y FP(=)23 b(\()p FO(ax)1150 3701 y FN(\003)1189 3735 y FO(x)c FP(+)f FO(c)p FP(\)\()p FO(bx)1521 3701 y FN(\003)1559 3735 y FO(x)h FP(+)f FO(d)p FP(\))1783 3701 y FN(\000)p FL(1)1873 3735 y FO(;)700 3860 y(a;)c(b;)g(c;)g(d)22 b FQ(2)h FJ(R)p FO(;)103 b(b)23 b(>)g FP(0)p FO(;)96 b(ad)19 b FQ(\000)f FO(bc)k FQ(6)p FP(=)h(0)p FO(:)369 b FP(\(2.11\))118 4048 y(According)39 b(to)g(Theorem)f(16,)k(in)e (order)e(to)h(describ)r(e)g(\014nite-dimensional)118 4147 y(represen)n(tations)30 b(of)h(relation)g(\(12\))o(,)i(one)e (needs)g(to)h(\014nd)g(cycles)f(of)g(the)h(dy-)p eop %%Page: 96 100 96 99 bop 118 100 a FP(96)485 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(namical)g(system)g(generated)g(b)n(y)g(the)h (mapping)1072 537 y FO(F)12 b FP(\()p FO(z)t FP(\))23 b(=)1364 481 y FO(az)f FP(+)c FO(c)p 1364 518 224 4 v 1364 594 a(bz)k FP(+)c FO(d)1598 537 y(:)742 b FP(\(2.12\))118 747 y(T)-7 b(o)27 b(do)h(that,)g(w)n(e)f(follo)n(w)g([267)o(].)38 b(First,)27 b(consider)g(\014xed)h(p)r(oin)n(ts)f(of)h(the)g(map-)118 846 y(ping.)59 b(If)36 b(\()p FO(d)24 b FQ(\000)f FO(a)p FP(\))711 816 y FL(2)771 846 y FP(+)g(4)p FO(bc)35 b FP(=)g(0,)h(then)g(there)f(exists)f(a)h(single)g(\014xed)g(p)r(oin)n(t) 118 946 y FO(\030)154 958 y FL(1)215 946 y FP(=)22 b(\()p FO(a)d FQ(\000)f FO(d)p FP(\))p FO(=)p FP(2)p FO(b)p FP(;)27 b(otherwise,)g(there)g(are)g(t)n(w)n(o)f(\014xed)i(p)r(oin)n (ts,)g FO(\030)2135 958 y FL(1)2172 946 y FP(,)g FO(\030)2259 958 y FL(2)2297 946 y FP(.)243 1046 y(If)33 b(there)f(exists)h(a)f (single)g(stationary)f(p)r(oin)n(t)i FO(\030)1751 1058 y FL(1)1789 1046 y FP(,)h(then)f FO(F)45 b FP(is)33 b(conjugate)118 1145 y(to)g(the)h(shift)g(in)g(the)g(complex)f(plane,)i FO(F)44 b FP(=)33 b FO(\036)1646 1115 y FN(\000)p FL(1)1758 1145 y FQ(\016)21 b FO(T)34 b FQ(\016)22 b FO(\036)p FP(,)35 b(where)e FO(\036)p FP(\()p FO(z)t FP(\))g(=)118 1245 y(1)p FO(=)p FP(\()p FO(z)22 b FQ(\000)d FO(\030)415 1257 y FL(1)452 1245 y FP(\),)30 b FO(T)12 b FP(\()p FO(w)r FP(\))25 b(=)g FO(w)d FP(+)c FO(l)r FP(,)29 b FO(l)d FP(=)f(2)p FO(b=)p FP(\()p FO(a)18 b FP(+)h FO(d)p FP(\).)41 b(W)-7 b(e)29 b(see)f(that,)i(in)f(this)g(case,)118 1345 y FO(\030)154 1357 y FL(1)223 1345 y FP(is)i(a)g(unique)h(p)r (erio)r(dic)f(p)r(oin)n(t.)48 b(Represen)n(tations)30 b(exist)i(only)f(if)g FO(\030)2349 1357 y FL(1)2416 1345 y FQ(\025)e FP(0;)118 1444 y(for)39 b FO(a)44 b FP(=)f FO(d)p FP(,)g FO(X)50 b FP(=)43 b(0)c(is)h(a)g(single)f(solution)g(of) 47 b(\(12\))o(,)c(for)d FO(\030)2120 1456 y FL(1)2201 1444 y FO(>)j FP(0)c(there)118 1544 y(exists)i(a)f(one-parameter)f (family)i(of)g(one-dimensional)f(represen)n(tations,)118 1643 y FO(X)29 b FP(=)23 b FO(\013)357 1581 y FQ(p)p 427 1581 74 4 v 427 1643 a FO(\030)463 1655 y FL(1)500 1643 y FP(,)28 b FQ(j)p FO(\013)p FQ(j)c FP(=)e(1.)243 1743 y(If)28 b(there)g(are)f(t)n(w)n(o)g(stationary)f(p)r(oin)n(ts,)i FO(\030)1539 1755 y FL(1)1576 1743 y FP(,)h FO(\030)1664 1755 y FL(2)1701 1743 y FP(,)f(one)g(can)f(construct)h(one-)118 1843 y(dimensional)33 b(represen)n(tations)e(quite)j(similarly)-7 b(,)34 b(if)g(one)e(or)h(b)r(oth)h(of)f(these)118 1942 y(p)r(oin)n(ts)28 b(are)e(non-negativ)n(e.)243 2042 y(If)36 b(there)h(are)e(no)h(stationary)f(p)r(oin)n(ts)i(\(they)f(are)g (conjugate)g(complex-)118 2142 y(v)-5 b(alued\),)28 b(write)1083 2360 y FO(\036)p FP(\()p FO(z)t FP(\))23 b(=)1360 2304 y FO(z)e FQ(\000)e FO(\030)1540 2316 y FL(1)p 1360 2341 218 4 v 1360 2417 a FO(z)i FQ(\000)e FO(\030)1540 2429 y FL(2)1587 2360 y FP(;)118 2584 y(then)37 b FO(F)50 b FP(=)37 b FO(\036)570 2554 y FN(\000)p FL(1)684 2584 y FQ(\016)24 b FO(T)35 b FQ(\016)24 b FO(\036)p FP(,)39 b(where)d FO(T)12 b(w)39 b FP(=)e FO(q)s(w)j FP(with)c FO(q)41 b FP(=)c(\()p FO(a)25 b FQ(\000)f FO(\030)2263 2596 y FL(1)2300 2584 y FO(b)p FP(\))p FO(=)p FP(\()p FO(a)g FQ(\000)118 2684 y FO(\030)154 2696 y FL(2)192 2684 y FO(b)p FP(\))37 b(=)g(\()p FO(d)24 b FQ(\000)g FO(\030)623 2696 y FL(2)661 2684 y FO(b)p FP(\))p FO(=)p FP(\()p FO(d)g FQ(\000)f FO(\030)994 2696 y FL(1)1032 2684 y FO(b)p FP(\).)62 b(No)n(w)36 b(it)g(is)g(easy)g(to)g(see)f(that) i(either)f FO(q)j FP(is)118 2783 y(the)33 b FO(n)p FP(-th)f(ro)r(ot)f (of)h(unit)n(y)h(for)e(some)h FO(n)p FP(,)h(and)f(all)g(p)r(oin)n(ts)g (are)f(p)r(erio)r(dic)h(with)118 2883 y(p)r(erio)r(d)26 b FO(n)p FP(,)g(or)e(there)i(are)e(no)i(p)r(erio)r(dic)f(p)r(oin)n(ts)h (at)f(all.)36 b(In)26 b(the)g(p)r(erio)r(dic)g(case,)118 2983 y(represen)n(tations)38 b(corresp)r(ond)f(to)j(orbits)e(with)i (all)f(non-negativ)n(e)f(p)r(oin)n(ts,)118 3082 y(and)26 b(these)g(represen)n(tations)e(are)h(constructed)h(according)e(to)i (Theorem)g(16.)243 3182 y(Notice)37 b(that)h(in)f(this)h(example)f(the) h(rule)f(ab)r(out)g(dimensions)g(of)h(rep-)118 3281 y(resen)n(tations) 31 b(established)h(b)n(y)f(the)i(Shark)n(o)n(vsky)c(theorem)j(do)r(es)g (not)g(hold:)118 3381 y(there)20 b(can)f(b)r(e)h(only)g (one-dimensional)e(and)i FO(n)p FP(-dimensional)e(irreducible)i(rep-) 118 3481 y(resen)n(tations.)118 3695 y FR(2.1.3)94 b (In\014nite-dimensional)28 b(represen)m(tations)118 3848 y FP(In)22 b(order)e(to)i(describ)r(e)f(the)h(general)e(case,)i(recall) f(that,)i(according)d(to)i(Prop)r(o-)118 3948 y(sition)28 b(29,)f(the)h(op)r(erator)e FO(U)37 b FP(is)28 b(a)f(cen)n(tered)g (partial)g(isometry)-7 b(.)37 b(W)-7 b(e)28 b(will)g(see)118 4048 y(that)k(in)g(the)g(irreducible)f(represen)n(tation)f(with)i (non-unitary)e FO(U)9 b FP(,)33 b(the)f(pair)118 4147 y FO(U)9 b FP(,)24 b FO(U)297 4117 y FN(\003)358 4147 y FP(is)g(again)e(irreducible;)i(since)f(all)g(irreducible)g(partial)f (isometries)g(can)p eop %%Page: 97 101 97 100 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f(systems)896 b FP(97)118 333 y(easily)25 b(b)r(e)h(describ)r(ed,)g(this)h(enables)e(us)h(to)f(giv)n(e)g(a)h (complete)g(description)f(of)118 432 y(all)i(irreducible)g(represen)n (tations)f(in)i(the)g(non-unitary)e(case.)243 532 y(In)f(the)h(unitary) f(case,)g(t)n(w)n(o)g(classes)f(of)h(represen)n(tations)f(can)h(arise:) 35 b(rep-)118 632 y(resen)n(tations)20 b(in)h(whic)n(h)g FO(U)30 b FP(acts)20 b(as)h(a)f(shift)i(in)f FO(l)1598 644 y FL(2)1656 632 y FP(\(in)h(this)f(case)f(the)i(sp)r(ectrum)118 731 y(of)34 b FO(C)284 701 y FL(2)356 731 y FP(lies)h(on)f(a)g(single)g (orbit\),)i(and)e(represen)n(tations)f(corresp)r(onding)f(to)118 831 y(non-trivial)g(ergo)r(dic)g(measures.)51 b(The)33 b(latter)g(class)f(of)h(represen)n(tations)e(is)118 931 y(to)r(o)f(complicated)f(to)h(b)r(e)g(classi\014ed)f(up)h(to)g(a)f (unitary)h(equiv)-5 b(alence)29 b(for)g(the)118 1030 y(momen)n(t;)f(ho)n(w)n(ev)n(er,)d(non-trivial)h(ergo)r(dic)g(measures) g(can)h(arise)g(only)g(if)g(the)118 1130 y(corresp)r(onding)19 b(dynamical)h(system)h(do)r(es)g(not)f(ha)n(v)n(e)g(a)h(measurable)e (section.)118 1280 y FR(1.)36 b FP(W)-7 b(e)28 b(start)f(with)h(the)g (description)f(of)h(cen)n(tered)f(partial)g(isometries.)118 1446 y FR(Theorem)40 b(18.)46 b FC(A)n(ny)37 b(irr)l(e)l(ducible)i(c)l (enter)l(e)l(d)f(p)l(artial)h(isometry)f(is)g(one)g(of)118 1546 y(the)30 b(fol)t(lowing:)197 1729 y FP(\(i\))42 b FC(a)30 b(one-dimensional)h(unitary)f(op)l(er)l(ator)h FO(U)h FP(=)22 b FO(\013)p FC(,)31 b FQ(j)p FO(\013)p FQ(j)23 b FP(=)g(1;)173 1896 y(\(ii\))43 b FC(a)30 b(unilater)l(al)g (shift)h(op)l(er)l(ator)f(in)g FO(l)1406 1908 y FL(2)1443 1896 y FC(,)g FO(U)9 b(e)1603 1908 y FM(k)1667 1896 y FP(=)22 b FO(e)1793 1908 y FM(k)q FL(+1)1918 1896 y FP(;)150 2063 y(\(iii\))43 b FC(an)34 b(adjoint)j(to)d(the)h(unilater)l(al)g (shift)h(op)l(er)l(ator)g(in)e FO(l)2012 2075 y FL(2)2050 2063 y FC(,)i FO(U)9 b(e)2216 2075 y FM(k)2288 2063 y FP(=)32 b FO(e)2424 2075 y FM(k)q FN(\000)p FL(1)2549 2063 y FC(,)326 2163 y FO(k)26 b(>)c FP(1)p FC(,)30 b FO(U)9 b(e)684 2175 y FL(1)744 2163 y FP(=)22 b(0;)153 2329 y(\(iv\))42 b FC(a)35 b(\014nite-dimensional)g(op)l(er)l(ator)h (in)e FJ(C)1576 2299 y FM(n)1662 2329 y FC(of)h(the)g(form)g FO(U)9 b(e)2217 2341 y FM(k)2289 2329 y FP(=)32 b FO(e)2425 2341 y FM(k)q FL(+1)2549 2329 y FC(,)326 2429 y FO(k)26 b FP(=)c(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n)k FQ(\000)h FP(1)p FC(,)29 b FO(U)9 b(e)1061 2441 y FM(n)1129 2429 y FP(=)22 b(0)p FC(,)30 b(for)h(some)f FO(n)23 b FP(=)f(1)p FC(,)30 b FP(2)p FC(,)g FO(:)14 b(:)g(:)27 b FC(.)118 2612 y(Pr)l(o)l(of.)43 b FP(W)-7 b(e)28 b(start)f(with)h(a)g(simple)f (fact.)118 2763 y FR(Prop)s(osition)c(36.)36 b FC(The)25 b(op)l(er)l(ators)g FO(U)1371 2733 y FM(k)1412 2763 y FP(\()p FO(U)1510 2733 y FN(\003)1548 2763 y FP(\))1580 2733 y FM(k)1621 2763 y FC(,)h FP(\()p FO(U)1770 2733 y FN(\003)1808 2763 y FP(\))1840 2733 y FM(l)1866 2763 y FO(U)1932 2733 y FM(l)1957 2763 y FC(,)g FO(k)s FC(,)f FO(l)g FP(=)d(1)p FC(,)k FP(2)p FC(,)e FO(:)14 b(:)g(:)27 b FC(,)118 2863 y(ar)l(e)j(pr)l(oje)l(ctions.)118 3030 y(Pr)l(o)l(of.)43 b FP(Since)30 b FO(U)9 b(U)729 3000 y FN(\003)767 3030 y FO(U)36 b FP(=)26 b FO(U)39 b FP(and)29 b(the)i(op)r(erators)d FO(U)1792 3000 y FN(\003)1830 3030 y FO(U)38 b FP(and)30 b FO(U)2155 3000 y FM(k)q FN(\000)p FL(1)2280 3030 y FP(\()p FO(U)2378 3000 y FN(\003)2417 3030 y FP(\))2449 3000 y FM(k)q FN(\000)p FL(1)118 3130 y FP(comm)n(ute,)e(w)n(e)f(ha)n(v)n(e)f(b)n(y)i(induction)g(that)320 3313 y FO(U)386 3279 y FM(k)426 3313 y FP(\()p FO(U)524 3279 y FN(\003)562 3313 y FP(\))594 3279 y FM(k)636 3313 y FO(U)702 3279 y FM(k)742 3313 y FP(\()p FO(U)840 3279 y FN(\003)878 3313 y FP(\))910 3279 y FM(k)975 3313 y FP(=)22 b FO(U)9 b(U)1194 3279 y FM(k)q FN(\000)p FL(1)1320 3313 y FP(\()p FO(U)1418 3279 y FN(\003)1456 3313 y FP(\))1488 3279 y FM(k)q FN(\000)p FL(1)1614 3313 y FO(U)1680 3279 y FN(\003)1718 3313 y FO(U)g(U)1850 3279 y FM(k)q FN(\000)p FL(1)1975 3313 y FP(\()p FO(U)2073 3279 y FN(\003)2111 3313 y FP(\))2143 3279 y FM(k)q FN(\000)p FL(1)2269 3313 y FO(U)2335 3279 y FN(\003)975 3451 y FP(=)22 b FO(U)9 b(U)1194 3416 y FN(\003)1232 3451 y FO(U)g(U)1364 3416 y FM(k)q FN(\000)p FL(1)1489 3451 y FP(\()p FO(U)1587 3416 y FN(\003)1625 3451 y FP(\))1657 3416 y FM(k)q FN(\000)p FL(1)1783 3451 y FO(U)1849 3416 y FM(k)q FN(\000)p FL(1)1975 3451 y FP(\()p FO(U)2073 3416 y FN(\003)2111 3451 y FP(\))2143 3416 y FM(k)q FN(\000)p FL(1)2269 3451 y FO(U)2335 3416 y FN(\003)975 3588 y FP(=)22 b FO(U)9 b(U)1194 3554 y FM(k)q FN(\000)p FL(1)1320 3588 y FP(\()p FO(U)1418 3554 y FN(\003)1456 3588 y FP(\))1488 3554 y FM(k)q FN(\000)p FL(1)1614 3588 y FO(U)1680 3554 y FN(\003)1741 3588 y FP(=)22 b FO(U)1894 3554 y FM(k)1935 3588 y FP(\()p FO(U)2033 3554 y FN(\003)2071 3588 y FP(\))2103 3554 y FM(k)2144 3588 y FO(:)118 3781 y FP(Similarly)-7 b(,)25 b(since)g FO(U)752 3750 y FN(\003)790 3781 y FO(U)9 b(U)922 3750 y FN(\003)982 3781 y FP(=)23 b FO(U)1136 3750 y FN(\003)1199 3781 y FP(and)i FO(U)9 b(U)1490 3750 y FN(\003)1552 3781 y FP(and)25 b(\()p FO(U)1809 3750 y FN(\003)1847 3781 y FP(\))1879 3750 y FM(k)q FN(\000)p FL(1)2005 3781 y FO(U)2071 3750 y FM(k)q FN(\000)p FL(1)2222 3781 y FP(comm)n(ute,)118 3880 y(w)n(e)i(get)h(that)g(\()p FO(U)657 3850 y FN(\003)695 3880 y FP(\))727 3850 y FM(k)768 3880 y FO(U)834 3850 y FM(k)902 3880 y FP(is)g(a)f(pro)5 b(jection.)p 2514 3880 4 57 v 2518 3827 50 4 v 2518 3880 V 2567 3880 4 57 v 243 4048 a(Denote)38 b(these)g(pro)5 b(jections)37 b(b)n(y)h FO(P)1378 4060 y FM(k)1459 4048 y FP(=)i(\()p FO(U)1662 4018 y FN(\003)1701 4048 y FP(\))1733 4018 y FM(k)1774 4048 y FO(U)1840 4018 y FM(k)1880 4048 y FP(,)h FO(P)1997 4060 y FN(\000)p FM(k)2131 4048 y FP(=)f FO(U)2302 4018 y FM(k)2342 4048 y FP(\()p FO(U)2440 4018 y FN(\003)2479 4048 y FP(\))2511 4018 y FM(k)2552 4048 y FP(,)118 4147 y FO(k)26 b FP(=)d(1,)k(2,)g FO(:)14 b(:)g(:)28 b FP(;)f FO(P)687 4159 y FL(0)748 4147 y FP(=)c FO(I)7 b FP(.)p eop %%Page: 98 102 98 101 bop 118 100 a FP(98)485 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FR(Prop)s(osition)j(37.)41 b FC(F)-6 b(or)30 b(al)t(l)h FO(k)25 b FQ(2)f FJ(Z)o FC(,)g(the)30 b(fol)t(lowing)j(r)l(elations)d(hold)1078 518 y FO(P)1131 530 y FM(k)1172 518 y FO(U)i FP(=)22 b FO(U)9 b(P)1467 530 y FM(k)q FL(+1)1592 518 y FO(:)748 b FP(\(2.13\))118 705 y FC(Pr)l(o)l(of.)43 b FP(Indeed,)28 b(for)f FO(k)f(>)d FP(0,)k(w)n(e)g(ha)n(v)n(e)342 891 y FO(P)395 903 y FM(k)436 891 y FO(U)32 b FP(=)23 b(\()p FO(U)711 856 y FN(\003)749 891 y FP(\))781 856 y FM(k)822 891 y FO(U)888 856 y FM(k)929 891 y FO(U)31 b FP(=)23 b(\()p FO(U)1203 856 y FN(\003)1241 891 y FP(\))1273 856 y FM(k)1314 891 y FO(U)1380 856 y FM(k)1421 891 y FO(U)9 b(U)1553 856 y FN(\003)1591 891 y FO(U)31 b FP(=)23 b FO(U)9 b(U)1899 856 y FN(\003)1937 891 y FP(\()p FO(U)2035 856 y FN(\003)2073 891 y FP(\))2105 856 y FM(k)2146 891 y FO(U)2212 856 y FM(k)2253 891 y FO(U)525 1028 y FP(=)23 b FO(U)9 b FP(\()p FO(U)777 994 y FN(\003)815 1028 y FP(\))847 994 y FM(k)q FL(+1)972 1028 y FO(U)1038 994 y FM(k)q FL(+1)1185 1028 y FP(=)23 b FO(U)9 b(P)1392 1040 y FM(k)q FL(+1)1517 1028 y FO(;)290 1166 y(P)343 1178 y FN(\000)p FM(k)436 1166 y FO(U)32 b FP(=)23 b FO(U)679 1132 y FM(k)719 1166 y FP(\()p FO(U)817 1132 y FN(\003)855 1166 y FP(\))887 1132 y FM(k)929 1166 y FO(U)31 b FP(=)23 b FO(U)9 b(U)1237 1132 y FM(k)q FN(\000)p FL(1)1362 1166 y FP(\()p FO(U)1460 1132 y FN(\003)1498 1166 y FP(\))1530 1132 y FM(k)q FN(\000)p FL(1)1657 1166 y FO(U)1723 1132 y FN(\003)1761 1166 y FO(U)525 1304 y FP(=)23 b FO(U)9 b(U)745 1270 y FN(\003)782 1304 y FO(U)g(U)914 1270 y FM(k)q FN(\000)p FL(1)1040 1304 y FP(\()p FO(U)1138 1270 y FN(\003)1176 1304 y FP(\))1208 1270 y FM(k)q FN(\000)p FL(1)1357 1304 y FP(=)23 b FO(U)9 b(U)1577 1270 y FM(k)q FN(\000)p FL(1)1702 1304 y FP(\()p FO(U)1800 1270 y FN(\003)1838 1304 y FP(\))1870 1270 y FM(k)q FN(\000)p FL(1)2019 1304 y FP(=)23 b FO(U)9 b(P)2226 1316 y FN(\000)p FM(k)q FL(+1)118 1489 y FP(and)40 b FO(P)345 1501 y FN(\000)p FL(1)435 1489 y FO(U)53 b FP(=)44 b(\()p FO(U)9 b(U)818 1459 y FN(\003)856 1489 y FP(\))p FO(U)54 b FP(=)44 b FO(U)9 b(I)51 b FP(=)44 b FO(U)9 b(P)1489 1501 y FL(0)1527 1489 y FP(,)43 b FO(P)1646 1501 y FL(0)1684 1489 y FO(U)53 b FP(=)44 b FO(I)7 b(U)54 b FP(=)44 b FO(U)9 b FP(\()p FO(U)2330 1459 y FN(\003)2368 1489 y FO(U)g FP(\))44 b(=)118 1589 y FO(U)9 b(P)237 1601 y FL(1)274 1589 y FP(.)p 2514 1589 4 57 v 2518 1536 50 4 v 2518 1589 V 2567 1589 4 57 v 243 1767 a(The)23 b(case)g(where)g FO(U)32 b FP(is)23 b(unitary)g(is)h(trivial.)35 b(Supp)r(ose)23 b(that)h(the)g(op)r(erator)118 1866 y FO(U)184 1836 y FN(\003)253 1866 y FP(has)30 b(a)h(non-trivial)f(k)n (ernel)g(\(the)h(case)f(of)h(a)g(non-trivial)e(k)n(ernel)h(of)h FO(U)40 b FP(is)118 1966 y(similar\).)35 b(Let)22 b FO(f)32 b FQ(2)24 b FP(k)n(er)12 b FO(U)939 1936 y FN(\003)977 1966 y FP(.)35 b(F)-7 b(or)22 b(ev)n(ery)g FO(k)k FP(=)c(1,)h(2,)g FO(:)14 b(:)g(:)27 b FP(,)d(consider)d(the)i(v)n(ector)118 2065 y(\()p FO(U)216 2035 y FN(\003)254 2065 y FP(\))286 2035 y FM(k)327 2065 y FO(U)393 2035 y FM(k)434 2065 y FO(f)9 b FP(.)37 b(The)27 b(follo)n(wing)g(situations)g(ma)n(y)g(o)r (ccur:)243 2167 y(a\))e(\()p FO(U)440 2136 y FN(\003)478 2167 y FP(\))510 2136 y FM(k)551 2167 y FO(U)617 2136 y FM(k)658 2167 y FO(f)32 b FP(=)22 b FO(f)34 b FP(for)25 b(all)h FO(k)f FP(=)e(1,)j(2,)f FO(:)14 b(:)g(:)27 b FP(.)37 b(Then)25 b(the)h(v)n(ector)e FO(f)2296 2179 y FL(0)2356 2167 y FP(=)f FO(f)34 b FP(is)118 2266 y(a)27 b(join)n(t)h(eigen)n(v)n(ector)d(of)j(a)f(comm)n(uting)g(family)h(\()p FO(P)1746 2278 y FM(k)1788 2266 y FP(\).)243 2367 y(b\))g(for)f(some)g FO(k)f(>)c FP(0,)28 b(the)g(follo)n(wing)e(conditions)h(hold:)705 2553 y(\()p FO(U)803 2519 y FN(\003)842 2553 y FP(\))874 2519 y FM(l)899 2553 y FO(U)965 2519 y FM(l)991 2553 y FO(f)k FP(=)23 b FO(f)t(;)180 b(l)24 b FP(=)f(1)p FO(;)14 b(:)g(:)g(:)f(;)h(k)21 b FQ(\000)d FP(1)p FO(;)1074 2691 y FP(\()p 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i(relation)d(\(2.13\))i(implies)g(that)g FO(f)2092 3197 y FL(0)2129 3185 y FP(,)g FO(U)9 b(f)2285 3197 y FL(0)2322 3185 y FP(,)26 b FO(U)2437 3155 y FL(2)2474 3185 y FO(f)2515 3197 y FL(0)2552 3185 y FP(,)118 3285 y FO(:)14 b(:)g(:)28 b FP(,)h(are)g(orthogonal)e(join)n(t)i(eigenspaces)f(of)h(the)g(family) h(\()p FO(P)2070 3297 y FM(k)2111 3285 y FP(\))g(and)f(can)f(b)r(e)118 3385 y(c)n(hosen)19 b(to)h(b)r(e)g(a)g(basis)f(of)h(the)g(space.)34 b(The)20 b(rest)f(of)h(the)g(pro)r(of)g(is)g(ob)n(vious.)p 2514 3385 V 2518 3332 50 4 v 2518 3385 V 2567 3385 4 57 v 118 3562 a FR(2.)40 b FP(T)-7 b(o)29 b(apply)f(this)i(theorem)e (to)h(the)g(description)f(of)h(irreducible)g(solutions)118 3662 y(of)34 b(\(2.1\),)28 b(w)n(e)f(need)g(the)h(follo)n(wing)f(fact.) 118 3832 y FR(Theorem)37 b(19.)43 b FC(L)l(et)34 b(the)h(p)l(air)h FP(\()p FO(X)r(;)14 b(X)1379 3802 y FN(\003)1416 3832 y FP(\))35 b FC(satisfying)43 b FP(\(2.1\))34 b FC(b)l(e)h(irr)l(e)l (ducible.)118 3932 y(If)f(one)f(of)h(the)f(op)l(er)l(ators)h FO(X)7 b FC(,)34 b FO(X)1186 3902 y FN(\003)1256 3932 y FC(has)g(a)g(non-zer)l(o)f(kernel,)i(then)d(the)i(p)l(air)118 4031 y FP(\()p FO(U;)14 b(U)310 4001 y FN(\003)348 4031 y FP(\))30 b FC(is)h(irr)l(e)l(ducible,)h(i.e.,)h(any)d(b)l(ounde)l(d)h (op)l(er)l(ator)g(c)l(ommuting)f(with)h FO(U)118 4131 y FC(and)f FO(U)345 4101 y FN(\003)413 4131 y FC(is)g(a)g(multiple)g (of)h(the)f(identity.)p eop %%Page: 99 103 99 102 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f(systems)896 b FP(99)118 333 y FC(Pr)l(o)l(of.)43 b FP(First,)29 b(consider)e(the)i(case)f(k)n(er)13 b FO(U)33 b FQ(6)p FP(=)24 b(0.)40 b(T)-7 b(ak)n(e)27 b(a)h(v)n(ector)g FO(e)2220 345 y FL(0)2281 333 y FQ(2)d FP(k)n(er)13 b FO(U)c FP(,)118 432 y(and)25 b(consider)f(v)n(ectors)g FO(e)917 444 y FM(k)980 432 y FP(=)f(\()p FO(U)1166 402 y FN(\003)1204 432 y FP(\))1236 402 y FM(k)1277 432 y FP(,)j FO(k)g FP(=)d(1,)i(2,)g FO(:)14 b(:)g(:)27 b FP(.)36 b(Since)26 b(k)n(er)12 b FO(U)32 b FP(=)23 b(k)n(er)12 b FO(C)6 b FP(,)118 532 y(eac)n(h)27 b FO(e)344 544 y FM(k)412 532 y FP(is)h(an)f(eigen)n(v)n(ector)e(of)j FO(C)1202 502 y FL(2)1267 532 y FP(with)g(the)g(eigen)n(v)-5 b(alue)27 b FO(F)2061 502 y FM(k)2102 532 y FP(\(0\).)243 635 y(W)-7 b(e)34 b(sho)n(w)f(that)h(these)g(v)n(ectors)e(form)h(a)h (basis)f(in)h(the)g(space.)55 b(Indeed,)118 734 y(the)35 b(linear)f(span)729 713 y(~)707 734 y FO(H)42 b FP(of)35 b(these)f(v)n(ectors)g(is)g(in)n(v)-5 b(arian)n(t)34 b(with)h(resp)r(ect)f(to)h FO(U)2537 704 y FN(\003)118 834 y FP(and)28 b FO(C)345 804 y FL(2)382 834 y FP(.)243 936 y(T)-7 b(ak)n(e)28 b(a)g(v)n(ector)g FO(e)805 948 y FM(k)871 936 y FQ(6)p FP(=)d(0)k(with)h(some)e FO(k)g(>)d FP(0,)30 b(and)e(assume)h(that)j(~)-45 b FO(e)2359 948 y FM(k)q FN(\000)p FL(1)2510 936 y FP(=)118 1036 y FO(U)9 b(e)223 1048 y FM(k)287 1036 y FP(=)23 b FO(U)9 b(U)507 1006 y FN(\003)544 1036 y FO(e)583 1048 y FM(k)q FN(\000)p FL(1)737 1036 y FP(is)27 b(not)h(in)1087 1015 y(~)1065 1036 y FO(H)7 b FP(.)37 b(Since)28 b FO(U)1484 1006 y FN(\003)1522 1036 y FO(U)9 b(U)1654 1006 y FN(\003)1715 1036 y FP(=)23 b FO(U)1869 1006 y FN(\003)1907 1036 y FP(,)28 b(w)n(e)f(get)h FO(U)2285 1006 y FN(\003)2326 1036 y FP(~)-46 b FO(e)2361 1048 y FM(k)q FN(\000)p FL(1)2510 1036 y FP(=)118 1136 y FO(e)157 1148 y FM(k)198 1136 y FP(;)23 b(therefore,)e(\()s(~)-45 b FO(e)681 1148 y FM(k)q FN(\000)p FL(1)811 1136 y FQ(\000)t FO(e)919 1148 y FM(k)q FN(\000)p FL(1)1045 1136 y FP(\))23 b FQ(2)g FP(k)n(er)13 b FO(U)1369 1105 y FN(\003)1430 1136 y FP(=)23 b(k)n(er)12 b FO(F)g FP(\()p FO(C)1804 1105 y FL(2)1842 1136 y FP(\),)23 b(whic)n(h)d(is)h(the)g(image)118 1235 y(of)33 b(the)h(pro)5 b(jection)32 b FO(E)828 1250 y FM(C)880 1233 y Fy(2)917 1235 y FP(\()p FO(F)1014 1205 y FN(\000)p FL(1)1103 1235 y FP(\(0\)\).)55 b(On)33 b(the)h(other)e (hand,)j(\()s(~)-45 b FO(e)2143 1247 y FM(k)q FN(\000)p FL(1)2291 1235 y FQ(\000)22 b FO(e)2417 1247 y FM(k)q FN(\000)p FL(1)2543 1235 y FP(\))118 1335 y(b)r(elongs)32 b(to)g FO(E)588 1350 y FM(C)640 1333 y Fy(2)676 1335 y FP(\(\001\))p FO(H)7 b FP(,)34 b(where)e(\001)f(=)g FO(F)1448 1305 y FN(\000)p FL(1)1537 1335 y FP(\()p FO(F)1634 1305 y FM(k)1675 1335 y FP(\(0\)\).)51 b(If)33 b(the)f(latter)g(v)n (ector)118 1434 y(is)25 b(non-zero,)f(this)i(w)n(ould)f(imply)g(that)h (b)r(oth)g FO(e)1598 1446 y FM(k)q FN(\000)p FL(1)1749 1434 y FP(and)i(~)-45 b FO(e)1947 1446 y FM(k)q FN(\000)p FL(1)2097 1434 y FP(b)r(elong)25 b(to)g(the)118 1534 y(k)n(ernel)j(of)g FO(U)525 1504 y FN(\003)591 1534 y FP(and)g(that)h FO(e)973 1546 y FM(k)1038 1534 y FP(=)24 b(0.)39 b(Therefore,)30 b(~)-45 b FO(e)1670 1546 y FM(k)q FN(\000)p FL(1)1820 1534 y FP(=)24 b FO(e)1948 1546 y FM(k)q FN(\000)p FL(1)2073 1534 y FP(,)29 b(and)2309 1513 y(~)2287 1534 y FO(H)i FP(=)24 b FO(H)7 b FP(.)118 1634 y(The)28 b(op)r(erator)e FO(U)36 b FP(is)28 b(adjoin)n(t)f(to)g (the)h(unilateral)f(shift)h(in)g FO(l)2009 1646 y FL(2)2046 1634 y FP(.)243 1736 y(No)n(w)e(consider)f(the)i(case)f(k)n(er)12 b FO(U)1260 1706 y FN(\003)1321 1736 y FQ(6)p FP(=)23 b(0.)36 b(F)-7 b(or)26 b(eac)n(h)g FO(k)f FQ(\025)e FP(0,)j(in)n(tro)r (duce)g(the)118 1836 y(subspace)h FO(H)534 1848 y FM(k)598 1836 y FP(=)c FO(E)747 1851 y FM(C)799 1834 y Fy(2)835 1836 y FP(\()p FO(F)932 1806 y FN(\000)p FM(k)1025 1836 y FP(\(0\)\))p FO(H)7 b FP(.)37 b(The)28 b(equalities)833 2025 y FO(U)9 b(E)960 2040 y FM(C)1012 2023 y Fy(2)1048 2025 y FP(\(\001\))24 b(=)e FO(E)1353 2040 y FM(C)1405 2023 y Fy(2)1442 2025 y FP(\()p FO(F)1539 1990 y FN(\000)p FL(1)1628 2025 y FP(\(\001\)\))p FO(U;)795 2159 y(E)856 2174 y FM(C)908 2158 y Fy(2)944 2159 y FP(\(\001\))p FO(U)1143 2125 y FN(\003)1205 2159 y FP(=)g FO(U)1358 2125 y FN(\003)1396 2159 y FO(E)1457 2174 y FM(C)1509 2158 y Fy(2)1546 2159 y FP(\()p FO(F)1643 2125 y FN(\000)p FL(1)1732 2159 y FP(\(\001\)\))466 b(\(2.14\))118 2348 y(for)21 b(all)f(measurable)g(\001)h(imply)g(that)g FO(U)9 b(H)1399 2360 y FM(k)1463 2348 y FP(=)23 b FO(H)1620 2360 y FM(k)q FL(+1)1745 2348 y FP(,)f FO(U)1856 2318 y FN(\003)1894 2348 y FO(H)1963 2360 y FM(k)q FL(+1)2111 2348 y FP(=)g FO(H)2267 2360 y FM(k)2308 2348 y FP(,)h FO(k)j FQ(\025)c FP(1,)118 2448 y(and)k(the)g(span)g(of)g(these)g (subspaces)e(is)i(an)g(in)n(v)-5 b(arian)n(t)25 b(subspace;)h(due)g(to) f(the)118 2547 y(irreducibilit)n(y)-7 b(,)27 b(it)h(is)g(the)g(whole)f (of)h FO(H)7 b FP(.)243 2650 y(Note)23 b(that)g(k)n(er)13 b FO(U)805 2620 y FN(\003)866 2650 y FP(=)23 b(k)n(er)12 b FO(F)g FP(\()p FO(C)1240 2620 y FL(2)1278 2650 y FP(\))24 b(=)e FO(H)1490 2662 y FL(1)1527 2650 y FP(.)36 b(First)23 b(w)n(e)g(sho)n(w)f(that)i FO(H)2348 2662 y FL(1)2408 2650 y FP(is)f(an)118 2750 y(eigenspace)k(of)g FO(C)684 2719 y FL(2)722 2750 y FP(,)h(i.e.,)f(there)h(exists)f(a)h(single)f(p)r (oin)n(t)h FO(\025)1937 2762 y FL(0)2002 2750 y FP(of)g(the)g(sp)r (ectrum)118 2849 y(of)38 b FO(C)288 2819 y FL(2)363 2849 y FP(suc)n(h)f(that)h FO(F)12 b FP(\()p FO(\025)895 2861 y FL(0)933 2849 y FP(\))40 b(=)g(0.)66 b(Indeed,)41 b(tak)n(e)c(an)n(y) g(measurable)f(subset)118 2949 y FO(\016)28 b FQ(\032)c FO(F)337 2919 y FN(\000)p FL(1)426 2949 y FP(\(0\).)39 b(Consider)28 b(the)h(subspaces)e FO(H)1542 2919 y FM(\016)1535 2972 y(k)1603 2949 y FP(=)d FO(E)1753 2964 y FM(C)1805 2947 y Fy(2)1842 2949 y FP(\()p FO(F)1939 2919 y FN(\000)p FM(k)2032 2949 y FP(\()p FO(\016)s FP(\)\))p FO(H)32 b FQ(\032)24 b FO(H)2427 2961 y FM(k)q FL(+1)2552 2949 y FP(,)118 3048 y FO(k)36 b FQ(\025)d FP(0.)54 b(Equalities)33 b(\(2.14\))g(imply)h(that)g(the)g(span)f(of)h(the)g(subspaces)f FO(H)2539 3018 y FM(\016)2532 3072 y(k)118 3148 y FP(is)i(an)f(in)n(v) -5 b(arian)n(t)34 b(subspace;)j(the)e(irreducibilit)n(y)f(then)h (implies)g(that)g(there)118 3248 y(exists)25 b(a)g(single)g(p)r(oin)n (t)h FO(\025)904 3260 y FL(0)941 3248 y FP(,)h FO(F)12 b FP(\()p FO(\025)1136 3260 y FL(0)1173 3248 y FP(\))24 b(=)e(0,)k(suc)n(h)f(that)h FO(E)1831 3263 y FM(C)1883 3246 y Fy(2)1919 3248 y FP(\()p FO(F)2016 3218 y FN(\000)p FL(1)2106 3248 y FP(\(0\))14 b FQ(n)g FO(\025)2330 3260 y FL(0)2367 3248 y FP(\))24 b(=)e(0.)243 3350 y(Th)n(us,)31 b FO(H)549 3362 y FL(1)617 3350 y FP(is)g(an)g(eigenspace)e(of)i FO(C)1395 3320 y FL(2)1432 3350 y FP(.)47 b(No)n(w)31 b(w)n(e)f(sho)n(w)g(that)h FO(H)2281 3362 y FL(2)2349 3350 y FP(is)g(also)118 3450 y(an)37 b(eigenspace.)64 b(Since)37 b FO(U)9 b(U)1068 3420 y FN(\003)1143 3450 y FP(comm)n(utes)36 b(with)i FO(F)12 b FP(\()p FO(C)1903 3420 y FL(2)1940 3450 y FP(\),)40 b FO(U)9 b(U)2167 3420 y FN(\003)2242 3450 y FP(maps)37 b FO(H)2538 3462 y FL(2)118 3550 y FP(in)n(to)25 b(itself,)i(and)e(is)g(a)g(pro)5 b(jection)24 b(on)h(it.)37 b(Actually)-7 b(,)26 b(since)f(k)n(er)12 b FO(U)2176 3519 y FN(\003)2237 3550 y FP(=)23 b FO(H)2394 3562 y FL(1)2431 3550 y FP(,)j(w)n(e)118 3649 y(conclude)31 b(that)h FO(U)9 b(U)779 3619 y FN(\003)848 3649 y FP(is)32 b(the)g(iden)n(tit)n(y)g(on)f FO(H)1582 3661 y FL(2)1619 3649 y FP(.)49 b(No)n(w)31 b(w)n(e)g(aply)g(similar)g(ar-)118 3749 y(gumen)n(ts)d(as)f(ab)r(o)n(v)n(e.)36 b(T)-7 b(ak)n(e)26 b(an)n(y)h(measurable)g FO(\016)f FQ(\032)d FO(F)1827 3719 y FN(\000)p FL(1)1916 3749 y FP(\()p FO(\025)1996 3761 y FL(0)2034 3749 y FP(\),)28 b(and)g(consider)118 3848 y(the)j(subspaces)e FO(H)722 3818 y FM(\016)715 3872 y(k)785 3848 y FP(=)e FO(E)938 3863 y FM(C)990 3847 y Fy(2)1027 3848 y FP(\()p FO(F)1124 3818 y FN(\000)p FM(k)1216 3848 y FP(\()p FO(\016)s FP(\)\))i FQ(\032)d FO(H)1541 3860 y FM(k)q FL(+2)1666 3848 y FP(,)31 b FO(k)f FQ(\025)d FP(0.)44 b(Relations)30 b(\(2.14\))118 3948 y(and)35 b FO(U)9 b(U)419 3918 y FN(\003)491 3948 y Fs(\026)34 b FO(H)629 3960 y FL(2)701 3948 y FP(=)g FO(I)42 b FP(imply)35 b(that)g(the)g(span)f(of)g FO(H)1833 3918 y FM(\016)1826 3972 y(k)1870 3948 y FP(,)i FO(k)h FQ(\025)e FP(0,)h(and)e FO(H)2447 3960 y FL(1)2519 3948 y FP(is)118 4048 y(an)28 b(in)n(v)-5 b(arian)n(t)28 b(subspace;)g(therefore,)g(b)n(y)h(the)g (irreducibilit)n(y)-7 b(,)28 b FO(H)2169 4060 y FL(2)2235 4048 y FP(is)h(also)e(an)118 4147 y(eigenspace)f(of)i FO(C)684 4117 y FL(2)722 4147 y FP(.)p eop %%Page: 100 104 100 103 bop 118 100 a FP(100)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)243 333 y FP(Con)n(tin)n(uing)38 b(this)h(pro)r(cess,)i(w) n(e)e(conclude)g(that)g(all)g FO(H)2057 345 y FM(k)2098 333 y FP(,)j FO(k)j FQ(\025)c FP(1,)h(are)118 432 y(eigenspaces)22 b(of)h FO(C)708 402 y FL(2)746 432 y FP(.)35 b(Also,)24 b(in)g(the)f(irreducible)g(case,)g(all)g(these)g(eigenspaces)118 532 y(are)k(one-dimensional;)f(therefore,)h FO(U)36 b FP(is)28 b(a)f(unilateral)g(shift)h(in)g FO(l)2190 544 y FL(2)2227 532 y FP(.)p 2514 532 4 57 v 2518 479 50 4 v 2518 532 V 2567 532 4 57 v 118 700 a FR(3.)55 b FP(No)n(w)33 b(w)n(e)h(apply)f(the)h(results)f(on)h(cen)n(tered)f(partial)g (isometries)g(to)g(the)118 799 y(description)g(of)g(op)r(erators)e (satisfying)i(\(2.1\))g(\(note)g(that)g(the)h(same)e(result)118 899 y(can)38 b(b)r(e)h(obtained)g(using)f([288)o(]\).)70 b(Since)39 b(w)n(e)g(are)e(in)n(terested)i(in)f(in\014nite-)118 999 y(dimensional)e(represen)n(tations,)h(only)f(isometries,)i(or)e (co-isometries)f(ma)n(y)118 1098 y(arise)27 b(in)g(the)h (in\014nite-dimensional)g(non-unitary)e(case.)118 1265 y FR(Theorem)43 b(20.)k FC(A)n(ny)39 b(in\014nite-dimensional)i(irr)l (e)l(ducible)f(r)l(epr)l(esentation)118 1365 y(of)33 b(r)l(elations)39 b FP(\(2.1\))o FC(,)33 b(for)f(which)h FP(k)n(er)13 b FO(X)26 b FQ([)20 b FP(k)n(er)13 b FO(X)1658 1335 y FN(\003)1722 1365 y FQ(6)p FP(=)26 b FQ(f)p FP(0)p FQ(g)p FC(,)k(fal)t(ls)j(into)f(one)g(of)118 1464 y(the)e(fol)t(lowing) i(classes)7 b FP(:)197 1631 y(\(i\))42 b FC(in\014nite-dimensional)31 b(in)e FO(l)1201 1643 y FL(2)1238 1631 y FP(:)615 1814 y FO(X)7 b(e)730 1826 y FL(1)789 1814 y FP(=)23 b(0)p FO(;)98 b(X)7 b(e)1155 1826 y FM(j)1212 1814 y FP(=)1300 1743 y Fz(p)p 1383 1743 84 4 v 71 x FO(\025)1431 1826 y FM(j)1466 1814 y FO(e)1505 1826 y FM(j)s FN(\000)p FL(1)1625 1814 y FO(;)183 b(j)28 b FP(=)23 b(2)p FO(;)14 b FP(3)p FO(;)g(:)g(:)g(:)f(;)1029 1950 y(\025)1077 1962 y FM(j)1135 1950 y FO(>)23 b FP(0)p FO(;)98 b(\025)1434 1962 y FM(j)1493 1950 y FP(=)22 b FO(F)1645 1916 y FM(j)s FN(\000)p FL(1)1765 1950 y FP(\(0\))326 2133 y(\()p FC(F)-6 b(o)l(ck)30 b(r)l(epr)l(esentation)6 b FP(\))p FC(,)173 2300 y FP(\(ii\))43 b FC(in\014nite)29 b(dimensional)j(in)d FO(l)1201 2312 y FL(2)1238 2300 y FP(:)828 2483 y FO(X)7 b(e)943 2495 y FM(j)1000 2483 y FP(=)1088 2412 y Fz(p)p 1171 2412 V 71 x FO(\025)1219 2495 y FM(j)1254 2483 y FO(e)1293 2495 y FM(j)s FL(+1)1412 2483 y FO(;)183 b(j)28 b FP(=)23 b(1)p FO(;)14 b FP(2)p FO(;)g(:)g(:)g(:)f(;)744 2619 y(\025)792 2631 y FM(j)850 2619 y FO(>)23 b FP(0)p FO(;)98 b(\025)1149 2631 y FM(j)1208 2619 y FQ(2)23 b FO(F)1351 2585 y FN(\000)p FM(j)1438 2619 y FP(\(0\))p FO(;)99 b(F)12 b FP(\()p FO(\025)1811 2631 y FM(j)s FL(+1)1930 2619 y FP(\))24 b(=)e FO(\025)2121 2631 y FM(j)326 2802 y FP(\()p FC(anti-F)-6 b(o)l(ck)30 b(r)l(epr)l(esentations)7 b FP(\))p FC(.)118 2969 y(Pr)l(o)l(of.)43 b FP(Indeed,)31 b(the)g(phase)e FO(U)39 b FP(of)30 b(the)h(op)r(erator)d FO(X)37 b FP(is)30 b(a)g(partial)f(isometry)118 3069 y(\(non-unitary\),)k(th)n(us)f(b)n(y)g(Theorem)f(19,)h(the)h(represen)n (tation)d(space)h(of)h(an)118 3168 y(irreducible)22 b FO(X)28 b FP(is)22 b(the)h(same)e(as)h(for)f(an)h(irreducible)g FO(U)9 b FP(.)34 b(Use)22 b(Theorem)g(18)f(to)118 3268 y(represen)n(t)29 b(the)i(op)r(erator)d FO(U)9 b FP(;)31 b(the)g(rest)e(of)i(the)f(pro)r(of)g(follo)n(ws)f(immediately)118 3368 y(from)e(\(2.5\).)p 2514 3368 4 57 v 2518 3315 50 4 v 2518 3368 V 2567 3368 4 57 v 118 3536 a FC(Example)36 b FP(13)p FC(.)j FP(\(Mapping)27 b(of)f(degree)f(t)n(w)n(o.)35 b(F)-7 b(o)r(c)n(k)26 b(and)g(an)n(ti-F)-7 b(o)r(c)n(k)25 b(represen-)118 3635 y(tations\).)36 b(F)-7 b(or)23 b(relation)h (\(2.2\))o(,)h(zero)e(is)h(a)g(stationary)f(p)r(oin)n(t;)j(therefore,)e (this)118 3735 y(relation)j(admits)g(no)h(F)-7 b(o)r(c)n(k)27 b(or)g(an)n(ti-F)-7 b(o)r(c)n(k)26 b(represen)n(tations.)243 3835 y(Ho)n(w)n(ev)n(er,)f(this)j(is)f(not)g(true)g(for)g(\(2.3\))o(;)h (b)r(elo)n(w,)f(w)n(e)g(lo)r(ok)f(at)h(suc)n(h)g(repre-)118 3934 y(sen)n(tations)g(for)g(the)h(relation)1022 4117 y FO(xx)1116 4083 y FN(\003)1178 4117 y FP(=)23 b(\()p FO(x)1345 4083 y FN(\003)1384 4117 y FO(x)c FQ(\000)f FO(q)s(I)7 b FP(\))p FO(:)692 b FP(\(2.15\))p eop %%Page: 101 105 101 104 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f(systems)854 b FP(101)118 333 y(As)21 b(follo)n(ws)f(from) g(the)h(previous)f(theorem,)h(w)n(e)g(need)g(to)f(consider)g (half-orbits)118 432 y(lying)27 b(in)h(the)g(p)r(ositiv)n(e)f(region,)g (and)g(coming)g(from)g(or)g(going)f(to)i(zero.)243 532 y(1.)35 b(F)-7 b(o)r(c)n(k)26 b(represen)n(tation.)34 b(The)26 b(image)e(of)32 b(0)26 b(under)f(the)h(action)f(of)h(p)r(o)n (w-)118 632 y(ers)i(of)h FO(F)42 b FP(is)28 b(alw)n(a)n(ys)g(p)r (ositiv)n(e)g(except)h(for)g(the)g(case)g FO(q)f FP(=)d(0.)41 b(Therefore,)28 b(for)118 731 y(an)n(y)21 b FO(q)26 b FQ(6)p FP(=)c(0,)h(one)d(can)h(construct)g(the)h(F)-7 b(o)r(c)n(k)20 b(represen)n(tation)g(of)28 b(\(2.15\))o(.)35 b(Ho)n(w-)118 831 y(ev)n(er,)24 b(for)f FO(q)j(<)d FQ(\000)p FP(1)p FO(=)p FP(4,)g(the)h(sequence)f FO(F)1369 801 y FM(n)1414 831 y FP(\(0\))h(is)g(un)n(b)r(ounded,)h(and)f(therefore,) 118 930 y(the)19 b(corresp)r(onding)e(represen)n(tation)h(is)g(un)n(b)r (ounded.)35 b(F)-7 b(or)18 b(all)h FO(q)26 b(>)c FQ(\000)p FP(1)p FO(=)p FP(4,)d(for)118 1030 y(whic)n(h)g(zero)f(is)g(not)h(a)f (p)r(erio)r(dic)h(p)r(oin)n(t,)i(there)d(exists)h(a)f(unique)h(b)r (ounded)g(F)-7 b(o)r(c)n(k)118 1130 y(represen)n(tation;)36 b(the)f(sp)r(ectrum)g(of)f FO(C)1380 1100 y FL(2)1453 1130 y FP(lies)g(b)r(et)n(w)n(een)g(zero)g(and)g(the)h(\014rst)118 1229 y(stationary)29 b(p)r(oin)n(t)i(for)f FO(q)h(<)c FP(0,)k(and)g(on)f(the)h(in)n(terv)-5 b(al)30 b([0)p FO(;)14 b(q)1996 1199 y FL(2)2032 1229 y FP(\))31 b(for)f FO(q)h FQ(2)d FP(\(0)p FO(;)14 b FP(2].)118 1329 y(Notice)32 b(that)g(for)f FO(q)i FP(=)d(1,)i(zero)f(is)g(a)h(p)r(erio)r(dic)f(p)r (oin)n(t)h(of)g(the)g(second)f(order,)118 1429 y(and)d(the)h(corresp)r (onding)d(represen)n(tation)g(is)i(t)n(w)n(o-dimensional.)37 b(Similarly)-7 b(,)118 1528 y(for)26 b(those)g(v)-5 b(alues)26 b(of)33 b FO(q)d FP(for)c(whic)n(h)g FO(q)k FP(is)c(a)g(p)r(erio)r(dic) h(p)r(oin)n(t)f(of)h(some)f(order)f FO(n)p FP(,)118 1628 y(the)j(corresp)r(onding)e(F)-7 b(o)r(c)n(k)27 b(represen)n(tation)f (is)h FO(n)p FP(-dimensional.)243 1727 y(F)-7 b(or)27 b FO(q)f(>)c FP(2,)28 b(the)g(F)-7 b(o)r(c)n(k)27 b(represen)n(tation)f (is)h(un)n(b)r(ounded.)243 1827 y(2.)72 b(An)n(ti-F)-7 b(o)r(c)n(k)39 b(represen)n(tations.)70 b(No)n(w)39 b(w)n(e)h(consider) e(p)r(ositiv)n(e)h(half-)118 1927 y(orbits)d(going)g(in)n(to)h(zero.)63 b(It)38 b(is)f(ob)n(vious)e(that)i(this)h(is)e(p)r(ossible)h(only)f (for)118 2026 y FO(q)j(>)c FP(0.)59 b(F)-7 b(or)35 b(0)g FO(<)h(q)i(<)e FP(1,)g(there)f(exists)g(a)g(unique)g(sequence)g(of)g(p) r(ositiv)n(e)118 2126 y(pre-images)24 b(of)i(zero,)g FO(F)888 2096 y FN(\000)p FM(k)980 2126 y FP(\(0\))d FQ(\000)-48 b(!)23 b FO(\025)1280 2138 y FL(2)1318 2126 y FP(,)j(where)g FO(\025)1654 2138 y FL(2)1718 2126 y FP(is)g(the)g(second)f(stationary)118 2226 y(p)r(oin)n(t)j(of)f(the)h (mapping.)243 2325 y(F)-7 b(or)38 b FO(q)44 b FP(=)d(1,)g(zero)d(is)g (a)g(p)r(erio)r(dic)h(p)r(oin)n(t)f(of)h(p)r(erio)r(d)f(2,)j(and)e (zero)e(has)118 2425 y(t)n(w)n(o)29 b(p)r(ositiv)n(e)h(pre-images,)e(0) i(and)f(2.)44 b(Ho)n(w)n(ev)n(er,)28 b(the)i(represen)n(tation)f(that) 118 2524 y(corresp)r(onds)40 b(to)h(0)g(is)g(t)n(w)n(o-dimensional,)i (and)e(w)n(e)g(again)f(ha)n(v)n(e)g(a)h(single)118 2624 y(in\014nite-dimensional)23 b(an)n(ti-F)-7 b(o)r(c)n(k)23 b(represen)n(tation.)34 b(Indeed,)24 b(one)g(can)f(easily)118 2724 y(see)k(that)h(there)g(are)e(at)i(least)f(coun)n(tably)g(man)n(y)g (in)n(v)n(erse)f(orbits.)243 2823 y(F)-7 b(or)18 b FO(q)27 b(>)22 b FP(1,)f(zero)d(has)h(t)n(w)n(o)g(p)r(ositiv)n(e)g(pre-images,) g(0)k FO(<)f(t)1989 2835 y FL(1)2050 2823 y FO(<)g(q)27 b(<)22 b(t)2318 2835 y FL(2)2378 2823 y FO(<)h(\025)2514 2835 y FL(2)2552 2823 y FP(,)118 2923 y(whic)n(h)34 b(generate)f(at)i (least)f(coun)n(tably)f(man)n(y)h(sequences)f(of)i(p)r(ositiv)n(e)f (pre-)118 3023 y(images.)118 3186 y FC(Pr)l(op)l(osition)43 b FP(38)p FC(.)h(F)-6 b(or)34 b FP(1)d FO(<)g(q)k FQ(\024)c FO(q)1260 3156 y FN(\003)1332 3186 y FC(ther)l(e)k(ar)l(e)f(c)l (ountably)h(many)g(ine)l(quiv-)118 3286 y(alent)i(irr)l(e)l(ducible)h (anti-F)-6 b(o)l(ck)37 b(r)l(epr)l(esentations.)60 b(F)-6 b(or)37 b FO(q)i(>)c(q)2120 3256 y FN(\003)2158 3286 y FC(,)k(ther)l(e)e(is)g(a)118 3386 y(c)l(ontinuum)29 b(of)h(such)g(r)l(epr)l(esentations.)118 3550 y(Pr)l(o)l(of.)43 b FP(First)30 b(mark)e(some)h(p)r(oin)n(ts)g(on)h(the)f(line.)43 b(Denote)30 b(b)n(y)f FO(\025)2169 3562 y FL(1)2207 3550 y FP(,)h FO(\025)2308 3562 y FL(2)2345 3550 y FP(,)g FO(\025)2446 3562 y FL(1)2510 3550 y FO(<)118 3649 y(\025)166 3661 y FL(2)204 3649 y FP(,)c(the)g(stationary)e(p)r(oin)n(ts)i(of)g FO(F)12 b FP(\()p FQ(\001)p FP(\);)27 b FO(\025)1378 3619 y FN(0)1378 3670 y FL(1)1439 3649 y FQ(6)p FP(=)22 b FO(\025)1574 3661 y FL(1)1638 3649 y FP(is)j(the)i(second)e (pre-image)f(of)118 3749 y FO(\025)166 3761 y FL(1)204 3749 y FP(,)29 b FO(F)12 b FP(\()p FO(\025)401 3719 y FN(0)401 3769 y FL(1)439 3749 y FP(\))24 b(=)h FO(\025)633 3761 y FL(1)670 3749 y FP(,)k(and)g FO(\025)933 3719 y FN(00)933 3769 y FL(1)1000 3749 y FO(<)24 b(\025)1137 3761 y FL(1)1204 3749 y FP(is)k(a)g(pre-image)f(of)i FO(\025)1886 3761 y FL(1)1952 3749 y FP(whic)n(h)f(is)h(di\013eren)n(t) 118 3848 y(from)f FO(\025)363 3860 y FL(1)430 3848 y FP(and)g FO(\025)640 3818 y FN(0)640 3869 y FL(1)706 3848 y FP(with)h(resp)r(ect)f(to)h FO(F)1349 3818 y FL(2)1386 3848 y FP(\()p FQ(\001)p FP(\),)g FO(F)1590 3818 y FL(2)1628 3848 y FP(\()p FO(\025)1708 3818 y FN(00)1708 3869 y FL(1)1751 3848 y FP(\))c(=)f FO(\025)1945 3860 y FL(1)1982 3848 y FP(.)40 b(Also)28 b(in)n(tro)r(duce)118 3948 y FO(\025)166 3960 y FQ(\003)235 3948 y FP(=)23 b FO(q)363 3918 y FL(2)423 3948 y FP(=)g FO(F)576 3918 y FL(2)613 3948 y FP(\(0\).)243 4048 y(Let)33 b FO(q)j FQ(\024)d FO(q)608 4018 y FN(\003)646 4048 y FP(.)54 b(Then)34 b FO(F)12 b FP(\()p FQ(\001)p FP(\))34 b(has)f(cycles)g(only)g(of)g (orders)f(2)2116 4018 y FM(k)2157 4048 y FP(,)j FO(k)g FQ(\025)e FP(1.)54 b FO(F)118 4147 y FP(maps)30 b(the)h(in)n(terv)-5 b(al)30 b([0)p FO(;)14 b(\025)937 4159 y FQ(\003)982 4147 y FP(])31 b(in)n(to)f(itself,)h(and)f(one)g(can)g(easily)g(see)g (that)g(the)p eop %%Page: 102 106 102 105 bop 118 100 a FP(102)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(n)n(um)n(b)r(er)19 b(of)g(in)n(v)n(erse)e (orbits)i(that)g(lea)n(v)n(e)f(this)h(in)n(terv)-5 b(al)18 b(do)r(es)h(not)g(in\015uence)g(the)118 432 y(total)31 b(cardinalit)n(y)f(of)h(the)h(in)n(v)n(erse)d(orbits,)j(pro)n(vided)e (it)i(is)f(in\014nite.)48 b(Th)n(us,)118 532 y(w)n(e)30 b(need)h(to)f(in)n(v)n(estigate)g(the)g(n)n(um)n(b)r(er)h(of)f(orbits)g (that)h(go)f(to)g(zero)f(and)i(lie)118 632 y(completely)d(inside)f([0)p FO(;)14 b(\025)921 644 y FQ(\003)967 632 y FP(].)243 731 y(Consider)36 b(t)n(w)n(o)g(in)n(terv)-5 b(als,)39 b FO(I)1176 701 y FL(1)1169 752 y(1)1252 731 y FP(=)f([)p FO(\025)1426 701 y FN(0)q(0)1426 752 y FL(1)1469 731 y FO(;)14 b(\025)1554 743 y FL(1)1592 731 y FP(])37 b(and)g FO(I)1866 701 y FL(1)1859 752 y(2)1942 731 y FP(=)h([)p FO(\025)2116 743 y FL(1)2154 731 y FO(;)14 b(\025)2239 701 y FN(0)2239 752 y FL(1)2277 731 y FP(].)65 b(They)118 831 y(are)31 b(in)n(v)-5 b(arian)n(t)31 b(with)h(resp)r(ect)g(to)g FO(F)1267 801 y FL(2)1304 831 y FP(\()p FQ(\001)p FP(\).)50 b(Indeed,)34 b(it)e(is)g(su\016cien)n(t)g(to)g(sho)n(w)118 931 y(that)j FO(\025)353 901 y FN(0)353 951 y FL(1)425 931 y FO(>)f(\025)572 943 y FQ(\003)618 931 y FP(;)k(if)d(not,)h(the)f (dynamical)f(system)g FO(F)1844 901 y FL(2)1916 931 y FP(on)g(the)h(in)n(terv)-5 b(al)34 b FO(I)2538 901 y FL(1)2531 951 y(2)118 1030 y FP(w)n(ould)24 b(ha)n(v)n(e)e(the)j(same)e (dynamics)g(as)h(the)g(quadratic)f(mapping)g(in)h(the)h(case)118 1130 y(of)36 b FO(q)41 b FQ(\025)c FP(2)e([261)o(],)k(and)d FO(F)946 1100 y FL(2)983 1130 y FP(\()p FQ(\001)p FP(\))h(w)n(ould)e (ha)n(v)n(e)g(cycles)h(of)g(an)n(y)g(p)r(erio)r(d,)i(whic)n(h)118 1230 y(con)n(tradicts)i(the)i(condition)f FO(q)49 b FQ(\024)c FO(q)1330 1199 y FN(\003)1368 1230 y FP(.)78 b(Notice)42 b(also)e(that)i(in)f(this)h(case,)118 1329 y FO(\025)166 1299 y FN(0)q(0)166 1350 y FL(1)241 1329 y FO(<)31 b FP(0.)52 b(Therefore,)32 b(there)h(are)f(no)g(orbits)g(of)h FO(F)1747 1299 y FL(2)1784 1329 y FP(\()p FQ(\001)p FP(\))h(whic)n(h)e (mo)n(v)n(e)g(inside)118 1429 y(b)r(oth)c FO(I)357 1399 y FL(1)350 1449 y(1)422 1429 y FP(and)f FO(I)626 1399 y FL(1)619 1449 y(2)664 1429 y FP(.)243 1529 y(No)n(w)21 b(w)n(e)h(in)n(v)n(estigate)f(the)h(set)g(of)g(orbits)g(of)g(the)g (mapping)g FO(F)2159 1499 y FL(2)2196 1529 y FP(\()p FQ(\001)p FP(\))h(on)f(eac)n(h)118 1628 y(of)36 b(the)h(in)n(terv)-5 b(als,)38 b FO(I)783 1598 y FL(1)776 1649 y(1)857 1628 y FP(and)e FO(I)1070 1598 y FL(1)1063 1649 y(2)1107 1628 y FP(.)64 b(On)36 b(eac)n(h)f(of)i(these)f(in)n(terv)-5 b(als,)38 b FO(F)2293 1598 y FL(2)2330 1628 y FP(\()p FQ(\001)p FP(\))f(has)118 1728 y(similar)28 b(dynamical)h(prop)r (erties)f(as)g(a)h(quadratic)e(mapping;)j(in)f(particular,)118 1828 y(for)f FO(F)311 1797 y FL(2)348 1828 y FP(\()p FQ(\001)p FP(\))i(there)e(are)f(cycles)h(only)g(of)h(orders)e(2)1628 1797 y FM(k)1668 1828 y FP(,)i FO(k)f FQ(\025)c FP(1.)39 b(Again,)29 b(there)f(are)118 1927 y(four)g(in)n(terv)-5 b(als,)27 b FO(I)691 1897 y FL(2)684 1951 y FM(k)729 1927 y FP(,)h FO(k)f FP(=)c(1,)28 b FO(:)14 b(:)g(:)28 b FP(,)g(4,)g(whic)n(h)g(are)f(in)n(v)-5 b(arian)n(t)27 b(with)i(resp)r(ect)f(to)118 2027 y FO(F)183 1997 y FL(4)220 2027 y FP(\()p FQ(\001)p FP(\),)23 b(and)e(whic)n(h)g(determine)g(the)g (cardinalit)n(y)f(of)g(orbits.)34 b(Con)n(tin)n(uing)21 b(this)118 2126 y(pro)r(cess,)32 b(w)n(e)g(get)g(either)g(a)g(\014nite) g(n)n(um)n(b)r(er)g(of)g(in)n(terv)-5 b(als)32 b(where)f FO(F)2289 2096 y FM(n)2335 2126 y FP(\()p FQ(\001)p FP(\))h(has)118 2226 y(only)e(one)g(orbit)g(going)f(to)h(zero)f(\(for)h FO(q)h(<)c(q)1532 2196 y FN(\003)1570 2226 y FP(\),)k(or)f(a)g(Can)n (tor)f(set)h(on)g(whic)n(h)118 2326 y FO(F)12 b FP(\()p FQ(\001)p FP(\))32 b(is)g(one-to-one)e(\(for)h FO(q)i FP(=)c FO(q)1167 2296 y FN(\003)1205 2326 y FP(\);)34 b(in)e(b)r(oth)g(cases,)g(w)n(e)f(ha)n(v)n(e)f(a)i(coun)n(table)118 2425 y(n)n(um)n(b)r(er)27 b(of)h(orbits)f(going)f(to)i(zero.)243 2525 y(T)-7 b(o)30 b(pro)n(v)n(e)f(that)i(for)f FO(q)h(>)d(q)1109 2495 y FN(\003)1178 2525 y FP(there)i(is)h(a)f(con)n(tin)n(uum)g(of)h (an)n(ti-F)-7 b(o)r(c)n(k)30 b(rep-)118 2625 y(resen)n(tations,)42 b(consider)d(\014rst)h(the)h(case)e(of)h FO(q)48 b FP(=)43 b FO(q)1818 2637 y FL(0)1900 2625 y FQ(\031)h FP(1)p FO(:)p FP(54,)e(suc)n(h)e(that)118 2724 y FO(\025)166 2736 y FQ(\003)241 2724 y FP(=)28 b FO(\025)382 2694 y FN(0)382 2745 y FL(1)419 2724 y FP(,)k(i.e.,)g FO(F)700 2694 y FL(2)737 2724 y FP(\(0\))d(=)f FO(\025)1013 2736 y FL(1)1050 2724 y FP(.)47 b(In)31 b(this)g(case,)g(the)g(in)n(terv)-5 b(als)30 b FO(I)2120 2694 y FL(1)2113 2745 y(1)2188 2724 y FP(and)h FO(I)2396 2694 y FL(1)2389 2745 y(2)2464 2724 y FP(are)118 2824 y(in)n(v)-5 b(arian)n(t)24 b(with)i(resp)r(ect)f(to)h FO(F)1098 2794 y FL(2)1135 2824 y FP(\()p FQ(\001)p FP(\),)g(and)g(on)f (eac)n(h)f(of)i(these)f(in)n(terv)-5 b(als,)25 b FO(F)2450 2794 y FL(2)2487 2824 y FP(\()p FQ(\001)p FP(\))118 2924 y(has)30 b(the)h(same)f(dynamics)g(as)g(the)g(quadratic)g(mapping)g (with)h FO(q)g FP(=)c(2.)45 b(One)118 3023 y(can)32 b(easily)g(see)h (that)g(in)g(this)g(case)e(the)i(set)g(of)g(pre-images)e(of)h(zero)g (under)118 3123 y FO(F)183 3093 y FL(2)220 3123 y FP(\()p FQ(\001)p FP(\))26 b(con)n(tains)e(an)h(in\014nite)h(binary)e(tree,)h (and)g(therefore,)g(an)f(uncoun)n(table)118 3223 y(n)n(um)n(b)r(er)j (of)h(paths)f(in)h(it.)243 3323 y(F)-7 b(or)38 b(an)g(arbitrary)e FO(q)45 b(>)c(q)1121 3292 y FN(\003)1159 3323 y FP(,)g(one)d(can)g(sho) n(w)g(using)g([261)o(])h(that)f(some)118 3422 y(iteration)25 b(of)h FO(F)12 b FP(\()p FQ(\001)p FP(\))26 b(p)r(ossesses)e(the)i (same)g(prop)r(ert)n(y)-7 b(,)25 b(and)g(therefore,)g(there)h(is)118 3522 y(an)h(uncoun)n(table)h(n)n(um)n(b)r(er)f(of)g(orbits)g(passing)g (through)g(zero.)p 2514 3522 4 57 v 2518 3469 50 4 v 2518 3522 V 2567 3522 4 57 v 118 3690 a FC(Example)37 b FP(14)p FC(.)k FP(\(Tw)n(o-parameter)25 b(unit)k(quan)n(tum)e (disk\).)37 b(F)-7 b(or)27 b(relations)g(re-)118 3790 y(lated)f(to)f(con)n(tin)n(uous)f(fractions,)h(w)n(e)g(consider)g(only) g(a)g(v)n(ery)f(sp)r(ecial)h(case)g(of)118 3889 y(the)j(mapping)803 4116 y FO(F)12 b FP(\()p FO(\025)p FP(\))24 b(=)1102 4060 y(\()p FO(q)e FP(+)c FO(\026)p FP(\))p FO(\025)h FP(+)f(1)g FQ(\000)g FO(q)k FQ(\000)c FO(\026)p 1102 4097 741 4 v 1276 4173 a(\026\025)h FP(+)f(1)g FQ(\000)g FO(\026)1853 4116 y(;)487 b FP(\(2.16\))p eop %%Page: 103 107 103 106 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f(systems)854 b FP(103)118 333 y(with)30 b(0)d FQ(\024)f FO(\026)h FQ(\024)f FP(1,)k(0)c FQ(\024)g FO(q)k FQ(\024)c FP(1,)k(\()p FO(\026;)14 b(q)s FP(\))27 b FQ(6)p FP(=)g(\(0)p FO(;)14 b FP(1\).)43 b(This)29 b(mapping)h(is)g(related)118 432 y(to)23 b(the)h(t)n(w)n(o-parameter)d (unit)j(quan)n(tum)f(disk)h(algebra)d(in)n(tro)r(duced)i(in)h([139)o (],)118 532 y(whic)n(h)k(is)f(generated)g(b)n(y)g(the)h(relation)589 701 y FO(q)s(z)t(z)715 667 y FN(\003)770 701 y FQ(\000)18 b FO(z)896 667 y FN(\003)933 701 y FO(z)27 b FP(=)22 b FO(q)g FQ(\000)c FP(1)g(+)g FO(\026)p FP(\(1)g FQ(\000)g FO(z)t(z)1682 667 y FN(\003)1719 701 y FP(\)\(1)g FQ(\000)g FO(z)1969 667 y FN(\003)2007 701 y FO(z)t FP(\))p FO(:)118 871 y FP(Putting)35 b FO(x)h FP(=)f FO(z)657 841 y FN(\003)694 871 y FP(,)i(w)n(e)e(come)g(to)g(a)f(relation)g(of)h(the)h(form)e (\(2.1\))h(with)g(the)118 970 y FO(F)12 b FP(\()p FQ(\001)p FP(\))35 b(in)n(tro)r(duced)g(in)f(this)h(w)n(a)n(y)-7 b(.)57 b(If)35 b FO(\026)g FP(=)f(0,)i(w)n(e)e(get)g(the)h FO(q)s FP(-CCR)g(relation)118 1070 y(considered)27 b(ab)r(o)n(v)n(e;)f (th)n(us)i(w)n(e)f(assume)g(that)h FO(\026)23 b FQ(6)p FP(=)f(0.)243 1170 y(The)41 b(mapping)g FO(F)12 b FP(\()p FQ(\001)p FP(\))42 b(is)f(one-to-one)f(and)h(p)r(ossesses)f(t)n(w)n(o)g (stationary)118 1269 y(p)r(oin)n(ts,)23 b FO(t)416 1281 y FL(1)476 1269 y FP(=)g(1)6 b FQ(\000)g FP(\(1)17 b FQ(\000)h FO(q)s FP(\))p FO(=\026)k FP(and)f FO(t)1228 1281 y FL(2)1288 1269 y FP(=)i(1,)f(that)g(corresp)r(ond)e(to)h(t)n(w)n (o)g(families)118 1369 y(of)28 b(one-dimensional)e(represen)n(tations.) 35 b(Consider)26 b(the)i(follo)n(wing)f(cases.)243 1469 y(1.)47 b(Let)32 b FO(q)g FP(=)d(1.)48 b(In)31 b(this)h(case,)f FO(t)1290 1481 y FL(1)1357 1469 y FP(=)e FO(t)1481 1481 y FL(2)1518 1469 y FP(,)j(and)f(b)r(esides)h(the)g(one-dimen-)118 1568 y(sional)20 b(family)-7 b(,)23 b(there)e(exists)g(a)g(unique)g (in\014nite-dimensional)g(b)r(ounded)h(rep-)118 1668 y(resen)n(tation)36 b(that)h(corresp)r(onds)e(to)i(the)h(sequence)e(of) h(the)h(pre-images)d(of)118 1767 y(zero,)27 b FO(\025)365 1779 y FM(k)429 1767 y FP(=)c FO(F)582 1737 y FN(\000)p FM(k)674 1767 y FP(\(0\))g(=)g FO(\026k)s(=)p FP(\(1)18 b(+)g FO(\026k)s FP(\),)28 b FO(k)d FQ(\025)e FP(1.)37 b(The)27 b(op)r(erator)f FO(z)31 b FP(is)862 1946 y FO(z)t(e)944 1958 y FM(k)1007 1946 y FP(=)1095 1873 y Fz(p)p 1178 1873 175 4 v 73 x FO(\025)1226 1958 y FM(k)q FN(\000)p FL(1)1366 1946 y FO(e)1405 1958 y FM(k)q FN(\000)p FL(1)1531 1946 y FO(;)824 2093 y(z)867 2059 y FN(\003)905 2093 y FO(e)944 2105 y FM(k)1007 2093 y FP(=)1095 2016 y Fz(p)p 1178 2016 90 4 v 77 x FO(\025)1226 2105 y FM(k)1281 2093 y FO(e)1320 2105 y FM(k)q FL(+1)1445 2093 y FO(;)180 b(k)25 b FQ(\025)e FP(1)p FO(:)243 2262 y FP(2.)75 b(Let)41 b(1)27 b FQ(\000)f FO(\026)45 b(<)f(q)k(<)d FP(1.)75 b(No)n(w)40 b(there)h(are)e(t)n(w)n(o)h(one-dimensional)118 2362 y(families,)i(and)d(the)g(represen)n(tation)f(corresp)r(onding)f (to)i(the)g(sequence)g(of)118 2462 y(pre-images)23 b(of)i(zero,)f FO(\025)867 2474 y FM(k)932 2462 y FP(=)e FO(F)1084 2432 y FN(\000)p FM(k)1177 2462 y FP(\(0\))h(=)g(\(1)13 b FQ(\000)g FO(q)1599 2432 y FN(\000)p FM(k)1703 2462 y FP(+)g FO(\026q)1871 2432 y FN(\000)p FL(\()p FM(k)q FL(+1\))p FM(=)p FL(2)2167 2462 y FP([)p FO(k)s FP(])2259 2439 y FN(p)p 2313 2439 33 3 v 2313 2475 a FM(q)2350 2462 y FP(\))p FO(=)p FP(\(1)g(+)118 2576 y FO(\026q)208 2546 y FN(\000)p FL(\()p FM(k)q FL(+1\))p FM(=)p FL(2)504 2576 y FP([)p FQ(\000)p FO(k)s FP(])661 2553 y FN(p)p 715 2553 V 715 2589 a FM(q)752 2576 y FP(\),)22 b FO(k)k FQ(\025)c FP(1)e(\(w)n(e)g(use)g(the)h(notation)f([)p FO(n)p FP(])1887 2588 y FM(q)1946 2576 y FP(=)j(\()p FO(q)2106 2546 y FM(n)2155 2576 y FQ(\000)t FO(q)2264 2546 y FN(\000)p FM(n)2360 2576 y FP(\))p FO(=)p FP(\()p FO(q)7 b FQ(\000)118 2684 y FO(q)158 2654 y FN(\000)p FL(1)247 2684 y FP(\)\).)38 b(The)28 b(represen)n(tation)e(acts)h(b)n (y)h(the)g(same)f(form)n(ula)g(with)h(the)g FO(\025)2409 2696 y FM(k)2478 2684 y FP(in-)118 2784 y(tro)r(duced)d(in)h(this)f(w)n (a)n(y)-7 b(.)35 b(But)26 b(b)r(esides)f(these)g(represen)n(tations,)f (there)h(exists)118 2884 y(another)31 b(family)g(of)h(b)r(ounded)f (represen)n(tations)f(suc)n(h)h(that)h(the)g(k)n(ernels)e(of)118 2983 y FO(z)h FP(and)c FO(z)392 2953 y FN(\003)458 2983 y FP(are)f(zero)h(\(see)g(b)r(elo)n(w\).)243 3083 y(3.)38 b(F)-7 b(or)27 b FO(q)g FP(=)c(1)c FQ(\000)f FO(\026)p FP(,)28 b(zero)f(is)h(a)g(\014xed)g(p)r(oin)n(t,)g(whic)n(h)g(corresp)r (onds)f(to)g(the)118 3182 y(trivial)g(represen)n(tation.)35 b(In)28 b(this)g(case,)f(there)g(exists)g(a)g(family)h(of)f(b)r(ounded) 118 3282 y(represen)n(tations)f(with)i(zero)e(k)n(ernels)h(of)g FO(z)k FP(and)d FO(z)1705 3252 y FN(\003)1770 3282 y FP(\(see)f(b)r(elo)n(w\).)243 3382 y(4.)68 b(F)-7 b(or)38 b(0)i FQ(\024)h FO(q)j(<)c FP(1)25 b FQ(\000)g FO(\026)p FP(,)41 b(the)e(v)-5 b(alue)38 b FO(t)1590 3394 y FL(1)1666 3382 y FP(is)g(negativ)n(e,)i(and)e(there)g(is)118 3481 y(again)19 b(only)g(one)g(family)h(of)g(one-dimensional)e(represen)n (tations)g(corresp)r(ond-)118 3581 y(ing)33 b(to)f(the)h(\014xed)g(p)r (oin)n(t)g FO(t)974 3593 y FL(2)1043 3581 y FP(=)e(1,)j(and)f(a)f (single)g(b)r(ounded)i(represen)n(tation)118 3681 y(corresp)r(onding)39 b(to)i(the)g(sequence)f(of)h(images)f(of)g(zero,)j FO(\025)2057 3693 y FM(k)2144 3681 y FP(=)h FO(F)2318 3650 y FM(k)2359 3681 y FP(\(0\))h(=)118 3780 y(\(1)18 b FQ(\000)g FO(q)333 3750 y FM(k)393 3780 y FQ(\000)g FO(\026q)566 3750 y FL(\()p FM(k)q FN(\000)p FL(1\))p FM(=)p FL(2)811 3780 y FP([)p FO(k)s FP(])903 3757 y FN(p)p 957 3757 V 957 3794 a FM(q)994 3780 y FP(\))p FO(=)p FP(\(1)g FQ(\000)g FO(\026q)1333 3750 y FL(\()p FM(k)q FN(\000)p FL(1\))p FM(=)p FL(2)1578 3780 y FP([)p FO(k)s FP(])1670 3757 y FN(p)p 1724 3757 V 1724 3794 a FM(q)1761 3780 y FP(\),)28 b FO(k)e FQ(\025)c FP(0,)819 3974 y FO(z)t(e)901 3986 y FM(k)964 3974 y FP(=)1052 3897 y Fz(p)p 1135 3897 90 4 v 77 x FO(\025)1183 3986 y FM(k)1238 3974 y FO(e)1277 3986 y FM(k)q FL(+1)1402 3974 y FO(;)781 4117 y(z)824 4083 y FN(\003)862 4117 y FO(e)901 4129 y FM(k)964 4117 y FP(=)1052 4044 y Fz(p)p 1135 4044 175 4 v 73 x FO(\025)1183 4129 y FM(k)q FN(\000)p FL(1)1323 4117 y FO(e)1362 4129 y FM(k)q FN(\000)p FL(1)1488 4117 y FO(;)179 b(k)26 b FQ(\025)d FP(1)p FO(:)p eop %%Page: 104 108 104 107 bop 118 100 a FP(104)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FR(4.)72 b FP(No)n(w,)42 b(it)e(remains)f(to)h (consider)e(the)i(case)f(of)h(the)g(unitary)f(op)r(erator)118 432 y FO(U)9 b FP(.)70 b(W)-7 b(e)39 b(distinguish)g(b)r(et)n(w)n(een)g (t)n(w)n(o)f(cases:)58 b(represen)n(tations)37 b(related)h(to)118 532 y(a)e(single)f(orbit,)i(and)f(represen)n(tations)e(related)h(to)h (a)f(non-trivial)g(ergo)r(dic)118 632 y(quasi-in)n(v)-5 b(arian)n(t)26 b(measure.)243 731 y(If)21 b(there)f(exists)g(a)g (measurable)f(section)h(of)h(the)g(dynamical)f(system,)h(then)118 831 y(an)n(y)33 b(ergo)r(dic)f(measure)g(is)h(concen)n(trated)f(on)h(a) g(single)f(orbit;)k(in)d(this)h(case)118 930 y(w)n(e)27 b(will)h(pro)n(vide)f(a)g(complete)g(description)g(of)h(the)g(represen) n(tations.)118 1091 y FR(Theorem)42 b(21.)k FC(L)l(et)38 b(the)h(op)l(er)l(ator)g FO(X)45 b FC(b)l(e)39 b(invertible,)j(and)d (the)g(dynami-)118 1191 y(c)l(al)f(system)g(p)l(ossess)g(a)g(me)l(asur) l(able)g(se)l(ction.)62 b(A)n(ny)38 b(in\014nite-dimensional)118 1290 y(irr)l(e)l(ducible)31 b(r)l(epr)l(esentation)f(of)48 b FP(\(2.1\))30 b FC(has)g(the)g(form)815 1479 y FO(X)7 b(e)930 1491 y FM(k)993 1479 y FP(=)1081 1402 y Fz(p)p 1164 1402 90 4 v 77 x FO(\025)1212 1491 y FM(k)1267 1479 y FO(e)1306 1491 y FM(k)q FN(\000)p FL(1)1431 1479 y FO(;)184 b(k)26 b FQ(2)d FJ(Z)p FO(;)493 b FP(\(2.17\))118 1655 y FC(wher)l(e)43 b FO(\025)413 1667 y FM(k)454 1655 y FC(,)j FO(k)j FQ(2)e FJ(Z)o FC(,)40 b(is)j(any)g(se)l(quenc)l(e)f(of) h(p)l(ositive)h(numb)l(ers)d(such)i(that)118 1755 y FO(F)12 b FP(\()p FO(\025)263 1767 y FM(k)305 1755 y FP(\))23 b(=)g FO(\025)496 1767 y FM(k)q FL(+1)621 1755 y FC(,)29 b FO(k)d FQ(2)d FJ(Z)p FC(.)32 b(Two)d(such)g(r)l(epr)l(esentations)g (ar)l(e)g(unitarily)g(e)l(quiv-)118 1854 y(alent)h(if)h(and)f(only)g (if)h(the)f(c)l(orr)l(esp)l(onding)h(se)l(quenc)l(es)e(c)l(oincide.)118 2015 y(Pr)l(o)l(of.)43 b FP(First)28 b(let)g(the)g(mapping)g FO(F)12 b FP(\()p FQ(\001)p FP(\))29 b(b)r(e)f(one-to-one)e(on)i(the)g (sp)r(ectrum)g(of)118 2115 y FO(C)183 2085 y FL(2)221 2115 y FP(.)35 b(In)21 b(this)h(case,)g(an)n(y)f(ergo)r(dic)f(quasi-in) n(v)-5 b(arian)n(t)19 b(measure)i(is)g(concen)n(trated)118 2214 y(on)34 b(a)f(single)g(orbit)g(of)h(the)g(dynamical)f(system,)i (and)f(is)f(equiv)-5 b(alen)n(t)34 b(to)f(an)118 2314 y(atomic)39 b(measure)f(concen)n(trated)g(at)h(p)r(oin)n(ts)g(of)g(the) h(orbit.)71 b(Cho)r(ose)39 b(an)n(y)118 2414 y(p)r(oin)n(t)25 b FO(\025)h FP(of)f(the)h(orbit,)f(and)g(tak)n(e)g(a)g(unit)g(eigen)n (v)n(ector)f FO(e)1908 2426 y FL(0)1970 2414 y FP(corresp)r(onding)f (to)118 2513 y FO(\025)p FP(.)53 b(W)-7 b(rite)34 b FO(e)518 2525 y FM(k)590 2513 y FP(=)e FO(U)753 2483 y FM(k)793 2513 y FO(e)832 2525 y FL(0)869 2513 y FP(,)i FO(k)h FQ(2)e FJ(Z)o FP(.)47 b(W)-7 b(e)33 b(claim)g(that)g(the)g(space)g (spanned)f(b)n(y)118 2613 y(these)c(v)n(ectors)e(is)h(in)n(v)-5 b(arian)n(t)27 b(under)g FO(X)34 b FP(and)28 b FO(X)1621 2583 y FN(\003)1658 2613 y FP(.)37 b(F)-7 b(or)27 b FO(k)f(<)d FP(0,)k(w)n(e)g(ha)n(v)n(e)343 2798 y FO(C)408 2764 y FL(2)445 2798 y FO(e)484 2810 y FM(k)548 2798 y FP(=)c FO(C)701 2764 y FL(2)738 2798 y FO(U)804 2764 y FM(k)845 2798 y FO(e)884 2810 y FL(0)944 2798 y FP(=)f FO(C)1096 2764 y FL(2)1134 2798 y FO(U)1200 2764 y FN(\003)1238 2757 y(j)p FM(k)q FN(j)1318 2798 y FO(e)1357 2810 y FL(0)548 2945 y FP(=)h FO(U)702 2911 y FN(\003)740 2905 y(j)p FM(k)q FN(j)820 2945 y FO(F)885 2911 y FN(j)p FM(k)q FN(j)965 2945 y FP(\()p FO(C)1062 2911 y FL(2)1100 2945 y FP(\))14 b FO(e)1185 2957 y FL(0)1245 2945 y FP(=)23 b FO(U)1399 2911 y FN(\003)1436 2905 y(j)p FM(k)q FN(j)1517 2945 y FO(F)1582 2911 y FN(j)p FM(k)q FN(j)1662 2945 y FP(\()p FO(\025)p FP(\))14 b FO(e)1827 2957 y FL(0)1888 2945 y FP(=)22 b FO(F)2040 2911 y FN(j)p FM(k)q FN(j)2121 2945 y FP(\()p FO(\025)p FP(\))14 b FO(e)2286 2957 y FM(k)2327 2945 y FO(:)118 3121 y FP(Also,)43 b(since)d FO(F)12 b FP(\()p FQ(\001)p FP(\))41 b(is)f(one-to-one,)i(w)n(e)e(ha)n (v)n(e)f(in)h(this)h(case)e(that)i FO(C)2363 3091 y FL(2)2400 3121 y FO(U)53 b FP(=)118 3221 y FO(U)9 b(F)249 3191 y FN(\000)p FL(1)338 3221 y FP(\()p FO(C)435 3191 y FL(2)473 3221 y FP(\),)28 b(whic)n(h)f(implies)h(for)f(eac)n(h)g FO(k)f(>)c FP(0:)453 3397 y FO(C)518 3363 y FL(2)555 3397 y FO(e)594 3409 y FM(k)658 3397 y FP(=)h FO(C)811 3363 y FL(2)848 3397 y FO(U)9 b(e)953 3409 y FM(k)q FN(\000)p FL(1)1101 3397 y FP(=)23 b FO(:)14 b(:)g(:)658 3535 y FP(=)23 b FO(C)811 3500 y FL(2)848 3535 y FO(U)914 3500 y FM(k)955 3535 y FO(e)994 3547 y FL(0)1054 3535 y FP(=)f FO(U)1207 3500 y FM(k)1248 3535 y FO(F)1313 3500 y FN(\000)p FM(k)1406 3535 y FP(\()p FO(C)1503 3500 y FL(2)1540 3535 y FP(\))14 b FO(e)1625 3547 y FL(0)1685 3535 y FP(=)23 b FO(U)1839 3500 y FM(k)1880 3535 y FO(F)1945 3500 y FN(\000)p FM(k)2037 3535 y FP(\()p FO(\025)p FP(\))14 b FO(e)2202 3547 y FL(0)658 3672 y FP(=)23 b FO(F)811 3638 y FN(\000)p FM(k)903 3672 y FP(\()p FO(\025)p FP(\))14 b FO(e)1068 3684 y FM(k)1110 3672 y FO(:)118 3848 y FP(Therefore,)41 b(all)d FO(e)697 3860 y FM(k)776 3848 y FP(are)g(eigen)n(v)-5 b(alues)38 b(of)g FO(C)1537 3818 y FL(2)1575 3848 y FP(,)k(and)c (hence,)k(of)d FO(C)6 b FP(.)71 b(In)38 b(the)118 3948 y(c)n(hosen)27 b(basis,)g(the)h(op)r(erator)e FO(X)34 b FP(acts)27 b(as)g(stated)g(in)h(the)g(theorem.)243 4048 y(In)j(the)g(general)e(case,)i(consider)e(a)i(comm)n(uting)f (family)h(of)f(self-adjoin)n(t)118 4147 y(op)r(erators,)40 b FO(C)582 4159 y FM(k)664 4147 y FP(=)i FO(U)837 4117 y FM(k)877 4147 y FO(C)942 4117 y FL(2)980 4147 y FO(U)1046 4117 y FN(\000)p FM(k)1138 4147 y FP(.)71 b(The)38 b(relations)g(are)g FO(C)1971 4159 y FM(k)2012 4147 y FO(U)2078 4117 y FN(\003)2157 4147 y FP(=)k FO(U)2330 4117 y FN(\003)2367 4147 y FO(C)2426 4159 y FM(k)q FL(+1)2552 4147 y FP(.)p eop %%Page: 105 109 105 108 bop 118 100 a FK(2.1.)36 b(One-dimensional)27 b(dynamical)f(systems)854 b FP(105)118 333 y(No)n(w)36 b(w)n(e)f(ha)n(v)n(e)g(a)g(one-to-one)g(action)g(of)h FJ(Z)29 b FP(on)36 b(the)g(in\014nite-dimensional)118 432 y(space)31 b FJ(R)398 402 y Fv(Z)446 432 y FP(.)48 b(Using)31 b(the)h(same)f(argumen)n(ts)f(as)g(ab)r(o)n(v)n(e,)h(one)g (concludes)g(that)118 532 y(the)40 b(join)n(t)g(sp)r(ectrum)f(of)h(the) g(comm)n(utativ)n(e)f(family)g(is)h(concen)n(trated)e(on)118 632 y(a)32 b(single)h(orbit,)g(the)g(represen)n(tation)e(space)h(is)h (generated)e(b)n(y)i FO(\016)s FP(-functions)118 731 y(concen)n(trated)20 b(at)i(p)r(oin)n(ts)f(of)h(the)g(orbit,)g FO(U)30 b FP(acts)21 b(as)g(a)g(shift,)j(and)d(the)h(form)n(ula)118 831 y(follo)n(ws.)p 2514 831 4 57 v 2518 778 50 4 v 2518 831 V 2567 831 4 57 v 118 996 a FC(R)l(emark)39 b FP(26)p FC(.)i FP(Notice)28 b(that)g(w)n(e)g(do)f(not)h(assume)g(here)f(that)h (the)h(dynamical)118 1096 y(system)20 b(is)f(one-to-one.)33 b(Lo)r(oking)19 b(closely)g(at)g(the)i(sp)r(ectrum)f(of)26 b FO(C)2209 1066 y FL(2)2246 1096 y FP(,)c(one)e(sees)118 1196 y(that)26 b(it)g(can)f(just)g(b)r(e)h(a)f(c)n(hain)g(of)g(p)r(oin) n(ts,)h(or)f(a)g(c)n(hain)f(\\glued")h(to)g(a)g(cycle)g(or)118 1295 y(stationary)h(p)r(oin)n(t.)36 b(In)28 b(the)f(case)f(of)h(the)h (mapping)e(on)h(the)h(sp)r(ectrum)f(b)r(eing)118 1395 y(one-to-one,)f(only)h(c)n(hains)g(ma)n(y)g(arise.)118 1527 y FC(Example)33 b FP(15)p FC(.)k FP(\(Mapping)24 b(of)f(degree)f(t)n(w)n(o.)35 b(Con)n(tin)n(ued:)f FO(q)26 b(<)d(q)2134 1497 y FN(\003)2172 1527 y FP(\).)36 b(Since)23 b(for)118 1626 y FO(q)j(<)d(q)309 1596 y FN(\003)347 1626 y FP(,)h(relation)e(\(2.3\))h(p)r(ossesses)f(only)h(a)f(\014nite)i (n)n(um)n(b)r(er)f(of)g(cycles,)g(the)h(set)118 1726 y(of)30 b(p)r(erio)r(dic)g(p)r(oin)n(ts)h(is)f(closed;)h(therefore,)f (the)g(corresp)r(onding)f(dynamical)118 1826 y(system)22 b(is)g(\\simple",)g(and)g(therefore,)g(all)g(in\014nite-dimensional)g (irreducible)118 1925 y(represen)n(tations)c(with)j(unitary)f FO(U)29 b FP(are)19 b(describ)r(ed)h(b)n(y)g(sequences)f(of)h(p)r (ositiv)n(e)118 2025 y(n)n(um)n(b)r(ers)31 b FO(\025)505 2037 y FM(k)546 2025 y FP(,)i FO(k)f FQ(2)e FJ(Z)p FP(,)c(suc)n(h)31 b(that)h FO(F)12 b FP(\()p FO(\025)1393 2037 y FM(k)1435 2025 y FP(\))30 b(=)f FO(\025)1639 2037 y FM(k)q FL(+1)1796 2025 y FP(for)i(all)g FO(k)j FP(according)c(to)118 2124 y(\(2.17\))o(.)118 2256 y FC(Example)43 b FP(16)p FC(.)i FP(\(Tw)n(o-parameter)32 b(unit)j(quan)n(tum)g(disk.)57 b(Non-degenerate)118 2356 y(case\).)j(F)-7 b(or)35 b(1)23 b FQ(\000)g FO(\026)37 b FQ(\024)e FO(q)40 b(<)c FP(1,)h(the)e (dynamical)g(system)h(\(2.16\))e(p)r(ossesses)118 2456 y(a)e(family)g(of)g(b)r(ounded)h(p)r(ositiv)n(e)e(orbits.)50 b(T)-7 b(ak)n(e)32 b FO(\025)1740 2468 y FL(0)1808 2456 y FQ(2)f FP(\()p FO(t)1956 2468 y FL(1)1993 2456 y FO(;)14 b FP(1\))32 b(as)g(an)g(initial)118 2555 y(p)r(oin)n(t;)c(then)g FO(H)i FP(=)23 b FO(l)759 2567 y FL(2)796 2555 y FP(\()p FJ(Z)o FP(\),)f(and)902 2745 y FO(z)t(e)984 2757 y FM(k)1047 2745 y FP(=)1135 2671 y Fz(p)p 1218 2671 175 4 v 74 x FO(\025)1266 2757 y FM(k)q FN(\000)p FL(1)1406 2745 y FO(e)1445 2757 y FM(k)q FN(\000)p FL(1)1570 2745 y FO(;)864 2891 y(z)907 2857 y FN(\003)944 2891 y FO(e)983 2903 y FM(k)1047 2891 y FP(=)1135 2815 y Fz(p)p 1218 2815 90 4 v 76 x FO(\025)1266 2903 y FM(k)1321 2891 y FO(e)1360 2903 y FM(k)1400 2891 y FO(;)180 b(k)26 b FQ(2)e FJ(Z)o FO(:)118 3072 y FP(where)295 3299 y FO(\025)343 3311 y FM(k)407 3299 y FP(=)505 3234 y FO(\025)553 3246 y FL(0)591 3234 y FP(\()p FO(q)663 3204 y FM(k)722 3234 y FP(+)18 b FO(\026q)895 3204 y FL(\()p FM(k)q FN(\000)p FL(1\))p FM(=)p FL(2)1140 3234 y FP([)p FO(k)s FP(])1232 3211 y FN(p)p 1287 3211 33 3 v 36 x FM(q)1323 3234 y FP(\))h(+)f(1)g FQ(\000)g FP(\()p FO(q)1672 3204 y FM(k)1732 3234 y FP(+)g FO(\026q)1905 3204 y FL(\()p FM(k)q FN(\000)p FL(1\))p FM(=)p FL(2)2150 3234 y FP([)p FO(k)s FP(])2242 3211 y FN(p)p 2296 3211 V 2296 3247 a FM(q)2333 3234 y FP(\))p 505 3280 1861 4 v 752 3357 a FO(\025)800 3369 y FL(0)837 3357 y FO(\026q)927 3333 y FL(\()p FM(k)q FN(\000)p FL(1\))p FM(=)p FL(2)1172 3357 y FP([)p FO(k)s FP(])1264 3334 y FN(p)p 1319 3334 33 3 v 37 x FM(q)1374 3357 y FP(+)g(1)g FQ(\000)g FO(\026q)1690 3333 y FL(\()p FM(k)q FN(\000)p FL(1\))p FM(=)p FL(2)1935 3357 y FP([)p FO(k)s FP(])2027 3334 y FN(p)p 2081 3334 V 2081 3371 a FM(q)2375 3299 y FO(:)118 3541 y FP(Tw)n(o)31 b(suc)n(h)g(represen)n (tations)e(corresp)r(onding)g(to)j FO(\025)1770 3553 y FL(0)1839 3541 y FP(and)f FO(\025)2052 3511 y FN(0)2052 3561 y FL(0)2121 3541 y FP(are)f(unitarily)118 3640 y(equiv)-5 b(alen)n(t)25 b(if)g(and)g(only)f(if)i FO(\025)1043 3652 y FL(0)1105 3640 y FP(and)f FO(\025)1312 3610 y FN(0)1312 3661 y FL(0)1375 3640 y FP(are)e(on)i(the)g(same)f(orbit.)36 b(Therefore,)118 3740 y(for)j(the)g(measurable)f(section)h(one)g(can)g (tak)n(e)f(an)n(y)h(in)n(terv)-5 b(al)38 b(of)i(the)f(form)118 3781 y Fz(\002)153 3848 y FO(\025)201 3860 y FL(0)238 3848 y FO(;)285 3808 y FL(\()p FM(q)r FL(+)p FM(\026)p FL(\))p FM(\025)499 3816 y Fy(0)533 3808 y FL(+1)p FN(\000)p FM(q)r FN(\000)p FM(\026)p 285 3829 509 4 v 395 3877 a(\026\025)474 3885 y Fy(0)507 3877 y FL(+1)p FN(\000)p FM(\026)803 3781 y Fz(\001)841 3848 y FP(.)243 3948 y(Also,)j(there)e (exists)f(a)h(family)g(of)f(un)n(b)r(ounded)i(p)r(ositiv)n(e)e(orbits)g (lying)118 4048 y(to)d(the)g(righ)n(t)e(of)i(1.)60 b(The)36 b(set)g(of)f(suc)n(h)g(orbits)g(also)g(p)r(ossesses)f(a)h(measur-)118 4147 y(able)i(section,)i(and)e(one)g(can)f(use)h(the)h(form)n(ula)e(ab) r(o)n(v)n(e)g(to)h(construct)f(the)p eop %%Page: 106 110 106 109 bop 118 100 a FP(106)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(corresp)r(onding)35 b(class)g(of)h (irreducible)g(represen)n(tations.)61 b(Ho)n(w)n(ev)n(er,)37 b(these)118 432 y(represen)n(tations)26 b(are)g(un)n(b)r(ounded.)243 549 y(If)g(the)h(dynamical)e(system)h(do)r(es)g(not)h(ha)n(v)n(e)e(a)g (measurable)g(section,)h(the)118 649 y(represen)n(tations)g(listed)i (in)g(the)g(theorem)f(ab)r(o)n(v)n(e)f(do)h(not)h(form,)f(in)h (general,)118 749 y(the)g(complete)g(list)f(of)h(irreducible)f (represen)n(tations.)118 866 y FC(Example)k FP(17)p FC(.)36 b FP(\(Mapping)21 b(of)h(degree)e(t)n(w)n(o.)34 b FO(q)27 b FQ(\025)22 b FO(q)1693 836 y FN(\003)1731 866 y FP(\).)36 b(F)-7 b(or)21 b(an)n(y)f FO(q)26 b FQ(\025)d FO(q)2306 836 y FN(\003)2344 866 y FP(,)g(there)118 966 y(exists)31 b(an)f(in)n(v)-5 b(arian)n(t)30 b(set)h FO(K)37 b FP(homeomorphic)29 b(to)i(a)g(Can)n(tor)e(set)i(where)f(the)118 1065 y(mapping)25 b(is)h(one-to-one)e(and)h(whic)n(h)h(carries)e(a)h(non-trivial)f(ergo)r (dic)h(quasi-)118 1165 y(in)n(v)-5 b(arian)n(t)26 b(probabilit)n(y)f (measure)h FO(\026)p FP(.)37 b(Consider)26 b(the)h(space)f FO(L)2088 1177 y FL(2)2124 1165 y FP(\()p FO(K)q(;)14 b(d\026)p FP(\),)28 b(and)118 1264 y(de\014ne)g FO(X)34 b FP(b)n(y)439 1408 y(\()p FO(X)7 b(f)i FP(\)\()p FO(\025)p FP(\))23 b(=)g FO(\025)900 1333 y Fz(p)p 983 1333 584 4 v 75 x FO(d\026)p FP(\()p FO(F)12 b FP(\()p FO(\025)p FP(\)\))p FO(=d\026)p FP(\()p FO(\025)p FP(\))k FO(f)9 b FP(\()p FO(F)j FP(\()p FO(\025)p FP(\)\))p FO(;)400 1553 y FP(\()p FO(X)508 1519 y FN(\003)546 1553 y FO(f)d FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(F)917 1519 y FN(\000)p FL(1)1006 1553 y FP(\()p FO(\025)p FP(\))1118 1478 y Fz(p)p 1202 1478 673 4 v 1202 1553 a FO(d\026)p FP(\()p FO(F)1392 1529 y FN(\000)p FL(1)1481 1553 y FP(\()p FO(\025)p FP(\)\))p FO(=d\026)p FP(\()p FO(\025)p FP(\))16 b FO(f)9 b FP(\()p FO(F)2035 1519 y FN(\000)p FL(1)2124 1553 y FP(\()p FO(\025)p FP(\)\))p FO(;)118 1696 y FP(where)40 b FO(F)12 b FP(\()p FO(\025)p FP(\))44 b(=)g(\()p FO(\025)27 b FQ(\000)f FO(q)s FP(\))971 1666 y FL(2)1009 1696 y FP(.)74 b(Then)40 b(the)h(op)r(erators)d(satisfy)h(the)i(relation)118 1796 y FO(X)7 b(X)270 1766 y FN(\003)352 1796 y FP(=)44 b(\()p FO(X)569 1766 y FN(\003)607 1796 y FO(X)34 b FQ(\000)26 b FO(q)s(I)7 b FP(\))916 1766 y FL(2)954 1796 y FP(,)44 b(the)d(represen)n(tation)e(is)i(irreducible,)i(but)f(an)n(y)118 1896 y(orbit)27 b(has)g(sp)r(ectral)g(measure)g(of)g FO(X)1275 1865 y FN(\003)1313 1896 y FO(X)34 b FP(equal)27 b(to)g(zero.)118 2013 y FC(R)l(emark)40 b FP(27)p FC(.)i FP(Notice)30 b(that)f(the)h(description)f(of)g(non-trivial)f(ergo)r (dic)h(mea-)118 2112 y(sures)e(can)h(b)r(e)h(a)e(v)n(ery)g(complicated) h(task;)g(also,)f(there)h(can)f(b)r(e)i(lots)f(of)g(uni-)118 2212 y(tarily)k(inequiv)-5 b(alen)n(t)33 b(irreducible)g(represen)n (tations)e(corresp)r(onding)g(to)h(the)118 2312 y(same)27 b(non-trivial)g(ergo)r(dic)f(measure.)118 2546 y FH(2.2)112 b(Some)41 b(classes)g(of)g FG(\003)p FH(-algebras)h(with)d(3)i(and)h(4) f(gen-)373 2663 y(erators)118 2844 y FP(The)34 b(metho)r(d)g(dev)n (elop)r(ed)f(in)h(Section)g(2.1)f(can)g(b)r(e)h(carried)e(o)n(v)n(er)g (to)h(other)118 2944 y(classes)k(of)i(op)r(erator)d(relations.)69 b(The)38 b(idea)g(b)r(ehind)i(this)e(is)h(to)f(consider)118 3044 y(some)32 b(comm)n(utativ)n(e)g(family)g(of)g(self-adjoin)n(t)g (op)r(erators,)g(and)g(to)g(see)h(ho)n(w)118 3143 y(other)27 b(op)r(erators)f(act)h(on)g(their)h(\(generalized\))f(eigen)n(v)n (ectors.)118 3352 y FR(2.2.1)94 b(Represen)m(tations)44 b(of)g(graded)h FO(so)p FP(\(3\))h FR(and)f(four-tuples)f(of)410 3452 y(pro)5 b(jections)32 b(satisfying)f(a)h(linear)f(relation)118 3605 y FP(In)24 b(this)g(section,)g(w)n(e)f(study)h(irreducible)e (represen)n(tations)g(of)i(a)f(graded)f(ana-)118 3705 y(logue)37 b(of)g(the)h(Lie)f(algebra)f FO(so)p FP(\(3\),)k(and)d(sho)n (w)g(ho)n(w)f(they)i(are)e(related)h(to)118 3804 y(four-tuples)20 b(of)g(pro)5 b(jections)19 b(satisfying)g(a)h(linear)g(relation)f(of)h (a)g(sp)r(ecial)g(t)n(yp)r(e.)243 3904 y(Consider)27 b(a)h(triple)g(of)h(b)r(ounded)g(self-adjoin)n(t)f(op)r(erators)e FO(A)p FP(,)j FO(B)t FP(,)g FO(C)35 b FP(sat-)118 4004 y(isfying)27 b(the)h(follo)n(wing)f(relations)575 4147 y FQ(f)p FO(A;)14 b(B)t FQ(g)22 b FP(=)h FO(C)q(;)97 b FQ(f)p FO(B)t(;)14 b(C)6 b FQ(g)23 b FP(=)f FO(A;)98 b FQ(f)p FO(C)q(;)14 b(A)p FQ(g)22 b FP(=)h FO(B)t(;)259 b FP(\(2.18\))p eop %%Page: 107 111 107 110 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(107)118 333 y(where)26 b FQ(f)p FO(A;)14 b(B)t FQ(g)23 b FP(=)f FO(AB)g FP(+)16 b FO(B)t(A)28 b FP(denotes)e(the)h(an)n(ticomm)n(utator)f(of)h(the)g(op)r(er-)118 432 y(ators)e FO(A)i FP(and)g FO(B)t FP(.)37 b(As)26 b(w)n(as)g(men)n(tioned)h(ab)r(o)n(v)n(e,)e(suc)n(h)h(triples)h(of)f (self-adjoin)n(t)118 532 y(op)r(erators)31 b(can)h(b)r(e)h(considered)e (as)h(represen)n(tations)f(of)h(the)h(graded)f FO(so)p FP(\(3\))118 632 y(algebra.)g(W)-7 b(e)19 b(describ)r(e)g(all)f(suc)n (h)h(irreducible)f(families)g(up)i(to)e(unitary)g(equiv-)118 731 y(alence.)243 833 y(On)34 b(the)h(other)f(hand,)i(consider)d(a)h (four-tuple)h(of)f(orthogonal)e(pro)5 b(jec-)118 933 y(tions,)32 b FO(P)402 945 y FL(1)440 933 y FP(,)g FO(P)548 945 y FL(2)586 933 y FP(,)g FO(P)694 945 y FL(3)732 933 y FP(,)g FO(P)840 945 y FL(2)909 933 y FP(that)g(are)e(connected)h(b)n (y)g(a)g(linear)g(relation)f(of)h(the)118 1033 y(form)859 1220 y FO(\013)14 b FP(\()p FO(P)1011 1232 y FL(1)1067 1220 y FP(+)k FO(P)1203 1232 y FL(2)1259 1220 y FP(+)g FO(P)1395 1232 y FL(3)1451 1220 y FP(+)g FO(P)1587 1232 y FL(4)1625 1220 y FP(\))23 b(=)g FO(I)7 b(:)118 1408 y FP(It)31 b(turns)g(out)g(that)g(the)g(unitary)f(description)g(of)h (suc)n(h)f(collections)g(of)h(pro-)118 1508 y(jections)d(is)f(closely)g (related)g(to)g(represen)n(tations)f(of)34 b(\(2.18\))o(.)118 1681 y FR(Theorem)i(22.)43 b FC(A)n(ny)34 b(irr)l(e)l(ducible)h(family) h(of)e(self-adjoint)i(op)l(er)l(ators)f(sat-)118 1781 y(isfying)44 b FP(\(2.18\))33 b FC(is)i(\014nite-dimensional.)55 b(F)-6 b(or)35 b(any)g FO(l)e FQ(\025)f FP(1)i FC(ther)l(e)h(exist)f (four)118 1881 y(irr)l(e)l(ducible)d(r)l(epr)l(esentations)f(of)h (dimension)g FP(2)p FO(l)r FC(,)e(and)i(\014ve)e(irr)l(e)l(ducible)j(r) l(ep-)118 1980 y(r)l(esentations)e(of)g(dimension)h FP(2)p FO(l)20 b FQ(\000)e FP(1)p FC(,)29 b(which)j(act)e(as)g(fol)t(lows.)243 2083 y FP(1)p FC(.)38 b(F)-6 b(our)29 b(r)l(epr)l(esentations)h(with)h (any)f(\014nite)f(dimension)i FP(dim)14 b FO(H)30 b FP(=)23 b FO(n)p FP(:)311 2305 y FO(Ae)412 2317 y FM(k)476 2305 y FP(=)g FO(\013)617 2317 y FL(1)668 2305 y FP(\()p FQ(\000)p FP(1\))839 2270 y FM(k)q FL(+1)964 2213 y Fz(\020)1013 2305 y FO(k)f FQ(\000)1171 2249 y FP(1)p 1171 2286 42 4 v 1171 2362 a(2)1222 2213 y Fz(\021)1285 2305 y FO(e)1324 2317 y FM(k)1365 2305 y FO(;)184 b(k)25 b FP(=)e(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n;)310 2487 y(B)t(e)416 2499 y FL(1)476 2487 y FP(=)574 2431 y FO(\013)627 2443 y FL(1)664 2431 y FO(\013)717 2443 y FL(2)755 2431 y FO(n)p 574 2468 231 4 v 668 2544 a FP(2)828 2487 y FO(e)867 2499 y FL(1)923 2487 y FQ(\000)1016 2431 y FO(\013)1069 2443 y FL(1)p 1016 2468 91 4 v 1040 2544 a FP(2)1130 2405 y Fz(p)p 1213 2405 231 4 v 82 x FO(n)1263 2463 y FL(2)1318 2487 y FQ(\000)k FP(1)c FO(e)1496 2499 y FL(2)1533 2487 y FO(;)307 2704 y(B)t(e)413 2716 y FM(k)476 2704 y FP(=)574 2647 y FO(\013)627 2659 y FL(1)678 2647 y FP(\()p FQ(\000)p FP(1\))849 2617 y FM(k)q FN(\000)p FL(1)p 574 2685 401 4 v 753 2761 a FP(2)998 2629 y Fz(p)p 1081 2629 480 4 v 75 x FO(n)1131 2680 y FL(2)1187 2704 y FQ(\000)k FP(\()p FO(k)j FQ(\000)e FP(1\))1524 2680 y FL(2)1574 2704 y FO(e)1613 2716 y FM(k)q FN(\000)p FL(1)557 2920 y FP(+)650 2864 y FO(\013)703 2876 y FL(1)740 2864 y FP(\()p FQ(\000)p FP(1\))911 2834 y FM(k)p 650 2901 302 4 v 780 2977 a FP(2)975 2845 y Fz(p)p 1058 2845 337 4 v 75 x FO(n)1108 2896 y FL(2)1164 2920 y FQ(\000)f FP(\()p FO(k)s FP(\))1357 2896 y FL(2)1408 2920 y FO(e)1447 2932 y FM(k)q FL(+1)1572 2920 y FO(;)183 b(k)26 b FP(=)d(2)p FO(;)14 b(:)g(:)g(:)f(;)h(n)k FQ(\000)g FP(1)p FO(;)302 3134 y(B)t(e)408 3146 y FM(n)476 3134 y FP(=)574 3078 y FO(\013)627 3090 y FL(1)664 3078 y FP(\()p FQ(\000)p FP(1\))835 3048 y FM(n)p FN(\000)p FL(1)p 574 3115 392 4 v 749 3191 a FP(2)989 3065 y FQ(p)p 1058 3065 235 4 v 69 x FP(2)p FO(n)g FQ(\000)g FP(1)13 b FO(e)1345 3146 y FM(n)p FN(\000)p FL(1)1475 3134 y FO(;)312 3334 y(C)6 b(e)416 3346 y FL(1)476 3334 y FP(=)574 3278 y FO(\013)627 3290 y FL(2)664 3278 y FO(n)p 574 3315 141 4 v 623 3391 a FP(2)738 3334 y FO(e)777 3346 y FL(1)832 3334 y FP(+)925 3278 y(1)p 925 3315 42 4 v 925 3391 a(2)991 3253 y Fz(p)p 1074 3253 231 4 v 81 x FO(n)1124 3310 y FL(2)1179 3334 y FQ(\000)18 b FP(1)13 b FO(e)1356 3346 y FL(2)1393 3334 y FO(;)308 3534 y(C)6 b(e)412 3546 y FM(k)476 3534 y FP(=)574 3478 y(1)p 574 3515 42 4 v 574 3591 a(2)639 3459 y Fz(p)p 722 3459 480 4 v 75 x FO(n)772 3510 y FL(2)828 3534 y FQ(\000)18 b FP(\()p FO(k)j FQ(\000)d FP(1\))1164 3510 y FL(2)1215 3534 y FO(e)1254 3546 y FM(k)q FN(\000)p FL(1)557 3734 y FP(+)650 3678 y(1)p 650 3715 42 4 v 650 3791 a(2)715 3659 y Fz(p)p 798 3659 337 4 v 75 x FO(n)848 3710 y FL(2)903 3734 y FQ(\000)g FP(\()p FO(k)s FP(\))1096 3710 y FL(2)1148 3734 y FO(e)1187 3746 y FM(k)q FL(+1)1311 3734 y FO(;)184 b(k)26 b FP(=)d(2)p FO(;)14 b(:)g(:)g(:)f(;)h(n)k FQ(\000)g FP(1)p FO(;)304 3934 y(C)6 b(e)408 3946 y FM(n)476 3934 y FP(=)574 3877 y(1)p 574 3915 42 4 v 574 3991 a(2)639 3864 y FQ(p)p 708 3864 235 4 v 70 x FP(2)p FO(n)18 b FQ(\000)g FP(1)c FO(e)996 3946 y FM(n)p FN(\000)p FL(1)1125 3934 y FO(;)184 b FC(wher)l(e)30 b FO(\013)1619 3946 y FL(1)1657 3934 y FC(,)g FO(\013)1765 3946 y FL(2)1825 3934 y FP(=)23 b FQ(\006)p FP(1)o(;)243 4147 y(2)p FC(.)37 b(A)n(n)28 b(additional)j(r)l(epr)l(esentation)e(for)h(e)l(ach)f(o)l (dd)h(value)f(of)h(the)e(dimen-)p eop %%Page: 108 112 108 111 bop 118 100 a FP(108)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(sion,)k FP(dim)14 b FO(H)30 b FP(=)22 b FO(n)h FP(=)g(2)p FO(l)c FQ(\000)f FP(1:)329 558 y FO(Ae)430 570 y FM(k)494 558 y FP(=)k(\()p FQ(\000)p FP(1\))752 524 y FM(k)q FL(+1)901 502 y FO(n)c FP(+)g(1)g FQ(\000)g FP(2)p FO(k)p 901 539 382 4 v 1071 615 a FP(2)1306 558 y FO(e)1345 570 y FM(k)1386 558 y FO(;)183 b(k)26 b FP(=)d(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n;)328 758 y(B)t(e)434 770 y FL(1)494 758 y FP(=)591 702 y(1)p 591 739 42 4 v 591 815 a(2)657 688 y FQ(p)p 726 688 193 4 v 70 x FO(n)k FQ(\000)g FP(1)13 b FO(e)971 770 y FL(2)1008 758 y FO(;)324 975 y(B)t(e)430 987 y FM(k)494 975 y FP(=)591 918 y(\()p FQ(\000)p FP(1\))762 888 y FM(k)p 591 955 212 4 v 676 1032 a FP(2)827 900 y Fz(p)p 910 900 659 4 v 75 x FP(\()p FO(k)21 b FQ(\000)d FP(1\)\()p FO(n)h FQ(\000)f FO(k)j FP(+)d(1\))c FO(e)1621 987 y FM(k)q FN(\000)p FL(1)574 1191 y FP(+)667 1135 y(\()p FQ(\000)p FP(1\))838 1105 y FM(k)q FL(+1)p 667 1172 296 4 v 794 1248 a FP(2)986 1116 y Fz(p)p 1069 1116 308 4 v 75 x FO(k)s FP(\()p FO(n)19 b FQ(\000)f FO(k)s FP(\))c FO(e)1430 1203 y FM(k)q FL(+1)1554 1191 y FO(;)184 b(k)26 b FP(=)c(2)p FO(;)14 b(:)g(:)g(:)g(;)g(n)k FQ(\000)g FP(1)p FO(;)320 1391 y(B)t(e)426 1403 y FM(n)494 1391 y FP(=)k FQ(\000)656 1335 y FP(1)p 656 1372 42 4 v 656 1448 a(2)721 1322 y FQ(p)p 790 1322 193 4 v 69 x FO(n)d FQ(\000)f FP(1)13 b FO(e)1036 1403 y FM(n)p FN(\000)p FL(1)1166 1391 y FO(;)330 1591 y(C)6 b(e)434 1603 y FL(1)494 1591 y FP(=)591 1535 y(1)p 591 1572 42 4 v 591 1648 a(2)657 1522 y FQ(p)p 726 1522 193 4 v 69 x FO(n)18 b FQ(\000)g FP(1)13 b FO(e)971 1603 y FL(2)1008 1591 y FO(;)326 1791 y(C)6 b(e)430 1803 y FM(k)494 1791 y FP(=)591 1735 y(1)p 591 1772 42 4 v 591 1848 a(2)657 1716 y Fz(p)p 740 1716 659 4 v 75 x FP(\()p FO(k)21 b FQ(\000)d FP(1\)\()p FO(n)h FQ(\000)f FO(k)j FP(+)d(1\))c FO(e)1451 1803 y FM(k)q FN(\000)p FL(1)574 1991 y FP(+)667 1935 y(1)p 667 1972 42 4 v 667 2048 a(2)732 1916 y Fz(p)p 815 1916 308 4 v 75 x FO(k)s FP(\()p FO(n)19 b FQ(\000)f FO(k)s FP(\))c FO(e)1176 2003 y FM(k)q FL(+1)1300 1991 y FO(;)184 b(k)26 b FP(=)c(2)p FO(;)14 b(:)g(:)g(:)g(;)g(n)k FQ(\000)g FP(1)p FO(;)322 2191 y(C)6 b(e)426 2203 y FM(n)494 2191 y FP(=)591 2134 y(1)p 591 2171 42 4 v 591 2247 a(2)657 2121 y FQ(p)p 726 2121 193 4 v 70 x FO(n)18 b FQ(\000)g FP(1)13 b FO(e)971 2203 y FM(n)p FN(\000)p FL(1)1101 2191 y FO(:)118 2402 y FC(Pr)l(o)l(of.)43 b FP(The)38 b(pro)r(of)e(of)i(the)f(theorem)g(is)h(essen)n(tially)e(based)h(on)g (the)g(same)118 2502 y(ideas)h(as)g(the)h(pro)r(of)f(for)g(non-graded)f FO(so)p FP(\(3\).)71 b(In)n(tro)r(duce)38 b(the)h(op)r(erators)118 2602 y FO(E)179 2614 y FL(0)251 2602 y FP(=)34 b FO(A)p FP(,)i FO(E)532 2614 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FL(2)1449 3282 y(1)1510 3261 y FQ(\000)g FO(E)1654 3273 y FL(0)1714 3261 y FP(=)23 b FO(E)1868 3227 y FL(2)1863 3282 y(0)1924 3261 y FP(+)18 b FO(E)2073 3227 y FL(2)2068 3282 y(2)2128 3261 y FP(+)g FO(E)2272 3273 y FL(0)118 3448 y FP(comm)n(utes)35 b(with)g FO(A)p FP(,)i FO(B)t FP(,)h FO(C)6 b FP(.)59 b(This)35 b(implies,)i(in)e(particular,)h(that)f(an)n(y)g(ir-)118 3548 y(reducible)e(represen)n(tation)f(is)h(b)r(ounded,)i(since)e FO(A)1757 3517 y FL(2)1795 3548 y FP(,)i FO(B)1920 3517 y FL(2)1957 3548 y FP(,)g FO(C)2080 3517 y FL(2)2151 3548 y FP(are)d(p)r(ositiv)n(e)118 3647 y(op)r(erators,)26 b(and)h(their)h(sum)f(is)h(a)f(m)n(ultiple)h(of)g(the)g(iden)n(tit)n(y) -7 b(.)243 3749 y(One)28 b(can)h(easily)g(see)f(from)h(relations)f (\(2.19\))h(that)g FO(E)1960 3761 y FL(1)2027 3749 y FP(maps)g(an)g(eigen-)118 3848 y(v)n(ector)c FO(e)405 3860 y FM(\025)475 3848 y FP(of)h FO(E)629 3860 y FL(0)693 3848 y FP(to)h(an)f(eigen)n(v)n(ector)e FO(e)1377 3860 y FL(1)p FN(\000)p FM(\025)1505 3848 y FP(,)j(and)g FO(E)1777 3860 y FL(2)1841 3848 y FP(maps)f FO(e)2096 3860 y FM(\025)2165 3848 y FP(in)n(to)h FO(e)2372 3860 y FN(\000)p FL(1)p FN(\000)p FM(\025)2552 3848 y FP(.)118 3948 y(Then)g(w)n(e)g(ha)n(v)n (e)f(a)g(dynamical)g(system)h(on)g FJ(R)33 b FP(generated)25 b(b)n(y)i(t)n(w)n(o)f(\015ips)h(with)118 4048 y(resp)r(ect)35 b(to)f(the)i(p)r(oin)n(ts)e(1)p FO(=)p FP(2,)i(and)f FQ(\000)p FP(1)p FO(=)p FP(2.)56 b(F)-7 b(or)35 b(an)f(irreducible)g (represen-)118 4147 y(tation,)e(the)f(sp)r(ectral)f(measure)g(of)g FO(E)1336 4159 y FL(0)1405 4147 y FP(m)n(ust)g(b)r(e)i(ergo)r(dic)d (with)j(resp)r(ect)e(to)p eop %%Page: 109 113 109 112 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(109)118 333 y(the)36 b(action)e(of)h(this)h (dynamical)e(system,)j(since,)g(otherwise,)g(an)n(y)d(ergo)r(dic)118 432 y(comp)r(onen)n(t)28 b(generates)f(an)h(in)n(v)-5 b(arian)n(t)27 b(subspace.)39 b(The)28 b(dynamical)g(system)118 532 y(p)r(ossesses)22 b(a)g(measurable)f(section,)j(i.e.,)g(a)e(set)h (that)g(meets)g(ev)n(ery)f(orbit)g(only)118 632 y(once.)61 b(F)-7 b(or)36 b(suc)n(h)f(a)h(set)g(one)f(can)h(tak)n(e)f([)p FQ(\000)p FP(1)p FO(=)p FP(2)p FO(;)14 b FP(1)p FO(=)p FP(2].)59 b(Then)36 b(an)n(y)f(ergo)r(dic)118 731 y(measure)27 b(is)g(concen)n(trated)g(at)g(a)g(single)g(orbit)h(of)f(some)g(p)r(oin) n(t.)243 833 y(Th)n(us,)i(the)g(sp)r(ectral)g(measure)f(of)h FO(E)1418 845 y FL(0)1484 833 y FP(is)g(discrete,)g(and)g(w)n(e)f(can)h (c)n(ho)r(ose)118 932 y(a)e(basis)g(consisting)g(of)g(its)h(eigen)n(v)n (ectors.)35 b(Then)28 b(w)n(e)f(ha)n(v)n(e:)311 1118 y FO(E)372 1130 y FL(0)410 1118 y FO(e)449 1130 y FM(\025)515 1118 y FP(=)c FO(\025e)690 1130 y FM(\025)734 1118 y FO(;)96 b(E)914 1130 y FL(1)952 1118 y FO(e)991 1130 y FM(\025)1057 1118 y FP(=)23 b FO(a)1189 1130 y FL(1)1226 1118 y FP(\()p FO(\025)p FP(\))14 b FO(e)1391 1130 y FL(1)p FN(\000)p FM(\025)1520 1118 y FO(;)97 b(E)1701 1130 y FL(2)1738 1118 y FO(e)1777 1130 y FM(\025)1844 1118 y FP(=)22 b FO(a)1975 1130 y FL(2)2012 1118 y FP(\()p FO(\025)p FP(\))14 b FO(e)2177 1130 y FN(\000)p FL(1)p FN(\000)p FM(\025)2358 1118 y FO(;)118 1305 y FP(where)32 b FO(\025)g FP(are)f(tak)n(en)h(from)g(some)f(orbit.)50 b(It)33 b(remains)e(to)h(\014nd)g(a)g(condition)118 1404 y(on)i FO(a)284 1416 y FL(1)321 1404 y FP(,)h FO(a)423 1416 y FL(2)494 1404 y FP(so)e(that)h(the)g(relation)f FO(E)1314 1374 y FL(2)1309 1425 y(1)1374 1404 y FQ(\000)22 b FO(E)1527 1374 y FL(2)1522 1425 y(2)1598 1404 y FP(=)33 b(2)p FO(E)1799 1416 y FL(0)1870 1404 y FP(w)n(ould)g(hold,)i(and)f(to) 118 1504 y(c)n(hec)n(k)c(whether)i(the)f(these)g(conditions)g(can)g(b)r (e)h(satis\014ed,)f(and)h(the)f(repre-)118 1604 y(sen)n(tation)d(is)h (irreducible)f(\(the)i(ergo)r(dicit)n(y)d(is)i(a)f(necessary)f (condition,)i(but)118 1703 y(not)f(su\016cien)n(t)f(in)h(general\).)243 1805 y(One)f(can)g(easily)f(c)n(hec)n(k)h(that)g(necessary)f(and)h (su\016cien)n(t)h(conditions)e(for)118 1904 y FO(a)162 1916 y FL(1)199 1904 y FP(,)i FO(a)294 1916 y FL(2)359 1904 y FP(to)f(form)h(a)f(represen)n(tation)f(are)g(the)i(follo)n (wing:)207 2091 y FO(a)251 2103 y FL(1)288 2091 y FP(\(1)18 b FQ(\000)h FO(\025)p FP(\))k(=)p 655 2018 195 4 v 23 w FO(a)699 2103 y FL(1)736 2091 y FP(\()p FO(\025)p FP(\))q FO(;)97 b(a)1013 2103 y FL(2)1050 2091 y FP(\()p FQ(\000)p FP(1)18 b FQ(\000)g FO(\025)p FP(\))24 b(=)p 1481 2018 V 22 w FO(a)1525 2103 y FL(2)1562 2091 y FP(\()p FO(\025)p FP(\))q FO(;)97 b(a)1839 2056 y FL(2)1839 2111 y(1)1877 2091 y FP(\()p FO(\025)p FP(\))19 b FQ(\000)f FO(a)2135 2056 y FL(2)2135 2111 y(2)2172 2091 y FP(\()p FO(\025)p FP(\))24 b(=)f(2)p FO(\025)118 2277 y FP(for)h(almost)g(all)g FO(\025)h FP(tak)n(en)f(with)h(resp)r(ect)f(to)h(the)f(sp)r(ectral)g (measure)g(of)g FO(E)2402 2289 y FL(0)2440 2277 y FP(.)36 b(In)118 2376 y(particular,)e(these)g(relations)f(imply)h(that)g(b)r (oth)h FO(a)1767 2346 y FL(2)1767 2397 y(1)1838 2376 y FP(and)f FO(a)2050 2346 y FL(2)2050 2397 y(2)2121 2376 y FP(are)f(uniquely)118 2476 y(determined)24 b(b)n(y)e(the)i(v)-5 b(alue)23 b(of)g FO(a)1141 2488 y FL(1)1202 2476 y FP(at)g(a)g(single)f (p)r(oin)n(t)i(of)f(a)g(non-zero)e(sp)r(ectral)118 2576 y(measure.)36 b(Actually)-7 b(,)28 b(there)f(exists)g FO(\036)d(>)f FP(0)k(suc)n(h)g(that)390 2762 y FO(a)434 2728 y FL(2)434 2782 y(1)471 2762 y FP(\()p FO(\025)p FP(\))d(=)f FQ(\000)p FP(\()p FO(\025)18 b FQ(\000)g FP(1)p FO(=)p FP(2\))1099 2728 y FL(2)1154 2762 y FP(+)g FO(\036;)97 b(a)1450 2728 y FL(2)1450 2782 y(2)1488 2762 y FP(\()p FO(\025)p FP(\))24 b(=)e FQ(\000)p FP(\()p FO(\025)d FP(+)f(1)p FO(=)p FP(2\))2116 2728 y FL(2)2171 2762 y FP(+)g FO(\036)118 2948 y FP(on)27 b(the)h(sp)r(ectrum)g(of)g FO(E)895 2960 y FL(0)932 2948 y FP(.)243 3050 y(But)34 b(the)h(latter)f(relations)g(cannot)g(hold)g(for)g(all)g(p)r(oin)n(ts)h (of)f(the)h(orbit,)118 3149 y(since)f(in)f(the)h(righ)n(t-hand)f(sides) g(w)n(e)g(ha)n(v)n(e)g(functions)h(decreasing)e(to)h FQ(\0001)p FP(,)118 3249 y(while)24 b(the)g(left-hand)g(sides)g(m)n (ust)g(b)r(e)g(non-negativ)n(e.)34 b(T)-7 b(o)23 b(a)n(v)n(oid)f(the)j (con)n(tra-)118 3348 y(diction,)e(w)n(e)d(need)h(to)g(demand)g(that,)h (on)f(the)g(highest)g(v)n(ector)e(\(the)j(v)n(ector)e FO(e)2532 3360 y FM(\025)118 3448 y FP(with)29 b(the)f(largest)f FO(\025)p FP(\),)i(the)g(op)r(erator)d FO(E)1392 3460 y FL(2)1458 3448 y FP(acts)i(as)f(zero,)h(and)g(on)f(the)i(lo)n(w)n (est)118 3548 y(v)n(ector,)f(the)g(op)r(erator)f FO(E)932 3560 y FL(1)998 3548 y FP(is)h(zero.)39 b(This)28 b(means)g(that)h (zero)r(es)e(of)i(b)r(oth)f(the)118 3647 y(functions)g FO(a)520 3659 y FL(1)585 3647 y FP(and)f FO(a)790 3659 y FL(2)855 3647 y FP(m)n(ust)h(b)r(elong)f(to)g(the)h(same)f(orbit.)243 3749 y(One)i(can)h(easily)f(c)n(hec)n(k)g(that)h(these)g(conditions)g (can)f(b)r(e)i(satis\014ed)e(only)118 3848 y(for)j(three)g(orbits,)h (corresp)r(onding)e(to)h(the)h(p)r(oin)n(ts)f(0,)h(and)g FQ(\006)p FP(1)p FO(=)p FP(2.)49 b(In)33 b(eac)n(h)118 3948 y(case,)h(only)f(a)g(discrete)f(n)n(um)n(b)r(er)h(of)g(v)-5 b(alues)33 b(of)g FO(\036)h FP(are)e(admissible,)j(one)d(for)118 4048 y(eac)n(h)26 b(dimension.)37 b(Also,)27 b(the)g(represen)n(tation) e(corresp)r(onding)g(to)i(the)g(orbit)118 4147 y(that)h(con)n(tains)f (zero)f(has)h(an)g(o)r(dd)h(dimension.)p eop %%Page: 110 114 110 113 bop 118 100 a FP(110)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)243 333 y FP(T)-7 b(o)20 b(complete)h(the)g(pro)r(of,)g (it)g(remains)f(to)h(notice)f(that)h(for)f(an)n(y)g(orbit)h(con-)118 432 y(taining)34 b(zero)e(there)i(corresp)r(onds)e(a)h(unique)h (irreducible)g(represen)n(tation,)118 532 y(while)c(the)g(orbits)g(of)f FQ(\006)p FP(1)p FO(=)p FP(2)g(carry)f(t)n(w)n(o)h(represen)n(tations)f (eac)n(h,)i(dep)r(ending)118 632 y(on)e(the)g(sign)g(of)f FO(a)687 644 y FL(1)725 632 y FP(\(1)p FO(=)p FP(2\))g(or)g FO(a)1088 644 y FL(2)1125 632 y FP(\()p FQ(\000)p FP(1)p FO(=)p FP(2\).)37 b(Finding)28 b(admissible)f(v)-5 b(alues)28 b(for)f FO(\036)118 731 y FP(and)33 b(restoring)f(the)i(op)r(erators)d FO(A)p FP(,)36 b FO(B)t FP(,)f FO(C)40 b FP(from)33 b FO(E)1768 743 y FL(0)1805 731 y FP(,)i FO(E)1924 743 y FL(1)1962 731 y FP(,)g FO(E)2081 743 y FL(2)2152 731 y FP(is)e(a)g(routine)118 831 y(calculation.)p 2514 831 4 57 v 2518 778 50 4 v 2518 831 V 2567 831 4 57 v 243 994 a(No)n(w)23 b(w)n(e)h(consider)g(another)f(problem)h(that)g(app)r (eared)g(to)g(b)r(e)g(related)g(to)118 1093 y(the)k(one)f(just)h (considered.)243 1193 y(The)j(problem)g(is)g(related)g(to)g(general)f (non-orthogonal)f(resolutions)h(of)118 1293 y(the)g(iden)n(tit)n(y)-7 b(.)41 b(Let)29 b FO(A)811 1305 y FL(1)849 1293 y FP(,)g FO(:)14 b(:)g(:)27 b FP(,)j FO(A)1140 1305 y FM(n)1214 1293 y FP(b)r(e)g(p)r(ositiv)n(e)e(self-adjoin)n(t)h(op)r(erators)e(in) i(a)118 1392 y(\014nite-dimensional)e(space)g FO(H)35 b FP(suc)n(h)27 b(that)1179 1518 y FM(n)1140 1543 y Fz(X)1139 1721 y FM(k)q FL(=1)1274 1622 y FO(A)1336 1634 y FM(k)1400 1622 y FP(=)c FO(I)7 b(:)118 1863 y FP(W)-7 b(riting)26 b(the)g(sp)r(ectral)f(decomp)r(osition)h(for)f(eac)n(h)g(of)h(the)g(op) r(erators)e FO(A)2390 1875 y FM(k)2431 1863 y FP(,)i(w)n(e)118 1963 y(see)h(that)h(the)g(latter)f(sum)h(can)f(b)r(e)h(rewritten)g(as) 843 2072 y FM(m)813 2097 y Fz(X)820 2276 y FM(l)p FL(=1)946 2176 y FO(\013)999 2188 y FM(l)1025 2176 y FO(P)1078 2188 y FM(l)1127 2176 y FP(=)22 b FO(I)7 b(;)180 b FP(0)23 b FO(<)g(\013)1666 2188 y FM(l)1714 2176 y FQ(\024)g FP(1)p FO(;)496 b FP(\(2.20\))118 2418 y(where)32 b FO(P)416 2430 y FM(l)475 2418 y FP(are)f(orthogonal)g(pro)5 b(jections.)51 b(If)33 b(all)f FO(\013)1775 2430 y FM(l)1832 2418 y FP(=)f(1,)j(the)f(pro)5 b(jections)118 2517 y(comm)n(ute,)28 b(and)f(w)n(e)g(get)h(an)f(ordinary)f(resolution)g(of)i(the)g(iden)n (tit)n(y)-7 b(.)243 2617 y(The)25 b(complexit)n(y)f(of)i(the)f (description)g(problem)f(for)h(families)g(of)g(pro)5 b(jec-)118 2717 y(tions)29 b(that)g(form)f(a)g(non-orthogonal)e (resolution)i(of)h(the)g(iden)n(tit)n(y)f(dep)r(ends)118 2816 y(on)f(the)h(n)n(um)n(b)r(er)g(of)f(the)h(pro)5 b(jections)27 b(in)g(\(2.20\).)243 2916 y(Tw)n(o)c(pro)5 b(jections)24 b(satisfying)h(\(2.20\))f(are)f(orthogonal;)g(therefore,) i FO(\013)2450 2928 y FL(1)2510 2916 y FP(=)118 3016 y FO(\013)171 3028 y FL(2)232 3016 y FP(=)d(1.)243 3115 y(A)39 b(triple)h(of)f(pro)5 b(jections)39 b(satisfying)h(\(2.20\))e (has)h(only)h(one-)e(or)h(t)n(w)n(o-)118 3215 y(dimensional)27 b(irreducible)g(represen)n(tations.)243 3314 y(The)35 b(problem)f(of)h(the)h(unitary)e(description)h(of)g(\014v)n(e)g(or)f (more)g(pro)5 b(jec-)118 3414 y(tions)25 b(satisfying)h(\(2.20\))o(,)f (is)g(v)n(ery)f(complicated)h(\(\\wild",)g(see)g(Section)g(3.1.3)118 3514 y(b)r(elo)n(w\).)243 3613 y(Consider)g(four)i(orthogonal)d(pro)5 b(jections,)26 b FO(P)1678 3625 y FL(1)1716 3613 y FP(,)h FO(P)1819 3625 y FL(2)1856 3613 y FP(,)g FO(P)1959 3625 y FL(3)1997 3613 y FP(,)g FO(P)2100 3625 y FL(4)2164 3613 y FP(that)g(satisfy)118 3713 y(a)g(sp)r(ecial)h(case)e(of)i(the)g (relation)e(\(2.20\))h(with)h(all)g FO(\013)1773 3725 y FM(l)1821 3713 y FP(=)23 b FO(\013)p FP(,)866 3880 y FO(\013)p FP(\()p FO(P)1004 3892 y FL(1)1060 3880 y FP(+)c FO(P)1197 3892 y FL(2)1252 3880 y FP(+)g FO(P)1389 3892 y FL(3)1445 3880 y FP(+)f FO(P)1581 3892 y FL(4)1618 3880 y FP(\))23 b(=)g FO(I)7 b(:)536 b FP(\(2.21\))118 4048 y(W)-7 b(e)25 b(study)g(for)f(whic)n(h)g FO(\013)h FP(solutions)f(exist,)h(and)g(giv)n(e)e(their)i(unitary)f(descrip-)118 4147 y(tion.)p eop %%Page: 111 115 111 114 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(111)118 333 y FR(Theorem)35 b(23.)43 b FC(Solutions)34 b(of)52 b FP(\(2.21\))32 b FC(exist)i(for)g(the)g (fol)t(lowing)i(values)e(of)118 432 y FO(\013)p FP(:)243 532 y(1)p FC(.)70 b FO(\013)43 b FP(=)594 499 y FL(1)p 594 513 34 4 v 594 561 a(2)637 532 y FC(.)71 b(Ther)l(e)42 b(ar)l(e)e(six)h(one-dimensional,)k(and)d(a)f(c)l(ontinuous)118 632 y(family)32 b(of)e(two-dimensional)i(r)l(epr)l(esentations)7 b FP(;)243 731 y(2)p FC(.)52 b FO(\013)32 b FP(=)553 698 y FL(1)p 553 712 V 553 760 a(2)618 731 y FQ(\000)733 698 y FL(1)p 714 712 70 4 v 714 760 a(4)p FM(k)794 731 y FC(,)k FO(k)e FQ(2)e FJ(N)t FC(.)59 b(Ther)l(e)35 b(is)g(one)g(irr)l (e)l(ducible)g(r)l(epr)l(esentation)118 831 y(with)30 b FP(dim)15 b FO(H)29 b FP(=)23 b(2)p FO(k)e FQ(\000)d FP(1;)243 930 y(3)p FC(.)52 b FO(\013)32 b FP(=)553 898 y FL(1)p 553 912 34 4 v 553 959 a(2)618 930 y FP(+)733 898 y FL(1)p 714 912 70 4 v 714 959 a(4)p FM(k)794 930 y FC(,)k FO(k)e FQ(2)e FJ(N)t FC(.)59 b(Ther)l(e)35 b(is)g(one)g(irr)l (e)l(ducible)g(r)l(epr)l(esentation)118 1030 y(with)30 b FP(dim)15 b FO(H)29 b FP(=)23 b(2)p FO(k)e FP(+)d(1;)243 1130 y(4)p FC(.)45 b FO(\013)28 b FP(=)538 1097 y FL(1)p 538 1111 34 4 v 538 1158 a(2)602 1130 y FQ(\000)757 1097 y FL(1)p 696 1111 154 4 v 696 1158 a(4)p FM(k)q FL(+2)860 1130 y FC(,)34 b FO(k)c FQ(2)e FJ(N)t FC(.)52 b(Ther)l(e)33 b(ar)l(e)g(four)f(irr)l(e)l(ducible)i(r)l(epr)l(esenta-)118 1229 y(tions)c(with)g FP(dim)14 b FO(H)30 b FP(=)23 b FO(k)s FP(;)243 1329 y(5)p FC(.)45 b FO(\013)28 b FP(=)538 1296 y FL(1)p 538 1310 34 4 v 538 1358 a(2)601 1329 y FP(+)757 1296 y FL(1)p 696 1310 155 4 v 696 1358 a(4)p FM(k)q FN(\000)p FL(2)861 1329 y FC(,)33 b FO(k)d FQ(2)e FJ(N)t FC(.)52 b(Ther)l(e)33 b(ar)l(e)g(four)f(irr)l(e)l(ducible)i(r)l (epr)l(esenta-)118 1429 y(tions)c(with)g FP(dim)14 b FO(H)30 b FP(=)23 b FO(k)s FC(.)118 1593 y(Pr)l(o)l(of.)43 b FP(In)n(tro)r(duce)28 b(self-adjoin)n(t)g(unitary)g(op)r(erators)e FO(R)1908 1605 y FM(i)1960 1593 y FP(=)e(2)p FO(P)2144 1605 y FM(i)2190 1593 y FQ(\000)19 b FO(I)7 b FP(,)28 b FO(i)c FP(=)g(1,)118 1692 y FO(:)14 b(:)g(:)28 b FP(,)f(4.)37 b(Then)932 1843 y FL(4)888 1868 y Fz(X)895 2045 y FM(i)p FL(=1)1022 1947 y FO(R)1085 1959 y FM(i)1136 1947 y FP(=)1234 1891 y(2)18 b FQ(\000)g FP(4)p FO(\013)p 1234 1928 238 4 v 1326 2004 a(\013)1495 1947 y(I)30 b FP(=)23 b(2)p FO(hI)7 b(;)118 2204 y FP(where)27 b(w)n(e)g(put)i FO(h)23 b FP(=)f(\(1)c FQ(\000)g FP(2)p FO(\013)p FP(\))p FO(=\013)p FP(.)243 2304 y(First)27 b(consider)g(the)h(case)e FO(\013)e FP(=)f(1)p FO(=)p FP(2)j(\()p FO(h)d FP(=)g(0\).)36 b(In)n(tro)r(duce) 27 b(the)h(elemen)n(ts)215 2484 y FO(X)284 2496 y FL(1)345 2484 y FP(=)22 b(\()p FO(R)527 2496 y FL(1)583 2484 y FP(+)c FO(R)729 2496 y FL(4)767 2484 y FP(\))p FO(=)p FP(2)p FO(;)96 b(X)1071 2496 y FL(2)1131 2484 y FP(=)22 b(\()p FO(R)1313 2496 y FL(2)1369 2484 y FP(+)d FO(R)1516 2496 y FL(4)1553 2484 y FP(\))p FO(=)p FP(2)p FO(;)96 b(X)1857 2496 y FL(3)1917 2484 y FP(=)23 b(\()p FO(R)2100 2496 y FL(3)2156 2484 y FP(+)18 b FO(R)2302 2496 y FL(4)2339 2484 y FP(\))p FO(=)p FP(2)p FO(:)118 2664 y FP(Then)386 2844 y FO(R)449 2856 y FL(1)509 2844 y FP(=)23 b FO(X)666 2856 y FL(1)721 2844 y FQ(\000)18 b FO(X)873 2856 y FL(2)929 2844 y FQ(\000)g FO(X)1081 2856 y FL(3)1118 2844 y FO(;)332 b(R)1536 2856 y FL(3)1597 2844 y FP(=)22 b FQ(\000)p FO(X)1818 2856 y FL(1)1873 2844 y FQ(\000)d FO(X)2026 2856 y FL(2)2081 2844 y FP(+)f FO(X)2233 2856 y FL(3)2270 2844 y FO(;)386 2969 y(R)449 2981 y FL(2)509 2969 y FP(=)23 b FQ(\000)p FO(X)731 2981 y FL(1)786 2969 y FP(+)18 b FO(X)938 2981 y FL(2)993 2969 y FQ(\000)h FO(X)1146 2981 y FL(3)1183 2969 y FO(;)267 b(R)1536 2981 y FL(4)1597 2969 y FP(=)22 b FO(X)1753 2981 y FL(1)1809 2969 y FP(+)c FO(X)1961 2981 y FL(2)2016 2969 y FP(+)g FO(X)2168 2981 y FL(3)2205 2969 y FO(;)118 3149 y FP(and)32 b(the)g(relation)f(holds)g (if)i(and)e(only)h(if)g FO(X)1543 3161 y FL(1)1580 3149 y FP(,)h FO(X)1705 3161 y FL(2)1742 3149 y FP(,)g FO(X)1867 3161 y FL(3)1936 3149 y FP(are)e(pairwise)f(an)n(ti-)118 3248 y(comm)n(uting)d(self-adjoin)n(t)g(op)r(erators)f(suc)n(h)h(that) 898 3428 y(\001)c(=)g FO(X)1154 3394 y FL(2)1147 3449 y(1)1209 3428 y FP(+)18 b FO(X)1368 3394 y FL(2)1361 3449 y(2)1423 3428 y FP(+)g FO(X)1582 3394 y FL(2)1575 3449 y(3)1641 3428 y FP(=)23 b FO(I)7 b(:)118 3609 y FP(Then)28 b(an)f(irreducible)g(represen)n(tation)f(is)i(either)f (one-dimensional,)g(one)g(of)118 3708 y FO(X)187 3720 y FM(i)238 3708 y FP(=)22 b FQ(\006)p FP(1,)27 b(and)g(the)h(others)f (are)g(zeros,)f(or)h(t)n(w)n(o-dimensional,)293 3934 y FO(X)362 3946 y FL(1)422 3934 y FP(=)c FO(a)568 3817 y Fz(\022)629 3883 y FP(1)115 b(0)629 3983 y(0)82 b FQ(\000)p FP(1)859 3817 y Fz(\023)934 3934 y FO(;)97 b(X)1123 3946 y FL(2)1183 3934 y FP(=)23 b FO(b)1321 3817 y Fz(\022)1382 3883 y FP(0)82 b(1)1382 3983 y(1)g(0)1548 3817 y Fz(\023)1623 3934 y FO(;)97 b(X)1812 3946 y FL(3)1872 3934 y FP(=)22 b FO(c)2009 3817 y Fz(\022)2096 3883 y FP(0)115 b FO(i)2070 3983 y FQ(\000)p FO(i)82 b FP(0)2288 3817 y Fz(\023)2363 3934 y FO(;)1037 4117 y(a)1081 4083 y FL(2)1136 4117 y FP(+)18 b FO(b)1255 4083 y FL(2)1311 4117 y FP(+)g FO(c)1430 4083 y FL(2)1490 4117 y FP(=)23 b(1)p FO(;)p eop %%Page: 112 116 112 115 bop 118 100 a FP(112)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(and)21 b(either)h FO(a)h(>)f FP(0,)h FO(b)f(>)h FP(0,)f FO(c)h FQ(2)h FJ(R)p FP(,)k(or)21 b FO(a)i FP(=)g(0,)f FO(b)h(>)f FP(0,)h FO(c)f(>)h FP(0,)f(or)f FO(a)i(>)g FP(0,)f FO(b)h FP(=)f(0,)118 432 y FO(c)27 b(>)f FP(0.)43 b(These)29 b(represen)n(tations)f(form)h(the)h(\014rst)g (family)g(in)g(the)g(statemen)n(t)118 532 y(of)e(the)g(theorem.)36 b(The)28 b(form)n(ulas)e(for)h FO(P)1399 544 y FM(k)1468 532 y FP(are)258 752 y FO(P)311 764 y FL(1)371 752 y FP(=)469 696 y(1)p 469 733 42 4 v 469 809 a(2)534 635 y Fz(\022)635 701 y FP(1)18 b(+)g FO(a)123 b FQ(\000)p FO(b)17 b FQ(\000)h FO(ic)595 801 y FQ(\000)p FO(b)g FP(+)g FO(ic)122 b FP(1)18 b FQ(\000)g FO(a)1211 635 y Fz(\023)1286 752 y FO(;)140 b(P)1502 764 y FL(2)1562 752 y FP(=)1660 696 y(1)p 1660 733 V 1660 809 a(2)1725 635 y Fz(\022)1794 701 y FP(1)18 b FQ(\000)g FO(a)90 b(b)18 b FQ(\000)g FO(ic)1786 801 y(b)g FP(+)g FO(ic)90 b FP(1)18 b(+)g FO(a)2273 635 y Fz(\023)2347 752 y FO(;)258 984 y(P)311 996 y FL(3)371 984 y FP(=)469 928 y(1)p 469 965 V 469 1041 a(2)534 867 y Fz(\022)635 934 y FP(1)g FQ(\000)g FO(a)123 b FQ(\000)p FO(b)17 b FP(+)h FO(ic)595 1033 y FQ(\000)p FO(b)g FQ(\000)g FO(ic)122 b FP(1)18 b(+)g FO(a)1211 867 y Fz(\023)1286 984 y FO(;)140 b(P)1502 996 y FL(4)1562 984 y FP(=)1660 928 y(1)p 1660 965 V 1660 1041 a(2)1725 867 y Fz(\022)1826 934 y FP(1)18 b(+)g FO(a)122 b(b)19 b FP(+)f FO(ic)1786 1033 y FQ(\000)p FO(b)g FQ(\000)g FO(ic)90 b FP(1)18 b FQ(\000)g FO(a)2337 867 y Fz(\023)2412 984 y FO(:)243 1212 y FP(F)-7 b(or)32 b FO(\013)h FQ(6)p FP(=)f(1)p FO(=)p FP(2)g(\()p FO(h)h FQ(6)p FP(=)f(0\),)i(in)n(tro)r(duce)f(the)h(elemen)n(ts)f FO(X)2017 1224 y FL(1)2054 1212 y FP(,)i FO(X)2181 1224 y FL(2)2218 1212 y FP(,)g FO(X)2345 1224 y FL(3)2415 1212 y FP(suc)n(h)118 1312 y(that)879 1509 y FO(X)948 1521 y FL(1)1008 1509 y FP(=)1129 1452 y(1)p 1105 1489 90 4 v 1105 1566 a(2)p FO(h)1218 1509 y FP(\()p FO(R)1313 1521 y FL(2)1369 1509 y FP(+)18 b FO(R)1515 1521 y FL(3)1571 1509 y FQ(\000)g FO(hI)7 b FP(\))p FO(;)879 1708 y(X)948 1720 y FL(2)1008 1708 y FP(=)1129 1652 y(1)p 1105 1689 V 1105 1765 a(2)p FO(h)1218 1708 y FP(\()p FO(R)1313 1720 y FL(1)1369 1708 y FP(+)18 b FO(R)1515 1720 y FL(3)1571 1708 y FQ(\000)g FO(hI)7 b FP(\))p FO(;)879 1908 y(X)948 1920 y FL(3)1008 1908 y FP(=)1129 1852 y(1)p 1105 1889 V 1105 1965 a(2)p FO(h)1218 1908 y FP(\()p FO(R)1313 1920 y FL(1)1369 1908 y FP(+)18 b FO(R)1515 1920 y FL(2)1571 1908 y FQ(\000)g FO(hI)7 b FP(\))p FO(:)118 2110 y FP(Then)754 2288 y FO(R)817 2300 y FL(1)877 2288 y FP(=)23 b FO(h)14 b FP(\()p FQ(\000)p FO(X)1193 2300 y FL(1)1248 2288 y FP(+)k FO(X)1400 2300 y FL(2)1455 2288 y FP(+)g FO(X)1607 2300 y FL(3)1663 2288 y FP(+)g(1)p FO(=)p FP(2\))p FO(;)754 2413 y(R)817 2425 y FL(2)877 2413 y FP(=)23 b FO(h)14 b FP(\()p FO(X)1128 2425 y FL(1)1183 2413 y FQ(\000)k FO(X)1335 2425 y FL(2)1391 2413 y FP(+)g FO(X)1543 2425 y FL(3)1598 2413 y FP(+)g(1)p FO(=)p FP(2\))p FO(;)754 2537 y(R)817 2549 y FL(3)877 2537 y FP(=)23 b FO(h)14 b FP(\()p FO(X)1128 2549 y FL(1)1183 2537 y FP(+)k FO(X)1335 2549 y FL(2)1391 2537 y FQ(\000)g FO(X)1543 2549 y FL(3)1598 2537 y FP(+)g(1)p FO(=)p FP(2\))p FO(;)754 2662 y(R)817 2674 y FL(4)877 2662 y FP(=)23 b FO(h)14 b FP(\()p FQ(\000)p FO(X)1193 2674 y FL(1)1248 2662 y FQ(\000)k FO(X)1400 2674 y FL(2)1455 2662 y FQ(\000)g FO(X)1607 2674 y FL(3)1663 2662 y FP(+)g(1)p FO(=)p FP(2\))p FO(;)118 2840 y FP(and)28 b(the)g(relation)e(\(2.21\))h(is)g(equiv)-5 b(alen)n(t)28 b(to)f(the)h(follo)n(wing)f(set)g(of)h(relations)385 3018 y FQ(f)p FO(X)496 3030 y FL(1)533 3018 y FO(;)14 b(X)639 3030 y FL(2)676 3018 y FQ(g)22 b FP(=)h FO(X)897 3030 y FL(3)934 3018 y FO(;)97 b FQ(f)p FO(X)1165 3030 y FL(2)1201 3018 y FO(;)14 b(X)1307 3030 y FL(3)1344 3018 y FQ(g)23 b FP(=)f FO(X)1565 3030 y FL(1)1602 3018 y FO(;)97 b FQ(f)p FO(X)1833 3030 y FL(3)1870 3018 y FO(;)14 b(X)1976 3030 y FL(1)2013 3018 y FQ(g)22 b FP(=)h FO(X)2234 3030 y FL(2)2271 3018 y FO(;)1023 3153 y(h)1071 3119 y FL(2)1121 3153 y FP(\(\001)c(+)f(1)p FO(=)p FP(4\))k(=)h(1)p FO(;)118 3331 y FP(where,)31 b(as)e(ab)r(o)n(v)n(e,)h(\001)d(=)g FO(X)1015 3301 y FL(2)1008 3352 y(1)1072 3331 y FP(+)20 b FO(X)1233 3301 y FL(2)1226 3352 y(2)1290 3331 y FP(+)f FO(X)1450 3301 y FL(2)1443 3352 y(3)1487 3331 y FP(.)45 b(No)n(w,)30 b(w)n(e)g(can)g(easily)f(describ)r(e)118 3431 y(all)35 b(irreducible)f(represen)n(tations)f(of)41 b(\(2.21\))34 b(in)h(terms)g(of)f(the)i(irreducible)118 3531 y(represen)n(tations)21 b(of)30 b(\(2.18\))23 b(with)g(one)g (extra)g(restriction)f FO(h)2006 3501 y FL(2)2057 3531 y FP(\(\001)10 b(+)g(1)p FO(=)p FP(4\))22 b(=)g(1.)243 3630 y(i.)42 b(Represen)n(tations)28 b(of)h(o)r(dd)g(dimension,)h (corresp)r(onding)d(to)j(the)f(orbit)118 3739 y(con)n(taining)h(zero.) 46 b(Let)31 b(dim)14 b FO(H)35 b FP(=)28 b FO(n)h FP(=)f(2)p FO(k)23 b FP(+)d(1,)31 b(then)h(\001)d(=)2099 3707 y FM(n)2140 3682 y Fy(2)2172 3707 y FN(\000)p FL(1)p 2099 3721 159 4 v 2161 3768 a(4)2267 3739 y FO(I)7 b FP(,)32 b(whic)n(h)118 3839 y(implies)37 b(that)g FO(h)i FP(=)f FQ(\006)p FP(2)p FO(=n)d FP(and)i FO(\013)i FP(=)1398 3806 y FL(1)p 1398 3820 34 4 v 1398 3868 a(2)1466 3839 y FQ(\000)1654 3806 y FL(1)p 1565 3820 211 4 v 1565 3868 a(2\()p FM(n)p FL(+1\))1785 3839 y FP(,)h(or)c FO(\013)j FP(=)2164 3806 y FL(1)p 2164 3820 34 4 v 2164 3868 a(2)2232 3839 y FP(+)2420 3806 y FL(1)p 2331 3820 212 4 v 2331 3868 a(2\()p FM(n)p FN(\000)p FL(1\))2552 3839 y FP(,)118 3948 y(whic)n(h)27 b(giv)n(e)g(cases)f(2)h(and)h(3)f(of)g(the)h (theorem.)36 b(The)28 b(pro)5 b(jections)26 b(are)g(three-)118 4048 y(diagonal)21 b(matrices)h(that)g(can)g(easily)g(b)r(e)h(restored) e(from)h(the)h(corresp)r(onding)118 4147 y(represen)n(tation)j(of)i (the)g(graded)e FO(so)p FP(\(3\).)p eop %%Page: 113 117 113 116 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(113)243 333 y(ii.)35 b(Represen)n(tations)20 b(with)j(an)e(arbitrary)f(dimension)i FO(n)p FP(,)h(corresp)r(onding) 118 432 y(to)j(the)h(orbits)e(con)n(taining)g(1)p FO(=)p FP(2,)g(or)h FQ(\000)p FP(1)p FO(=)p FP(2.)34 b(No)n(w)26 b(\001)d(=)g(\()p FO(n)1969 402 y FL(2)2022 432 y FQ(\000)15 b FP(1)p FO(=)p FP(4\))f FO(I)7 b FP(,)25 b(whic)n(h)118 532 y(implies)31 b FO(h)e FP(=)f FQ(\006)p FP(1)p FO(=n)p FP(,)i(and)h FO(\013)d FP(=)1174 499 y FL(1)p 1174 513 34 4 v 1174 561 a(2)1238 532 y FQ(\000)1438 499 y FL(1)p 1333 513 244 4 v 1333 561 a(2\(2)p FM(n)p FL(+1\))1586 532 y FP(,)k(or)e FO(\013)f FP(=)1931 499 y FL(1)p 1931 513 34 4 v 1931 561 a(2)1995 532 y FQ(\000)2195 499 y FL(1)p 2090 513 244 4 v 2090 561 a(2\(2)p FM(n)p FL(+1\))2343 532 y FP(.)47 b(This)118 641 y(giv)n(es)27 b(cases)f(4)h(and)h(5.)p 2514 641 4 57 v 2518 588 50 4 v 2518 641 V 2567 641 4 57 v 118 804 a FC(R)l(emark)39 b FP(28)p FC(.)i FP(One)27 b(can)g(consider)g(the)h(follo)n(wing)e(relation)910 971 y FO(P)963 983 y FL(1)1019 971 y FP(+)18 b FO(P)1155 983 y FL(2)1211 971 y FP(+)g FO(P)1347 983 y FL(3)1403 971 y FP(+)g FO(P)1539 983 y FL(4)1600 971 y FP(=)23 b FO(Z)q(;)594 b FP(\(2.22\))118 1138 y(where)34 b FO(P)418 1150 y FM(i)481 1138 y FP(are)g(orthogonal)e(pro)5 b(jections,)35 b(and)g FO(Z)40 b FP(comm)n(utes)34 b(with)h(them.)118 1238 y(Since,)45 b(for)40 b(an)h(irreducible)f(represen)n(tation,)j (the)e(cen)n(tral)f(elemen)n(t)h FO(Z)47 b FP(is)118 1337 y(scalar,)30 b FO(Z)j FP(=)28 b FO(\013I)7 b FP(,)32 b(w)n(e)e(can)g(apply)g(the)h(latter)f(theorem)g(to)h(the)f (description)118 1437 y(of)23 b(irreducible)f(represen)n(tations)e(of) 29 b(\(2.22\))o(.)36 b(Indeed,)23 b(the)g(set)g(of)g(irreducible)118 1536 y(represen)n(tations)d(of)28 b(\(2.22\))21 b(consists)g(of)g (represen)n(tations)f(of)28 b(\(2.21\))21 b(with)h(all)118 1636 y FO(\013)p FP(,)27 b(i.e.,)f(all)f(represen)n(tations)f(of)h(the) h(graded)f FO(so)p FP(\(3\))h(algebra)d(\(2.18\),)j(and)f(all)118 1736 y(represen)n(tations)38 b(of)h(triples)g(of)h(an)n(ti-comm)n (uting)e(self-adjoin)n(t)h(op)r(erators)118 1835 y(with)28 b(the)g(sum)g(of)f(squares)g(equal)g(to)g(the)h(iden)n(tit)n(y)-7 b(.)118 2048 y FR(2.2.2)94 b(Represen)m(tations)25 b(of)h(a)h(class)f (of)g(quadratic)i(algebras)e(with)410 2148 y(three)32 b(generators)118 2301 y FP(Consider)22 b(an)h(algebra)e(with)j(three)f (generators)e FO(X)7 b FP(,)23 b FO(Y)c FP(,)24 b FO(Z)29 b FP(and)23 b(the)g(relations)984 2468 y FO(X)7 b(Y)37 b FQ(\000)18 b FO(q)s(Y)g(X)30 b FP(=)22 b FO(\026Y)5 b(;)992 2593 y(Z)h(X)25 b FQ(\000)18 b FO(q)s(X)7 b(Z)28 b FP(=)22 b FO(\026Z)q(;)945 2717 y(\013Y)d(Z)24 b FQ(\000)18 b FO(\014)t(Z)6 b(Y)42 b FP(=)22 b FO(P)12 b FP(\()p FO(X)7 b FP(\))p FO(;)118 2884 y FP(where)27 b FO(q)s FP(,)h FO(\026)p FP(,)g FO(\013)p FP(,)g FO(\014)f FQ(2)d FJ(C)15 b FP(,)33 b(and)28 b FO(P)12 b FP(\()p FQ(\001)p FP(\))28 b(is)f(a)h(quadratic)e(p)r(olynomial)h(in)h FO(X)7 b FP(.)243 2984 y(In)18 b(what)h(follo)n(ws,)g(w)n(e)g(assume)e (that)i FO(\026)p FP(,)i FO(q)26 b FQ(2)d FJ(R)p FP(,)k FO(q)f FQ(6)p FP(=)d FQ(\006)p FP(1,)c(and)g(the)g(algebra)118 3084 y(is)j(equipp)r(ed)g(with)g(an)f(in)n(v)n(olution)g(de\014ned)h (on)f(the)h(generators)d(b)n(y)i FO(X)2328 3053 y FN(\003)2389 3084 y FP(=)h FO(X)7 b FP(,)118 3183 y FO(Y)185 3153 y FN(\003)246 3183 y FP(=)23 b FO(Z)6 b FP(.)34 b(Notice)23 b(that)g(the)g(ideal)f(generated)f(b)n(y)h(the)h(relations)e(is)i(a)f FQ(\003)p FP(-ideal.)118 3283 y(With)28 b(suc)n(h)e(an)g(in)n(v)n (olution,)g(the)h(in)n(tro)r(duced)g(family)g(of)f(relations)g (includes)118 3383 y(the)21 b(Lie)g(algebra)d FO(su)p FP(\(2\))j(\(for)f FO(q)26 b FP(=)d(1\),)f(and)e(man)n(y)g(of)h(its)f (deformations)g(whic)n(h)118 3482 y(ha)n(v)n(e)27 b(arisen)f(in)i (recen)n(t)f(pap)r(ers)g(in)h(ph)n(ysics.)243 3582 y(F)-7 b(or)29 b FO(q)i FQ(6)p FP(=)c(1,)j(one)g(can)g(rewrite)g(the)g (relation)g(with)g FO(\026)e FP(=)f(0;)k(just)g(replace)118 3681 y FO(X)38 b FP(with)32 b FO(X)27 b FP(+)21 b FO(\025I)39 b FP(for)31 b(an)g(appropriate)e(v)-5 b(alue)32 b(of)f FO(\025)p FP(.)49 b(In)31 b(what)h(follo)n(ws,)f(w)n(e)118 3781 y(assume)c(that)h FO(\026)23 b FP(=)g(0.)243 3881 y(In)n(tro)r(ducing)k(self-adjoin)n(t)g(generators,)e FO(A)f FP(=)e FO(X)7 b FP(,)27 b FO(B)h FP(=)2033 3848 y FL(1)p 2033 3862 34 4 v 2033 3909 a(2)2076 3881 y FP(\()p FO(Y)37 b FP(+)18 b FO(Z)6 b FP(\),)28 b FO(C)h FP(=)128 3948 y FL(1)p 128 3962 V 128 4009 a(2)171 3980 y FP(\()p FO(Y)38 b FQ(\000)18 b FO(Z)6 b FP(\),)27 b(the)h(relations)f(will)g (tak)n(e)g(the)h(form)621 4147 y(\(1)18 b(+)g FO(q)s FP(\)[)p FO(A;)c(B)t FP(])24 b(=)f FQ(\000)p FO(i)14 b FP(\(1)j FQ(\000)h FO(q)s FP(\))p FQ(f)p FO(A;)c(C)6 b FQ(g)p FO(;)p eop %%Page: 114 118 114 117 bop 118 100 a FP(114)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)623 333 y FP(\(1)18 b(+)g FO(q)s FP(\)[)p FO(A;)c(C)6 b FP(])24 b(=)f FO(i)14 b FP(\(1)k FQ(\000)g FO(q)s FP(\))p FQ(f)p FO(A;)c(B)t FQ(g)p FO(;)488 467 y FQ(\000)p FO(i)g FP(\()p FO(\013)k FQ(\000)g FO(\014)t FP(\)[)p FO(B)t(;)c(C)6 b FP(])20 b(+)e(\()p FO(\013)h FP(+)f FO(\014)t FP(\)\()p FO(B)1552 433 y FL(2)1609 467 y FQ(\000)g FO(C)1757 433 y FL(2)1794 467 y FP(\))24 b(=)1947 435 y FL(1)p 1947 449 34 4 v 1947 496 a(2)1990 467 y FO(P)12 b FP(\()p FO(A)p FP(\))p FO(:)243 657 y FP(No)n(w)34 b(w)n(e)g(turn)h(to)g(the)g(study)g(of)g(irreducible)f (represen)n(tations)f(of)h(the)118 757 y(in)n(tro)r(duced)18 b(algebras,)h(i.e.,)h(the)f(op)r(erators)e FO(X)7 b FP(,)20 b FO(Y)37 b FP(that)19 b(satisfy)f(the)h(relations)614 944 y FO(X)7 b(Y)41 b FP(=)22 b FO(q)s(Y)d(X)r(;)97 b(\013)14 b(Y)19 b(Y)1364 910 y FN(\003)1421 944 y FP(+)f FO(\014)g(Y)1636 910 y FN(\003)1674 944 y FO(Y)42 b FP(=)22 b FO(P)12 b FP(\()p FO(X)7 b FP(\))p FO(:)284 b FP(\(2.23\))243 1134 y(Using)30 b(the)g(argumen)n(ts)f(that)i(are)e(quite)i(similar)e (to)i(those)f(used)g(in)g(the)118 1234 y(previous)20 b(section,)j(one)e(sees)f(that)i FO(Y)40 b FP(maps)21 b(eigenspaces)f(of)i(the)f(self-adjoin)n(t)118 1333 y(op)r(erator)28 b FO(X)35 b FP(in)n(to)30 b(eigenspaces,)e(and)h(the)h(corresp)r (onding)d(dynamical)i(sys-)118 1433 y(tem)37 b(is)f FO(\025)i FQ(7!)f FO(q)s(\025)p FP(;)k(ho)n(w)n(ev)n(er,)36 b(w)n(e)g(need)g(not) h(assume)e(that)i FO(X)42 b FP(is)36 b(p)r(ositiv)n(e,)118 1533 y(and)h(w)n(e)f(ha)n(v)n(e)g(no)g(restrictions)g(on)g(the)i(k)n (ernels)d(of)i(the)g(op)r(erators.)63 b(The)118 1632 y(dynamical)38 b(system)h FO(\025)j FQ(7!)f FO(q)s(\025)e FP(p)r(ossesses)f(a)g(measurable)f(section,)k(there-)118 1732 y(fore,)31 b(the)g(sp)r(ectral)f(measure)f(of)i FO(X)37 b FP(in)30 b(an)n(y)g(irreducible)g(represen)n(tation)f(is)118 1831 y(concen)n(trated)e(on)g(a)g(single)g(orbit.)243 1933 y(In)n(tro)r(duce)g(the)h(set)585 2175 y FO(M)k FP(=)785 2033 y Fz(\()852 2118 y FP([1)p FO(;)14 b FQ(j)p FO(q)s FQ(j)p FP(\))19 b FQ([)f FP(\()p FQ(\000j)p FO(q)s FQ(j)p FO(;)c FQ(\000)p FP(1])k FQ([)h(f)p FP(0)p FQ(g)p FO(;)81 b FQ(j)p FO(q)s FQ(j)24 b FO(>)e FP(1)p FO(;)852 2238 y FP(\()p FQ(j)p FO(q)s FQ(j)p FO(;)14 b FP(1])19 b FQ([)f FP([)p FQ(\000)p FP(1)p FO(;)c FQ(\000j)p FO(q)s FQ(j)p FP(\))k FQ(\\)h(f)p FP(0)p FQ(g)p FO(;)81 b FQ(j)p FO(q)s FQ(j)24 b FO(<)e FP(1)p FO(:)118 2432 y(M)36 b FP(is)28 b(a)f(measurable)f(section)i(of)f(the)h(dynamical)f(system.) 243 2534 y(If)34 b(the)g(orbit)f(consists)g(of)h(the)g(zero)e(p)r(oin)n (t,)k(w)n(e)d(ha)n(v)n(e)g FO(X)39 b FP(=)33 b(0,)i(and)e(w)n(e)118 2634 y(ha)n(v)n(e)21 b(the)h(relation)f FO(\013Y)e(Y)927 2604 y FN(\003)972 2634 y FP(+)7 b FO(\014)t(Y)1162 2604 y FN(\003)1200 2634 y FO(Y)42 b FP(=)22 b FO(P)12 b FP(\(0\))p FO(I)29 b FP(whic)n(h)22 b(is)g(either)g(the)g FO(q)s FP(-plane,)118 2734 y(or)29 b(the)h FO(q)s FP(-CCR)g(relation)f (considered)g(in)h(Section)g(1.4.2.)42 b(In)30 b(what)g(follo)n(ws,)118 2833 y(w)n(e)d(study)h(the)g(represen)n(tations)e(corresp)r(onding)f (to)j(non-zero)e(orbits.)118 3006 y FR(Theorem)44 b(24.)j FC(Unitary)41 b(r)l(epr)l(esentations)f(c)l(orr)l(esp)l(onding)i(to)f (the)f(orbit)118 3106 y FO(O)181 3118 y FM(\025)225 3106 y FC(,)30 b FO(\025)24 b FQ(2)f FO(M)k FQ(n)18 b(f)p FP(0)p FQ(g)p FC(,)29 b(may)h(b)l(e)g(describ)l(e)l(d)h(in)f(terms)f (of)i(the)f(fol)t(lowing)7 b FP(:)171 3298 y(\()p FC(a)f FP(\))43 b FC(in\014nite)29 b(se)l(quenc)l(es)g FO(y)1024 3310 y FM(k)1088 3298 y FO(>)23 b FP(0)p FC(,)29 b FO(k)d FQ(2)e FJ(Z)o FC(,)g(for)31 b(which)1025 3485 y FO(\013)14 b(y)1136 3451 y FL(2)1133 3506 y FM(k)1192 3485 y FP(+)k FO(\014)g(y)1384 3451 y FL(2)1381 3506 y FM(k)q FL(+1)1529 3485 y FP(=)23 b FO(P)12 b FP(\()p FO(q)1754 3451 y FM(k)1795 3485 y FO(\025)p FP(\))488 b(\(2.24\))326 3673 y FC(holds)40 b(for)f(al)t(l)g FO(k)j FQ(2)d FJ(Z)32 b FP(\()p FC(as)39 b(a)g(rule,)i(these)e(r)l(epr)l(esentations)f(form)h(a)326 3772 y(c)l(ontinuous)29 b(family)i(indexe)l(d)f(by)h FO(\025)p FP(\);)176 3948 y(\()p FC(b)5 b FP(\))43 b FC(in\014nite)29 b(se)l(quenc)l(es)g FO(y)1024 3960 y FM(k)1065 3948 y FC(,)h FO(k)c FQ(\025)d FO(l)r FC(,)29 b(with)h FO(l)i FC(\014xe)l(d,)d(such)h(that)g FO(y)2211 3960 y FM(l)2259 3948 y FP(=)23 b(0)p FC(,)29 b(and)326 4048 y FO(y)367 4060 y FM(k)436 4048 y FO(>)f FP(0)p FC(,)33 b FO(k)e(>)d(l)r FC(,)33 b(and)42 b FP(\(2.24\))32 b FC(holds)i(for)f(al)t(l)h FO(k)d FQ(\025)d FO(l)34 b FP(\()p FC(r)l(epr)l(esentations)326 4147 y(with)c(the)g(lowest)g (weight)8 b FP(\);)p eop %%Page: 115 119 115 118 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(115)177 333 y(\()p FC(c)5 b FP(\))42 b FC(in\014nite)31 b(se)l(quenc)l(es)g FO(y)1028 345 y FM(k)1068 333 y FC(,)h FO(k)d FQ(\024)c FO(l)r FC(,)32 b(with)g FO(l)h FC(\014xe)l(d,)e(such)h(that)f(for)h FO(y)2366 345 y FM(l)2417 333 y FP(=)26 b(0)p FC(,)326 432 y(and)32 b FO(y)530 444 y FM(k)597 432 y FO(>)27 b FP(0)p FC(,)32 b FO(k)d(<)e(l)r FC(,)32 b(and)41 b FP(\(2.24\))31 b FC(holds)i(for)g(al)t(l)g FO(k)c FQ(\024)d FO(l)34 b FP(\()p FC(r)l(epr)l(esenta-)326 532 y(tions)29 b(with)i(the)f(highest)h(weight)8 b FP(\);)169 711 y(\()p FC(d)h FP(\))42 b FC(\014nite)32 b(se)l(quenc)l(es)h FO(y)959 723 y FM(k)999 711 y FC(,)i FO(l)30 b FQ(\024)f FO(k)j FQ(\024)c FO(m)p FC(,)34 b(with)g FO(l)g FC(and)g FO(m)f FC(\014xe)l(d,)h(such)f(that)326 811 y FO(y)367 823 y FM(l)419 811 y FP(=)26 b FO(y)551 823 y FM(m)641 811 y FP(=)g(0)p FC(,)32 b(and)h FO(y)1036 823 y FM(k)1103 811 y FO(>)26 b FP(0)32 b FC(for)g FO(l)d(<)d(k)k(<)c(m)p FC(,)33 b(and)41 b FP(\(2.24\))31 b FC(holds)i(for)326 910 y(al)t(l)d FO(l)25 b FQ(\024)e FO(k)i FQ(\024)e FO(m)30 b FP(\()p FC(\014nite-dimensional)h(r)l(epr)l(esentations)7 b FP(\))p FC(.)118 1106 y(In)31 b(series)40 b FP(\()p FC(a)6 b FP(\))p FC(,)34 b(the)e(r)l(epr)l(esentations)g(ar)l(e)g(unb)l (ounde)l(d.)44 b(In)31 b(series)40 b FP(\()p FC(b)5 b FP(\))32 b FC(and)118 1206 y FP(\()p FC(c)5 b FP(\))p FC(,)34 b(the)e(r)l(epr)l(esentations)g(may)h(b)l(e)f(b)l(ounde)l(d)g (or)h(unb)l(ounde)l(d.)45 b(Besides)34 b(the)118 1305 y(mentione)l(d)c(r)l(epr)l(esentations,)h(ther)l(e)f(c)l(an)g(stil)t(l) g(b)l(e)g(one-dimensional)i(r)l(epr)l(e-)118 1405 y(sentations)e FO(Y)43 b FP(=)24 b(0)p FC(,)30 b FO(P)12 b FP(\()p FO(X)7 b FP(\))24 b(=)f(0)p FC(,)31 b(and)f(the)h(r)l(epr)l(esentations)f(c)l (orr)l(esp)l(onding)118 1505 y(to)g(the)g(zer)l(o)g(orbit.)118 1681 y(Pr)l(o)l(of.)43 b FP(Using)28 b(the)h(same)f(argumen)n(ts)e(as)i (in)h(the)f(previous)f(section,)h(w)n(e)g(see)118 1780 y(that)c(the)g(sp)r(ectrum)f(of)h FO(X)30 b FP(lies)23 b(in)h FQ(f)p FO(q)1295 1750 y FM(k)1335 1780 y FO(\025;)14 b(k)26 b FQ(2)e FJ(Z)o FQ(g)p FP(,)18 b(where)23 b FO(\025)g FQ(6)p FP(=)g(0)g(is)g(an)h(initial)118 1880 y(p)r(oin)n(t)35 b(from)e FO(M)9 b FP(,)36 b(and)e(the)h(space)e FO(H)41 b FP(is)34 b(a)g(direct)g(sum)h(of)f(its)g(eigenspaces)118 1979 y FO(H)187 1991 y FM(k)267 1979 y FP(corresp)r(onding)j(to)i(the)g (eigen)n(v)-5 b(alues)37 b FO(\025)1567 1991 y FM(k)1651 1979 y FP(=)k FO(q)1797 1949 y FM(k)1838 1979 y FO(\025)p FP(.)71 b(The)39 b(op)r(erator)e FO(Y)118 2079 y FP(maps)c FO(H)410 2091 y FM(k)q FN(\000)p FL(1)570 2079 y FP(in)n(to)g FO(H)813 2091 y FM(k)854 2079 y FP(;)k(write)c FO(Y)1180 2091 y FM(k)1230 2079 y FP(:)d FO(H)1352 2091 y FM(k)q FN(\000)p FL(1)1511 2079 y FQ(\000)-49 b(!)33 b FO(H)1712 2091 y FM(k)1787 2079 y FP(for)g(the)h(corresp)r(onding)118 2179 y(restrictions.)46 b(F)-7 b(rom)31 b(the)h(second)e(relation)h(in) g(\(2.23\))o(,)h(w)n(e)f(ha)n(v)n(e)f FO(\013)15 b(Y)2344 2191 y FM(k)2385 2179 y FO(Y)2451 2149 y FN(\003)2433 2202 y FM(k)2510 2179 y FP(+)118 2278 y FO(\014)j(Y)250 2248 y FN(\003)231 2302 y FM(k)q FL(+1)356 2278 y FO(Y)404 2290 y FM(k)q FL(+1)553 2278 y FP(=)k FO(P)12 b FP(\()p FO(q)777 2248 y FM(k)818 2278 y FO(\025)p FP(\))i FO(I)7 b FP(.)243 2381 y(W)-7 b(e)41 b(will)h(sho)n(w)f(that)g(all)g FO(H)1181 2393 y FM(k)1264 2381 y FP(are)f(either)h(zero)g(or)f (one-dimensional.)118 2481 y(Indeed,)25 b FO(Y)456 2493 y FM(k)497 2481 y FO(Y)564 2451 y FN(\003)545 2504 y FM(k)626 2481 y FP(and)f FO(Y)851 2451 y FN(\003)832 2504 y FM(k)q FL(+1)957 2481 y FO(Y)1005 2493 y FM(k)q FL(+1)1155 2481 y FP(are)f(comm)n(uting)h(self-adjoin)n(t)f(op)r (erators)g(in)118 2580 y FO(H)187 2592 y FM(k)228 2580 y FP(.)58 b(T)-7 b(ak)n(e)34 b(an)h(in)n(v)-5 b(arian)n(t)33 b(subspace)h FO(H)1425 2550 y FL(0)1418 2604 y FM(k)1497 2580 y FQ(\032)h FO(H)1666 2592 y FM(k)1707 2580 y FP(,)i(then)e(the)g (image)f(of)h FO(H)2538 2550 y FL(0)2531 2604 y FM(k)118 2680 y FP(under)29 b(the)h(action)e(of)h FO(Y)19 b FP(,)30 b FO(Y)1032 2650 y FN(\003)1100 2680 y FP(and)f FO(X)35 b FP(is)29 b(in)n(v)-5 b(arian)n(t)28 b(in)i FO(H)7 b FP(.)41 b(Th)n(us,)29 b FO(H)2345 2692 y FM(k)2415 2680 y FP(do)r(es)118 2780 y(not)24 b(ha)n(v)n(e)e(prop)r(er)h(subspaces.)34 b(Th)n(us,)24 b(the)g(op)r(erators)e FO(Y)1902 2792 y FM(k)1943 2780 y FO(Y)2009 2750 y FN(\003)1991 2803 y FM(k)2071 2780 y FP(and)i FO(Y)2295 2750 y FN(\003)2277 2803 y FM(k)q FL(+1)2402 2780 y FO(Y)2450 2792 y FM(k)q FL(+1)118 2889 y FP(are)k(scalar,)g(and)h(w)n(e)g(obtain)g(the)h (relations)e FO(\013)14 b FQ(j)p FO(y)1681 2901 y FM(k)1722 2889 y FQ(j)1745 2859 y FL(2)1802 2889 y FP(+)19 b FO(\014)f FQ(j)p FO(y)2015 2901 y FM(k)q FL(+1)2140 2889 y FQ(j)2163 2859 y FL(2)2226 2889 y FP(=)25 b FO(P)12 b FP(\()p FO(q)2453 2859 y FM(k)2494 2889 y FO(\025)p FP(\))118 2989 y(for)35 b(all)g FO(k)s FP(.)59 b(P)n(assing)33 b(to)i(a)g(unitarily)f(equiv)-5 b(alen)n(t)35 b(represen)n(tation,)h(w)n(e)e(can)118 3088 y(assume)k(that)g FO(y)647 3100 y FM(k)728 3088 y FQ(\025)j FP(0.)68 b(T)-7 b(o)38 b(complete)g(the)g(pro)r(of)g(one)g (can)g(notice)g(that)118 3188 y(the)g(subspaces)661 3126 y Fz(L)753 3213 y FM(k)q FN(\025)p FM(l)881 3188 y FO(H)950 3200 y FM(k)1029 3188 y FP(and)1201 3126 y Fz(L)1293 3213 y FM(k)q FN(\024)p FM(l)1421 3188 y FO(H)1490 3200 y FM(k)1569 3188 y FP(are)e(in)n(v)-5 b(arian)n(t)37 b(if)h(and)g(only)f(if)118 3287 y FO(y)159 3299 y FM(l)207 3287 y FP(=)23 b(0.)p 2514 3287 4 57 v 2518 3235 50 4 v 2518 3287 V 2567 3287 4 57 v 118 3480 a FC(R)l(emark)35 b FP(29)p FC(.)i FP(Consider)22 b(a)h(generalization)e(of)i(the)g(giv)n (en)g(class)f(of)h(quadratic)118 3579 y(algebras.)32 b(Replace)20 b(the)g(second)f(relation)f(in)i(\(2.23\))f(with)h(a)g (general)e(second-)118 3679 y(order)26 b(relation)h(connecting)g FO(X)7 b FP(,)27 b FO(Y)19 b FP(,)28 b FO(Y)1365 3649 y FN(\003)1403 3679 y FP(,)504 3868 y FO(a)548 3880 y FL(11)619 3868 y FO(X)695 3834 y FL(2)750 3868 y FP(+)18 b FO(a)877 3880 y FL(22)947 3868 y FO(Y)1014 3834 y FL(2)1069 3868 y FP(+)g FO(a)1196 3880 y FL(33)1267 3868 y FP(\()p FO(Y)1365 3834 y FN(\003)1404 3868 y FP(\))1436 3834 y FL(2)1492 3868 y FP(+)g FO(a)1619 3880 y FL(12)1689 3868 y FO(X)7 b(Y)36 b FP(+)18 b FO(a)1976 3880 y FL(21)2046 3868 y FO(Y)h(X)620 3993 y FP(+)f FO(a)747 4005 y FL(13)817 3993 y FO(X)7 b(Y)959 3959 y FN(\003)1015 3993 y FP(+)18 b FO(a)1142 4005 y FL(31)1213 3993 y FO(Y)1279 3959 y FN(\003)1336 3993 y FP(+)g FO(a)1463 4005 y FL(23)1533 3993 y FO(Y)h(Y)1666 3959 y FN(\003)1723 3993 y FP(+)f FO(a)1850 4005 y FL(32)1920 3993 y FO(Y)1987 3959 y FN(\003)2025 3993 y FO(Y)786 4117 y FP(+)g FO(a)913 4129 y FL(1)950 4117 y FO(X)25 b FP(+)18 b FO(a)1171 4129 y FL(2)1208 4117 y FO(Y)37 b FP(+)18 b FO(a)1420 4129 y FL(3)1457 4117 y FO(Y)1524 4083 y FN(\003)1580 4117 y FP(+)g FO(aI)30 b FP(=)23 b(0)p FO(:)p eop %%Page: 116 120 116 119 bop 118 100 a FP(116)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(One)20 b(can)g(sho)n(w)f(that,)j(in)f(an)n(y) e(represen)n(tation,)h(the)h(latter)f(relation)f(is)h(equiv-)118 432 y(alen)n(t)27 b(to)h(the)g(follo)n(wing)e(system:)528 613 y FO(a)572 625 y FL(11)642 613 y FO(X)718 579 y FL(2)773 613 y FP(+)18 b FO(a)900 625 y FL(1)938 613 y FO(X)24 b FP(+)18 b FO(aI)26 b FP(+)18 b FO(a)1347 625 y FL(23)1417 613 y FO(Y)h(Y)1550 579 y FN(\003)1607 613 y FP(+)f FO(a)1734 625 y FL(32)1804 613 y FO(Y)1871 579 y FN(\003)1909 613 y FO(Y)42 b FP(=)22 b(0)p FO(;)196 738 y(a)240 750 y FL(12)310 738 y FO(X)7 b(Y)36 b FP(+)18 b FO(a)597 750 y FL(21)667 738 y FO(Y)h(X)25 b FP(+)18 b FO(a)955 750 y FL(2)992 738 y FO(Y)42 b FP(=)22 b(0)p FO(;)97 b(a)1375 750 y FL(13)1445 738 y FO(X)7 b(Y)1587 703 y FN(\003)1644 738 y FP(+)18 b FO(a)1771 750 y FL(31)1841 738 y FO(Y)1907 703 y FN(\003)1946 738 y FO(X)24 b FP(+)18 b FO(a)2166 750 y FL(3)2203 738 y FO(Y)2270 703 y FN(\003)2331 738 y FP(=)23 b(0)p FO(;)905 873 y(a)949 885 y FL(22)1019 873 y FO(Y)1086 838 y FL(2)1146 873 y FP(=)f(0)p FO(;)97 b(a)1439 885 y FL(22)1509 873 y FO(Y)1576 838 y FL(2)1636 873 y FP(=)23 b(0)p FO(:)118 1088 y FR(2.2.3)94 b(Op)s(erator)33 b(relations)e(connected)i(with)f(a)h(dynamical)f(sys-)410 1188 y(tem)e(on)i(a)g(plane)118 1341 y FP(Let)g FO(A)e FP(=)f FO(A)519 1311 y FN(\003)557 1341 y FP(,)k FO(X)7 b FP(,)32 b FO(X)820 1311 y FN(\003)889 1341 y FP(b)r(e)f(b)r(ounded)h (op)r(erators)e(that)i(satisfy)f(relations)f(of)118 1441 y(the)e(form:)1015 1622 y FO(AX)h FP(=)23 b FO(X)7 b(F)1392 1634 y FL(1)1429 1622 y FP(\()p FO(A)p FP(\))p FO(;)964 1746 y(X)1040 1712 y FN(\003)1077 1746 y FO(X)29 b FP(=)23 b FO(F)1316 1758 y FL(2)1354 1746 y FP(\()p FO(A;)14 b(X)7 b(X)1637 1712 y FN(\003)1674 1746 y FP(\))p FO(;)647 b FP(\(2.25\))118 1927 y(where)29 b FO(F)413 1939 y FL(1)450 1927 y FP(\()p FQ(\001)p FP(\))9 b(:)30 b FJ(R)h FQ(!)26 b FJ(R)p FP(,)36 b FO(F)959 1939 y FL(2)996 1927 y FP(\()p FQ(\001)p FO(;)14 b FQ(\001)p FP(\))9 b(:)29 b FJ(R)1258 1897 y FL(2)1327 1927 y FQ(!)d FJ(R)35 b FP(are)29 b(measurable)e (mappings.)42 b(It)118 2027 y(follo)n(ws)30 b(from)h(the)h(\014rst)e (relation)h(in)g(\(2.25\))f(that)i(the)f(op)r(erators)f FO(A)p FP(,)i FO(X)7 b(X)2538 1997 y FN(\003)118 2126 y FP(comm)n(ute)28 b(and,)f(hence,)h FO(F)967 2138 y FL(2)1004 2126 y FP(\()p FO(A;)14 b(X)7 b(X)1287 2096 y FN(\003)1325 2126 y FP(\))28 b(is)f(w)n(ell)g(de\014ned.)118 2258 y FC(R)l(emark)38 b FP(30)p FC(.)i FP(Instead)26 b(of)g(assuming)f(that)i(the)g(op)r(erator)d FO(A)j FP(is)f (self-adjoin)n(t,)118 2358 y(one)e(can)f(consider)g(the)i(case)e(of)h (a)g(unitary)f(or)g(normal)h(op)r(erator)e(as)h(w)n(ell.)36 b(In)118 2458 y(this)26 b(case,)f(w)n(e)g(consider)f(the)i(mapping)f FO(F)1449 2470 y FL(1)1496 2458 y FP(:)i FJ(T)c FQ(\000)-49 b(!)23 b FJ(T)p FP(,)j(and)f FO(F)2064 2470 y FL(2)2111 2458 y FP(:)i FJ(T)14 b FQ(\002)f FJ(R)29 b FQ(\000)-48 b(!)23 b FJ(R)118 2557 y FP(for)32 b(the)g(unitary)g(op)r(erator)e FO(A)p FP(,)k(and)e FO(F)1372 2569 y FL(1)1419 2557 y FP(:)d FJ(C)52 b FQ(\000)-49 b(!)31 b FJ(C)15 b FP(,)39 b(and)32 b FO(F)2027 2569 y FL(2)2074 2557 y FP(:)d FJ(C)43 b FQ(\002)21 b FJ(R)37 b FQ(\000)-49 b(!)31 b FJ(R)118 2657 y FP(in)39 b(the)g(normal)f(case.)69 b(The)38 b(mapping)h FO(F)50 b FP(in)n(tro)r(duced)39 b(b)r(elo)n(w,)i(de\014nes)d(a)118 2757 y(dynamical)30 b(system)h(on)g FJ(T)20 b FQ(\002)g FJ(R)37 b FP(or)30 b FJ(C)42 b FQ(\002)20 b FJ(R)p FP(,)38 b(resp)r(ectiv)n(ely)-7 b(.)47 b(All)31 b(statemen)n(ts)118 2856 y(ab)r(out)23 b(represen)n(tations)e(of)i(relations)f(\(2.25\))h (also)f(hold)h(true)g(in)g(these)g(cases)118 2956 y(\(of)29 b(course,)e(no)n(w,)h(the)g(sp)r(ectrum)g(of)h FO(A)f FP(b)r(elongs)g(to)g(the)g(circle)g(or)f(complex)118 3055 y(plane\).)243 3188 y(As)21 b(b)r(efore,)h(let)g(us)f(consider)f (the)i(p)r(olar)e(decomp)r(osition)h(of)g(the)h(op)r(erator)118 3287 y FO(X)194 3257 y FN(\003)255 3287 y FP(=)g FO(U)9 b(C)d FP(,)28 b(where)f FO(C)i FP(=)23 b FO(C)1005 3257 y FN(\003)1066 3287 y FP(=)g(\()p FO(X)7 b(X)1338 3257 y FN(\003)1375 3287 y FP(\))1407 3257 y FL(1)p FM(=)p FL(2)1512 3287 y FP(,)27 b FO(U)37 b FP(is)27 b(a)g(partial)g(isometry) f(suc)n(h)118 3387 y(that)i(k)n(er)13 b FO(U)32 b FP(=)22 b(k)n(er)13 b FO(C)6 b FP(.)37 b(Using)28 b(the)g(relations)e(\(2.25\)) h(one)g(can)g(obtain)580 3568 y FO(AU)708 3534 y FN(\003)769 3568 y FP(=)c FO(U)923 3534 y FN(\003)961 3568 y FO(F)1014 3580 y FL(1)1051 3568 y FP(\()p FO(A)p FP(\))p FO(;)98 b(C)1363 3534 y FL(2)1401 3568 y FO(U)1467 3534 y FN(\003)1528 3568 y FP(=)22 b FO(U)1681 3534 y FN(\003)1719 3568 y FO(F)1772 3580 y FL(2)1810 3568 y FP(\()p FO(A;)14 b(C)2006 3534 y FL(2)2044 3568 y FP(\))p FO(;)264 b FP(\(2.26\))118 3749 y(and)34 b FO(U)44 b FP(is)34 b(a)g(cen)n(tered)g(partial)g (isometry)g(with)h(k)n(er)12 b FO(U)1898 3719 y FN(\003)1971 3749 y FP(=)34 b(k)n(er)13 b FO(F)2248 3761 y FL(2)2285 3749 y FP(\()p FO(A;)h(C)2481 3719 y FL(2)2519 3749 y FP(\).)118 3848 y(Con)n(v)n(ersely)-7 b(,)30 b(an)n(y)h(triple)g(of)h (the)g(op)r(erators)d FO(A)h FP(=)f FO(A)1810 3818 y FN(\003)1848 3848 y FP(,)j FO(C)k FQ(\025)29 b FP(0,)j(and)f(a)g(cen-) 118 3948 y(tered)39 b(partial)g(isometry)g FO(U)48 b FP(satisfying)39 b(\(2.26\))g(and)g(suc)n(h)g(that)h(k)n(er)13 b FO(U)51 b FP(=)118 4048 y(k)n(er)13 b FO(C)6 b FP(,)33 b(and)f(k)n(er)13 b FO(U)721 4018 y FN(\003)789 4048 y FP(=)30 b(k)n(er)12 b FO(F)1061 4060 y FL(2)1099 4048 y FP(\()p FO(A;)i(C)1295 4018 y FL(2)1333 4048 y FP(\),)33 b(de\014ne)g(a)e(represen)n(tation)f FO(A)p FP(,)k FO(X)i FP(=)118 4147 y FO(C)6 b(U)249 4117 y FN(\003)315 4147 y FP(of)28 b(the)f(relations)g(\(2.25\).)p eop %%Page: 117 121 117 120 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(117)243 333 y(Let)28 b FO(F)37 b FP(=)24 b(\()p FO(F)656 345 y FL(1)694 333 y FO(;)14 b(F)784 345 y FL(2)822 333 y FP(\))9 b(:)28 b FJ(R)968 303 y FL(2)1036 333 y FQ(!)d FJ(R)1198 303 y FL(2)1241 333 y FP(.)40 b(F)-7 b(or)27 b FO(k)h FQ(2)d FJ(N)t FP(,)35 b(w)n(e)28 b(will)h(denote)f(b)n(y)g FO(F)2446 303 y FM(k)2487 333 y FP(\()p FQ(\001)p FP(\))118 432 y(the)35 b FO(k)s FP(-th)f(iteration)f(of)h FO(F)12 b FP(\()p FQ(\001)p FP(\))35 b(and,)h(for)d FO(\025)i FQ(2)f FJ(R)p FP(,)42 b FO(n)33 b FP(=)h(1,)h(2,)h(b)n(y)e FO(F)2272 402 y FM(k)2260 453 y(n)2312 432 y FP(\()p FO(\025)p FP(\))h(the)118 532 y FO(n)p FP(-th)28 b(co)r(ordinate)e(of)i FO(F)870 502 y FM(k)910 532 y FP(\()p FO(\025)p FP(\).)243 632 y(Analogously)-7 b(,)19 b(the)h(relations)e(\(2.25\))h(corresp)r (ond)f(to)h(a)h(t)n(w)n(o-dimensional)118 731 y(dynamical)26 b(system,)g FO(F)12 b FP(\()p FQ(\001)p FP(\))24 b(:)f FJ(R)1089 701 y FL(2)1155 731 y FQ(\000)-48 b(!)23 b FJ(R)1332 701 y FL(2)1375 731 y FP(.)36 b(The)27 b(p)r(ossibilit)n(y)f (of)g(classifying)f(all)118 831 y(irreducible)c(represen)n(tations)f (of)i(the)g(relations)e(dep)r(ends)i(on)g(the)g(prop)r(erties)118 930 y(of)28 b(the)g(dynamical)f(system.)118 1089 y FR(Prop)s(osition)h (39.)39 b FC(L)l(et)d FP(\()p FO(A)23 b FP(=)g FO(A)1211 1059 y FN(\003)1249 1089 y FO(;)14 b(X)7 b FP(\))28 b FC(b)l(e)g(a)h(r)l(epr)l(esentation)f(of)46 b FP(\(2.25\))28 b FC(on)118 1189 y(a)e(sp)l(ac)l(e)h FO(H)7 b FC(.)37 b(Then)26 b FO(H)33 b FC(c)l(an)26 b(b)l(e)g(de)l(c)l(omp)l(ose)l(d)h (into)f(ortho)l(gonal)h(subsp)l(ac)l(es)g FO(H)2538 1201 y FL(1)118 1288 y FC(and)33 b FO(H)351 1300 y FL(2)389 1288 y FC(,)h(invariant)f(with)h(r)l(esp)l(e)l(ct)e(to)h FO(A)p FC(,)h FO(X)7 b FC(,)33 b FO(X)1701 1258 y FN(\003)1771 1288 y FC(such)g(that)g(the)g(phase)h FO(U)118 1388 y FC(of)d FO(X)36 b FC(is)30 b(unitary)g(in)f FO(H)872 1400 y FL(1)939 1388 y FC(and)39 b FP(k)n(er)13 b FO(U)27 b FQ([)19 b FP(k)n(er)13 b FO(U)1583 1358 y FN(\003)1644 1388 y FQ(6)p FP(=)22 b FQ(f)p FP(0)p FQ(g)29 b FC(in)g FO(H)2056 1400 y FL(2)2094 1388 y FC(.)243 1547 y FP(Similarly)24 b(to)h(the)g(case)f(of)h(relation)f(\(2.1\),)i(irreducible)e(represen)n (tations)118 1646 y(of)31 b(\(2.25\))f(in)h FO(H)628 1658 y FL(2)695 1646 y FP(can)g(b)r(e)g(completely)f(describ)r(ed.)46 b(There)30 b(is)h(a)f(corresp)r(on-)118 1746 y(dence)c(b)r(et)n(w)n (een)g(irreducible)f(represen)n(tations)f(and)i(orbits)f(of)h(the)g (dynami-)118 1845 y(cal)21 b(system)g(going)g(through)f(a)i(p)r(oin)n (t)f(with)h(zero)f(second)f(co)r(ordinate.)34 b(More-)118 1945 y(o)n(v)n(er,)27 b(since)i FO(C)590 1915 y FL(2)652 1945 y FQ(\025)24 b FP(0,)29 b(the)g(sp)r(ectral)f(measure)g(of)g(the)h (pair)f(\()p FO(A)p FP(,)i FO(C)2240 1915 y FL(2)2277 1945 y FP(\))f(is)g(con-)118 2045 y(cen)n(trated)38 b(on)g(that)h(part) f(of)g(the)h(orbit)g(where)e(the)i(second)f(co)r(ordinates)118 2144 y(are)26 b(non-negativ)n(e.)35 b(Namely)-7 b(,)28 b(w)n(e)f(ha)n(v)n(e)f(the)h(follo)n(wing)f(description)h(of)g(irre-) 118 2244 y(ducible)h(represen)n(tations.)118 2403 y FR(Prop)s(osition) 40 b(40.)46 b FC(A)n(ny)37 b(irr)l(e)l(ducible)i(r)l(epr)l(esentation) 44 b FP(\()p FO(A;)14 b(X)7 b FP(\))38 b FC(of)56 b FP(\(2.25\))118 2502 y FC(such)32 b(that)f FP(k)n(er)13 b FO(X)26 b FQ([)20 b FP(k)n(er)13 b FO(X)976 2472 y FN(\003)1040 2502 y FQ(6)p FP(=)26 b FQ(f)p FP(0)p FQ(g)k FC(is)i(unitarily)h(e)l (quivalent)f(to)f(one)h(of)h(the)118 2602 y(fol)t(lowing)7 b FP(:)243 2701 y(\(i\))p FC(.)39 b FO(H)30 b FP(=)22 b FJ(C)634 2671 y FM(n)686 2701 y FC(,)30 b FO(n)23 b FQ(2)g FJ(N)t FC(,)607 3060 y FO(A)h FP(=)780 2819 y Fz(0)780 2965 y(B)780 3015 y(B)780 3065 y(B)780 3118 y(@)853 2874 y FO(\025)1727 2915 y Fp(0)985 2974 y FO(F)1038 2986 y FL(1)1075 2974 y FP(\()p FO(\025;)14 b FP(0\))1354 3070 y FC(.)1389 3095 y(.)1424 3120 y(.)1096 3245 y Fp(0)382 b FO(F)1602 3202 y FL(\()p FM(n)p FN(\000)p FL(1\))1590 3267 y(1)1784 3245 y FP(\()p FO(\025;)14 b FP(0\))1975 2819 y Fz(1)1975 2965 y(C)1975 3015 y(C)1975 3065 y(C)1975 3118 y(A)2062 3060 y FO(;)594 3592 y(X)30 b FP(=)780 3325 y Fz(0)780 3472 y(B)780 3521 y(B)780 3571 y(B)780 3621 y(B)780 3674 y(@)973 3378 y FP(0)1709 3419 y Fp(0)853 3533 y FO(F)906 3545 y FL(2)944 3533 y FP(\()p FO(\025;)14 b FP(0\))1223 3475 y FC(.)1257 3500 y(.)1292 3525 y(.)1223 3630 y(.)1257 3654 y(.)1292 3680 y(.)1604 3688 y FP(0)965 3804 y Fp(0)381 b FO(F)1470 3761 y FL(\()p FM(n)p FN(\000)p FL(1\))1458 3826 y(2)1652 3804 y FP(\()p FO(\025;)14 b FP(0\))84 b(0)1968 3325 y Fz(1)1968 3472 y(C)1968 3521 y(C)1968 3571 y(C)1968 3621 y(C)1968 3674 y(A)2055 3592 y FO(;)118 3974 y FC(wher)l(e)31 b FO(\025)f FC(b)l(elongs)g(to)g(the)g (set)262 4147 y FO(\033)309 4159 y FM(n)378 4147 y FP(=)23 b FQ(f)p FO(\025)g FQ(2)g FJ(R)29 b FQ(j)23 b FO(F)851 4113 y FM(k)839 4168 y FL(2)892 4147 y FP(\()p FO(\025;)14 b FP(0\))23 b FO(>)g FP(0)p FO(;)k(k)f FP(=)d(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n)k FQ(\000)g FP(1)p FO(;)27 b(F)1977 4113 y FM(n)1965 4168 y FL(2)2022 4147 y FP(\()p FO(\025;)14 b FP(0\))24 b(=)f(0)p FQ(g)p FP(;)p eop %%Page: 118 122 118 121 bop 118 100 a FP(118)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)243 333 y FP(\(ii\))p FC(.)39 b FO(H)30 b FP(=)22 b FO(l)628 345 y FL(2)665 333 y FP(\()p FJ(N)5 b FP(\))p FC(,)514 526 y FO(Ae)615 538 y FM(k)679 526 y FP(=)23 b FO(F)832 483 y FL(\()p FM(k)q FN(\000)p FL(1\))820 549 y(1)1009 526 y FP(\()p FO(\025;)14 b FP(0\))g FO(e)1253 538 y FM(k)1294 526 y FO(;)99 b(X)7 b(e)1531 538 y FM(k)1594 526 y FP(=)22 b FO(F)1746 492 y FM(k)1734 547 y FL(2)1787 526 y FP(\()p FO(\025;)14 b FP(0\))g FO(e)2031 538 y FM(k)q FL(+1)2156 526 y FO(;)118 704 y FC(wher)l(e)31 b FO(\025)f FC(b)l(elongs)g(to)g(the)g(set)f FO(\033)1129 716 y FN(1)1223 704 y FP(=)23 b FQ(f)p FO(\025)g FQ(2)g FJ(R)29 b FQ(j)23 b FO(F)1696 674 y FM(k)1684 725 y FL(2)1737 704 y FP(\()p FO(\025;)14 b FP(0\))23 b FO(>)g FP(0)p FO(;)k(k)f FQ(2)d FJ(N)t FQ(g)p FP(;)243 804 y(\(iii\))p FC(.)39 b FO(H)30 b FP(=)23 b FO(l)652 816 y FL(2)689 804 y FP(\()p FJ(N)t FP(\))p FC(,)760 981 y FO(Ae)861 993 y FM(k)925 981 y FP(=)g FO(\025)1061 993 y FM(k)1102 981 y FO(e)1141 993 y FM(k)1182 981 y FO(;)99 b(X)7 b(e)1419 993 y FM(k)1481 981 y FP(=)23 b FO(\026)1619 993 y FM(k)q FN(\000)p FL(1)1745 981 y FO(e)1784 993 y FM(k)q FN(\000)p FL(1)1909 981 y FO(;)118 1159 y FC(wher)l(e)36 b FO(\025)406 1171 y FM(k)481 1159 y FP(=)d FO(F)632 1171 y FL(1)670 1159 y FP(\()p FO(\025)750 1171 y FM(k)q FL(+1)876 1159 y FP(\))p FC(,)k FO(\026)1020 1171 y FM(k)1095 1159 y FP(=)c FO(F)1246 1171 y FL(2)1283 1159 y FP(\()p FO(\025)1363 1171 y FM(k)q FL(+1)1489 1159 y FO(;)14 b(\026)1576 1171 y FM(k)q FL(+1)1701 1159 y FP(\))p FC(,)37 b FO(\026)1845 1171 y FL(1)1916 1159 y FP(=)c(0)p FC(,)k(and)f FO(\026)2335 1171 y FM(k)2410 1159 y FO(>)d FP(0)p FC(,)118 1259 y FO(k)26 b FP(=)d(2)p FC(,)29 b FO(:)14 b(:)g(:)28 b FC(.)118 1421 y(R)l(emark)38 b FP(31)p FC(.)i FP(Note)26 b(that)h(not)f(all)g (\014nite-dimensional)g(represen)n(tations)f(are)118 1521 y(necessarily)j(related)h(to)g(cycles)g(of)h(the)g(dynamical)f (system)g(\(see)h(examples)118 1620 y(b)r(elo)n(w)d(in)h(this)g (section\).)243 1751 y(The)34 b(p)r(ossibilit)n(y)h(of)g(the)g (description)f(of)h(irreducible)f(represen)n(tations)118 1851 y(in)29 b FO(H)285 1863 y FL(1)351 1851 y FP(dep)r(ends)g(on)f (top)r(ological)f(prop)r(erties)g(of)i(the)f(t)n(w)n(o-dimensional)f (dy-)118 1950 y(namical)g(system.)243 2050 y(Belo)n(w,)f(w)n(e)i (assume)e(that)i(the)g(mapping)g FO(F)12 b FP(\()p FQ(\001)p FP(\))28 b(is)f(one-to-one.)118 2212 y FR(Prop)s(osition)44 b(41.)k FC(If)41 b(the)g(dynamic)l(al)h(system)f FO(F)12 b FP(\()p FQ(\001)p FP(\))d(:)33 b FJ(R)2023 2182 y FL(2)2109 2212 y FQ(!)43 b FJ(R)2289 2182 y FL(2)2373 2212 y FC(has)e(a)118 2312 y(me)l(asur)l(able)c(se)l(ction,)h(then)d(any)h(irr)l(e)l(ducible) i(r)l(epr)l(esentation)e(is)g(unitarily)118 2411 y(e)l(quivalent)30 b(to)g(one)g(of)h(the)e(fol)t(lowing)7 b FP(:)243 2511 y(\(i\))p FC(.)39 b FO(H)30 b FP(=)22 b FJ(C)634 2481 y FM(n)686 2511 y FC(,)30 b FO(n)23 b FQ(2)g FJ(N)t FC(,)550 2874 y FO(A)g FP(=)723 2632 y Fz(0)723 2779 y(B)723 2828 y(B)723 2878 y(B)723 2931 y(@)796 2688 y FO(\025)1682 2729 y Fp(0)927 2787 y FO(F)980 2799 y FL(1)1018 2787 y FP(\()p FO(\025;)14 b(\026)p FP(\))1305 2884 y FC(.)1340 2909 y(.)1375 2934 y(.)905 3059 y Fp(0)524 b FO(F)1553 3016 y FL(\()p FM(n)p FN(\000)p FL(1\))1541 3081 y(1)1735 3059 y FP(\()p FO(\025;)14 b(\026)p FP(\))1935 2632 y Fz(1)1935 2779 y(C)1935 2828 y(C)1935 2878 y(C)1935 2931 y(A)2021 2874 y FO(;)537 3406 y(X)29 b FP(=)723 3139 y Fz(0)723 3285 y(B)723 3335 y(B)723 3385 y(B)723 3435 y(B)723 3488 y(@)920 3192 y FP(0)924 b FO(e)1925 3161 y FM(i')1996 3192 y FO(\026)796 3346 y(F)849 3358 y FL(2)886 3346 y FP(\()p FO(\025;)14 b(\026)p FP(\))1174 3288 y FC(.)1209 3313 y(.)1243 3338 y(.)1174 3443 y(.)1209 3468 y(.)1243 3493 y(.)1559 3501 y FP(0)912 3618 y Fp(0)385 b FO(F)1421 3575 y FL(\()p FM(n)p FN(\000)p FL(1\))1409 3640 y(2)1603 3618 y FP(\()p FO(\025;)14 b(\026)p FP(\))144 b(0)2046 3139 y Fz(1)2046 3285 y(C)2046 3335 y(C)2046 3385 y(C)2046 3435 y(C)2046 3488 y(A)2133 3406 y FO(;)118 3792 y FC(wher)l(e)31 b FO(\025)p FC(,)f FO(\026)g FC(b)l(elong)g(to)g (the)g(set)151 3970 y FO(\033)198 3982 y FM(n)266 3970 y FP(=)23 b FQ(f)p FO(\025)g FQ(2)g FJ(R)29 b FQ(j)23 b FO(F)739 3935 y FM(k)727 3990 y FL(2)780 3970 y FP(\()p FO(\025;)14 b(\026)p FP(\))24 b FO(>)f FP(0)p FO(;)k(k)f FP(=)d(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(n)k FQ(\000)g FP(1)p FO(;)27 b(F)1888 3935 y FN(\016)p FM(n)1967 3970 y FP(\()p FO(\025;)14 b(\026)p FP(\))24 b(=)f(\()p FO(\025;)14 b(\026)p FP(\))p FQ(g)p FO(;)118 4147 y(')23 b FQ(2)h FP([0)p FO(;)14 b FP(2)p FO(\031)s FP(\);)p eop %%Page: 119 123 119 122 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(119)243 333 y(\(ii\))p FC(.)39 b FO(H)30 b FP(=)22 b FO(l)628 345 y FL(2)665 333 y FP(\()p FJ(Z)p FP(\))p FC(,)760 499 y FO(Ae)861 511 y FM(k)925 499 y FP(=)h FO(\025)1061 511 y FM(k)1102 499 y FO(e)1141 511 y FM(k)1182 499 y FO(;)99 b(X)7 b(e)1419 511 y FM(k)1481 499 y FP(=)23 b FO(\026)1619 511 y FM(k)q FN(\000)p FL(1)1745 499 y FO(e)1784 511 y FM(k)q FN(\000)p FL(1)1909 499 y FO(;)118 666 y FC(wher)l(e)31 b FO(\025)401 678 y FM(k)465 666 y FP(=)23 b FO(F)606 678 y FL(1)643 666 y FP(\()p FO(\025)723 678 y FM(k)q FL(+1)849 666 y FP(\))p FC(,)30 b FO(\026)986 678 y FM(k)1050 666 y FP(=)23 b FO(F)1191 678 y FL(2)1228 666 y FP(\()p FO(\025)1308 678 y FM(k)q FL(+1)1434 666 y FO(;)14 b(\026)1521 678 y FM(k)q FL(+1)1646 666 y FP(\))p FC(,)30 b(and)h FO(\026)1945 678 y FM(k)2009 666 y FO(>)22 b FP(0)p FC(,)30 b FO(k)c FQ(2)d FJ(Z)o FC(.)118 820 y(R)l(emark)35 b FP(32)p FC(.)i FP(If)25 b(there)e(exists)g(an)g(ergo)r(dic)g(quasi-in)n(v)-5 b(arian)n(t)21 b(measure)i(whic)n(h)118 919 y(is)30 b(not)f(concen)n (trated)g(on)g(a)h(single)f(orbit,)h(then)g(one)f(can)h(construct)f (factor)118 1019 y(represen)n(tations)22 b(of)i(the)g(relation)f(whic)n (h)h(are)f(not)h(of)g(t)n(yp)r(e)g(I,)g(pro)n(vided)f(that)118 1118 y(all)k(second)g(co)r(ordinates)g(of)g(the)h(p)r(oin)n(ts)g(of)f (the)h(orbit)f(are)g(p)r(ositiv)n(e.)243 1245 y(In)37 b(the)g(follo)n(wing)f(subsections)g(w)n(e)g(consider)g(t)n(w)n(o)g (examples)g(of)h(rela-)118 1345 y(tions)30 b(from)g(this)h(class:)41 b(represen)n(tations)28 b(of)j(real)e(forms)h(of)g(Witten's)h(\014rst) 118 1444 y(deformation,)e(and)g(represen)n(tations)e(of)i(the)h(Skly)n (anin)f(algebra)e(in)j(the)f(de-)118 1544 y(generate)35 b(case)g(\(they)i(corresp)r(ond)e(to)h(represen)n(tations)e(of)i(the)h (quan)n(tum)118 1643 y FO(sl)182 1655 y FL(2)247 1643 y FP(group\).)118 1857 y FR(2.2.4)94 b(Represen)m(tation)30 b(of)h(real)g(forms)f(of)h(Witten's)f(\014rst)h(defor-)410 1956 y(mation)118 2109 y(1.)49 b FP(Studying)32 b(the)g(Jones)f(p)r (olynomials,)h(their)g(generalizations)d(and)j(their)118 2209 y(connections)24 b(with)i(\\v)n(ertex)d(mo)r(dels")i(in)g(t)n(w)n (o-dimensional)e(statistical)i(me-)118 2309 y(c)n(hanics,)h(Witten)i (\(see)e([301)o(]\))h(in)n(tro)r(duced)f(Hopf)h(algebra)d(deformations) i(of)118 2408 y(the)i(univ)n(ersal)e(en)n(v)n(eloping)g(algebra)g(of)i FO(su)p FP(\(2\).)36 b(There)27 b(is)g(a)h(family)f(of)h(asso-)118 2508 y(ciativ)n(e)h(algebras)e(that)j(dep)r(end)g(on)f(a)g(real)f (parameter)g FO(p)p FP(.)42 b(These)29 b(algebras)118 2608 y(are)19 b(de\014ned)i(b)n(y)f(the)h(generators)d FO(E)1228 2620 y FL(0)1265 2608 y FP(,)k FO(E)1371 2620 y FL(+)1426 2608 y FP(,)g FO(E)1532 2620 y FN(\000)1609 2608 y FP(and)e(the)h(follo)n(wing)e(relations:)849 2807 y FO(p)14 b(E)966 2819 y FL(0)1003 2807 y FO(E)1064 2819 y FL(+)1138 2807 y FQ(\000)k FO(p)1263 2773 y FN(\000)p FL(1)1365 2807 y FO(E)1426 2819 y FL(+)1482 2807 y FO(E)1543 2819 y FL(0)1603 2807 y FP(=)23 b FO(E)1752 2819 y FL(+)1807 2807 y FO(;)786 2990 y FP([)p FO(E)870 3002 y FL(+)926 2990 y FO(;)14 b(E)1024 3002 y FN(\000)1080 2990 y FP(])23 b(=)g FO(E)1275 3002 y FL(0)1331 2990 y FQ(\000)18 b FP(\()p FO(p)g FQ(\000)g FO(p)1631 2955 y FN(\000)p FL(1)1720 2990 y FP(\))c FO(E)1832 2955 y FL(2)1827 3010 y(0)1870 2990 y FO(;)854 3172 y(p)g(E)971 3184 y FN(\000)1027 3172 y FO(E)1088 3184 y FL(0)1144 3172 y FQ(\000)k FO(p)1269 3138 y FN(\000)p FL(1)1372 3172 y FO(E)1433 3184 y FL(0)1470 3172 y FO(E)1531 3184 y FN(\000)1611 3172 y FP(=)k FO(E)1759 3184 y FN(\000)1816 3172 y FO(:)524 b FP(\(2.27\))243 3350 y(In)24 b(item)h(2)f(w)n(e)f(in)n(tro)r(duce)h(a)g(class)f(of)h FQ(\003)p FP(-algebra)e(structures)i(in)g(Witten's)118 3450 y(\014rst)f(deformation.)34 b(In)23 b(items)g(3)f(an)h(4)f(w)n(e)g (giv)n(e)g(a)h(description)f(of)h(irreducible)118 3550 y(represen)n(tations)30 b(of)i(these)h FQ(\003)p FP(-algebras)c(on)j(a) f(Hilb)r(ert)i(space.)50 b(W)-7 b(e)33 b(use)e(the)118 3649 y(metho)r(d)39 b(of)g(\\dynamical)f(relations")f(dev)n(elop)r(ed)h (ab)r(o)n(v)n(e)f(in)i(this)g(section.)118 3749 y(Notice,)27 b(that)f(some)g(un)n(b)r(ounded)h(represen)n(tations)d(arise)h(in)i(a)f (natural)f(w)n(a)n(y)118 3848 y(here;)i(ho)n(w)n(ev)n(er,)e(for)i(un)n (b)r(ounded)h(op)r(erators)d(w)n(e)i(restrict)f(ourselv)n(es)g(with)h (a)118 3948 y(description)21 b(of)f(a)h(certain)f(class)g(of)h (represen)n(tations,)g(while)g(in)g(the)g(b)r(ounded)118 4048 y(case,)26 b(w)n(e)h(giv)n(e)f(a)g(complete)h(unitary)f (description)g(of)h(irreducible)f(represen-)118 4147 y(tations.)p eop %%Page: 120 124 120 123 bop 118 100 a FP(120)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FR(2.)43 b FP(Real)30 b(forms.)43 b(Witten's)31 b(\014rst)e(deformation)h(is)f(a)h(family)g(of)g(asso)r (ciativ)n(e)118 432 y(algebras)j FO(A)511 444 y FM(p)585 432 y FP(giv)n(en)i(b)n(y)g(generators)e(and)i(relations)f(\(2.27\),)j FO(p)f FQ(2)g FP(\(0)p FO(;)14 b FP(1\))35 b(is)118 532 y(a)e(parameter.)54 b FO(A)703 544 y FL(1)774 532 y FP(=)33 b FO(su)p FP(\(2\);)k FO(A)1187 544 y FL(0)1258 532 y FP(has)c(the)i(follo)n(wing)d(relations)h FO(E)2323 544 y FL(+)2378 532 y FO(E)2439 544 y FL(0)2510 532 y FP(=)118 632 y FO(E)179 644 y FL(0)217 632 y FO(E)278 644 y FL(+)356 632 y FP(=)23 b FO(E)510 601 y FL(2)505 652 y(0)570 632 y FP(=)g(0.)36 b(W)-7 b(e)26 b(consider)f(in)n(v)n(olutions)g(in)h FO(A)1800 644 y FM(p)1839 632 y FP(,)g(whic)n(h)g(are)f(obtained)118 731 y(from)j(in)n(v)n(olutions)f(in)i(the)g(free)f(algebra,)e(and)j (reduce)e(the)i(linear)f(subspace)118 831 y(generated)f(b)n(y)g(the)h (relations)e(\(2.27\))o(.)243 930 y(W)-7 b(e)31 b(will)g(sa)n(y)e(that) i(in)n(v)n(olutions)f(are)g(equiv)-5 b(alen)n(t)30 b(if)i(the)f (corresp)r(onding)118 1030 y(real)c(forms)g(are)f(isomorphic.)118 1193 y FR(Lemma)31 b(8.)42 b FC(Ther)l(e)32 b(ar)l(e)f(two)g(ine)l (quivalent)h(involutions)f(in)g(Witten)-8 b('s)31 b(\014rst)118 1292 y(deformation)6 b FP(:)938 1500 y FO(E)1004 1466 y FN(\003)999 1520 y FL(0)1065 1500 y FP(=)23 b FO(E)1214 1512 y FL(0)1252 1500 y FO(;)98 b(E)1439 1466 y FN(\003)1434 1520 y FL(+)1513 1500 y FP(=)23 b FO(E)1662 1512 y FN(\000)1718 1500 y FO(;)622 b FP(\(2.28\))913 1682 y FO(E)979 1648 y FN(\003)974 1703 y FL(0)1040 1682 y FP(=)23 b FO(E)1189 1694 y FL(0)1226 1682 y FO(;)99 b(E)1414 1648 y FN(\003)1409 1703 y FL(+)1488 1682 y FP(=)22 b FQ(\000)p FO(E)1701 1694 y FN(\000)1757 1682 y FO(:)583 b FP(\(2.29\))118 1876 y FR(3.)69 b FP(Relations)38 b(\(2.27\))g(with)h(the)g(in)n(v)n (olution)e(giv)n(en)h(b)n(y)h(\(2.28\))f(or)g(\(2.29\))118 1976 y(ha)n(v)n(e)24 b(the)i(form)f(\(2.25\))o(.)36 b(The)26 b(corresp)r(onding)d(dynamical)i(system)g(on)g FJ(R)2450 1946 y FL(2)2519 1976 y FP(is)118 2076 y(generated)i(b)n(y)411 2254 y FO(F)12 b FP(\()p FO(x;)i(y)s FP(\))23 b(=)g(\()p FO(p)853 2220 y FN(\000)p FL(1)942 2254 y FP(\(1)18 b(+)g FO(p)1159 2220 y FN(\000)p FL(1)1248 2254 y FO(x)p FP(\))p FO(;)c(q)s FP(\()p FO(q)s(y)22 b FQ(\000)c FO(x)h FP(+)f(\()p FO(p)h FQ(\000)f FO(p)1989 2220 y FN(\000)p FL(1)2078 2254 y FP(\))p FO(x)2157 2220 y FL(2)2195 2254 y FP(\)\))p FO(;)118 2432 y FP(where)27 b FO(q)f FP(=)d(1)k(\()p FO(q)f FP(=)d FQ(\000)p FP(1\))k(for)g(the)h(\014rst)g(\(second\))f (real)g(form.)243 2532 y(It)19 b(is)f(a)g(di\016cult)i(problem)e(to)g (\014nd)i(p)r(ositiv)n(e)e(orbits)g(of)g(a)h(t)n(w)n(o-dimensional)118 2631 y(nonlinear)32 b(dynamical)g(system.)52 b(W)-7 b(e)33 b(ha)n(v)n(e)f(a)n(v)n(oided)f(these)i(di\016culties)g(b)n(y)118 2731 y(using)27 b(the)h(Casimir)f(elemen)n(t,)751 2909 y FO(C)810 2921 y FM(p)871 2909 y FP(=)c FO(p)1001 2875 y FN(\000)p FL(1)1090 2909 y FO(E)1151 2921 y FL(+)1206 2909 y FO(E)1267 2921 y FN(\000)1342 2909 y FP(+)18 b FO(p)c(E)1542 2921 y FN(\000)1598 2909 y FO(E)1659 2921 y FL(+)1733 2909 y FP(+)k FO(E)1882 2875 y FL(2)1877 2930 y(0)1919 2909 y FO(:)118 3088 y FP(F)-7 b(or)29 b(an)n(y)g(irreducible)g(represen)n(tation)f FO(\031)s FP(,)i(w)n(e)f(ha)n(v)n(e)g FO(\031)s FP(\()p FO(C)1948 3100 y FM(p)1987 3088 y FP(\))e(=)f FO(\026I)7 b FP(,)30 b(where)f FO(\026)118 3187 y FP(is)f(a)f(complex)g(n)n(um)n(b)r(er.)243 3287 y(W)-7 b(e)28 b(will)f(w)n(ork)g(with)h(the)g(follo)n(wing)e (system:)634 3494 y FO(E)695 3506 y FL(0)733 3494 y FO(E)794 3506 y FL(+)872 3494 y FP(=)d FO(E)1021 3506 y FL(+)1076 3494 y FO(f)9 b FP(\()p FO(E)1219 3506 y FL(0)1257 3494 y FP(\))p FO(;)97 b(E)1475 3460 y FN(\003)1470 3515 y FL(+)1525 3494 y FO(E)1586 3506 y FL(+)1665 3494 y FP(=)22 b FO(G)1817 3506 y FM(\027)1859 3494 y FP(\()p FO(E)1952 3506 y FL(0)1990 3494 y FP(\))p FO(;)318 b FP(\(2.30\))118 3684 y(where)27 b FO(\027)i FP(=)22 b FO(\026p)p FP(,)920 3892 y FO(f)9 b FP(\()p FO(X)e FP(\))22 b(=)h FO(p)1262 3857 y FN(\000)p FL(1)1351 3892 y FP(\(1)18 b(+)g FO(p)1568 3857 y FN(\000)p FL(1)1657 3892 y FO(x)p FP(\))p FO(;)706 4074 y(G)771 4086 y FM(\027)812 4074 y FP(\()p FO(y)s FP(\))24 b(=)1132 4018 y FO(q)p 1041 4055 222 4 v 1041 4131 a FP(1)18 b(+)g FO(p)1226 4107 y FL(2)1287 4074 y FP(\()p FQ(\000)p FO(y)j FQ(\000)d FO(p)1571 4040 y FN(\000)p FL(1)1660 4074 y FO(y)1704 4040 y FL(2)1759 4074 y FP(+)g FO(\027)5 b(I)i FP(\))p FO(:)p eop %%Page: 121 125 121 124 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(121)118 333 y FR(Lemma)26 b(9.)38 b FC(F)-6 b(or)28 b(any)f(irr)l(e)l(ducible)h(r)l(epr)l(esentation)g FO(\031)i FC(of)e(the)f(r)l(e)l(al)g(form)h FO(A)2536 345 y FM(p)118 432 y FC(ther)l(e)33 b(is)f(a)h(unique)f FO(\027)37 b FP(\()p FO(\027)d FQ(\025)27 b FP(0)32 b FC(for)h(the)f(\014rst)g(r)l(e)l(al)h(form)6 b FP(\))p FC(,)34 b(such)f(that)f FO(\031)k FC(is)c(a)118 532 y(r)l(epr)l (esentation)e(of)48 b FP(\(2.30\))o FC(.)243 632 y(F)-6 b(or)22 b(an)g(arbitr)l(ary)h FO(\027)k FP(\()p FO(\027)i FQ(\025)22 b FP(0)g FC(for)h(the)f(\014rst)f(r)l(e)l(al)h(form)6 b FP(\))p FC(,)26 b(every)c(irr)l(e)l(ducible)118 731 y(r)l(epr)l(esentation)30 b(of)48 b FP(\(2.30\))29 b FC(with)i FP(dim)14 b FO(\031)26 b(>)d FP(1)29 b FC(is)h(a)g(r)l(epr)l (esentation)g(of)h FO(A)2443 743 y FM(p)2481 731 y FC(.)243 885 y FP(The)25 b(dynamical)g(system)g(corresp)r(onding)f(to)h (relations)g(\(2.30\))f(is,)i(actu-)118 984 y(ally)-7 b(,)27 b(one-dimensional,)g(linear,)g(and)g(dep)r(ends)h(on)f(one)h (real)e(parameter.)243 1084 y(Ev)n(ery)33 b(irreducible)h(represen)n (tation)f(of)i(\(2.30\))e(is)i(determined)g(b)n(y)f(the)118 1184 y(subset)28 b(\001)23 b FQ(\032)g FJ(R)607 1153 y FL(2)650 1184 y FP(,)864 1351 y(\001)g(=)g FQ(f)p FP(\()p FO(\025)1166 1363 y FM(k)1207 1351 y FO(;)14 b(\026)1294 1363 y FM(k)1335 1351 y FP(\))p FO(;)g(j)28 b(<)23 b(k)j(<)c(J)8 b FQ(g)p FO(;)118 1518 y FP(where)23 b FO(\025)402 1530 y FM(k)q FL(+1)551 1518 y FP(=)f FO(f)9 b FP(\()p FO(\025)768 1530 y FM(k)810 1518 y FP(\),)24 b FO(\026)939 1530 y FM(k)q FL(+1)1087 1518 y FP(=)f FO(G)1240 1530 y FM(\027)1282 1518 y FP(\()p FO(\025)1362 1530 y FM(k)1403 1518 y FP(\),)i FO(\026)1533 1530 y FM(k)1597 1518 y FQ(\025)e FP(0,)h FO(\026)1824 1530 y FM(k)1888 1518 y FP(=)f(0)g(for)g FO(k)j FP(=)d FO(j)15 b FP(+)c(1)22 b FO(>)118 1617 y FQ(\0001)30 b FP(and)h FO(k)g FP(=)d FO(J)g FQ(\000)20 b FP(1)28 b FO(<)g FP(+)p FQ(1)p FP(;)k FO(j)5 b FP(,)31 b FO(J)39 b FP(are)30 b(in)n(teger)g(or)f(in\014nities;)k FO(l)2245 1629 y FL(2)2282 1617 y FP(\(\001\))f(is)e(a)118 1717 y(Hilb)r(ert)e(space)f(with)h(an)g(orthonormal)d(base)i FQ(f)p FO(e)1672 1732 y FL(\()p FM(\025)1737 1741 y Fw(k)1773 1732 y FM(;\026)1833 1741 y Fw(k)1869 1732 y FL(\))1909 1717 y FP(:)g(\()p FO(\025)2039 1729 y FM(k)2081 1717 y FO(;)14 b(\026)2168 1729 y FM(k)2209 1717 y FP(\))23 b FQ(2)g FP(\001)p FQ(g)p FP(,)467 1927 y FO(T)12 b FP(\()p FO(E)621 1939 y FL(0)658 1927 y FP(\))i FO(e)743 1942 y FL(\()p FM(\025)808 1951 y Fw(k)844 1942 y FM(;\026)904 1951 y Fw(k)940 1942 y FL(\))993 1927 y FP(=)23 b FO(\025)1129 1939 y FM(k)1170 1927 y FO(e)1209 1942 y FL(\()p FM(\025)1274 1951 y Fw(k)1311 1942 y FM(;\026)1371 1951 y Fw(k)1407 1942 y FL(\))1437 1927 y FO(;)449 2109 y(T)12 b FP(\()p FO(E)603 2121 y FL(+)658 2109 y FP(\))i FO(e)743 2124 y FL(\()p FM(\025)808 2133 y Fw(k)844 2124 y FM(;\026)904 2133 y Fw(k)940 2124 y FL(\))993 2109 y FP(=)23 b FO(\026)1131 2066 y FL(1)p FM(=)p FL(2)1131 2134 y FM(k)q FL(+1)1256 2109 y FO(e)1295 2124 y FL(\()p FM(\025)1360 2133 y Fw(k)q Fy(+1)1467 2124 y FM(;\026)1527 2133 y Fw(k)q Fy(+1)1634 2124 y FL(\))1665 2109 y FO(;)180 b(j)28 b(<)22 b(k)f FP(+)d(1)23 b FO(<)g(J)o(;)448 2292 y(T)12 b FP(\()p FO(E)602 2304 y FN(\000)658 2292 y FP(\))i FO(e)743 2307 y FL(\()p FM(\025)808 2316 y Fw(k)844 2307 y FM(;\026)904 2316 y Fw(k)940 2307 y FL(\))993 2292 y FP(=)23 b FO(\026)1131 2249 y FL(1)p FM(=)p FL(2)1131 2317 y FM(k)1235 2292 y FO(e)1274 2307 y FL(\()p FM(\025)1339 2316 y Fw(k)q Fx(\000)p Fy(1)1449 2307 y FM(;\026)1509 2316 y Fw(k)q Fx(\000)p Fy(1)1619 2307 y FL(\))1649 2292 y FO(;)180 b(j)28 b(<)22 b(k)g FQ(\000)c FP(1)p FO(;)295 2475 y(T)12 b FP(\()p FO(E)449 2487 y FL(+)503 2475 y FP(\))i FO(e)588 2490 y FL(\()p FM(\025)653 2498 y Fw(J)t Fx(\000)p Fy(1)767 2490 y FM(;\026)827 2498 y Fw(J)t Fx(\000)p Fy(1)940 2490 y FL(\))993 2475 y FP(=)23 b(0)p FO(;)97 b(T)12 b FP(\()p FO(E)1397 2487 y FN(\000)1452 2475 y FP(\))i FO(e)1537 2490 y FL(\()p FM(\025)1602 2498 y Fw(j)r Fy(+1)1704 2490 y FM(;\026)1764 2498 y Fw(j)r Fy(+1)1866 2490 y FL(\))1919 2475 y FP(=)23 b(0)p FO(:)118 2696 y FR(4.)36 b FP(Classi\014cation)27 b(of)g(represen)n(tations.)118 2850 y FR(Theorem)f(25.)37 b FC(Every)28 b(irr)l(e)l(ducible)f(r)l(epr) l(esentation)g(of)g(the)g(\014rst)e(r)l(e)l(al)i(form)118 2949 y(is)j(b)l(ounde)l(d.)243 3049 y FP(1.)41 b FC(F)-6 b(or)31 b(every)h(non-ne)l(gative)f(inte)l(ger)g FO(m)g FC(ther)l(e)g(is)g(a)h(r)l(epr)l(esentation)f(of)118 3149 y(dimension)g FO(m)18 b FP(+)g(1)30 b FC(with)729 3355 y FO(\027)e FP(=)896 3299 y FO(p)p 896 3336 42 4 v 896 3412 a FP(4)961 3263 y Fz(\020\020)1070 3299 y FP(\(1)18 b FQ(\000)g FO(p)1287 3269 y FL(2)p FM(m)1383 3299 y FP(\)\(1)h(+)f FO(p)1633 3269 y FL(2)1670 3299 y FP(\))p 1070 3336 633 4 v 1070 3412 a(\(1)g(+)g FO(p)1287 3388 y FL(2)p FM(m)1383 3412 y FP(\)\(1)h FQ(\000)f FO(p)1633 3388 y FL(2)1670 3412 y FP(\))1712 3263 y Fz(\021)1762 3280 y FL(2)1818 3355 y FQ(\000)g FP(1)1943 3263 y Fz(\021)1992 3355 y FO(;)664 3571 y FP(\001)733 3583 y FM(\027)798 3571 y FP(=)23 b FQ(f)p FO(f)9 b FP(\()p FO(k)s(;)14 b(x)1140 3583 y FL(1)1176 3571 y FP(\))p FO(;)g FQ(\000)p FP(1)23 b FO(<)f(k)k FQ(\024)d FO(m)18 b FP(+)g(1)p FQ(g)p FP(;)243 3749 y(2.)52 b FC(Ther)l(e)35 b(is)g(a)g(family)h(of)f (one-dimensional)h(r)l(epr)l(esentations)7 b FP(:)48 b FO(E)2441 3761 y FL(0)2510 3749 y FP(=)118 3848 y FO(p)p FP(\()p FO(p)234 3818 y FL(2)290 3848 y FQ(\000)18 b FP(1\))447 3818 y FN(\000)p FL(1)536 3848 y FC(,)35 b FO(E)657 3860 y FL(+)743 3848 y FP(=)30 b FO(\025)p FC(,)36 b FO(E)1008 3860 y FN(\000)1095 3848 y FP(=)1194 3826 y(\026)1191 3848 y FO(\025)p FP(,)f FC(wher)l(e)g FO(\025)f FC(is)h(a)f(c)l(omplex)g(numb)l(er,)h FO(\027)h FP(=)118 3948 y FO(p)160 3918 y FL(3)197 3948 y FP(\(1)18 b FQ(\000)g FO(p)414 3918 y FL(2)451 3948 y FP(\))483 3918 y FN(\000)p FL(2)573 3948 y FP(;)243 4048 y(3.)62 b FC(F)-6 b(or)38 b(every)h FO(\027)k FQ(2)38 b FP([)p FO(p)1003 4018 y FL(3)1040 4048 y FP(\(1)19 b FQ(\000)f FO(p)1258 4018 y FL(2)1295 4048 y FP(\))1327 4018 y FN(\000)p FL(2)1416 4048 y FP(;)c(+)p FQ(1)p FP(\))38 b FC(ther)l(e)g(is)g(a)g(r)l(epr)l (esentation)118 4147 y(with)30 b(the)g(highest)h(weight,)g FP(\001)1065 4159 y FM(\027)1130 4147 y FP(=)23 b FQ(f)p FO(f)9 b FP(\()p FO(k)s(;)14 b(x)1472 4159 y FL(2)1508 4147 y FP(\))p FO(;)g(k)27 b(<)22 b FP(1)p FQ(g)p FC(.)p eop %%Page: 122 126 122 125 bop 118 100 a FP(122)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)243 333 y FP(4.)62 b FC(F)-6 b(or)38 b(every)h FO(\027)k FQ(2)38 b FP([)p FO(p)1003 303 y FL(3)1040 333 y FP(\(1)19 b FQ(\000)f FO(p)1258 303 y FL(2)1295 333 y FP(\))1327 303 y FN(\000)p FL(2)1416 333 y FP(;)c(+)p FQ(1)p FP(\))38 b FC(ther)l(e)g(is)g(a)g(r)l(epr)l(esentation)118 432 y(with)30 b(the)g(lowest)h(weight,)g FP(\001)1036 444 y FM(\027)1100 432 y FP(=)23 b FQ(f)p FO(f)9 b FP(\()p FO(k)s(;)14 b(x)1442 444 y FL(1)1479 432 y FP(\))p FO(;)g(k)26 b(<)c FP(1)p FQ(g)p FC(.)243 596 y FP(In)29 b(the)h(theorem)f(w)n(e)f (used)i(the)f(notation)g FO(f)9 b FP(\()p FO(k)s(;)14 b(x)p FP(\))26 b(=)2015 563 y FL(1)p 1982 577 99 4 v 1982 626 a FM(p)2016 609 y Fy(2)p Fw(k)2091 596 y FP(\()p FO(x)20 b FP(+)2284 559 y FM(p)2318 534 y Fy(2)p Fw(k)2383 559 y FN(\000)p FL(1)p 2284 577 184 4 v 2300 625 a FM(p)2334 608 y Fy(2)2367 625 y FN(\000)p FL(1)2478 596 y FO(p)p FP(\);)118 713 y FO(g)158 725 y FM(\027)199 713 y FP(\()p FO(x)p FP(\))k(=)f FQ(\000)p FO(x)8 b FQ(\000)g FO(p)657 683 y FN(\000)p FL(1)746 713 y FO(x)793 683 y FL(2)838 713 y FP(+)g FO(\027)d FP(,)24 b FO(x)1051 725 y FL(1)1112 713 y FO(<)e(x)1246 725 y FL(2)1307 713 y FP(are)f(ro)r(ots)h(of)g(the) h(equation)f FO(g)2247 725 y FM(\027)2288 713 y FP(\()p FO(x)p FP(\))i(=)e(0.)118 856 y FR(Theorem)32 b(26.)41 b FC(Ther)l(e)32 b(ar)l(e)g(b)l(ounde)l(d)f(and)h(unb)l(ounde)l(d)e (irr)l(e)l(ducible)j(r)l(epr)l(e-)118 956 y(sentations)c(of)h(the)f(se) l(c)l(ond)g(r)l(e)l(al)g(form)h(in)f(an)g(in\014nite-dimensional)h (Hilb)l(ert)118 1055 y(sp)l(ac)l(e,)40 b(which)f(ar)l(e)e(given,)j(exc) l(ept)c(for)i(the)f(one-dimensional)i(r)l(epr)l(esenta-)118 1155 y(tion,)30 b(by)h(the)f(fol)t(lowing)7 b FP(:)220 1298 y(1)p FO(:)41 b FC(b)l(ounde)l(d)30 b(r)l(epr)l(esentations)g (with)g(the)g(highest)h(weight)g(such)e(that)525 1453 y FO(\027)g FQ(2)23 b FP([)p FQ(\000)p FO(p=)p FP(4;)14 b FO(p)966 1419 y FL(3)1002 1453 y FP(\(1)k FQ(\000)g FO(p)1219 1419 y FL(2)1256 1453 y FP(\))1288 1419 y FN(\000)p FL(2)1377 1453 y FP(\))p FO(;)99 b FP(\001)1600 1465 y FM(\027)1665 1453 y FP(=)22 b FQ(f)p FO(f)9 b FP(\()p FO(k)s(;)14 b(x)2006 1465 y FL(1)2043 1453 y FP(\))p FO(;)g(k)26 b(<)d FP(1)p FQ(g)p FP(;)220 1635 y(2)p FO(:)41 b FC(unb)l(ounde)l(d)29 b(r)l(epr)l(esentations)7 b FP(:)358 1789 y(\()p FO(a)p FP(\))42 b FC(a)30 b(family)i(with)e(the)g(highest)h (weight)g(such)f(that)812 1944 y FO(\027)e FQ(2)c FP(\()p FQ(\000)p FO(p=)p FP(4;)i(0\))p FO(;)99 b FP(\001)1497 1956 y FM(\027)1561 1944 y FP(=)23 b FQ(f)p FO(f)9 b FP(\()p FO(k)s(;)14 b(x)1903 1956 y FL(2)1940 1944 y FP(\))p FO(;)g(k)26 b(<)c FP(1)p FQ(g)p FP(;)508 2098 y FC(a)30 b(family)i(with)e(the)g(highest)h(weight)g(such)f(that)601 2252 y FO(\027)e FQ(2)23 b FP(\()p FO(p)822 2218 y FL(3)860 2252 y FP(\(1)18 b FQ(\000)g FO(p)1077 2218 y FL(2)1114 2252 y FP(\))1146 2218 y FN(\000)p FL(2)1235 2252 y FP(;)c(+)p FQ(1)p FP(\))p FO(;)99 b FP(\001)1643 2264 y FM(\027)1708 2252 y FP(=)22 b FQ(f)p FO(f)9 b FP(\()p FO(k)s(;)14 b(x)2049 2264 y FL(1)2086 2252 y FP(\))p FO(;)g(k)26 b(>)d FQ(\000)p FP(1)p FQ(g)p FP(;)367 2418 y(\()p FO(b)p FP(\))41 b FC(a)30 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FO(q)26 b(>)d FP(0;)g FO(X)1585 4018 y FN(\003)1646 4048 y FP(=)f FQ(\000)p FO(Y)40 b FP(for)21 b FO(q)26 b(<)d FP(0,)f FO(k)2291 4018 y FN(\003)2352 4048 y FP(=)h FO(k)2486 4018 y FN(\000)p FL(1)118 4147 y FP(for)k FO(q)f FQ(2)e FJ(T)p FP(;)p eop %%Page: 125 129 125 128 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(125)243 333 y FO(su)330 345 y FM(q)366 333 y FP(\(1)p FO(;)14 b FP(1\))36 b(:)54 b FO(k)710 303 y FN(\003)786 333 y FP(=)37 b FO(k)s FP(,)i FO(X)1072 303 y FN(\003)1147 333 y FP(=)e FQ(\000)p FO(Y)55 b FP(for)36 b FO(q)k(>)e FP(0;)i FO(X)1914 303 y FN(\003)1989 333 y FP(=)d FO(Y)55 b FP(for)36 b FO(q)41 b(<)c FP(0,)118 432 y FO(k)164 402 y FN(\003)225 432 y FP(=)23 b FO(k)359 402 y FN(\000)p FL(1)475 432 y FP(for)28 b FO(q)e FQ(2)d FJ(T)p FP(;)243 532 y FO(sl)307 544 y FM(q)343 532 y FP(\(2)p FO(;)14 b FJ(R)p FP(\))37 b(:)44 b FO(X)720 502 y FN(\003)786 532 y FP(=)29 b FO(X)7 b FP(,)31 b FO(Y)1077 502 y FN(\003)1143 532 y FP(=)e FO(Y)19 b FP(,)32 b FO(k)1405 502 y FN(\003)1471 532 y FP(=)d FO(k)1611 502 y FN(\000)p FL(1)1700 532 y FP(,)j(if)f FO(q)h FQ(2)d FJ(R)p FP(;)39 b FO(k)2149 502 y FN(\003)2216 532 y FP(=)28 b FO(k)2355 502 y FN(\000)p FL(1)2475 532 y FP(for)118 632 y FO(q)e FQ(2)e FJ(R)p FP(.)243 731 y(F)-7 b(or)27 b FO(q)f FQ(2)d FJ(T)p FP(,)k(the)h FQ(\003)p FP(-algebra)d FO(su)1229 743 y FM(q)1265 731 y FP(\(1)p FO(;)14 b FP(1\))27 b(is)h(isomorphic)e(to)i FO(su)2170 743 y FM(q)2206 731 y FP(\(2\).)118 863 y FC(R)l(emark)33 b FP(34)p FC(.)j FP(In)21 b([170)o(])h FQ(\003)p FP(-Hopf)f(algebra)e(structures)i(on)g FO(U)1963 875 y FM(q)2000 863 y FP(\()p FO(sl)r FP(\(2\)\))g(are)g (stud-)118 963 y(ied.)46 b(In)31 b(our)e(list,)j(the)f(in)n(v)n (olution)e(is)i(compatible)f(with)h(the)g(Hopf)g(algebra)118 1062 y(structure)f(in)g(the)h(cases)e(of)h FO(su)1117 1074 y FM(q)1153 1062 y FP(\(2\),)h FO(su)1400 1074 y FM(q)1437 1062 y FP(\(1)p FO(;)14 b FP(1\))29 b(with)i FO(q)g FQ(2)c FJ(R)p FP(,)37 b(and)31 b FO(sl)2336 1074 y FM(q)2372 1062 y FP(\(2)p FO(;)14 b FJ(R)p FP(\))118 1162 y(with)28 b FO(q)e FQ(2)e FJ(T)p FP(.)118 1293 y FR(5.)39 b FP(No)n(w)29 b(w)n(e)f(go)g(to)g(the)h(study)g(of)g FQ(\003)p FP(-represen)n(tations)c(of)k(real)f(forms)g(of)g(the)118 1393 y(Skly)n(anin)f(algebra)f(and)h(the)h(corresp)r(onding)e(real)h (forms)g(of)g FO(U)2128 1405 y FM(q)2165 1393 y FP(\()p FO(sl)r FP(\(2\)\).)243 1493 y(First)j(notice)h(that)g(the)h(relations) d(in)j FB(F)f FP(are)f(homogeneous,)g(therefore,)118 1592 y(for)25 b(an)n(y)g FO(\025)f FQ(2)f FJ(R)p FP(,)33 b FO(\025)23 b FQ(6)p FP(=)g(0,)i(the)i(op)r(erators)c(\()p FO(S)1497 1604 y FM(i)1525 1592 y FP(\))j(form)g(represen)n(tations)e (of)h FB(F)h FP(if)118 1692 y(and)g(only)g(if)g(so)g(do)f(the)i(op)r (erators)d(\()p FO(\025S)1386 1704 y FM(i)1414 1692 y FP(\).)37 b(The)26 b(same)f(holds)h(for)f FO(U)2279 1704 y FM(q)2316 1692 y FP(\()p FO(sl)r FP(\(2\)\),)118 1792 y(and)d(w)n(e)g(agree)e(to)i(iden)n(tify)h(represen)n(tations)d(\()p FO(\025S)1696 1804 y FM(i)1725 1792 y FP(\))i(with)h(di\013eren)n(t)f FO(\025)i FQ(6)p FP(=)e(0)g(of)118 1891 y FB(F)r FP(,)h(and)h(\()p FO(k)s(;)14 b(X)r(;)g(Y)k FP(\))25 b(and)f(\()p FQ(\000)p FO(k)s(;)14 b(X)r(;)g(Y)j FP(\))25 b(of)f FO(U)1442 1903 y FM(q)1478 1891 y FP(\()p FO(sl)r FP(\(2\)\),)h(as)f(w)n(ell)f(as)h (the)g(unitarily)118 1991 y(equiv)-5 b(alen)n(t)27 b(ones.)118 2123 y FC(R)l(emark)40 b FP(35)p FC(.)i FP(In)29 b(the)h(case)e FO(J)33 b FP(=)26 b(0,)j(eac)n(h)f(irreducible)g(represen)n(tation)g (of)h FB(F)118 2222 y FP(is)g(either)f(one-dimensional)f(with)i FO(S)1290 2234 y FL(0)1352 2222 y FP(=)24 b(0,)29 b(or)e(is)i(suc)n(h)f (that)h FO(S)2142 2234 y FL(0)2204 2222 y FP(=)2303 2190 y FL(1)p 2303 2204 34 4 v 2303 2251 a(2)2346 2222 y FO(I)7 b FP(,)29 b(and)118 2322 y FO(S)169 2334 y FL(1)206 2322 y FP(,)f FO(S)308 2334 y FL(2)345 2322 y FP(,)g FO(S)447 2334 y FL(3)512 2322 y FP(form)f(an)g(irreducible)g FQ(\003)p FP(-represen)n(tation)e(of)j FO(sl)r FP(\(2\).)243 2454 y(Belo)n(w,)21 b(w)n(e)g(will)h(study)f(irreducible)g FQ(\003)p FP(-represen)n(tations)d(of)k(the)f(Skly)n(anin)118 2553 y(algebra,)28 b(and)i(of)f(the)h(deformed)f(en)n(v)n(eloping)f (algebra)g FO(U)1963 2565 y FM(q)1999 2553 y FP(\()p FO(sl)r FP(\(2\)\).)43 b(W)-7 b(e)30 b(will)118 2653 y(see)d(that)h(in)g(the)g(case)f FB(F)904 2665 y FL(1)939 2653 y FP(,)h(and)f FO(su)1238 2665 y FM(q)1274 2653 y FP(\(2\))h(with)g FO(J)j(<)23 b FP(0,)k(and)g FQ(j)p FO(q)s FQ(j)d FP(=)e(1)27 b(whic)n(h)h(is)118 2752 y(not)f(a)f(ro)r(ot) f(of)i(1,)f(the)h(description)f(problem)g(includes)g(suc)n(h)g(a)g (problem)g(for)118 2852 y(the)e(irrational)e(rotation)h(algebra,)f(and) i(is,)g(therefore,)g(rather)e(complicated.)243 2952 y(Let)30 b(us)g(b)r(egin)g(with)h FO(J)36 b(>)27 b FP(0;)k(in)f(this)h(case,)f FO(q)g FQ(2)e FJ(R)p FP(,)37 b FQ(j)p FO(q)s FQ(j)27 b FO(>)g FP(1,)k FO(A)2284 2964 y FL(1)2321 2952 y FP(,)g FO(A)2437 2964 y FL(2)2475 2952 y FP(,)g FO(k)118 3051 y FP(are)f(self-adjoin)n(t,)h(and)g FO(X)955 3021 y FN(\003)1021 3051 y FP(=)d FO(Y)50 b FP(for)30 b FO(J)37 b FQ(\024)28 b FP(1)j(\()p FO(q)g(>)e FP(1\),)i(and)g FO(X)2154 3021 y FN(\003)2220 3051 y FP(=)d FQ(\000)p FO(Y)49 b FP(for)118 3151 y FO(J)31 b(>)23 b FP(1)k(\()p FO(q)f(<)d FQ(\000)p FP(1\).)118 3315 y FR(Theorem)34 b(27.)42 b FC(If)50 b FO(J)35 b FP(=)27 b(1)p FC(,)33 b(then)f FB(F)1315 3327 y FL(1)1382 3315 y FC(has)h(the)g(fol)t(lowing)h(irr)l(e)l (ducible)g(r)l(ep-)118 3414 y(r)l(esentations)7 b FP(:)220 3578 y(1)p FO(:)41 b FC(one-dimensional)9 b FP(:)51 b FO(A)1059 3590 y FL(1)1130 3578 y FP(=)33 b FQ(\006)p FP(1)p FC(,)k FO(A)1459 3590 y FL(2)1530 3578 y FP(=)c(1)p FC(,)k FO(X)j FP(=)34 b(0)p FC(;)k FO(A)2107 3590 y FL(1)2178 3578 y FP(=)33 b FO(A)2338 3590 y FL(2)2410 3578 y FP(=)g(0)p FC(,)326 3678 y FO(X)c FQ(2)23 b FJ(T)p FP(;)220 3843 y(2)p FO(:)41 b(n)p FC(-dimensional)31 b FP(\()p FO(n)23 b FP(=)g(2)p FO(;)14 b FP(3)p FO(;)g(:)g(:)g(:)e FP(\))p FC(:)594 4023 y FO(A)656 4035 y FL(1)694 4023 y 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3085 y FP(,)36 b FO(X)584 3055 y FN(\003)621 3085 y FB(H)697 3055 y FN(?)696 3106 y FL(3)785 3085 y FQ(\032)e FB(H)959 3097 y FL(0)1017 3085 y FQ(\010)22 b FB(H)1179 3097 y FL(1)1215 3085 y FP(.)56 b(Set)34 b FB(H)1519 3042 y FL(\(1\))1518 3107 y(1)1640 3085 y FP(=)f FQ(f)p FO(f)42 b FQ(2)34 b FB(H)2027 3097 y FL(1)2096 3085 y FQ(j)f FO(X)7 b(f)42 b FQ(2)34 b FB(H)2475 3097 y FL(2)2510 3085 y FQ(g)p FP(,)118 3202 y FB(H)194 3159 y FL(\()p FM(n)p FL(+1\))193 3225 y(1)410 3202 y FP(=)j FQ(f)p FO(f)44 b FQ(2)37 b FB(H)806 3214 y FL(1)879 3202 y FQ(j)g FO(X)7 b(f)t(;)14 b(:)g(:)g(:)26 b(;)14 b(X)1333 3172 y FM(n)1378 3202 y FO(f)45 b FQ(2)37 b FB(H)1631 3214 y FL(0)1667 3202 y FO(;)27 b(X)1793 3172 y FM(n)p FL(+1)1922 3202 y FO(f)45 b FQ(2)37 b FB(H)2175 3214 y FL(2)2211 3202 y FQ(g)p FP(,)g FO(n)g FQ(2)g FJ(N)t FP(;)118 3320 y FB(H)194 3276 y FL(\()p FN(1)p FL(\))193 3342 y(1)338 3320 y FP(=)24 b FQ(f)p FO(f)31 b FQ(2)24 b FB(H)695 3332 y FL(1)754 3320 y FQ(j)g FO(X)877 3289 y FM(n)922 3320 y FO(f)32 b FQ(2)24 b FB(H)1149 3332 y FL(0)1184 3320 y FO(;)k(n)c FQ(2)f FJ(N)t FQ(g)p FP(,)34 b FB(H)1616 3276 y FL(\()p FN(1)p FL(\))1615 3342 y(2)1760 3320 y FP(=)24 b FQ(f)p FO(f)31 b FQ(2)24 b FB(H)2117 3332 y FL(2)2176 3320 y FQ(j)g FP(\()p FO(X)2331 3289 y FN(\003)2369 3320 y FP(\))2401 3289 y FM(n)2446 3320 y FO(f)32 b FQ(2)118 3419 y FB(H)193 3431 y FL(0)229 3419 y FO(;)c(n)22 b FQ(2)i FJ(N)t FQ(g)p FP(.)42 b(Then:)197 3611 y(\(i\))g(the)37 b(subspace)f FB(H)910 3581 y FL(\()p FM(n)p FL(\))1044 3611 y FP(=)i FQ(\010)1212 3581 y FM(n)1212 3633 y(j)s FL(=0)1330 3611 y FO(X)1406 3581 y FM(j)1441 3611 y FB(H)1517 3568 y FL(\()p FM(n)p FL(\))1516 3634 y(1)1649 3611 y FP(is)f(in)n(v)-5 b(arian)n(t)35 b(and)i(the)g(irre-)326 3739 y(ducibilit)n(y)31 b(implies)g(that)g(dim)15 b FO(X)1395 3709 y FM(n)1439 3739 y FB(H)1515 3696 y FL(\()p FM(n)p FL(\))1514 3762 y(1)1639 3739 y FP(=)29 b(1,)i FO(j)j FP(=)28 b(1,)j(2,)g FO(:)14 b(:)g(:)28 b FP(;)k(there-)326 3839 y(fore)27 b(dim)14 b FB(H)718 3809 y FL(\()p FM(n)p FL(\))837 3839 y FP(=)22 b FO(n)c FP(+)g(1;)173 4030 y(\(ii\))43 b(the)38 b(subspace)853 4008 y(~)837 4030 y FB(H)912 4042 y FL(1)988 4030 y FP(=)j FQ(\010)1159 4000 y FN(1)1159 4051 y FM(n)p FL(=0)1288 4030 y FO(X)1364 4000 y FM(n)1408 4030 y FB(H)1484 3987 y FL(\()p FN(1)p FL(\))1483 4052 y(1)1643 4030 y FP(is)d(in)n(v)-5 b(arian)n(t)37 b(and)h(the)h(irre-)326 4147 y(ducibilit)n(y)28 b(implies)f(that)h(dim)15 b FO(X)1385 4117 y FM(n)1429 4147 y FB(H)1505 4104 y FL(\()p FN(1)p FL(\))1504 4169 y(1)1649 4147 y FP(=)23 b(1,)k FO(n)c FQ(2)g FJ(N)t FP(;)p eop %%Page: 127 131 127 130 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(127)150 336 y(\(iii\))43 b(the)29 b(subspace)834 314 y(~)818 336 y FB(H)893 348 y FL(2)954 336 y FP(=)24 b FQ(\010)1108 306 y FN(1)1108 357 y FM(n)p FL(=0)1237 336 y FO(X)1313 306 y FM(n)1357 336 y FB(H)1433 293 y FL(\()p FN(1)p FL(\))1432 358 y(2)1583 336 y FP(is)29 b(in)n(v)-5 b(arian)n(t)27 b(and)i(in)g(the)g(irre-)326 453 y(ducible)f(case,)e(dim)15 b FO(X)1039 423 y FM(n)1083 453 y FB(H)1159 410 y FL(\()p FN(1)p FL(\))1158 476 y(2)1303 453 y FP(=)23 b(1,)k FO(n)c FQ(2)g FJ(N)t FP(.)118 630 y(On)33 b(the)g(in)n(v)-5 b(arian)n(t)32 b(subspace)1133 608 y(~)1116 630 y FB(H)1191 642 y FL(0)1259 630 y FP(=)g FQ(f)p FO(f)40 b FQ(2)32 b FB(H)1641 642 y FL(0)1709 630 y FQ(j)g FO(X)1840 600 y FM(n)1884 630 y FO(f)41 b FQ(2)32 b FB(H)2128 642 y FL(0)2164 630 y FO(;)c FP(\()p FO(X)2323 600 y FN(\003)2360 630 y FP(\))2392 600 y FM(n)2438 630 y FO(f)40 b FQ(2)118 730 y FB(H)193 742 y FL(0)229 730 y FP(;)28 b FO(n)i FQ(2)g FJ(N)t FQ(g)p FP(,)39 b(the)32 b(op)r(erator)e FO(X)38 b FP(is)32 b(normal,)g(and)g(irreducible)f (represen)n(ta-)118 829 y(tions)d(are)e(one-dimensional.)243 930 y(Finally)20 b(w)n(e)h(note)g(that)g(for)f(an)n(y)g(represen)n (tation,)h(w)n(e)f(ha)n(v)n(e)g(the)h(follo)n(wing)118 1029 y(decomp)r(osition:)663 1275 y FB(H)h FP(=)877 1171 y FN(1)848 1196 y Fz(M)848 1372 y FM(n)p FL(=1)987 1275 y FB(H)1063 1241 y FL(\()p FM(n)p FL(\))1177 1275 y FQ(\010)1277 1252 y FP(~)1260 1275 y FB(H)1335 1287 y FL(0)1389 1275 y FQ(\010)1489 1252 y FP(~)1472 1275 y FB(H)1547 1287 y FL(1)1601 1275 y FQ(\010)1701 1252 y FP(~)1684 1275 y FB(H)1759 1287 y FL(2)1814 1275 y FQ(\010)c FB(H)1972 1287 y FL(3)2007 1275 y FO(;)118 1530 y FP(whic)n(h)28 b(completes)f(the)h(pro)r(of.)p 2514 1530 4 57 v 2518 1478 50 4 v 2518 1530 V 2567 1530 4 57 v 118 1701 a FR(Theorem)44 b(28.)j FC(F)-6 b(or)41 b FO(q)k FQ(2)e FJ(R)p FC(,)49 b FQ(j)p FO(q)s FQ(j)43 b FO(>)f FP(1)p FC(,)g(the)f FQ(\003)p FC(-algebr)l(a)g FO(su)2125 1713 y FM(q)2161 1701 y FP(\(2\))f FC(has)i(the)118 1801 y(fol)t(lowing)32 b(irr)l(e)l(ducible)f(r)l(epr)l(esentations)37 b FP(\()p FO(n)23 b FQ(2)h FJ(N)t FP(\):)343 1985 y FO(k)s(f)430 1997 y FM(j)488 1985 y FP(=)e FQ(j)p FO(q)s FQ(j)661 1951 y FL(\(1)p FN(\000)p FM(n)p FL(\))p FM(=)p FL(2)911 1985 y FO(q)951 1951 y FM(j)s FN(\000)p FL(1)1071 1985 y FO(f)1112 1997 y FM(j)1146 1985 y FO(;)99 b(X)7 b(f)1385 1997 y FM(j)1442 1985 y FP(=)23 b(\()p FQ(j)p FP([)p FO(j)5 b FP(])1670 1997 y FM(q)1721 1985 y FP([)p FO(n)18 b FQ(\000)g FO(j)5 b FP(])1957 1997 y FM(q)1994 1985 y FQ(j)p FP(\))2049 1951 y FL(1)p FM(=)p FL(2)2153 1985 y FO(f)2194 1997 y FM(j)s FL(+1)2313 1985 y FO(;)1088 2109 y(j)28 b FP(=)22 b(1)p FO(;)14 b FP(2)p FO(;)g(:)g(:)g(:)27 b(;)14 b(n)118 2293 y FP(\()p FC(we)30 b(use)g(a)g(standar)l(d)g (notation)g FP([)p FO(m)p FP(])1280 2305 y FM(q)1340 2293 y FP(=)23 b(\()p FO(q)1500 2263 y FM(m)1582 2293 y FQ(\000)18 b FO(q)1705 2263 y FN(\000)p FM(m)1820 2293 y FP(\))p FO(=)p FP(\()p FO(q)j FQ(\000)d FO(q)2107 2263 y FN(\000)p FL(1)2196 2293 y FP(\)\))p FC(.)118 2461 y(Pr)l(o)l(of.)43 b FP(Apply)f(Prop)r(ositions)d(40)h(and)g(41)g(to)h (the)g(self-adjoin)n(t)g(op)r(erator)118 2561 y FO(A)27 b FP(=)g FO(k)33 b FP(and)d(the)g(map)g FO(F)12 b FP(\()p FO(l)r(;)i(\025)p FP(\))27 b(=)g(\()p FO(q)s(l)r(;)14 b(\025)20 b FQ(\000)g FP(\()p FO(l)1579 2531 y FL(2)1577 2581 y(1)1634 2561 y FQ(\000)e FO(l)1744 2525 y FN(\000)p FL(2)1742 2583 y(2)1833 2561 y FP(\))p FO(=)p FQ(j)p FO(q)j FQ(\000)d FO(q)2111 2531 y FN(\000)p FL(1)2200 2561 y FQ(j)p FP(\).)45 b(W)-7 b(e)30 b(see)118 2660 y(that)265 2877 y FO(F)330 2842 y FM(n)375 2877 y FP(\()p FO(l)r(;)14 b(\025)p FP(\))23 b(=)g FO(\025)c FQ(\000)f FP([)p FO(n)p FP(])908 2889 y FM(q)968 2820 y FO(l)995 2790 y FL(2)1032 2820 y FO(q)1072 2790 y FM(n)p FN(\000)p FL(1)1220 2820 y FQ(\000)g FO(l)1330 2790 y FN(\000)p FL(2)1419 2820 y FO(q)1459 2790 y FL(1)p FN(\000)p FM(n)p 968 2858 622 4 v 1120 2934 a FQ(j)p FO(q)k FQ(\000)c FO(q)1325 2910 y FN(\000)p FL(1)1414 2934 y FQ(j)1622 2877 y(!)23 b(\0001)p FO(;)180 b(n)23 b FQ(!)g(\0061)p FO(:)118 3105 y FP(It)28 b(means)f(that)h(only)f(the)h (\014nite-dimensional)f(case)g(is)h(realized.)p 2514 3105 4 57 v 2518 3052 50 4 v 2518 3105 V 2567 3105 4 57 v 243 3275 a(No)n(w)f(w)n(e)g(consider)g(the)h(case)e(where)h FO(J)k(>)23 b FP(1.)118 3443 y FR(Prop)s(osition)36 b(44.)44 b FC(F)-6 b(or)34 b FO(J)40 b(>)31 b FP(0)p FC(,)36 b FO(J)j FQ(6)p FP(=)31 b(1)p FC(,)36 b(the)f FQ(\003)p FC(-algebr)l(a)g FB(F)2091 3455 y FL(1)2160 3443 y FC(has)g(the)g(fol-) 118 3543 y(lowing)c(irr)l(e)l(ducible)g(r)l(epr)l(esentations)7 b FP(:)220 3711 y(1)p FO(:)41 b FC(one-dimensional)9 b FP(:)39 b FO(A)1047 3723 y FL(1)1108 3711 y FP(=)23 b FO(A)1258 3723 y FL(2)1318 3711 y FP(=)g(0)p FO(;)43 b(X)29 b FQ(2)24 b FJ(T)p FP(;)220 3879 y(2)p FO(:)41 b(n)p FC(-dimensional)32 b FP(\()p FO(n)23 b FQ(2)h FJ(N)t FP(\):)41 b FO(A)1265 3891 y FL(1)1325 3879 y FP(=)23 b FO(k)s FC(,)h FO(A)1570 3891 y FL(2)1631 3879 y FP(=)e FQ(\006)p FO(k)1829 3849 y FN(\000)p FL(1)1918 3879 y FP(;)j FO(k)s FC(,)f FO(X)7 b FC(,)23 b FO(Y)42 b FC(gener)l(ate)326 3979 y(an)29 b(irr)l(e)l(ducible)j(r)l(epr)l(esentation)e(of)g FO(su)1573 3991 y FM(q)1609 3979 y FP(\(2\);)220 4147 y(3)p FO(:)41 b FC(in\014nite-dimensional)9 b FP(:)p eop %%Page: 128 132 128 131 bop 118 100 a FP(128)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)374 333 y FP(\()p FO(i)p FP(\))41 b FO(A)570 345 y FL(1)608 333 y FO(f)649 345 y FM(n)717 333 y FP(=)22 b FO(q)844 303 y FN(\000)p FM(n)941 333 y FO(f)982 345 y FM(n)1027 333 y FC(,)30 b FO(A)1144 345 y FL(2)1205 333 y FP(=)23 b(0)p FC(,)671 581 y FO(X)7 b(f)788 593 y FM(n)855 581 y FP(=)942 464 y Fz(\022)1013 525 y FP([)p FO(n)p FP(])1109 537 y FM(q)1160 525 y FO(q)1200 494 y FL(1)p FN(\000)p FM(n)p 1013 562 317 4 v 1013 638 a FQ(j)p FO(q)22 b FQ(\000)c FO(q)1218 614 y FN(\000)p FL(1)1307 638 y FQ(j)1340 464 y Fz(\023)1401 481 y FL(1)p FM(=)p FL(2)1506 581 y FO(f)1547 593 y FM(n)p FN(\000)p FL(1)1676 581 y FO(;)184 b(n)23 b FP(=)g(0)p FO(;)14 b FP(1)p FO(;)g FP(2)p FO(;)g(:)g(:)g(:)e FP(;)345 825 y(\()p FO(ii)p FP(\))41 b FO(A)570 837 y FL(1)631 825 y FP(=)23 b(0)p FC(,)29 b FO(A)877 837 y FL(2)915 825 y FO(f)956 837 y FM(n)1024 825 y FP(=)22 b FO(q)1151 794 y FN(\000)p FM(n)1248 825 y 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FL(2)1356 1583 y FO(;)28 b(\025)19 b FQ(\000)f FP(\()p FO(l)1616 1549 y FL(2)1614 1603 y(1)1671 1583 y FQ(\000)g FO(l)1781 1549 y FL(2)1779 1603 y(2)1818 1583 y FP(\))p FO(=)p FQ(j)p FO(q)j FQ(\000)d FO(q)2096 1549 y FN(\000)p FL(1)2186 1583 y FQ(j)2209 1516 y Fz(\001)2247 1583 y FO(:)118 1758 y FP(The)31 b(orbits)f(with)h FO(l)748 1770 y FL(1)785 1758 y FO(l)810 1770 y FL(2)875 1758 y FQ(6)p FP(=)d(0)i(giv)n(e)g(the)h(case)f(2\);)i(the)f(orbits)f(with)h FO(l)2270 1770 y FL(1)2336 1758 y FP(=)c(0)k(or)118 1858 y FO(l)143 1870 y FL(2)203 1858 y FP(=)23 b(0)k(giv)n(e)g(the)h(cases)e (1\))i(and)f(3\).)p 2514 1858 4 57 v 2518 1805 50 4 v 2518 1858 V 2567 1858 4 57 v 243 2022 a(No)n(w)i(consider)h(the)g(case) g FO(J)35 b(<)27 b FP(0;)k(no)n(w)f FO(q)h FP(=)c FO(e)1729 1992 y FM(i\021)1820 2022 y FQ(2)h FJ(T)p FP(,)i(0)d FO(<)g(\021)k(<)c(\031)s FP(,)k(and)118 2122 y FO(A)180 2092 y FN(\003)180 2142 y FL(1)242 2122 y FP(=)22 b FO(A)391 2134 y FL(2)429 2122 y FO(;)41 b(k)539 2092 y 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FP(=)c(1)p FO(;)96 b(l)841 3034 y FL(1)901 3022 y FP(=)23 b FO(a)1033 3034 y FL(1)1070 3022 y FO(;)97 b(l)1215 3034 y FM(p)1276 3022 y FP(=)23 b FO(l)1389 3034 y FM(p)p FN(\000)p FL(2)1530 3022 y FP(+)18 b FO(a)1657 3034 y FM(p)1695 3022 y FO(l)1720 3034 y FM(p)1754 3042 y Fy(1)1791 3022 y FO(;)180 b(p)23 b FQ(\025)f FP(2)p FO(:)118 3197 y FP(Then)32 b(de\014ne)h(an)e (increasing)g(sequence)g(of)h(in)n(tegers)f(\(\014nite)i(if)39 b FO(q)c FP(is)d(a)g(ro)r(ot)118 3297 y(of)c(1\):)423 3472 y FO(r)462 3438 y FL(\(1\))460 3492 y FM(m)575 3472 y FP(=)23 b FO(m;)179 b(m)23 b FP(=)g(1)p FO(;)14 b FP(2)p FO(;)g(:)g(:)g(:)f(;)h(a)1471 3484 y FL(1)1508 3472 y FO(;)422 3613 y(r)461 3579 y FL(\()p FM(p)p FL(\))459 3633 y FM(m)575 3613 y FP(=)23 b FO(l)688 3625 y FM(p)p FN(\000)p FL(2)829 3613 y FP(+)18 b FO(m)c(l)1024 3625 y FM(p)p FN(\000)p FL(1)1147 3613 y FO(;)180 b(m)23 b FP(=)g(1)p FO(;)14 b FP(2)p FO(;)g(:)g(:)g(:)26 b(;)14 b(a)1896 3625 y FM(p)1934 3613 y FP(;)97 b FO(p)23 b FQ(\025)f FP(2)p FO(:)118 3788 y FP(F)-7 b(or)28 b(example,)g(when)h FO(\021)e FP(=)d FO(\031)s(=)-5 b(N)9 b FP(,)29 b(these)f(n)n(um)n(b)r (ers)g(are)g(1,)g(2,)g FO(:)14 b(:)g(:)27 b FP(,)i FO(N)9 b FP(;)29 b(when)118 3888 y FO(\021)d FP(=)d FO(\031)s FP(\()355 3819 y FQ(p)p 425 3819 42 4 v 425 3888 a FP(5)18 b FQ(\000)g FP(1\))p FO(=)p FP(2,)26 b(they)i(are)f(the)h(Fib)r(onacci) f(sequence)g(1,)g(2,)h(3,)f(5,)g FO(:)14 b(:)g(:)28 b FP(.)118 4048 y FR(Theorem)c(29.)36 b FC(F)-6 b(or)34 b FO(q)26 b FQ(2)d FJ(T)h FC(such)h(that)33 b FO(q)28 b FC(is)d(not)f(a)h(r)l(o)l(ot)g(of)h FP(1)p FC(,)f(the)g FQ(\003)p FC(-algebr)l(a)118 4147 y FO(su)205 4159 y FM(q)241 4147 y FP(\(2\))30 b FC(has)h(the)e(fol)t(lowing)k(irr)l(e)l (ducible)e(r)l(epr)l(esentations)7 b FP(:)p eop %%Page: 129 133 129 132 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(129)220 336 y(1)p FO(:)41 b FC (\014nite-dimensional)30 b(with)h(dimension)g FO(n)23 b FP(=)f FO(r)1781 293 y FL(\()p FM(p)p FL(\))1779 346 y FM(m)1872 336 y FC(,)30 b FO(p)23 b FQ(2)h FJ(N)t FP(:)763 526 y FO(k)s(f)850 538 y FM(j)907 526 y FP(=)f FO(i)14 b FP(exp)1164 459 y Fz(\000)1202 526 y FO(i)1242 492 y FM(c)1276 526 y FP(\(\(1)19 b FQ(\000)f FO(n)p FP(\))p FO(=)p FP(2)f(+)h FO(j)24 b FQ(\000)18 b FP(1\))p FO(\021)s FP(\))c FO(f)2096 538 y FM(j)2130 526 y FO(;)733 671 y(X)7 b(f)850 683 y FM(j)907 671 y FP(=)995 596 y Fz(p)p 1078 596 634 4 v 75 x FP(\()p FQ(\000)p FP(1\))1249 647 y FM(c)1296 671 y FP([)p FO(j)e FP(])p FO(q)17 b FP([)p FO(n)i FQ(\000)f FO(j)5 b FP(])p FO(q)16 b(f)1766 683 y FM(j)s FL(+1)1885 671 y FO(;)455 b FP(\(2.32\))326 860 y FC(wher)l(e)25 b FO(c)e FP(=)f(0)p FC(,)k FP(1)e FC(is)h(such)f(that)h(the)f(expr)l(ession)h(under)g(the)f(squar)l(e)h (r)l(o)l(ot)326 960 y(is)30 b(p)l(ositive)6 b FP(;)220 1140 y(2)p FO(:)41 b FC(two)30 b(de)l(gener)l(ate)g (in\014nite-dimensional)9 b FP(:)39 b FC(for)31 b FO(j)d FP(=)22 b(1)p FC(,)30 b FP(2)p FC(,)g FO(:)14 b(:)g(:)27 b FC(,)862 1329 y FO(k)s(f)949 1341 y FM(j)1007 1329 y FP(=)c FO(i)14 b FP(exp)o(\()p FO(i)p FP(\()p FO(\021)s(=)p FP(2)k(+)g(\()p FO(j)23 b FQ(\000)18 b FP(1\))p FO(\021)s FP(\)\))c FO(f)1995 1341 y FM(j)2031 1329 y FO(;)833 1454 y(X)7 b(f)950 1466 y FM(j)1007 1454 y FP(=)23 b([)p FO(j)5 b FP(])p FO(q)17 b(f)1275 1466 y FM(j)s FL(+1)1393 1454 y FP(;)326 1643 y FC(and)849 1833 y FO(k)s(f)936 1845 y FM(j)993 1833 y FP(=)23 b FO(i)14 b FP(exp)o(\()p FQ(\000)p FO(i)p FP(\()p FO(\021)s(=)p FP(2)k(+)g(\()p FO(j)23 b FQ(\000)18 b FP(1\))p FO(\021)s FP(\)\))c FO(f)2046 1845 y FM(j)819 1957 y FO(X)7 b(f)936 1969 y FM(j)993 1957 y FP(=)23 b([)p FO(j)g FQ(\000)18 b FP(1])p FO(q)f(f)1404 1969 y FM(j)s FN(\000)p FL(1)1523 1957 y FP(;)220 2187 y(3)p FO(:)41 b FC(non-de)l(gener)l(ate)25 b(in\014nite-dimensional,)j (which)f(ar)l(e)f(r)l(epr)l(esentations)326 2286 y(of)k(the)g(irr)l (ational)h(r)l(otation)f(algebr)l(a)i FB(A)1591 2298 y FM(\021)1631 2286 y FC(.)118 2463 y(Pr)l(o)l(of.)43 b FP(No)n(w)30 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b(to)f([\()p FO(n)19 b FQ(\000)f FO(j)5 b FP(\))p FO(\021)s(=\031)s FP(])17 b(+)f([)p FO(j)5 b(\021)t(=\031)s FP(])23 b(=)f(const.)36 b(The)27 b(n)n(um)n(b)r(ers)g (satisfying)118 3260 y(this)h(condition)f(are)g FO(n)c FP(=)g FO(r)983 3217 y FL(\()p FM(p)p FL(\))981 3269 y FM(m)1074 3260 y FP(.)243 3363 y(Let)g FO(X)463 3333 y FN(\003)523 3363 y FP(=)g FO(U)677 3293 y FQ(p)p 746 3293 68 4 v 70 x FO(B)k FP(and)c FO(U)32 b FP(b)r(e)24 b(unitary)-7 b(.)35 b(If)23 b(the)g(cen)n(tral)g(elemen)n(t)g FO(C)29 b FP(=)23 b FO(cI)7 b FP(,)118 3462 y(then)28 b FO(c)23 b(>)g FP(1)p FO(=)p FP(2)j(and)i(the)g(op)r(erators)e FO(B)h FP(=)c FO(cI)i FP(+)18 b(\()p FO(e)1708 3432 y FN(\000)p FM(i\021)1824 3462 y FO(k)1870 3432 y FL(2)1926 3462 y FP(+)g FO(e)2048 3432 y FM(i\021)2111 3462 y FO(k)2157 3432 y FN(\000)p FL(2)2246 3462 y FP(\))p FO(=)p FP(4,)27 b FO(k)s FP(,)h FO(U)118 3562 y FP(form)34 b(a)f(represen)n(tation)f (of)i(the)g(irrational)e(rotation)h(algebra)f FB(A)2252 3574 y FM(\021)2293 3562 y FP(:)49 b FO(U)9 b(k)36 b FP(=)118 3662 y FO(e)157 3632 y FL(\()p FM(i\021)r FL(\))273 3662 y FO(k)s(U)9 b FP(.)p 2514 3662 4 57 v 2518 3609 50 4 v 2518 3662 V 2567 3662 4 57 v 118 3855 a FR(Theorem)34 b(30.)42 b FC(L)l(et)31 b FO(q)g FQ(2)d FJ(T)j FC(b)l(e)i(a)f(r)l(o)l (ot)g(of)48 b FP(1)p FC(,)32 b FO(\021)f FP(=)c FO(\031)s(M)9 b(=)-5 b(N)9 b FC(.)46 b(Then)33 b FO(su)2433 3867 y FM(q)2469 3855 y FP(\(2\))118 3955 y FC(has)26 b(the)f(fol)t(lowing)i (irr)l(e)l(ducible)g(r)l(epr)l(esentations)k FP(\()p FC(al)t(l)c(\014nite-dimensional)9 b FP(\):)220 4147 y(1)p FO(:)41 b FC(or)l(dinary)7 b FP(:)40 b FO(n)23 b FP(=)f FO(r)899 4104 y FL(\()p FM(p)p FL(\))897 4157 y FM(m)990 4147 y FC(,)30 b FP(1)23 b FQ(\024)g FO(n)g FQ(\024)f FO(N)9 b FP(;)30 b FO(k)i FC(and)f FO(X)36 b FC(ar)l(e)30 b(as)g(in)59 b FP(\(2.32\))o(;)p eop %%Page: 130 134 130 133 bop 118 100 a FP(130)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)217 333 y FP(2)p FC(.)42 b(c)l(ontinuous)36 b(series)7 b FP(:)54 b FO(n)37 b FP(=)f FO(N)46 b FC(for)39 b FO(M)46 b FC(even,)39 b(and)f FO(n)f FP(=)f(2)p FO(N)46 b FC(for)38 b FO(M)326 432 y FC(o)l(dd)9 b FP(:)665 614 y FO(k)s(f)752 626 y FM(j)810 614 y FP(=)22 b(exp\()p FO(\036)d FP(+)f(\()p FO(j)24 b FQ(\000)18 b FP(1\))p FO(\021)s FP(\))c FO(f)1585 626 y FM(j)1620 614 y FO(;)636 821 y(X)7 b(f)753 833 y FM(j)810 821 y FP(=)897 704 y Fz(\022)958 821 y FO(h)19 b FQ(\000)1118 764 y FP(sin\(2)p FO(\036)f FP(+)g(\()p FO(j)24 b FQ(\000)18 b FP(1\))p FO(\021)s FP(\))p 1118 801 650 4 v 1363 878 a(sin)13 b FO(\021)1791 821 y FP([)p FO(j)5 b FP(])1876 833 y FM(q)1913 704 y Fz(\023)1974 721 y FL(1)p FM(=)p FL(2)2078 821 y FO(f)2119 833 y FM(j)s FL(+1)2238 821 y FO(;)956 1002 y(j)28 b FP(=)23 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n)k FQ(\000)g FP(1)p FO(;)626 1147 y(X)7 b(f)743 1159 y FM(n)810 1147 y FP(=)22 b FO(E)963 1113 y FM(i\026)1031 1072 y FQ(p)p 1101 1072 48 4 v 1101 1147 a FO(h)o(f)1189 1159 y FL(1)1226 1147 y FO(;)326 1329 y FC(wher)l(e)29 b FO(\036)24 b FQ(2)f FO(\034)33 b FP(=)22 b([\()p FO(\031)g FP(+)c FO(\021)s FP(\))q FO(=)p FP(2)d FQ(\000)h FO(\031)s(=)p FP(2)p FO(n)o(;)e FP(\()p FO(\031)22 b FP(+)c FO(\021)s FP(\))p FO(=)p FP(2)e(+)f FO(\031)s(=)p FP(2)p FO(N)8 b FP(\))p FC(,)30 b FO(e)2325 1299 y FM(i\026)2416 1329 y FQ(2)23 b FJ(T)p FC(.)118 1494 y(Pr)l(o)l(of.)43 b FP(The)32 b(mapping)g FO(F)12 b FP(\()p FQ(\001)p FP(\))33 b(is)f(de\014ned)g(for)f(all)h FO(q)i FQ(2)c FJ(T)i FP(in)g(the)h(same) e(w)n(a)n(y)-7 b(.)118 1594 y(The)24 b(di\013erence)g(from)g(the)h (previous)e(case)g(is)h(that)h(eac)n(h)e(orbit)h(no)n(w)f(is)h(cyclic) 118 1693 y(with)k(the)g(p)r(erio)r(d)g FO(n)23 b FP(=)f(min)q(\()p FO(l)11 b FP(:)27 b FO(e)1165 1663 y FM(il\021)1273 1693 y FP(=)c(1\).)243 1793 y(Note)39 b(that)h(the)g(dynamical)f(system)g (in)h(this)g(situation)f(has)g(a)g(Borel)118 1893 y(section)30 b FO(\034)g FQ(\002)20 b FJ(R)603 1905 y FL(+)664 1893 y FP(.)45 b(It)31 b(means)f(that)g(eac)n(h)g(irreducible)g(represen)n (tation)e(with)118 1992 y(unitary)f FO(U)37 b FP(corresp)r(onds)25 b(to)j(a)f(single)g(orbit.)p 2514 1992 4 57 v 2518 1939 50 4 v 2518 1992 V 2567 1992 4 57 v 118 2158 a FR(Theorem)36 b(31.)44 b FC(F)-6 b(or)34 b FO(J)40 b(<)30 b FP(0)p FC(,)36 b(the)e FQ(\003)p FC(-algebr)l(a)h FB(F)1694 2170 y FL(1)1763 2158 y FC(has)g(the)g(fol)t(lowing)i(irr)l(e-)118 2258 y(ducible)31 b(r)l(epr)l(esentations)7 b FP(:)220 2423 y(1)p FO(:)41 b FC(one-dimensional)9 b FP(:)39 b FO(A)1047 2435 y FL(1)1108 2423 y FP(=)23 b FO(A)1258 2435 y FL(2)1318 2423 y FP(=)g(0)p FC(,)29 b FO(X)h FQ(2)23 b FJ(T)p FP(;)220 2589 y(2)p FO(:)41 b(A)388 2601 y FL(1)450 2589 y FP(=)24 b FO(k)s FC(,)31 b FO(A)703 2601 y FL(2)765 2589 y FP(=)24 b FO(k)900 2559 y FN(\000)p FL(1)989 2589 y FP(;)31 b FO(k)s(;)44 b(X)r(;)g(Y)50 b FC(gener)l(ate)30 b(an)h(irr)l(e)l(ducible)h(r)l(epr)l(esen-)326 2688 y(tation)e(of)g FO(su)752 2700 y FM(q)788 2688 y FP(\(2\))p FC(.)118 2854 y(Pr)l(o)l(of.)43 b FP(Recall)35 b(that)h FO(A)886 2824 y FN(\003)886 2874 y FL(1)960 2854 y FP(=)g FO(A)1123 2866 y FL(2)1196 2854 y FP(and)f FO(A)1427 2824 y FN(\003)1427 2874 y FL(1)1466 2854 y FO(A)1528 2866 y FL(1)1601 2854 y FP(=)h FO(A)1764 2866 y FL(1)1801 2854 y FO(A)1863 2824 y FN(\003)1863 2874 y FL(1)1937 2854 y FP(comm)n(ute)g(with)g FO(X)118 2953 y FP(and)g FO(X)364 2923 y FN(\003)401 2953 y FP(,)i(so)d(the)i(irreducible)e(represen)n(tation)f(with)i (degenerate)f FO(A)2377 2965 y FL(1)2450 2953 y FP(can)118 3053 y(only)27 b(b)r(e)h(trivial.)p 2514 3053 V 2518 3000 50 4 v 2518 3053 V 2567 3053 4 57 v 118 3219 a FR(6.)38 b FP(No)n(w)28 b(consider)f(represen)n(tations)g FB(F)1378 3231 y FL(2)1413 3219 y FP(.)39 b(Recall)28 b(that)h(in)f(this)h(case)e FO(J)32 b(>)24 b FP(0,)118 3318 y FO(q)34 b FQ(2)e FJ(R)p FP(,)40 b FQ(j)p FO(q)s FQ(j)31 b FO(>)g FP(1,)i FO(A)766 3330 y FL(1)803 3318 y FP(,)h FO(A)922 3330 y FL(2)960 3318 y FP(,)g FO(k)h FP(are)c(self-adjoin)n(t,)j(and)e 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FL(2)2296 3969 y(1)2361 3948 y FQ(\000)26 b FO(A)2514 3918 y FL(2)2514 3969 y(2)2552 3948 y FP(.)118 4048 y(Hence)i FB(H)22 b FP(=)h FB(H)625 4060 y FL(0)679 4048 y FQ(\010)18 b FB(H)837 4060 y FL(1)891 4048 y FQ(\010)g FB(H)1049 4060 y FL(2)1104 4048 y FQ(\010)g FB(H)1262 4060 y FL(3)1297 4048 y FP(,)28 b(where)f(all)g FB(H)1778 4060 y FM(j)1839 4048 y FP(are)g(in)n(v)-5 b(arian)n(t)27 b(and:)243 4147 y FO(A)305 4159 y FL(1)365 4147 y FP(=)c FO(A)515 4159 y FL(2)575 4147 y FP(=)g(0,)k FO(X)34 b FP(is)28 b(normal)e(in)i FB(H)1395 4159 y FL(0)1431 4147 y FP(;)p eop %%Page: 131 135 131 134 bop 118 100 a FK(2.2.)36 b(Algebras)26 b(with)j(3)e(and)g(4)g (generators)956 b FP(131)243 333 y FO(A)305 345 y FL(1)365 333 y FP(=)23 b FO(X)529 303 y FN(\003)589 333 y FP(=)g(0,)k FO(A)831 303 y FL(2)831 353 y(2)892 333 y FP(=)22 b FQ(\000)p FO(X)1120 303 y FN(\003)1157 333 y FO(X)30 b FQ(\))23 b FO(A)1424 345 y FL(2)1485 333 y FP(=)f FO(X)30 b FP(=)22 b(0)27 b(in)h FB(H)1999 345 y FL(1)2035 333 y FP(;)243 433 y FO(A)305 445 y FL(2)365 433 y FP(=)23 b FO(X)29 b FP(=)23 b(0,)k FO(A)793 403 y FL(2)793 454 y(1)854 433 y FP(=)22 b FQ(\000)p FO(X)7 b(X)1158 403 y FN(\003)1218 433 y FQ(\))23 b FO(A)1386 445 y FL(1)1446 433 y FP(=)g FO(X)1610 403 y FN(\003)1671 433 y FP(=)f(0)27 b(in)h FB(H)1999 445 y FL(2)2035 433 y FP(;)243 534 y FO(X)h FP(=)23 b FO(X)505 504 y FN(\003)565 534 y FP(=)g(0,)k FO(A)807 504 y FL(2)807 555 y(1)868 534 y FP(=)22 b FO(A)1017 504 y FL(2)1017 555 y(2)1082 534 y FP(in)28 b FB(H)1254 546 y FL(3)1290 534 y FP(.)p 2514 534 4 57 v 2518 481 50 4 v 2518 534 V 2567 534 4 57 v 118 709 a FR(Theorem)38 b(32.)44 b FC(F)-6 b(or)36 b FO(q)h FQ(2)d FJ(R)p FC(,)44 b FQ(j)p FO(q)s FQ(j)34 b FO(>)f FP(1)p FC(,)k(the)f FQ(\003)p FC(-algebr)l(a)g FO(su)2056 721 y FM(q)2092 709 y FP(\(1)p FO(;)14 b FP(1\))35 b FC(has)i(the)118 809 y(fol)t(lowing)32 b(irr)l(e)l(ducible)f(r)l(epr)l(esentations)7 b FP(:)220 978 y(1)p FO(:)41 b FC(one-dimensional)9 b FP(:)39 b FO(k)26 b FP(=)d(1)p FC(,)30 b FO(X)f FP(=)23 b(0;)220 1148 y(2)p FO(:)41 b FC(with)30 b(the)g(lowest)g(weight)8 b FP(:)766 1333 y FO(k)s(f)853 1345 y FM(j)911 1333 y FP(=)22 b FO(l)16 b(q)1079 1299 y FM(j)s FN(\000)p FL(1)1199 1333 y FO(f)1240 1345 y FM(j)1274 1333 y FO(;)737 1540 y(X)7 b(f)854 1552 y FM(j)911 1540 y FP(=)998 1423 y Fz(\022)1059 1540 y FP([)p FO(j)e FP(])1144 1552 y FM(q)1205 1484 y FO(l)1232 1454 y FL(2)1269 1484 y FO(q)1309 1454 y FM(j)s FN(\000)p FL(1)1447 1484 y FQ(\000)18 b FO(l)1557 1454 y FN(\000)p FL(2)1646 1484 y FO(q)1686 1454 y FL(1)p FN(\000)p FM(j)p 1205 1521 601 4 v 1347 1597 a FQ(j)p FO(q)j FQ(\000)d FO(q)1551 1573 y FN(\000)p FL(1)1641 1597 y FQ(j)1816 1423 y Fz(\023)1877 1440 y FL(1)p FM(=)p FL(2)1981 1540 y FO(f)2022 1552 y FM(j)s FL(+1)2141 1540 y FO(;)326 1770 y FC(wher)l(e)30 b FO(l)25 b(>)d FP(1)p FC(,)30 b FO(j)e FP(=)23 b(1)p FC(,)29 b FP(2)p FC(,)h FO(:)14 b(:)g(:)43 b FP(;)220 1940 y(3)p FO(:)e FC(with)30 b(the)g(highest)h(weight)8 b FP(:)825 2125 y FO(k)s(f)912 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432 y FQ(\000)c FO(l)1701 402 y FL(2)1699 453 y(2)1738 432 y FP(\))p FO(=)p FQ(j)p FO(q)j FQ(\000)d FO(q)2016 402 y FN(\000)p FL(1)2106 432 y FQ(j)p FP(\).)53 b(Then)33 b(the)118 532 y(orbits)26 b(with)h FO(l)566 544 y FL(1)603 532 y FO(l)628 544 y FL(2)688 532 y FQ(6)p FP(=)c(0)j(giv)n(e)g(the)h (case)e(2\),)i(the)g(orbits)f(with)h FO(l)2044 544 y FL(1)2104 532 y FP(=)c(0)j(or)g FO(l)2386 544 y FL(2)2446 532 y FP(=)c(0)118 632 y(giv)n(e)27 b(the)h(cases)e(1\),)i(3\),)f(4\),) h(5\).)243 733 y(Once)g(again,)h(for)f(the)i(non-degenerate)d(case)i (with)g(unitary)g FO(U)9 b FP(,)29 b(w)n(e)g(use)118 832 y(the)f(existence)f(of)h(a)f(Borel)g(section)g(for)g(this)h (dynamical)f(system.)p 2514 832 4 57 v 2518 779 50 4 v 2518 832 V 2567 832 4 57 v 118 1010 a FC(R)l(emark)48 b FP(37)p FC(.)e FP(Represen)n(tations)36 b(corresp)r(onding)f(to)i (the)h(cases)e(2\){5\))g(are)118 1109 y(un)n(b)r(ounded.)118 1358 y FH(2.3)112 b(Represen)m(tations)37 b(of)h Fr(q)t FH(-deformed)g Fr(U)10 b Fl(\()p 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FL(1)307 2097 y FP(,)h FO(I)394 2109 y FL(2)432 2097 y FP(,)f FO(I)518 2109 y FL(3)584 2097 y FP(satisfying)g(the)h(relations:)867 2282 y FO(q)907 2248 y FL(1)p FM(=)p FL(2)1012 2282 y FO(I)1048 2294 y FL(1)1085 2282 y FO(I)1121 2294 y FL(2)1178 2282 y FQ(\000)18 b FO(q)1301 2248 y FN(\000)p FL(1)p FM(=)p FL(2)1457 2282 y FO(I)1493 2294 y FL(2)1531 2282 y FO(I)1567 2294 y FL(1)1627 2282 y FP(=)23 b FO(I)1751 2294 y FL(3)1789 2282 y FO(;)867 2423 y(q)907 2389 y FL(1)p FM(=)p FL(2)1012 2423 y FO(I)1048 2435 y FL(2)1085 2423 y FO(I)1121 2435 y FL(3)1178 2423 y FQ(\000)18 b FO(q)1301 2389 y FN(\000)p FL(1)p FM(=)p FL(2)1457 2423 y FO(I)1493 2435 y FL(3)1531 2423 y FO(I)1567 2435 y FL(2)1627 2423 y FP(=)23 b FO(I)1751 2435 y FL(1)1789 2423 y FO(;)867 2564 y(q)907 2530 y FL(1)p FM(=)p FL(2)1012 2564 y FO(I)1048 2576 y FL(3)1085 2564 y FO(I)1121 2576 y FL(1)1178 2564 y FQ(\000)18 b FO(q)1301 2530 y FN(\000)p FL(1)p FM(=)p FL(2)1457 2564 y FO(I)1493 2576 y FL(1)1531 2564 y FO(I)1567 2576 y FL(3)1627 2564 y FP(=)23 b FO(I)1751 2576 y FL(2)1789 2564 y FO(:)551 b FP(\(2.33\))118 2750 y(Note)44 b(that)h(the)f(Lie)g(algebras)e FO(sl)r FP(\(2)p FO(;)14 b FJ(C)g FP(\))51 b(and)44 b FO(so)p FP(\(3)p FO(;)14 b FJ(C)g FP(\))51 b(are)43 b(isomorphic.)118 2849 y(Ho)n(w)n(ev)n(er,)27 b(the)j(quan)n(tum)f(algebra)e FO(U)1323 2861 y FM(q)1359 2849 y FP(\()p FO(sl)r FP(\(2)p FO(;)14 b FJ(C)g FP(\))q(\))35 b(di\013ers)29 b(from)f(the)i(algebra) 118 2949 y FO(U)175 2961 y FM(q)211 2949 y FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)i FP(\)\).)243 3050 y(Let)40 b(us)g(describ)r(e)g(in)n(v)n(olutions)f(on)h(the)g(algebra)f FO(U)1936 3062 y FM(q)1972 3050 y FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)h FP(\))q(\).)80 b(It)41 b(is)118 3150 y(clear)27 b(that)i(an)f(in)n(v)n(olution)g(in)g(an)g(algebra)f(with)i (generators)d(and)i(relations)118 3249 y(is)g(completely)f(de\014ned)h (b)n(y)f(its)h(v)-5 b(alues)28 b(on)f(the)h(generators.)35 b(An)28 b(in)n(v)n(olution)118 3349 y(ma)n(y)21 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FQ(\000)18 b FO(q)1826 497 y FN(\000)p FL(1)p FM(=)p FL(2)1982 527 y FO(I)2018 539 y FL(1)2056 527 y FO(I)2092 539 y FL(2)2130 527 y FP(\))p FO(;)85 b(q)26 b(>)d FP(0)p FO(;)1296 646 y(q)1336 616 y FL(1)p FM(=)p FL(2)1440 646 y FO(I)1476 658 y FL(2)1514 646 y FO(I)1550 658 y FL(1)1606 646 y FQ(\000)18 b FO(q)1729 616 y FN(\000)p FL(1)p FM(=)p FL(2)1885 646 y FO(I)1921 658 y FL(1)1959 646 y FO(I)1995 658 y FL(2)2033 646 y FO(;)214 b(q)26 b(<)d FP(0)p FO(;)243 834 y FP(3\))29 b FO(q)d FQ(2)e FJ(R)p FC(,)208 1084 y FO(I)251 1050 y FN(\003)244 1105 y FL(1)312 1084 y FP(=)f FQ(\000)p FO(I)501 1096 y FL(1)538 1084 y FO(;)99 b(I)703 1050 y FN(\003)696 1105 y FL(2)764 1084 y FP(=)23 b FQ(\000)p FO(I)953 1096 y FL(2)990 1084 y FO(;)14 b(I)1070 1050 y FN(\003)1063 1105 y FL(3)1131 1084 y FP(=)1219 943 y Fz(\()1286 1028 y FP(\()p FO(q)1358 998 y FL(1)p FM(=)p FL(2)1462 1028 y FO(I)1498 1040 y FL(2)1536 1028 y FO(I)1572 1040 y FL(1)1628 1028 y FQ(\000)k FO(q)1751 998 y FN(\000)p FL(1)p FM(=)p 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FL(1)2113 1511 y FO(I)2149 1523 y FL(2)2187 1511 y FO(:)118 1726 y FR(2.3.2)94 b(Represen)m(tations)31 b(of)g FO(U)1276 1738 y FM(q)1313 1726 y FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)h FP(\)\))118 1879 y(W)-7 b(e)25 b(will)g(study)g(b)r(ounded)g FQ(\003)p FP(-represen)n(tations)c(of)k (the)g(algebra)e FO(U)2192 1891 y FM(q)2228 1879 y FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)h FP(\))q(\))118 1978 y(on)23 b(a)g(Hilb)r(ert)h(space)f FO(H)7 b FP(.)35 b(W)-7 b(e)24 b(restrict)f(ourselv)n(es)f(to)h(a)g(study)h(of)f(represen)n (ta-)118 2078 y(tions)28 b(of)g(the)h FQ(\003)p FP(-algebra)d (de\014ned)i(b)n(y)g(the)h(in)n(v)n(olution)e FO(I)1901 2048 y FN(\003)1894 2099 y FL(1)1964 2078 y FP(=)d FQ(\000)p FO(I)2154 2090 y FL(1)2191 2078 y FP(,)k FO(I)2285 2048 y FN(\003)2278 2099 y FL(2)2348 2078 y FP(=)23 b FQ(\000)p FO(I)2537 2090 y FL(2)118 2178 y FP(\(for)i FO(q)h FQ(2)d FJ(R)31 b FP(and)25 b FQ(j)p FO(q)s FQ(j)e FP(=)g(1,)i FO(q)h FQ(6)p FP(=)c FQ(\006)p FP(1\),)j(whic)n(h,)h(for)e FO(q)i FP(=)d(1,)i(corresp)r(onds)e(to)h(the)118 2277 y(compact)j(real)g(form)g(of)h FO(so)p FP(\(3\).)37 b(Henceforth)27 b(w)n(e)h(denote)f(it)h(b)n(y)g FO(U)2218 2289 y FM(q)2254 2277 y FP(\()p FO(so)p FP(\(3\)\).)243 2377 y(W)-7 b(e)28 b(assume)f(the)g(follo)n(wing)g(notations)278 2600 y([)p FO(x)p FP(])371 2612 y FM(q)431 2600 y FP(=)529 2544 y FO(q)569 2514 y FM(x)629 2544 y FQ(\000)18 b FO(q)752 2514 y FN(\000)p FM(x)p 529 2581 318 4 v 552 2657 a FO(q)j FQ(\000)d FO(q)733 2633 y FN(\000)p FL(1)856 2600 y FO(;)97 b(d)1019 2612 y FM(q)1056 2600 y FP(\()p FO(m)p FP(\))23 b(=)1827 2544 y(1)p 1314 2581 1069 4 v 1314 2658 a(\()p FO(q)1386 2634 y FM(m)1468 2658 y FP(+)18 b FO(q)1591 2634 y FN(\000)p FM(m)1706 2658 y FP(\)\()p FO(q)1810 2634 y FM(m)p FL(+1)1976 2658 y FP(+)g FO(q)2099 2634 y FN(\000)p FL(\()p FM(m)p FL(+1\))2350 2658 y FP(\))2392 2600 y FO(:)118 2826 y FP(Let)25 b FO(E)325 2838 y FM(A)379 2826 y FP(\()p FQ(\001)p FP(\))g(denote)g(the)g(sp)r(ectral)f(measure)f 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FO(q)e(>)d FP(0)p FC(,)118 3872 y(is)h (\014nite-dimensional.)48 b(F)-6 b(or)33 b(any)g FO(n)27 b FQ(\025)h FP(1)p FC(,)33 b(irr)l(e)l(ducible)h(r)l(epr)l(esentations) f(in)118 3971 y FO(H)d FP(=)23 b FJ(C)359 3941 y FM(n)439 3971 y FC(ar)l(e)31 b(unitarily)f(e)l(quivalent)g(to)g(the)g(fol)t (lowing)i(one)6 b FP(:)500 4147 y FO(J)546 4159 y FL(1)584 4147 y FO(e)623 4159 y FM(k)686 4147 y FP(=)23 b([)p FO(k)e FQ(\000)d FP(\()p FO(n)h FP(+)f(1\))p FO(=)p FP(2])1309 4159 y FM(q)1344 4147 y FO(e)1383 4159 y FM(k)1424 4147 y FO(;)p eop %%Page: 136 140 136 139 bop 118 100 a FP(136)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)500 450 y FO(J)546 462 y FL(2)584 450 y FO(e)623 462 y FM(k)686 450 y FP(=)774 255 y Fz(8)774 329 y(>)774 354 y(<)774 504 y(>)774 529 y(:)848 333 y FO(\013)901 345 y FL(1)938 333 y FO(e)977 345 y FL(2)1014 333 y FO(;)621 b(k)26 b FP(=)d(1)p FO(;)848 453 y(\013)901 465 y FM(k)942 453 y FO(e)981 465 y FM(k)q FL(+1)1124 453 y FP(+)18 b FO(\013)1260 465 y FM(k)q FN(\000)p FL(1)1386 453 y FO(e)1425 465 y FM(k)q FN(\000)p FL(1)1550 453 y FO(;)85 b FP(2)23 b FQ(\024)f FO(k)k FQ(\024)d FO(n)18 b FQ(\000)g FP(1)p FO(;)848 572 y(\013)901 584 y FM(n)p FN(\000)p FL(1)1031 572 y FO(e)1070 584 y FM(n)p FN(\000)p FL(1)1200 572 y FO(;)435 b(k)26 b FP(=)d FO(n;)118 769 y FC(wher)l(e)31 b FO(\013)406 781 y FM(k)470 769 y FP(=)22 b(\()p FO(d)632 781 y FM(q)669 769 y FP(\()p FO(k)g FQ(\000)c FP(\()p FO(n)h FP(+)f(1\))p FO(=)p FP(2\)[)p FO(k)s FP(])1315 781 y FM(q)1350 769 y FP([)p FO(n)h FQ(\000)f FO(k)s FP(])1594 781 y FM(q)1630 769 y FP(\))1662 739 y FL(1)p FM(=)p FL(2)1767 769 y FC(.)118 932 y(Pr)l(o)l(of.)43 b FP(The)21 b(pro)r(of)f(of)h(the)g(theorem)g(is)f(based)h(on)f(the)h (tec)n(hnique)g(of)g(semilin-)118 1031 y(ear)f(relations)f(dev)n(elop)r (ed)h(in)h(Section)g(1.3)f(and)g(the)h(tec)n(hnique)g(of)g(dynamical) 118 1131 y(systems.)243 1231 y(Let)38 b FO(q)44 b FP(=)c FO(e)627 1200 y FM(\033)671 1231 y FP(,)h FO(\033)j FQ(2)d FJ(R)p FP(,)47 b(and)38 b FO(J)1264 1243 y FL(1)1301 1231 y FP(,)j FO(J)1411 1243 y FL(2)1487 1231 y FP(b)r(e)d(self-adjoin) n(t)g(op)r(erators)e(in)i(a)118 1330 y(Hilb)r(ert)33 b(space)e FO(H)39 b FP(satisfying)32 b(relations)f(\(2.35\))o ({\(2.36\))o(.)50 b(Equation)32 b(\(2.35\))118 1430 y(is)c(linear)e (with)j(resp)r(ect)e(to)g FO(J)1053 1442 y FL(2)1118 1430 y FP(with)h(the)g(corresp)r(onding)e(binary)h(relation:)392 1608 y(\000)c(=)g FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(j)g FP(\010\()p FO(t;)14 b(s)p FP(\))23 b FQ(\021)g FO(t)1207 1574 y FL(2)1262 1608 y FQ(\000)18 b FP(\()p FO(q)k FP(+)c FO(q)1559 1574 y FN(\000)p FL(1)1648 1608 y FP(\))c FO(ts)19 b FP(+)f FO(s)1904 1574 y FL(2)1960 1608 y FQ(\000)g FP(1)k(=)h(0)p FQ(g)p FO(:)118 1786 y FP(Let)705 1965 y FO(F)758 1977 y FL(1)796 1965 y FP(\()p FO(s)p FP(\))g(=)g FO(s)14 b FP(cosh)f FO(\033)21 b FP(+)1386 1877 y Fz(p)p 1469 1877 483 4 v 88 x FO(s)1508 1941 y FL(2)1559 1965 y FP(sinh)1707 1928 y FL(2)1758 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140 bop 118 100 a FK(2.3.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-deforemd)f FO(U)9 b FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(R)p FP(\)\))631 b(137)118 333 y(Let)33 b FO(E)333 345 y FL(1)402 333 y FP(=)e(\([)p FO(A)615 345 y FL(1)653 333 y FO(;)14 b(J)736 345 y FL(2)773 333 y FP(])22 b(+)f FO(J)950 345 y FL(2)987 333 y FP(\))p FO(=)p FP(2,)34 b FO(E)1226 303 y FN(\003)1221 353 y FL(1)1295 333 y FP(=)d(\([)p FO(A)1508 345 y FL(1)1546 333 y FO(;)14 b(J)1629 345 y FL(2)1666 333 y FP(])22 b FQ(\000)g FO(J)1844 345 y FL(2)1881 333 y FP(\))p FO(=)p FP(2.)51 b(Then)33 b(one)f(can)118 432 y(c)n(hec)n(k)27 b(that)547 585 y FO(A)609 597 y FL(1)646 585 y FO(E)707 597 y FL(1)768 585 y FP(=)22 b FO(E)916 597 y FL(1)954 585 y FP(\()p FO(A)1048 597 y FL(1)1104 585 y FP(+)c FO(I)7 b FP(\))p FO(;)97 b(A)1444 597 y FL(1)1482 585 y FO(E)1548 551 y FN(\003)1543 605 y FL(1)1609 585 y FP(=)23 b FO(E)1763 551 y FN(\003)1758 605 y FL(1)1801 585 y FP(\()p FO(A)1895 597 y FL(1)1951 585 y FQ(\000)18 b FO(I)7 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FC(wher)l(e)31 b FO(\013)406 2756 y FM(k)470 2744 y FP(=)22 b(\()p FO(d)632 2756 y FN(\000)p FM(q)721 2744 y FP(\()p FO(k)g FQ(\000)c FP(\()p FO(n)g FP(+)g(1\))p FO(=)p FP(2\))c([)p FO(k)s FP(])1380 2756 y FN(\000)p FM(q)1482 2744 y FP([)p FO(n)k FQ(\000)g FO(k)s FP(])1725 2756 y FN(\000)p FM(q)1814 2744 y FP(\))1846 2714 y FL(1)p FM(=)p FL(2)1950 2744 y FC(.)118 2908 y(Pr)l(o)l(of.)43 b FP(The)30 b(pro)r(of)f(essen)n (tially)g(go)r(es)g(in)h(the)g(same)f(w)n(a)n(y)g(as)g(that)h(of)g (Theo-)118 3008 y(rem)j(33.)51 b(Let)33 b FO(q)i FP(=)c FQ(\000)p FO(e)874 2978 y FM(\033)918 3008 y FP(,)j FO(\033)h FQ(2)d FJ(R)p FP(,)40 b(and)33 b FO(J)1474 3020 y FL(1)1511 3008 y FP(,)h FO(J)1614 3020 y FL(2)1684 3008 y FP(b)r(e)g(self-adjoin) n(t)e(op)r(erators)118 3107 y(satisfying)23 b(\(2.35\))o({\(2.36\))o(.) 35 b(Let)22 b(\000)h(=)g FQ(f)p FP(\010\()p FO(t;)14 b(s)p FP(\))22 b FQ(\021)h FO(t)1726 3077 y FL(2)1771 3107 y FQ(\000)8 b FP(\()p FO(q)j FP(+)d FO(q)2037 3077 y FN(\000)p FL(1)2124 3107 y FP(\))p FO(ts)g FP(+)g FO(s)2345 3077 y FL(2)2404 3107 y FP(=)23 b(1)p FQ(g)118 3207 y FP(b)r(e)32 b(the)h(c)n(haracteristic)c(binary)i(relation)g(corresp)r (onding)f(to)i(\(2.35\).)49 b(Con-)118 3307 y(sidering)19 b(the)h(same)f(parameterization)f FO(s)23 b FP(=)f(sinh)14 b FO(\033)s(x)p FP(\()p FO(s)p FP(\))r FO(=)p FP(sinh)f FO(\033)26 b FQ(\021)d FP([)p FO(x)p FP(\()p FO(s)p FP(\)])2462 3319 y FN(\000)p FM(q)2552 3307 y FP(,)118 3406 y FO(x)p FP(\()p FO(s)p FP(\))h FQ(2)g FJ(R)p FP(,)i(w)n(e)18 b(get)h(\000)k(=)f FQ(f)p FP(\()p FQ(\000)p FP([)p FO(x)p FP(+1])1218 3418 y FN(\000)p FM(q)1307 3406 y FO(;)14 b FP([)p FO(x)p FP(])1437 3418 y FN(\000)p FM(q)1526 3406 y FP(\))p FO(;)g FP(\()p FQ(\000)p FP([)p FO(x)p FQ(\000)p FP(1])1892 3418 y FN(\000)p FM(q)1981 3406 y FO(;)g FP([)p FO(x)p FP(])2111 3418 y FN(\000)p FM(q)2200 3406 y FP(\))23 b FQ(j)g FO(x)h FQ(2)f FJ(R)p FQ(g)p FP(.)118 3506 y(As)i(b)r(efore,)h FO(J)555 3518 y FL(1)592 3506 y FP(,)g FO(J)687 3518 y FL(2)750 3506 y FP(satisfy)f(\(2.35\))f (if)i(and)f(only)g(if)h FO(J)1777 3518 y FL(2)1840 3506 y FP(is)f(concen)n(trated)f(on)h(\000)118 3605 y(with)j(resp)r(ect)g (to)f FO(J)739 3617 y FL(1)776 3605 y FP(,)h(i.e.,)163 3786 y FO(E)224 3798 y FM(J)261 3806 y Fy(1)297 3786 y FP(\(\001\))p FO(J)476 3798 y FL(2)514 3786 y FO(E)575 3798 y FM(J)612 3806 y Fy(1)649 3786 y FP(\(\001)750 3752 y FN(0)774 3786 y FP(\))23 b(=)g(0)p FO(;)96 b FP(for)27 b(an)n(y)g(\001)p FO(;)14 b FP(\001)1537 3752 y FN(0)1584 3786 y FQ(2)23 b FA(B)p FP(\()p FJ(R)q FP(\))p FO(;)48 b FP(\001)18 b FQ(\002)g FP(\001)2164 3752 y FN(0)2206 3786 y FQ(\\)h FP(\000)k(=)g FJ(?)p FO(;)118 3967 y FP(whic)n(h)28 b(is)f(equiv)-5 b(alen)n(t)27 b(to)138 4147 y FO(E)199 4159 y FM(A)249 4167 y Fy(1)286 4147 y FP(\(\001\))p FO(J)465 4159 y FL(2)503 4147 y FO(E)564 4159 y FM(A)614 4167 y Fy(1)651 4147 y FP(\(\001)752 4113 y FN(0)775 4147 y FP(\))d(=)e(0)p FO(;)97 b FP(for)27 b(an)n(y)g(\001)p FO(;)14 b FP(\001)1539 4113 y FN(0)1585 4147 y FQ(2)24 b FA(B)p FP(\()p FJ(R)p FP(\))q FO(;)47 b FP(\001)19 b FQ(\002)f FP(\001)2166 4113 y FN(0)2208 4147 y FQ(\\)g FP(\000)2333 4113 y FN(0)2380 4147 y FP(=)k FJ(?)p FO(;)p eop %%Page: 139 143 139 142 bop 118 100 a FK(2.3.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-deforemd)f FO(U)9 b FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(R)p FP(\)\))631 b(139)118 333 y(where)29 b FO(J)406 345 y FL(1)468 333 y FP(=)d(sinh)13 b FO(\033)s(A)832 345 y FL(1)870 333 y FO(=)p FP(sinh)h FO(\033)32 b FP(and)d(\000)1368 303 y FN(0)1417 333 y FP(=)c FQ(f)p FP(\()p FQ(\000)p FP(\()p FO(x)19 b FP(+)g(1\))p FO(;)14 b(x)p FP(\))p FO(;)g FP(\()p FQ(\000)p FP(\()p FO(x)21 b FQ(\000)e FP(1\))p FO(;)14 b(x)p FP(\))26 b FQ(j)118 432 y FO(x)f FQ(2)g FJ(R)p FQ(g)31 b FP(=)24 b FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))25 b FQ(j)g FO(t)801 402 y FL(2)857 432 y FP(+)19 b(2)p FO(ts)f FP(+)h FO(s)1193 402 y FL(2)1255 432 y FP(=)24 b(1)p FQ(g)p FP(.)39 b(F)-7 b(rom)29 b(this)f(and)h (Theorem)f(8,)g(w)n(e)118 532 y(conclude)f(that)h FO(A)701 544 y FL(1)739 532 y FP(,)g FO(J)836 544 y FL(2)901 532 y FP(satisfy)f(the)h(relation)792 690 y FO(A)854 656 y FL(2)854 711 y(1)891 690 y FO(J)937 702 y FL(2)993 690 y FP(+)18 b(2)p FO(A)1180 702 y FL(1)1217 690 y FO(J)1263 702 y FL(2)1300 690 y FO(A)1362 702 y FL(1)1418 690 y FP(+)g FO(J)1547 702 y FL(2)1585 690 y FO(A)1647 656 y FL(2)1647 711 y(1)1707 690 y FP(=)23 b FO(J)1841 702 y FL(2)1878 690 y FO(:)118 849 y FP(Let)29 b FO(E)329 861 y FL(1)391 849 y FP(=)24 b(\()p FQ(f)p FO(A)616 861 y FL(1)653 849 y FO(;)14 b(J)736 861 y FL(2)773 849 y FQ(g)19 b FP(+)f FO(J)963 861 y FL(2)1000 849 y FP(\))p FO(=)p FP(2,)28 b FO(E)1228 861 y FL(2)1290 849 y FP(=)c FQ(\000)p FP(\()p FQ(f)p FO(A)1580 861 y FL(1)1617 849 y FO(;)14 b(J)1700 861 y FL(2)1737 849 y FQ(g)19 b(\000)g FO(J)1928 861 y FL(2)1965 849 y FP(\))p FO(=)p FP(2.)39 b(It)28 b(is)h(easy)e(to)118 948 y(sho)n(w)g(that)h FO(E)565 960 y FL(1)625 948 y FP(=)23 b FO(E)779 918 y FN(\003)774 969 y FL(1)817 948 y FP(,)28 b FO(E)929 960 y FL(2)989 948 y FP(=)23 b FO(E)1143 918 y FN(\003)1138 969 y FL(2)1181 948 y FP(,)28 b FO(J)1278 960 y FL(2)1338 948 y FP(=)23 b FO(E)1487 960 y FL(1)1543 948 y FP(+)18 b FO(E)1687 960 y FL(2)1752 948 y FP(and)488 1107 y FO(A)550 1119 y FL(1)587 1107 y FO(E)648 1119 y FL(1)709 1107 y FP(=)k FQ(\000)p FO(E)922 1119 y FL(1)959 1107 y FP(\()p FO(A)1053 1119 y FL(1)1110 1107 y FQ(\000)c FO(I)7 b FP(\))p FO(;)97 b(A)1450 1119 y FL(1)1487 1107 y FO(E)1548 1119 y FL(1)1609 1107 y FP(=)23 b FQ(\000)p FO(E)1823 1119 y FL(1)1860 1107 y FP(\()p FO(A)1954 1119 y FL(1)2010 1107 y FP(+)18 b FO(I)7 b FP(\))p FO(;)260 1284 y(E)326 1250 y FL(2)321 1304 y(1)377 1284 y FP(cosh)o(\()p FO(\033)s FP(\()p FO(A)710 1296 y FL(1)767 1284 y FP(+)18 b FO(I)7 b FP(\)\))24 b(=)e FO(E)1134 1250 y FL(2)1129 1304 y(2)1185 1284 y FP(cosh\()p FO(\033)s FP(\()p FO(A)1519 1296 y FL(1)1576 1284 y FQ(\000)c FP(1\)\))g FQ(\000)1876 1228 y FP(sinh)c FO(\033)s(A)2150 1240 y FL(1)p 1876 1265 312 4 v 1898 1341 a FP(2)g(sinh)g FO(\033)2198 1284 y(:)142 b FP(\(2.39\))118 1470 y(Con)n(v)n(ersely)-7 b(,)34 b(an)n(y)f(represen)n(tation)f FO(A)1337 1482 y FL(1)1375 1470 y FP(,)j FO(E)1494 1482 y FL(1)1532 1470 y FP(,)h FO(E)1652 1482 y FL(2)1723 1470 y FP(of)41 b(\(2.39\))33 b(de\014nes)h(a)g(rep-)118 1570 y(resen)n(tation)40 b FO(J)605 1582 y FL(1)689 1570 y FP(=)46 b(sinh)14 b FO(\033)s(A)1074 1582 y FL(1)1112 1570 y FO(=)g FP(sinh)f FO(\033)s FP(,)45 b FO(J)1493 1582 y FL(2)1577 1570 y FP(=)h FO(E)1749 1582 y FL(1)1814 1570 y FP(+)27 b FO(E)1967 1582 y FL(2)2046 1570 y FP(of)42 b(the)g(algebra)118 1669 y FO(U)175 1681 y FM(q)211 1669 y FP(\()p FO(so)p FP(\(3\)\).)j(Moreo)n(v)n(er,)28 b(there)i(is)g(a)f (one-to-one)g(corresp)r(ondence)f(b)r(et)n(w)n(een)118 1769 y(irreducible)f(and)h(unitary)f(equiv)-5 b(alen)n(t)27 b(represen)n(tations)f(of)h(b)r(oth)h(ob)5 b(jects.)243 1869 y(W)-7 b(e)29 b(can)g(no)n(w)g(pro)r(ceed)g(analogously)e(as)i(in) g(the)h(pro)r(of)f(of)g(Theorem)g(22)118 1968 y(on)e(represen)n (tations)f(of)h(graded)f FO(so)p FP(\(3\).)37 b(By)27 b(the)h(same)f(argumen)n(ts,)f(w)n(e)h(can)118 2068 y(c)n(ho)r(ose)h(a) i(basis)f(consisting)f(of)i(eigen)n(v)n(ectors)d(of)j(the)g(op)r (erator)d FO(A)2249 2080 y FL(1)2317 2068 y FP(for)i(an)n(y)118 2168 y(irreducible)e(represen)n(tation)f FO(A)1134 2180 y FL(1)1171 2168 y FP(,)i FO(E)1283 2180 y FL(1)1321 2168 y FP(,)f FO(E)1432 2180 y FL(2)1498 2168 y FP(in)h FO(H)7 b FP(.)36 b(Then)28 b(w)n(e)f(ha)n(v)n(e)304 2326 y FO(A)366 2338 y FL(1)404 2326 y FO(e)443 2338 y FM(\025)509 2326 y FP(=)22 b FO(\025)14 b(e)697 2338 y FM(\025)741 2326 y FO(;)97 b(E)922 2338 y FL(1)959 2326 y FO(e)998 2338 y FM(\025)1065 2326 y FP(=)22 b FO(a)1196 2338 y FL(1)1233 2326 y FP(\()p FO(\025)p FP(\))14 b FO(e)1398 2338 y FL(1)p FN(\000)p FM(\025)1527 2326 y FO(;)97 b(E)1708 2338 y FL(2)1746 2326 y FO(e)1785 2338 y FM(\025)1851 2326 y FP(=)23 b FO(a)1983 2338 y FL(2)2020 2326 y FP(\()p FO(\025)p FP(\))14 b FO(e)2185 2338 y FN(\000)p FL(1)p FN(\000)p FM(\025)2366 2326 y FO(;)118 2501 y FP(where)36 b FO(\025)i FP(b)r(elongs)e(to)g(the)h(orbit)g(\012)h(=)g FQ(f)p FO(F)1543 2458 y FL(\()p FM(k)q FL(\))1531 2523 y(1)1635 2501 y FO(F)1700 2458 y FL(\()p FM(m)p FL(\))1688 2523 y(2)1815 2501 y FP(\()p FO(\025)p FP(\))p FO(;)14 b(k)s(;)g(m)39 b FQ(2)f FJ(Z)2314 2513 y FL(+)2363 2501 y FQ(g)e FP(and)118 2600 y FO(F)171 2612 y FL(1)209 2600 y FP(\()p FO(\025)p FP(\))30 b(=)f(1)20 b FQ(\000)h FO(\025)p FP(,)33 b FO(F)750 2612 y FL(2)787 2600 y FP(\()p FO(\025)p FP(\))e(=)e FQ(\000)p FP(1)20 b FQ(\000)g FO(\025)p FP(.)49 b(The)31 b(conditions)g(for)g FO(A)2124 2612 y FL(1)2162 2600 y FP(,)h FO(E)2278 2612 y FL(1)2316 2600 y FP(,)g FO(E)2432 2612 y FL(2)2501 2600 y FP(to)118 2700 y(satisfy)27 b(relation)g(\(2.39\))g(are)f(the)i(follo)n(wing:)594 2858 y FO(a)638 2870 y FL(1)675 2858 y FP(\(1)18 b FQ(\000)g FO(\025)p FP(\))24 b(=)p 1042 2786 195 4 v 23 w FO(a)1086 2870 y FL(1)1123 2858 y FP(\()p FO(\025)p FP(\))q FO(;)97 b(a)1400 2870 y FL(2)1437 2858 y FP(\()p FQ(\000)p FP(1)18 b FQ(\000)g FO(\025)p FP(\))23 b(=)p 1868 2786 V 23 w FO(a)1912 2870 y FL(2)1949 2858 y FP(\()p FO(\025)p FP(\))q FO(;)177 2993 y FQ(j)p FO(a)244 3005 y FL(1)282 2993 y FP(\()p FO(\025)p FP(\))p FQ(j)417 2959 y FL(2)469 2993 y FP(cosh)13 b FO(\033)s FP(\()p FO(\025)19 b FP(+)f(1\))23 b(=)g FQ(j)p FO(a)1124 3005 y FL(2)1161 2993 y FP(\()p FO(\025)p FP(\))p FQ(j)1296 2959 y FL(2)1348 2993 y FP(cosh)13 b FO(\033)s FP(\()p FO(\025)20 b FQ(\000)e FP(1\))g FQ(\000)g FP(sinh)c FO(\033)s(\025=)p FP(2)g(sinh)g FO(\033)n(:)118 3152 y FP(Similarly)21 b(to)h(the)g(case)f(of)g(the)h(graded)f FO(so)p FP(\(3\),)i(w)n(e)e(see)g(that)h(the)g(last)g(relation)118 3251 y(cannot)j(hold)g(for)f(an)n(y)g(p)r(oin)n(t)h(of)g(the)h(orbit)e (\012)h(and)g(there)g(exists)g(the)g(highest)118 3351 y(v)n(ector)i(on)h(whic)n(h)g FO(E)783 3363 y FL(2)849 3351 y FP(acts)g(as)f(zero)h(and)g(the)g(lo)n(w)n(est)f(v)n(ector)g(on) h(whic)n(h)g(the)118 3451 y(op)r(erator)38 b FO(E)526 3463 y FL(1)604 3451 y FP(is)i(zero.)73 b(This)40 b(implies)h(that)f (the)g(only)g(orbits)g(satisfying)118 3550 y(these)33 b(conditions)f(are)g(those)h(whic)n(h)f(con)n(tain)g(0)h(and)g FQ(\006)p FP(1)p FO(=)p FP(2.)50 b(Using)33 b(these)118 3650 y(conditions)27 b(one)g(can)h(easily)e(get)i(the)g(statemen)n(t.)p 2514 3650 4 57 v 2518 3597 50 4 v 2518 3650 V 2567 3650 4 57 v 118 3811 a FR(3.)36 b FP(Represen)n(tations)26 b(of)i FO(U)985 3823 y FM(q)1021 3811 y FP(\()p FO(so)p FP(\(3\)\),)h FO(q)i FP(is)c(a)g(ro)r(ot)g(of)h(unit)n(y)-7 b(.)243 3911 y(Let)29 b FO(q)f FP(=)c FO(e)586 3881 y FM(i\033)654 3911 y FP(,)29 b FO(\033)g FQ(2)c FP(\()p FQ(\000)p FO(\031)s(;)14 b(\031)s FP(\).)41 b(If)29 b FO(q)j FP(is)d(a)f(ro)r(ot)g(of)h(unit)n(y)-7 b(,)29 b(then)g FO(\033)g FP(=)24 b FO(\031)s(k)s(=n)p FP(,)118 4085 y(where)j FO(k)s(=n)g FP(is)h(an)f(irreducible)g(fraction.)36 b(Let)28 b FO(s)23 b FP(=)1770 3943 y Fz(\()1837 4028 y FO(n;)125 b(k)86 b FP(is)27 b(ev)n(en)o FO(;)1837 4148 y FP(2)p FO(n;)83 b(k)j FP(is)27 b(o)r(dd)p FO(:)p eop %%Page: 140 144 140 143 bop 118 100 a FP(140)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)243 333 y FP(In)g(what)g(follo)n(ws,)f(w)n(e)g(denote)h(b) n(y)g FO(I)1387 345 y FL(1)1451 333 y FP(and)g FO(I)1648 345 y FL(2)1713 333 y FP(the)g(iden)n(tit)n(y)g(1)17 b FQ(\002)f FP(1,)27 b(2)17 b FQ(\002)f FP(2)118 432 y(matrices)27 b(resp)r(ectiv)n(ely)-7 b(.)118 576 y FR(Theorem)35 b(35.)43 b FC(L)l(et)33 b FO(\033)g FP(=)d FO(\031)s(k)s(=n)p FC(,)k FO(\033)f FQ(6)p FP(=)d FO(\031)s(l)r FC(.)49 b(A)n(ny)33 b(irr)l(e)l(ducible)i(r)l(epr)l(esenta-)118 675 y(tions)30 b(of)h FO(U)479 687 y FM(q)515 675 y FP(\()p FO(so)p FP(\(3\)\))f FC(is)g(unitarily)h(e)l(quivalent)f(to)g(one)g(of) g(the)g(fol)t(lowing)7 b FP(:)243 775 y(1)p FC(.)38 b FO(H)30 b FP(=)22 b FJ(C)588 745 y FM(s)630 775 y FC(,)30 b FO(J)731 787 y FL(1)768 775 y FO(e)807 787 y FM(m)893 775 y FP(=)23 b([)p FO(a)18 b FP(+)g FO(m)p FP(])1245 787 y FM(q)1295 775 y FO(e)1334 787 y FM(m)1397 775 y FC(,)398 1062 y FO(J)444 1074 y FL(2)481 1062 y FO(e)520 1074 y FM(m)606 1062 y FP(=)693 867 y Fz(8)693 941 y(>)693 966 y(<)693 1116 y(>)693 1141 y(:)767 945 y FO(\013)820 957 y FL(0)871 945 y FO(e)910 957 y FL(1)966 945 y FP(+)g FO(e)1088 915 y FM(i\036)1155 945 y FO(\013)1208 957 y FM(s)p FN(\000)p FL(1)1342 945 y FO(e)1381 957 y FM(s)p FN(\000)p FL(1)1501 945 y FO(;)221 b(m)22 b FP(=)h(0)p FO(;)767 1065 y(\013)820 1077 y FM(m)897 1065 y FO(e)936 1077 y FM(m)p FL(+1)1101 1065 y FP(+)18 b FO(\013)1237 1077 y FM(m)p FN(\000)p FL(1)1399 1065 y FO(e)1438 1077 y FM(m)p FN(\000)p FL(1)1586 1065 y FO(;)136 b FP(1)22 b FQ(\024)h FO(m)g FQ(\024)f FO(s)d FQ(\000)f FP(2)p FO(;)767 1184 y(\013)820 1196 y FM(s)p FN(\000)p FL(2)955 1184 y FO(e)994 1196 y FM(s)p FN(\000)p FL(2)1132 1184 y FP(+)g FO(e)1254 1154 y FN(\000)p FM(i\036)1373 1184 y FO(\013)1426 1196 y FM(s)p FN(\000)p FL(1)1561 1184 y FO(e)1600 1196 y FL(0)1637 1184 y FO(;)85 b(m)22 b FP(=)h FO(s)18 b FQ(\000)h FP(1)p FO(;)118 1341 y FC(wher)l(e)389 1524 y FO(\013)442 1536 y FM(m)529 1524 y FP(=)616 1407 y Fz(\022)862 1468 y FO(b)14 b FP(cos)f FO(a\033)k FP(cos)o(\()p FO(a)i FP(+)f(1\))p FO(\033)p 687 1505 1046 4 v 687 1581 a FP(cos\()p FO(a)g FP(+)g FO(m)h FP(+)f(1\))p FO(\033)f FP(cos)o(\()p FO(a)h FP(+)g FO(m)p FP(\))p FO(\033)694 1780 y FQ(\000)1010 1724 y FP(sin)13 b FO(m\033)18 b FP(sin)o(\(2)p FO(a)h FP(+)f FO(m)g FP(+)g(1\))p FO(\033)p 787 1761 1319 4 v 787 1843 a FP(4)c(sin)944 1808 y FL(2)995 1843 y FO(\033)j FP(cos\()p FO(a)h FP(+)g FO(m)h FP(+)f(1\))p FO(\033)f FP(cos)o(\()p FO(a)h FP(+)g FO(m)p FP(\))p FO(\033)2115 1663 y Fz(\023)2176 1680 y FL(1)p FM(=)p FL(2)2280 1780 y FO(;)118 1993 y FC(the)35 b(p)l(air)h FP(\()p FO(a;)14 b(b)p FP(\))34 b FC(b)l(elongs)i(to)e(the)h(set)f FQ(f)p FP(\()p FO(a;)14 b(b)p FP(\))32 b FQ(2)j FO(M)c FQ(\002)21 b FJ(R)1920 1963 y FL(+)2013 1993 y FQ(j)32 b FO(\013)2121 2005 y FM(m)2216 1993 y FO(>)g FP(0)p FO(;)27 b(m)32 b FP(=)118 2092 y(0)p FO(;)14 b(:)g(:)g(:)f(;)h(s)24 b FQ(\000)g FP(1)p FQ(g)p FC(,)39 b FO(\036)f FQ(2)f FP([0)p FO(;)14 b FP(2)p FO(\031)s FP(\))p FC(,)40 b(and)47 b FO(\033)s(M)f FP(=)37 b([)p FQ(\000)p FO(\031)s(=)p FP(2)p FO(;)14 b(\031)s(=)p FP(2])22 b FQ(n)i(f)p FP(\()p FO(\031)s FP(\(2)p FO(l)i FP(+)d(1\))h(+)118 2192 y FO(m\033)s FP(\))p FO(=)p FP(2)f FQ(j)g FO(l)r(;)14 b(m)22 b FQ(2)i FJ(Z)o FQ(g)o FP(;)243 2311 y(2)p FC(.)38 b FO(H)30 b FP(=)22 b FJ(C)588 2281 y FM(n)639 2311 y FC(,)31 b FO(k)h FC(is)e(o)l(dd,)h FO(J)1083 2323 y FL(1)1121 2311 y FO(e)1160 2323 y FM(m)1246 2311 y FP(=)22 b([)p FO(a)d FP(+)f FO(m)p FP(])1598 2323 y FM(q)1634 2311 y FO(e)1673 2323 y FM(m)1736 2311 y FC(,)417 2598 y FO(J)463 2610 y FL(2)501 2598 y FO(e)540 2610 y FM(m)625 2598 y FP(=)713 2403 y Fz(8)713 2477 y(>)713 2502 y(<)713 2652 y(>)713 2677 y(:)787 2481 y FP(\()p FQ(\000)p FP(1\))958 2451 y FM(i)985 2481 y FO(\025)c(e)1086 2493 y FL(1)1142 2481 y FP(+)k FO(\013)1278 2493 y FL(1)1329 2481 y FO(e)1368 2493 y FL(2)1405 2481 y FO(;)286 b(m)23 b FP(=)g(1)p FO(;)787 2601 y(\013)840 2613 y FM(m)917 2601 y FO(e)956 2613 y FM(m)p FL(+1)1121 2601 y FP(+)18 b FO(\013)1257 2613 y FM(m)p FN(\000)p FL(1)1419 2601 y FO(e)1458 2613 y FM(m)p FN(\000)p FL(1)1606 2601 y FO(;)85 b FP(2)23 b FQ(\024)f FO(m)h FQ(\024)g FO(n)18 b FQ(\000)g FP(1)p FO(;)787 2720 y(\013)840 2732 y FM(n)p FN(\000)p FL(1)984 2720 y FO(e)1023 2732 y FM(n)p FN(\000)p FL(1)1171 2720 y FP(+)g(\()p FQ(\000)p FP(1\))1425 2690 y FM(j)1460 2720 y FO(\025)c(e)1561 2732 y FM(n)1606 2720 y FO(;)85 b(m)23 b FP(=)g FO(n;)118 2877 y FC(wher)l(e)476 3068 y FO(\013)529 3080 y FM(m)615 3068 y FP(=)703 2951 y Fz(\022)1295 3012 y FP(sin)1397 2977 y FL(2)1448 3012 y FO(m\033)p 774 3049 V 774 3132 a FP(4)14 b(sin)931 3097 y FL(2)982 3132 y FO(\033)k FP(sin\()p FO(m)g FQ(\000)g FP(1)p FO(=)p FP(2\))p FO(\033)e FP(sin\()p FO(m)j FP(+)f(1)p FO(=)p FP(2\))p FO(\033)780 3329 y FQ(\000)g FO(\025)911 3295 y FL(2)1327 3273 y FP(sin)1429 3238 y FL(2)1466 3273 y FP(\()p FO(\033)s(=)p FP(2\))p 973 3310 1046 4 v 973 3386 a(sin\()p FO(m)g FQ(\000)g FP(1)p FO(=)p FP(2\))p FO(\033)f FP(sin)o(\()p FO(m)i FP(+)f(1)p FO(=)p FP(2\))p FO(\033)2028 3212 y Fz(\023)2089 3229 y FL(1)p FM(=)p FL(2)2194 3329 y FO(;)118 3537 y(a)32 b FP(=)g FQ(\000)p FO(\031)s(=)p FP(\(2)p FO(\033)s FP(\))22 b FQ(\000)g FP(1)p FO(=)p FP(2)p FC(,)35 b FO(\025)g FC(b)l(elongs)g(to)g(the)g (set)f FQ(f)p FO(\025)e FQ(2)h FJ(R)1917 3507 y FL(+)2010 3537 y FQ(j)i FO(\013)2121 3549 y FM(m)2216 3537 y FO(>)d FP(0)p FO(;)27 b(m)32 b FP(=)118 3637 y(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(n)k FQ(\000)g FP(1)p FQ(g)p FC(,)29 b FO(i)p FC(,)h FO(j)e FP(=)23 b(0)p FC(,)30 b FP(1;)243 3756 y(3)p FC(.)38 b FO(H)30 b FP(=)22 b FJ(C)588 3725 y FL(2)p FM(n)673 3756 y FC(,)30 b FO(k)i FC(is)e(o)l(dd,)853 4032 y FO(J)899 4044 y FL(1)959 4032 y FP(=)1047 3841 y Fz(0)1047 3987 y(B)1047 4040 y(@)1120 3904 y FO(\025)1168 3916 y FL(1)1205 3904 y FO(I)1241 3916 y FL(2)1612 3904 y FP(0)1367 4001 y FC(.)1401 4026 y(.)1436 4051 y(.)1179 4159 y FP(0)328 b FO(\025)1597 4171 y FM(n)1643 4159 y FO(I)1679 4171 y FL(2)1717 3841 y Fz(1)1717 3987 y(C)1717 4040 y(A)1803 4032 y FO(;)p eop %%Page: 141 145 141 144 bop 118 100 a FK(2.3.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-deforemd)f FO(U)9 b FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(R)p FP(\)\))631 b(141)606 525 y FO(J)652 537 y FL(2)712 525 y FP(=)800 258 y Fz(0)800 405 y(B)800 454 y(B)800 504 y(B)800 554 y(B)800 607 y(@)912 319 y FO(Y)960 331 y FL(1)1120 319 y FO(\013)1173 331 y FL(1)1210 319 y FO(I)1246 331 y FL(2)872 474 y FO(\013)925 486 y FL(1)963 474 y FO(I)999 486 y FL(2)1181 474 y FP(0)1448 416 y FC(.)1482 441 y(.)1517 466 y(.)1154 571 y(.)1189 596 y(.)1224 621 y(.)1448 571 y(.)1482 596 y(.)1517 621 y(.)1707 629 y FO(\013)1760 641 y FM(n)p FN(\000)p FL(1)1890 629 y FO(I)1926 641 y FL(2)1367 729 y FO(\013)1420 741 y FM(n)p FN(\000)p FL(1)1550 729 y FO(I)1586 741 y FL(2)1792 729 y FO(Y)1840 741 y FL(2)1964 258 y Fz(1)1964 405 y(C)1964 454 y(C)1964 504 y(C)1964 554 y(C)1964 607 y(A)2050 525 y FO(;)118 898 y FC(wher)l(e)31 b FO(\025)401 910 y FM(m)487 898 y FP(=)23 b([)p FO(a)18 b FP(+)g FO(m)p FP(])839 910 y FM(q)876 898 y FC(,)30 b FO(a)23 b FP(=)g FQ(\000)p FO(\031)s(=)p FP(\(2)p FO(\033)s FP(\))18 b FQ(\000)g FP(1)p FO(=)p FP(2)p FC(,)528 1120 y FO(Y)576 1132 y FL(1)637 1120 y FP(=)725 1003 y Fz(\022)786 1070 y FO(\025)119 b FP(0)789 1169 y(0)86 b FQ(\000)p FO(\025)1030 1003 y Fz(\023)1105 1120 y FO(;)99 b(Y)1275 1132 y FL(2)1335 1120 y FP(=)23 b FO(\025)1485 1003 y Fz(\022)1546 1070 y FP(cos)13 b FO(')128 b FP(sin)13 b FO(')1551 1169 y FP(sin)h FO(')88 b FQ(\000)14 b FP(cos)e FO(')2066 1003 y Fz(\023)2141 1120 y FO(;)118 1343 y(\013)171 1355 y FM(m)262 1343 y FC(is)29 b(the)f(same)h(as)g(in)f FP(2)p FC(,)h FO(\025)f FC(b)l(elongs)h(to)f(the)h(set)e FQ(f)p FO(\025)c FQ(2)h FJ(R)1965 1313 y FL(+)2049 1343 y FQ(j)f FO(\013)2148 1355 y FM(m)2234 1343 y FO(>)g FP(0)p FO(;)k(m)c FP(=)118 1442 y(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(n)k FQ(\000)g FP(1)p FQ(g)p FC(,)29 b FO(')24 b FQ(2)f FP(\(0)p FO(;)14 b(\031)s FP(\);)243 1564 y(4)p FC(.)38 b FO(H)30 b FP(=)22 b FJ(C)588 1534 y FM(n)p FL(+1)723 1564 y FC(,)31 b FO(k)h FC(is)e(o)l(dd,)h FO(J)1167 1576 y FL(1)1205 1564 y FO(e)1244 1576 y FM(m)1330 1564 y FP(=)22 b([)p FO(a)d FP(+)f FO(m)p FP(])1682 1576 y FM(q)1718 1564 y FO(e)1757 1576 y FM(m)1820 1564 y FC(,)503 1866 y FO(J)549 1878 y FL(2)586 1866 y FO(e)625 1878 y FM(m)711 1866 y FP(=)798 1670 y Fz(8)798 1745 y(>)798 1770 y(<)798 1919 y(>)798 1944 y(:)872 1749 y FO(\013)925 1761 y FL(1)963 1749 y FO(e)1002 1761 y FL(2)1039 1749 y FO(;)709 b(m)23 b FP(=)g(1)p FO(;)872 1869 y(\013)925 1881 y FM(m)988 1869 y FO(e)1027 1881 y FM(m)p FL(+1)1193 1869 y FP(+)18 b FO(\013)1329 1881 y FM(m)p FN(\000)p FL(1)1477 1869 y FO(e)1516 1881 y FM(m)p FN(\000)p FL(1)1664 1869 y FO(;)84 b FP(2)23 b FQ(\024)g FO(m)g FQ(\024)f FO(n;)872 1988 y(\013)925 2000 y FM(n)971 1988 y FO(e)1010 2000 y FM(n)1054 1988 y FO(;)694 b(m)23 b FP(=)g FO(n)18 b FP(+)g(1)p FO(;)118 2186 y FC(wher)l(e)24 b FO(a)f FP(=)g FQ(\000)p FO(\031)s(=)p FP(\(2)p FO(\033)s FP(\))5 b FQ(\000)g FP(1)p FC(,)24 b FO(\013)1033 2198 y FL(1)1094 2186 y FP(=)e FO(\013)1234 2198 y FM(n)1303 2186 y FP(=)1390 2117 y FQ(p)p 1459 2117 42 4 v 69 x FP(2)14 b FQ(j)p FO(q)8 b FQ(\000)d FO(q)1693 2156 y FN(\000)p FL(1)1782 2186 y FQ(j)1805 2156 y FN(\000)p FL(1)1880 2186 y FC(,)25 b FO(\013)1983 2198 y FM(m)2070 2186 y FP(=)d FQ(j)p FO(q)g FQ(\000)c FO(q)2362 2156 y FN(\000)p FL(1)2451 2186 y FQ(j)2474 2139 y FN(\000)p FL(1)2549 2186 y FC(,)118 2286 y FO(m)23 b FP(=)g(2)p FC(,)29 b FO(:)14 b(:)g(:)28 b FC(,)i FO(n)18 b FQ(\000)g FP(1;)243 2408 y(5)p FC(.)38 b FO(H)30 b FP(=)22 b FJ(C)588 2377 y FL(2)p FM(n)673 2408 y FC(,)30 b FO(k)i FC(is)e(o)l(dd,)225 2777 y FO(J)271 2789 y FL(1)332 2777 y FP(=)419 2585 y Fz(0)419 2731 y(B)419 2784 y(@)492 2648 y FO(\025)540 2660 y FL(1)578 2648 y FO(I)614 2660 y FL(1)1026 2648 y FP(0)739 2745 y FC(.)774 2770 y(.)808 2795 y(.)551 2903 y FP(0)328 b FO(\025)969 2915 y FM(n)p FL(+1)1099 2903 y FO(I)1135 2915 y FL(1)1173 2585 y Fz(1)1173 2731 y(C)1173 2784 y(A)1259 2777 y FO(;)99 b(J)1427 2789 y FL(2)1487 2777 y FP(=)1575 2510 y Fz(0)1575 2656 y(B)1575 2706 y(B)1575 2756 y(B)1575 2806 y(B)1575 2859 y(@)1680 2571 y FP(0)115 b FO(X)1913 2541 y FN(\003)1906 2592 y FL(1)1648 2726 y FO(X)1717 2738 y FL(1)1873 2726 y FP(0)2043 2668 y FC(.)2078 2693 y(.)2112 2718 y(.)1846 2823 y(.)1881 2848 y(.)1915 2873 y(.)2043 2823 y(.)2078 2848 y(.)2112 2873 y(.)2230 2881 y FO(X)2306 2851 y FN(\003)2299 2901 y FM(n)2033 2980 y FO(X)2102 2992 y FM(n)2266 2980 y FP(0)2344 2510 y Fz(1)2344 2656 y(C)2344 2706 y(C)2344 2756 y(C)2344 2806 y(C)2344 2859 y(A)2431 2777 y FO(;)118 3150 y FC(wher)l(e)31 b FO(\025)401 3162 y FM(m)487 3150 y FP(=)23 b([)p FO(m)18 b FQ(\000)g FP(1)g FQ(\000)g FO(\031)s(=)p FP(\(2)p FO(\033)s FP(])1154 3162 y FM(q)1191 3150 y FC(,)287 3377 y FO(X)356 3389 y FL(1)416 3377 y FP(=)504 3260 y Fz(\022)565 3258 y FQ(p)p 634 3258 V 69 x FP(2)13 b FQ(j)p FO(q)22 b FQ(\000)c FO(q)894 3297 y FN(\000)p FL(1)983 3327 y FQ(j)1006 3297 y FN(\000)p FL(1)1109 3327 y FP(cos)13 b FO(')569 3363 y FQ(p)p 638 3363 V 69 x FP(2)h FQ(j)p FO(q)21 b FQ(\000)d FO(q)898 3402 y FN(\000)p FL(1)988 3432 y FQ(j)1011 3402 y FN(\000)p FL(1)1114 3432 y FP(sin)13 b FO(')1288 3260 y Fz(\023)1363 3377 y FO(;)99 b(X)1554 3389 y FM(n)1622 3377 y FP(=)23 b(\()1742 3305 y FQ(p)p 1811 3305 V 72 x FP(2)13 b FQ(j)p FO(q)22 b FQ(\000)c FO(q)2071 3343 y FN(\000)p FL(1)2160 3377 y FQ(j)2183 3343 y FN(\000)p FL(1)2272 3377 y FO(;)c FP(0\))p FO(;)118 3607 y(X)187 3619 y FM(m)273 3607 y FP(=)23 b FQ(j)p FO(q)e FQ(\000)d FO(q)565 3577 y FN(\000)p FL(1)654 3607 y FQ(j)677 3577 y FN(\000)p FL(1)767 3607 y FO(I)803 3619 y FL(2)840 3607 y FC(,)30 b FO(m)23 b FP(=)g(2)p FC(,)30 b FO(:)14 b(:)g(:)27 b FC(,)j FO(n)19 b FQ(\000)f FP(1)p FC(,)29 b FO(')24 b FQ(2)f FP(\(0)p FO(;)14 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y FO(e)1313 4045 y FM(m)p FN(\000)p FL(1)1461 4033 y FO(;)227 b FP(2)23 b FQ(\024)g FO(m)g FQ(\024)f FP(\()p FO(n)d FQ(\000)f FP(1\))p FO(=)p FP(2)p FO(;)642 4153 y(\013)695 4168 y FL(\()p FM(n)p FN(\000)p FL(1\))p FM(=)p FL(2)958 4153 y FO(e)997 4168 y FL(\()p FM(n)p FN(\000)p FL(1\))p FM(=)p FL(2)1246 4153 y FO(;)442 b(m)23 b FP(=)g(\()p FO(n)c FP(+)f(1\))p FO(=)p FP(2)p FO(;)p eop %%Page: 142 146 142 145 bop 118 100 a FP(142)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 334 y FC(wher)l(e)32 b FO(a)25 b FP(=)g FO(\031)s(=)p FP(\(2)p FO(\033)s FP(\))20 b FQ(\000)f FP(1)p FO(=)p FP(2)p FC(,)30 b FO(\013)1099 346 y FM(m)1187 334 y FP(=)25 b FQ(j)p FO(q)d FQ(\000)c FO(q)1482 304 y FN(\000)p FL(1)1571 334 y FQ(j)1594 287 y FN(\000)p FL(1)1683 334 y FC(,)32 b FO(m)25 b FP(=)g(1)p FC(,)31 b FO(:)14 b(:)g(:)27 b FC(,)32 b FP(\()p FO(n)20 b FQ(\000)f FP(3\))p FO(=)p FP(2)p FC(,)118 434 y FO(\013)171 449 y FL(\()p FM(n)p FN(\000)p FL(1\))p FM(=)p FL(2)443 434 y FP(=)531 365 y FQ(p)p 600 365 42 4 v 69 x FP(2)14 b FQ(j)p FO(q)21 b FQ(\000)d FO(q)860 404 y FN(\000)p FL(1)949 434 y FQ(j)972 404 y FN(\000)p FL(1)1062 434 y FC(,)30 b FO(i)22 b FP(=)h(0)p FC(,)30 b FP(1)p FC(,)f FO(j)f FP(=)23 b(0)p FC(,)30 b FP(1;)243 558 y(7)p FC(.)38 b FO(H)30 b FP(=)22 b FJ(C)588 527 y FM(n)639 558 y FC(,)31 b FO(k)h FC(is)e(even,)232 938 y FO(J)278 950 y FL(1)339 938 y FP(=)426 746 y Fz(0)426 892 y(B)426 945 y(@)499 810 y FO(\025)547 822 y FL(1)585 810 y FO(I)621 822 y FL(2)1030 810 y FP(0)746 907 y FC(.)781 931 y(.)815 957 y(.)558 1064 y FP(0)329 b FO(\025)977 1076 y FM(p)p FL(+1)1099 1064 y FO(I)1135 1076 y FL(1)1173 746 y Fz(1)1173 892 y(C)1173 945 y(A)1260 938 y FO(;)98 b(J)1427 950 y FL(2)1488 938 y FP(=)1575 671 y Fz(0)1575 818 y(B)1575 867 y(B)1575 917 y(B)1575 967 y(B)1575 1020 y(@)1668 731 y FO(Y)121 b(X)1913 701 y FN(\003)1906 752 y FL(1)1648 886 y FO(X)1717 898 y FL(1)1873 886 y FP(0)2040 828 y FC(.)2074 853 y(.)2109 878 y(.)1846 983 y(.)1881 1008 y(.)1916 1033 y(.)2040 983 y(.)2074 1008 y(.)2109 1033 y(.)2224 1041 y FO(X)2300 1011 y FN(\003)2293 1062 y FM(p)2034 1143 y FO(X)2103 1155 y FM(p)2260 1143 y FP(0)2337 671 y Fz(1)2337 818 y(C)2337 867 y(C)2337 917 y(C)2337 967 y(C)2337 1020 y(A)2424 938 y FO(;)118 1329 y FC(wher)l(e)31 b FO(p)22 b FP(=)h(\()p FO(n)c FQ(\000)f FP(1\))p FO(=)p FP(2)p FC(,)29 b FO(\025)949 1341 y FM(m)1035 1329 y FP(=)23 b(\()p FQ(\000)p FP(1\))1294 1299 y FM(j)1328 1329 y FP([)p FO(\031)s(=)p FP(\(2)p FO(\033)t FP(\))18 b FQ(\000)g FP(1)p FO(=)p FP(2)f(+)h FO(m)p FP(])2023 1341 y FM(q)2060 1329 y FC(,)202 1561 y FO(Y)41 b FP(=)23 b FQ(j)p FO(q)f FQ(\000)c FO(q)584 1527 y FN(\000)p FL(1)673 1561 y FQ(j)696 1527 y FN(\000)p FL(1)799 1444 y Fz(\022)860 1511 y FP(cos)13 b FO(')127 b FP(sin)14 b FO(')865 1610 y FP(sin)f FO(')88 b FQ(\000)14 b FP(cos)f FO(')1380 1444 y Fz(\023)1455 1561 y FO(;)99 b(X)1646 1573 y FM(p)1707 1561 y FP(=)23 b(\()1827 1488 y FQ(p)p 1896 1488 V 73 x FP(2)13 b FQ(j)p FO(q)22 b FQ(\000)c FO(q)2156 1527 y FN(\000)p FL(1)2245 1561 y FQ(j)2268 1527 y FN(\000)p FL(1)2357 1561 y FO(;)c FP(0\))p FO(;)118 1796 y(X)187 1808 y FM(m)273 1796 y FP(=)23 b FQ(j)p FO(q)e FQ(\000)d FO(q)565 1765 y FN(\000)p FL(1)654 1796 y FQ(j)677 1765 y FN(\000)p FL(1)767 1796 y FO(I)803 1808 y FL(2)840 1796 y FC(,)30 b FO(m)23 b FP(=)g(1)p FC(,)30 b FO(:)14 b(:)g(:)27 b FC(,)j FO(p)19 b FQ(\000)f FP(1)p FC(,)29 b FO(')24 b FQ(2)f FP(\(0)p FO(;)14 b(\031)s FP(\))p FC(,)31 b FO(j)d FP(=)22 b(0)p FC(,)30 b FP(1;)243 1919 y(8)p FC(.)38 b FO(H)30 b FP(=)22 b FJ(C)588 1889 y FM(p)633 1919 y FC(,)30 b FO(p)23 b(<)f(n)30 b FC(if)g FO(k)j FC(is)d(o)l(dd,)h FO(p)23 b(<)g FP(\()p FO(n)18 b FP(+)g(1\))p FO(=)p FP(2)29 b FC(if)h FO(k)j FC(is)d(even,)317 2099 y FO(J)363 2111 y FL(1)400 2099 y FO(e)439 2111 y FM(m)525 2099 y FP(=)23 b(\()p FQ(\000)p FP(1\))784 2064 y FM(i)825 2099 y FP([)p FO(a)18 b FP(+)g FO(m)p FP(])1089 2111 y FM(q)1140 2099 y FO(e)1179 2111 y FM(m)1242 2099 y FO(;)317 2365 y(J)363 2377 y FL(2)400 2365 y FO(e)439 2377 y FM(m)525 2365 y FP(=)613 2170 y Fz(8)613 2245 y(>)613 2270 y(<)613 2419 y(>)613 2444 y(:)687 2245 y FO(\013)740 2257 y FL(1)791 2245 y FO(e)830 2257 y FL(2)885 2245 y FP(+)g(\()p FQ(\000)p FP(1\))1139 2215 y FM(j)1315 2208 y FL(sin)11 b FM(p\033)p 1184 2226 431 4 v 1184 2274 a FL(2)g(sin\()p FM(\033)r(=)p FL(2\))h(sin)f FM(\033)1638 2245 y FO(e)1677 2257 y FL(1)1714 2245 y FO(;)85 b(m)23 b FP(=)g(1)p FO(;)687 2372 y(\013)740 2384 y FM(m)817 2372 y FO(e)856 2384 y FM(m)p FL(+1)1021 2372 y FP(+)18 b FO(\013)1157 2384 y FM(m)p FN(\000)p FL(1)1319 2372 y FO(e)1358 2384 y FM(m)p FN(\000)p FL(1)1506 2372 y FO(;)293 b FP(2)23 b FQ(\024)f FO(m)h FQ(\024)g FO(p)18 b FQ(\000)g FP(1)p FO(;)687 2492 y(\013)740 2504 y FM(p)p FN(\000)p FL(1)877 2492 y FO(e)916 2504 y FM(p)p FN(\000)p FL(1)1039 2492 y FO(;)760 b(m)23 b FP(=)g FO(p;)118 2673 y FC(wher)l(e)439 2905 y FO(\013)492 2917 y FM(m)578 2905 y FP(=)666 2788 y Fz(\022)972 2848 y FP(sin\()p FO(m)18 b FQ(\000)g FO(l)r FP(\))p FO(\033)f FP(sin\()p FO(m)i FP(+)f FO(l)r FP(\))p FO(\033)p 737 2886 1319 4 v 737 2968 a FP(4)c(sin)894 2933 y FL(2)945 2968 y FO(\033)k FP(sin\()p FO(m)g FQ(\000)g FP(1)p FO(=)p FP(2\))p FO(\033)e FP(sin\()p FO(m)j FP(+)f(1)p FO(=)p FP(2\))p FO(\033)2065 2788 y Fz(\023)2126 2805 y FL(1)p FM(=)p FL(2)2231 2905 y FO(;)118 3139 y(a)23 b FP(=)g FO(\031)s(=)p FP(\(2)p FO(\033)s FP(\))18 b FQ(\000)g FP(1)p FO(=)p FP(2)p FC(,)29 b FO(p)23 b FQ(2)g(f)p FO(p)g FQ(2)g FJ(Z)1192 3151 y FL(+)1264 3139 y FQ(j)g FO(\013)1363 3151 y FM(m)1449 3139 y FO(>)g FP(0)p FO(;)k FP(1)c FQ(\024)f FO(m)h(<)g(p)p FQ(g)p FC(,)29 b FO(i)p FC(,)h FO(j)e FP(=)23 b(0)p FC(,)29 b FP(1;)243 3263 y(9)p FC(.)38 b FO(H)30 b FP(=)22 b FJ(C)588 3233 y FM(p)633 3263 y FC(,)30 b FO(J)734 3275 y FL(1)771 3263 y FO(e)810 3275 y FM(m)896 3263 y FP(=)22 b([)p FO(a)d FP(+)f FO(m)p FP(])1248 3275 y FM(q)1298 3263 y FO(e)1337 3275 y FM(m)1400 3263 y FC(,)435 3574 y FO(J)481 3586 y FL(2)519 3574 y FO(e)558 3586 y FM(m)643 3574 y FP(=)731 3379 y Fz(8)731 3454 y(>)731 3479 y(<)731 3628 y(>)731 3653 y(:)805 3458 y FO(\013)858 3470 y FL(1)895 3458 y FO(e)934 3470 y FL(2)971 3458 y FO(;)710 b(m)23 b FP(=)g(1)p FO(;)805 3578 y(\013)858 3590 y FM(m)921 3578 y FO(e)960 3590 y FM(m)p FL(+1)1125 3578 y FP(+)18 b FO(\013)1261 3590 y FM(m)p FN(\000)p FL(1)1409 3578 y FO(e)1448 3590 y FM(m)p FN(\000)p FL(1)1596 3578 y FO(;)85 b FP(2)23 b FQ(\024)f FO(m)h FQ(\024)g FO(p)18 b FQ(\000)g FP(1)p FO(;)805 3697 y(\013)858 3709 y FM(l)p FN(\000)p FL(1)968 3697 y FO(e)1007 3709 y FM(l)p FN(\000)p FL(1)1118 3697 y FO(;)563 b(m)23 b FP(=)g FO(p;)118 3879 y FC(wher)l(e)433 4110 y FO(\013)486 4122 y FL(1)546 4110 y FP(=)634 3993 y Fz(\022)695 4110 y FQ(\000)908 4054 y FP(sin\()p FO(a)18 b FP(+)g(1\))p FO(\033)p 770 4091 681 4 v 770 4167 a FP(2)c(sin)e FO(\033)18 b FP(cos)o(\()p FO(a)h FP(+)f(2\))p FO(\033)s FP(\))1460 3993 y Fz(\023)1521 4010 y FL(1)p FM(=)p FL(2)1625 4110 y FO(;)p eop %%Page: 143 147 143 146 bop 118 100 a FK(2.3.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-deforemd)f FO(U)9 b FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(R)p FP(\)\))631 b(143)407 393 y FO(\013)460 405 y FM(m)546 393 y FP(=)634 276 y Fz(\022)695 393 y FQ(\000)992 337 y FP(sin)14 b FO(m\033)j FP(sin\(2)p FO(a)h FP(+)g FO(m)h FP(+)f(1\))p FO(\033)p 770 374 1319 4 v 770 456 a FP(4)c(sin)927 421 y FL(2)978 456 y FO(\033)j FP(cos)o(\()p FO(a)i FP(+)f FO(m)p FP(\))p FO(\033)f FP(cos)o(\()p FO(a)i FP(+)f FO(m)g FP(+)g(1\))p FO(\033)2098 276 y Fz(\023)2159 293 y FL(1)p FM(=)p FL(2)2263 393 y FO(;)118 630 y(m)35 b FQ(6)p FP(=)f(1)p FC(,)k(the)f(p)l(air)g FP(\()p FO(a;)14 b(p)p FP(\))36 b FC(b)l(elongs)h(to)g(the)f(set)g FQ(f)p FP(\()p FO(a;)14 b(p)p FP(\))34 b FQ(2)i FJ(R)29 b FQ(\002)23 b FJ(Z)2239 642 y FL(+)2323 630 y FQ(j)35 b FO(\033)s(a)g FQ(6)p FP(=)128 697 y FM(\031)p 128 711 41 4 v 132 758 a FL(2)188 730 y FQ(\000)9 b FO(l)r(\033)j FP(+)d FO(\031)s(r)r FC(,)27 b FP([)p FO(p)p FP(])651 742 y FM(q)687 730 y FP([2)p FO(a)9 b FP(+)g FO(p)g FP(+)g(1])1069 742 y FM(q)1127 730 y FP(=)22 b(0)p FC(,)27 b FP([)p FO(a)9 b FP(+)g FO(m)p FP(])1554 742 y FM(q)1613 730 y FQ(6)p FP(=)22 b([)p FO(a)9 b FP(+)g FO(n)p FP(])1923 742 y FM(q)1960 730 y FC(,)26 b FP(1)d FQ(\024)f FO(m)h(<)g(n)g FQ(\024)g FO(p)p FC(,)118 829 y FO(\013)171 841 y FM(m)257 829 y FO(>)g FP(0)p FO(;)14 b(m)22 b FP(=)h(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(p)k FQ(\000)g FP(1)p FO(;)c(r)25 b FQ(2)f FJ(Z)o FQ(g)p FC(.)243 954 y FP(10)p FC(.)37 b FO(H)30 b FP(=)23 b FJ(C)630 924 y FL(1)673 954 y FC(,)30 b FO(J)774 966 y FL(1)834 954 y FP(=)23 b(0)p FC(,)30 b FO(J)1065 966 y FL(2)1125 954 y FP(=)23 b(0)p FC(.)118 1119 y(Pr)l(o)l(of.)43 b FP(Let)24 b FO(J)569 1131 y FL(1)629 1119 y FP(=)f FO(J)771 1089 y FN(\003)763 1140 y FL(1)809 1119 y FP(,)i FO(J)903 1131 y FL(2)963 1119 y FP(=)e FO(J)1105 1089 y FN(\003)1097 1140 y FL(2)1167 1119 y FP(b)r(e)g(op)r(erators)f(on)h(a)h(Hilb)r(ert)g(space)f (satisfy-)118 1219 y(ing)h(\(2.35\){\(2.36\))e(and)h(\000)g(=)g FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))23 b FQ(j)g FP(\010\()p FO(t;)14 b(s)p FP(\))23 b FQ(\021)g FO(t)1715 1189 y FL(2)1763 1219 y FQ(\000)11 b FP(\()p FO(q)j FP(+)d FO(q)2038 1189 y FN(\000)p FL(1)2127 1219 y FP(\))j FO(ts)d FP(+)g FO(s)2368 1189 y FL(2)2416 1219 y FQ(\000)g FP(1)p FQ(g)118 1318 y FP(the)26 b(c)n(haracteristic)e(binary)i(relation)f(corresp)r (onding)f(to)h(\(2.35\).)36 b(By)26 b(Theo-)118 1418 y(rem)h(8,)930 1600 y FO(E)991 1612 y FM(J)1028 1620 y Fy(1)1065 1600 y FP(\(\001\))p FO(J)1244 1612 y FL(2)1282 1600 y FO(E)1343 1612 y FM(J)1380 1620 y Fy(1)1416 1600 y FP(\(\001)1517 1566 y FN(0)1541 1600 y FP(\))d(=)e(0)p FO(;)614 b FP(\(2.40\))118 1782 y(for)37 b(an)n(y)g(\001,)j(\001)623 1752 y FN(0)686 1782 y FQ(2)g FA(B)p FP(\()p FJ(R)q FP(\),)46 b(\001)26 b FQ(\002)e FP(\001)1295 1752 y FN(0)1344 1782 y FQ(\\)h FP(\000)40 b(=)f FJ(?)p FP(.)66 b(Denote)38 b(b)n(y)f FO(S)2245 1794 y FL(0)2320 1782 y FP(the)h(set)118 1882 y FQ(f)p FO(s)32 b FQ(2)h FJ(R)38 b FQ(j)32 b FP(\()p FO(q)26 b FQ(\000)21 b FO(q)687 1852 y FN(\000)p FL(1)777 1882 y FP(\))809 1852 y FL(2)846 1882 y FO(s)885 1852 y FL(2)944 1882 y FP(+)h(4)32 b FO(<)g FP(0)p FQ(g)p FP(,)i(and)f(b)n(y)f FO(S)1681 1894 y FL(1)1752 1882 y FP(its)h(complemen)n(t.)54 b(Then)118 1981 y FO(R)17 b FQ(\002)g FO(S)331 1993 y FL(0)385 1981 y FQ(\\)g FP(\000)23 b(=)f FJ(?)p FP(,)27 b(whic)n(h)g(implies)g(that)g FO(J)1477 1993 y FL(2)1514 1981 y FO(E)1575 1993 y FM(J)1612 2001 y Fy(1)1648 1981 y FP(\()p FO(S)1731 1993 y FL(0)1769 1981 y FP(\))p FO(H)j FP(=)23 b(0.)36 b(Hence,)27 b FO(H)2427 1993 y FL(0)2487 1981 y FP(:=)118 2081 y FO(E)179 2093 y FM(J)216 2101 y Fy(1)253 2081 y FP(\()p FO(S)336 2093 y FL(0)373 2081 y FP(\))p FO(H)46 b FP(is)38 b(in)n(v)-5 b(arian)n(t)38 b(with)h(resp)r(ect)f(to)h FO(J)1628 2093 y FL(1)1665 2081 y FP(,)i FO(J)1775 2093 y FL(2)1851 2081 y FP(and)e(an)n(y)f(irreducible)118 2181 y(represen)n(tation)26 b(in)i FO(H)827 2193 y FL(0)892 2181 y FP(is)f(one-dimensional)g(and)g (giv)n(en)g(b)n(y)751 2363 y FO(J)797 2375 y FL(1)857 2363 y FP(=)c(\()p FO(\025)p FP(\))p FO(;)98 b(J)1224 2375 y FL(2)1284 2363 y FP(=)23 b(\(0\))p FO(;)180 b(\025)23 b FQ(2)h FO(S)1882 2375 y FL(0)1919 2363 y FO(:)118 2545 y FP(But,)32 b(b)n(y)f(\(2.36\),)h FO(\025)d FP(=)g(0)h(whic)n(h)h(do)r (es)g(not)g(b)r(elong)g(to)g FO(S)1948 2557 y FL(0)1985 2545 y FP(.)48 b(Therefore,)30 b(the)118 2644 y(sp)r(ectrum)e(of)g FO(J)622 2656 y FL(1)687 2644 y FP(b)r(elongs)f(to)h FO(S)1138 2656 y FL(1)1175 2644 y FP(.)37 b(Consider)27 b(the)h(follo)n(wing)f(parameteriza-)118 2744 y(tion)f(of)g FO(S)431 2756 y FL(1)469 2744 y FP(:)36 b FO(\025)23 b FP(=)g(sin)14 b FO(x\033)s(=)g FP(sin)g FO(\033)s FP(,)27 b FO(x)c FQ(2)h FJ(R)p FP(.)42 b(Let)26 b FB(O)p FP(\()p FQ(f)p FO(\025)p FQ(g)p FP(\))g(b)r(e)h(the)f(tra)5 b(jectory)25 b(of)118 2843 y(the)g(p)r(oin)n(t)g FO(\025)h FP(with)f(resp)r(ect)g (to)f(\000,)i(i.e.,)g(the)f(minimal)g(subset)g FO(M)31 b FQ(\032)23 b FJ(R)31 b FP(whic)n(h)118 2943 y(con)n(tains)i FO(\025)i FP(and)e(satis\014es)h(the)g(condition)g(\()p FJ(R)29 b FQ(n)22 b FO(M)9 b FP(\))22 b FQ(\002)h FO(M)31 b FQ(\\)23 b FP(\000)34 b(=)f FJ(?)p FP(,)i(and)118 3043 y FO(M)29 b FQ(\002)20 b FP(\()p FJ(R)27 b FQ(n)19 b FO(M)9 b FP(\))21 b FQ(\\)f FP(\000)28 b(=)g FJ(?)p FP(.)45 b(In)30 b(our)g(case,)g FB(O)p FP(\()p FQ(f)p FO(\025)p FQ(g)p FP(\))e(=)f FQ(f)p FP(sin\()p FO(x)21 b FP(+)f FO(k)s FP(\))p FO(\033)s(=)14 b FP(sin)f FO(\033)32 b FQ(j)118 3142 y FO(k)26 b FQ(2)d FJ(Z)p FQ(g)o FP(.)243 3242 y(If)k(\()p FO(J)403 3254 y FL(1)441 3242 y FO(;)14 b(J)524 3254 y FL(2)561 3242 y FP(\))28 b(is)f(irreducible,)g(then)h (the)f(measure)f FO(E)1853 3254 y FM(J)1890 3262 y Fy(1)1927 3242 y FP(\()p FQ(\001)p FP(\))i(is)f(ergo)r(dic)f(with)118 3342 y(resp)r(ect)36 b(to)g(\000,)j(i.e.,)g(either)d FO(E)1108 3354 y FM(J)1145 3362 y Fy(1)1181 3342 y FP(\()p FO(M)9 b FP(\))38 b(=)f(0)f(or)g FO(E)1725 3354 y FM(J)1762 3362 y Fy(1)1798 3342 y FP(\()p FO(M)9 b FP(\))38 b(=)f FO(I)44 b FP(for)35 b(an)n(y)h(set)118 3441 y FO(M)41 b FP(whic)n(h)33 b(is)f(in)n(v)-5 b(arian)n(t)31 b(with)i(resp)r(ect)f (to)h(\000.)51 b(In)33 b(fact,)h(if)f(it)g(w)n(ere)e(not)h(the)118 3541 y(case,)d(w)n(e)f(w)n(ould)h(conclude)g(that)g FO(E)1271 3553 y FM(J)1308 3561 y Fy(1)1344 3541 y FP(\()p FO(M)9 b FP(\))p FO(H)36 b FP(or)29 b FO(E)1768 3553 y FM(J)1805 3561 y Fy(1)1841 3541 y FP(\()p FO(M)9 b FP(\))p FO(H)36 b FP(is)29 b(a)g(subspace)118 3640 y(in)n(v)-5 b(arian)n(t)22 b(with)i(resp)r(ect)g(to)f FO(J)1071 3652 y FL(1)1108 3640 y FP(,)i FO(J)1202 3652 y FL(2)1239 3640 y FP(.)36 b(Moreo)n(v)n(er,)21 b(one)i(can)g(easily)g(c)n(hec)n(k)g(that)118 3740 y(there)32 b(exists)f(a)g(measurable)f(section)i(of)f(\()p FO(S)1542 3752 y FL(1)1580 3740 y FO(;)14 b FP(\000\),)32 b(i.e.,)h(a)f(set)f(whic)n(h)h(meets)118 3840 y(ev)n(ery)37 b(tra)5 b(jectory)36 b(only)h(once.)66 b(Th)n(us)38 b(an)n(y)f(ergo)r (dic)f(measure)h(is)h(concen-)118 3939 y(trated)24 b(on)h(a)f(single)g (tra)5 b(jectory)23 b(of)i(some)f(p)r(oin)n(t,)h(and,)g(hence)g(the)g (sp)r(ectrum)118 4039 y(of)38 b(the)g(op)r(erator)e FO(J)767 4051 y FL(1)842 4039 y FP(is)h(discrete)h(and)f(concen)n(trated)g(on)g (a)g(tra)5 b(jectory)36 b(if)118 4147 y(\()p FO(J)196 4159 y FL(1)234 4147 y FO(;)14 b(J)317 4159 y FL(2)354 4147 y FP(\))29 b(is)f(an)h(irreducible)f(pair.)39 b(Let)28 b FO(H)1452 4159 y FL(1)1514 4147 y FP(=)d FO(E)1665 4159 y FM(A)1719 4147 y FP(\()p FB(O)p FP(\()p FQ(f)1894 4107 y FL(sin\()p FM(\031)r(=)p FL(2\))p 1894 4128 242 4 v 1948 4176 a(sin)11 b FM(\033)2145 4147 y FQ(g)p FP(\)\))p FO(H)c FP(,)29 b FO(H)2448 4159 y FL(2)2510 4147 y FP(=)p eop %%Page: 144 148 144 147 bop 118 100 a FP(144)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FO(E)179 345 y FM(A)233 333 y FP(\()p FB(O)p FP(\()p FQ(f\000)473 293 y FL(sin\()p FM(\031)r(=)p FL(2\))p 473 314 242 4 v 527 362 a(sin)11 b FM(\033)724 333 y FQ(g)p FP(\)\))p FO(H)c FP(,)34 b(if)f FO(k)i FP(is)e(ev)n(en,)g(and)f FO(H)1664 345 y FL(1)1732 333 y FP(=)f FO(E)1889 345 y FM(A)1943 333 y FP(\()p FB(O)p FP(\()p FQ(f)2118 293 y FL(sin\()p FM(\031)r(=)p FL(2\))p 2118 314 V 2172 362 a(sin)11 b FM(\033)2370 333 y FQ(g)p FP(\)\))p FO(H)c FP(,)118 454 y FO(H)187 466 y FL(2)247 454 y FP(=)23 b FO(E)396 466 y FM(A)450 454 y FP(\()p FB(O)p FP(\()p FQ(f)625 414 y FL(sin\(\()p FM(\031)r FN(\000)p FM(\033)r FL(\))p FM(=)p FL(2\))p 625 435 387 4 v 751 483 a(sin)11 b FM(\033)1021 454 y FQ(g)p FP(\)\))p FO(H)c FP(,)28 b(if)g FO(k)i FP(is)e(o)r(dd,)g FO(H)1743 466 y FL(3)1803 454 y FP(=)23 b(\()p FO(H)1992 466 y FL(1)2048 454 y FQ(\010)18 b FO(H)2200 466 y FL(2)2237 454 y FP(\))2269 424 y FN(?)2325 454 y FP(.)243 554 y(An)n(y)35 b(tra)5 b(jectory)33 b(can)i(b)r(e)h(describ)r(ed)f(geometrically)f(in) h(the)h(follo)n(wing)118 654 y(w)n(a)n(y:)j(if)d FO(t)p FP(,)29 b FO(s)d FQ(2)f FB(O)p FP(\()p FQ(f)p FO(\025)p FQ(g)p FP(\))k(and)g(\()p FO(t;)14 b(s)p FP(\))26 b FQ(2)g FP(\000,)j(then)h(w)n(e)e(dra)n(w)g(an)h(edge)2331 636 y Fo(r)p 2331 637 125 4 v 100 w(r)2319 719 y FM(t)96 b(s)2526 654 y FP(if)118 811 y FO(t)29 b FQ(6)p FP(=)f FO(s)p FP(,)33 b(and)d(a)h(lo)r(op)829 793 y Fo(r)p 829 795 4 4 v 825 790 V 820 785 V 816 781 V 812 776 V 809 772 V 806 767 V 804 763 V 801 759 V 800 755 V 798 751 V 797 748 V 796 744 V 796 741 V 796 738 V 797 734 V 797 731 V 799 728 V 800 726 V 802 723 V 804 720 V 804 720 V 807 718 V 809 716 V 812 714 V 814 712 V 817 711 V 819 710 V 822 709 V 824 708 V 827 708 V 829 708 V 832 708 V 834 708 V 837 709 V 839 710 V 842 711 V 844 712 V 847 714 V 849 716 V 852 718 V 854 720 V 829 795 V 834 790 V 839 785 V 843 781 V 846 776 V 850 772 V 853 767 V 855 763 V 857 759 V 859 755 V 860 751 V 862 748 V 862 744 V 863 741 V 862 738 V 862 734 V 861 731 V 860 728 V 858 726 V 857 723 V 854 720 V 817 876 a FM(t)902 811 y FP(if)h FO(t)d FP(=)f FO(s)p FP(.)47 b(Then)32 b(w)n(e)e(ha)n(v)n(e)g(the)i (follo)n(wing)e(t)n(yp)r(es)h(of)118 911 y(tra)5 b(jectories.)243 1011 y(An)n(y)31 b(tra)5 b(jectory)30 b(of)h(the)h(p)r(oin)n(t)f FO(t)39 b(=)-51 b FQ(2)29 b FB(O)p FP(\()p FQ(f)p FP(1)p FO(=)p FP(sin)13 b FO(\033)s FQ(g)p FP(\))21 b FQ([)g FB(O)p FP(\()p FQ(f\000)p FP(1)p FO(=)p FP(sin)12 b FO(\033)t FQ(g)p FP(\))20 b FQ([)118 1110 y FB(O)p FP(\()p FQ(f)p FP(cos)o(\()p FO(\033)s(=)p FP(2\))p FO(=)p FP(sin)13 b FO(\033)t FQ(g)p FP(\))p FQ([)p FB(O)p FP(\()p FQ(f\000)p FP(cos)n(\()p FO(\033)s(=)p FP(2\))p FO(=)p FP(sin)g FO(\033)t FQ(g)p FP(\))18 b(is)g(a)g(cycle)g(of)h(length)f FO(s)p FP(,)j(i.e.,)118 1210 y(the)f(follo)n(wing)e(graph:)886 1192 y Fo(r)p 886 1194 125 4 v 100 w(r)265 b(r)p 1177 1194 V -149 w(r)p 886 1194 4 4 v 890 1192 V 894 1190 V 899 1189 V 903 1187 V 907 1186 V 911 1184 V 915 1183 V 919 1181 V 924 1180 V 928 1179 V 932 1177 V 936 1176 V 940 1175 V 944 1174 V 948 1172 V 953 1171 V 957 1170 V 961 1169 V 965 1168 V 969 1167 V 973 1166 V 977 1165 V 982 1164 V 986 1163 V 990 1162 V 994 1162 V 998 1161 V 1002 1160 V 1007 1159 V 1011 1159 V 1015 1158 V 1019 1157 V 1023 1157 V 1027 1156 V 1031 1156 V 1036 1155 V 1040 1155 V 1044 1154 V 1048 1154 V 1052 1154 V 1056 1153 V 1061 1153 V 1065 1153 V 1069 1153 V 1073 1152 V 1077 1152 V 1081 1152 V 1085 1152 V 1090 1152 V 1094 1152 V 1098 1152 V 1102 1152 V 1106 1152 V 1110 1152 V 1114 1152 V 1119 1153 V 1123 1153 V 1127 1153 V 1131 1153 V 1135 1154 V 1139 1154 V 1144 1154 V 1148 1155 V 1152 1155 V 1156 1156 V 1160 1156 V 1164 1157 V 1168 1157 V 1173 1158 V 1177 1159 V 1181 1159 V 1185 1160 V 1189 1161 V 1193 1162 V 1197 1162 V 1202 1163 V 1206 1164 V 1210 1165 V 1214 1166 V 1218 1167 V 1222 1168 V 1227 1169 V 1231 1170 V 1235 1171 V 1239 1172 V 1243 1174 V 1247 1175 V 1251 1176 V 1256 1177 V 1260 1179 V 1264 1180 V 1268 1181 V 1272 1183 V 1276 1184 V 1281 1186 V 1285 1187 V 1289 1189 V 1293 1190 V 1297 1192 V 1301 1194 V 850 1267 a FM(\025)889 1275 y Fy(1)975 1267 y FM(\025)1014 1275 y Fy(2)1266 1267 y FM(\025)1305 1275 y Fw(s)1038 1196 y FO(:)14 b(:)g(:)1343 1210 y FP(,)21 b(where)e FO(\025)1667 1222 y FM(m)1753 1210 y FP(=)k(sin\(\()p FO(x)c FP(+)f FO(m)p FP(\))p FO(\033)s FP(\))q FO(=)p FP(sin)13 b FO(\033)t FP(,)118 1355 y FO(x\033)27 b FQ(2)c FP([)p FQ(\000)p FO(\031)s(=)p FP(2)p FO(;)14 b(\031)s(=)p FP(2])j FQ(n)h(f)p FP(\()p FO(\031)s FP(\(2)p FO(l)i FP(+)e(1\))g(+)g FO(m\033)s FP(\))p FO(=)p FP(2)23 b FQ(j)g FO(m;)14 b(l)24 b FQ(2)g FJ(Z)o FQ(g)o FP(,)243 1455 y(The)h(tra)5 b(jectories)24 b(of)i(the)g(p)r(oin)n(ts)g FQ(f\006)p FP(1)p FO(=)p FP(sin)12 b FO(\033)s FQ(g)p FP(,)26 b FQ(f\006)p FP(cos)n(\()p FO(\033)s(=)p FP(2\))p FO(=)p FP(sin)13 b FO(\033)t FQ(g)25 b FP(are)118 1554 y(of)j(the)g(form:)243 1654 y(a\))f(if)h FO(k)e FP(=)d(2\(2)p FO(p)17 b FQ(\000)h FP(1\),)641 1785 y Fo(r)p 641 1787 V 636 1782 V 632 1777 V 628 1773 V 624 1768 V 621 1764 V 618 1759 V 615 1755 V 613 1751 V 611 1747 V 610 1743 V 609 1740 V 608 1736 V 608 1733 V 608 1729 V 608 1726 V 609 1723 V 610 1720 V 612 1717 V 614 1715 V 616 1712 V 616 1712 V 618 1710 V 621 1708 V 623 1706 V 626 1704 V 628 1703 V 631 1702 V 633 1701 V 636 1700 V 638 1700 V 641 1700 V 643 1700 V 646 1700 V 648 1701 V 651 1702 V 653 1703 V 656 1704 V 658 1706 V 661 1708 V 663 1710 V 666 1712 V 641 1787 V 646 1782 V 650 1777 V 654 1773 V 658 1768 V 661 1764 V 664 1759 V 666 1755 V 669 1751 V 670 1747 V 672 1743 V 673 1740 V 674 1736 V 674 1733 V 674 1729 V 673 1726 V 673 1723 V 671 1720 V 670 1717 V 668 1715 V 666 1712 V 641 1787 125 4 v 99 w(r)141 b(r)p 931 1787 V 100 w(r)517 1858 y FL(cos)o(\()p FM(\033)r(=)p FL(2\))p 517 1879 249 4 v 574 1926 a(sin)11 b FM(\033)947 1898 y FQ(\000)1072 1865 y FL(1)p 1021 1879 134 4 v 1021 1926 a(sin)g FM(\033)793 1789 y FO(:)j(:)g(:)1637 1785 y Fo(r)p 1637 1787 4 4 v 1632 1782 V 1628 1777 V 1624 1773 V 1620 1768 V 1617 1764 V 1614 1759 V 1611 1755 V 1609 1751 V 1607 1747 V 1606 1743 V 1605 1740 V 1604 1736 V 1604 1733 V 1604 1729 V 1604 1726 V 1605 1723 V 1606 1720 V 1608 1717 V 1610 1715 V 1612 1712 V 1612 1712 V 1615 1710 V 1617 1708 V 1620 1706 V 1622 1704 V 1625 1703 V 1627 1702 V 1630 1701 V 1632 1700 V 1635 1700 V 1637 1700 V 1640 1700 V 1642 1700 V 1645 1701 V 1647 1702 V 1649 1703 V 1652 1704 V 1654 1706 V 1657 1708 V 1659 1710 V 1662 1712 V 1637 1787 V 1642 1782 V 1646 1777 V 1650 1773 V 1654 1768 V 1657 1764 V 1660 1759 V 1663 1755 V 1665 1751 V 1667 1747 V 1668 1743 V 1669 1740 V 1670 1736 V 1670 1733 V 1670 1729 V 1670 1726 V 1669 1723 V 1668 1720 V 1666 1717 V 1664 1715 V 1662 1712 V 1637 1787 125 4 v 100 w(r)141 b(r)p 1928 1787 V 99 w(r)1471 1898 y FQ(\000)1546 1858 y FL(cos)o(\()p FM(\033)r(=)p FL(2\))p 1545 1879 249 4 v 1602 1926 a(sin)12 b FM(\033)2036 1865 y FL(1)p 1985 1879 134 4 v 1985 1926 a(sin)f FM(\033)1789 1789 y FO(:)j(:)g(:)243 2005 y FP(b\))28 b(if)g FO(k)e FP(=)c(4)p FO(p)p FP(,)641 2136 y Fo(r)p 641 2138 4 4 v 636 2133 V 632 2128 V 628 2124 V 624 2119 V 621 2115 V 618 2111 V 615 2106 V 613 2102 V 611 2098 V 610 2095 V 609 2091 V 608 2087 V 608 2084 V 608 2081 V 608 2077 V 609 2074 V 610 2071 V 612 2069 V 614 2066 V 616 2063 V 616 2063 V 618 2061 V 621 2059 V 623 2057 V 626 2055 V 628 2054 V 631 2053 V 633 2052 V 636 2051 V 638 2051 V 641 2051 V 643 2051 V 646 2051 V 648 2052 V 651 2053 V 653 2054 V 656 2055 V 658 2057 V 661 2059 V 663 2061 V 666 2063 V 641 2138 V 646 2133 V 650 2128 V 654 2124 V 658 2119 V 661 2115 V 664 2111 V 666 2106 V 669 2102 V 670 2098 V 672 2095 V 673 2091 V 674 2087 V 674 2084 V 674 2081 V 673 2077 V 673 2074 V 671 2071 V 670 2069 V 668 2066 V 666 2063 V 641 2138 125 4 v 99 w(r)141 b(r)p 931 2138 V 100 w(r)474 2249 y FQ(\000)549 2209 y FL(cos\()p FM(\033)r(=)p FL(2\))p 549 2230 249 4 v 606 2278 a(sin)11 b FM(\033)947 2249 y FQ(\000)1072 2216 y FL(1)p 1021 2230 134 4 v 1021 2278 a(sin)g FM(\033)793 2140 y FO(:)j(:)g(:)1637 2136 y Fo(r)p 1637 2138 4 4 v 1632 2133 V 1628 2128 V 1624 2124 V 1620 2119 V 1617 2115 V 1614 2111 V 1611 2106 V 1609 2102 V 1607 2098 V 1606 2095 V 1605 2091 V 1604 2087 V 1604 2084 V 1604 2081 V 1604 2077 V 1605 2074 V 1606 2071 V 1608 2069 V 1610 2066 V 1612 2063 V 1612 2063 V 1615 2061 V 1617 2059 V 1620 2057 V 1622 2055 V 1625 2054 V 1627 2053 V 1630 2052 V 1632 2051 V 1635 2051 V 1637 2051 V 1640 2051 V 1642 2051 V 1645 2052 V 1647 2053 V 1649 2054 V 1652 2055 V 1654 2057 V 1657 2059 V 1659 2061 V 1662 2063 V 1637 2138 V 1642 2133 V 1646 2128 V 1650 2124 V 1654 2119 V 1657 2115 V 1660 2111 V 1663 2106 V 1665 2102 V 1667 2098 V 1668 2095 V 1669 2091 V 1670 2087 V 1670 2084 V 1670 2081 V 1670 2077 V 1669 2074 V 1668 2071 V 1666 2069 V 1664 2066 V 1662 2063 V 1637 2138 125 4 v 100 w(r)141 b(r)p 1928 2138 V 99 w(r)1513 2209 y FL(cos\()p FM(\033)r(=)p FL(2\))p 1513 2230 249 4 v 1570 2278 a(sin)11 b FM(\033)2036 2216 y FL(1)p 1985 2230 134 4 v 1985 2278 a(sin)g FM(\033)1789 2140 y FO(:)j(:)g(:)243 2356 y FP(c\))28 b(if)g FO(k)e FP(=)c(2)p FO(p)c FQ(\000)g FP(1,)641 2488 y Fo(r)p 641 2489 4 4 v 636 2484 V 632 2480 V 628 2475 V 624 2470 V 621 2466 V 618 2462 V 615 2458 V 613 2453 V 611 2450 V 610 2446 V 609 2442 V 608 2439 V 608 2435 V 608 2432 V 608 2429 V 609 2426 V 610 2423 V 612 2420 V 614 2417 V 616 2415 V 616 2415 V 618 2412 V 621 2410 V 623 2408 V 626 2407 V 628 2405 V 631 2404 V 633 2403 V 636 2403 V 638 2402 V 641 2402 V 643 2402 V 646 2403 V 648 2403 V 651 2404 V 653 2405 V 656 2407 V 658 2408 V 661 2410 V 663 2412 V 666 2415 V 641 2489 V 646 2484 V 650 2480 V 654 2475 V 658 2470 V 661 2466 V 664 2462 V 666 2458 V 669 2453 V 670 2450 V 672 2446 V 673 2442 V 674 2439 V 674 2435 V 674 2432 V 673 2429 V 673 2426 V 671 2423 V 670 2420 V 668 2417 V 666 2415 V 641 2489 125 4 v 99 w(r)141 b(r)p 931 2489 V 100 w(r)p 1056 2489 4 4 v 1051 2484 V 1047 2480 V 1043 2475 V 1039 2470 V 1036 2466 V 1033 2462 V 1030 2458 V 1028 2453 V 1026 2450 V 1025 2446 V 1024 2442 V 1023 2439 V 1023 2435 V 1023 2432 V 1023 2429 V 1024 2426 V 1025 2423 V 1027 2420 V 1029 2417 V 1031 2415 V 1031 2415 V 1033 2412 V 1036 2410 V 1038 2408 V 1041 2407 V 1043 2405 V 1046 2404 V 1048 2403 V 1051 2403 V 1053 2402 V 1056 2402 V 1058 2402 V 1061 2403 V 1063 2403 V 1066 2404 V 1068 2405 V 1071 2407 V 1073 2408 V 1076 2410 V 1078 2412 V 1081 2415 V 1056 2489 V 1061 2484 V 1065 2480 V 1069 2475 V 1073 2470 V 1076 2466 V 1079 2462 V 1082 2458 V 1084 2453 V 1086 2450 V 1087 2446 V 1088 2442 V 1089 2439 V 1089 2435 V 1089 2432 V 1089 2429 V 1088 2426 V 1087 2423 V 1085 2420 V 1083 2417 V 1081 2415 V 517 2560 a FL(cos)o(\()p FM(\033)r(=)p FL(2\))p 517 2581 249 4 v 574 2629 a(sin)11 b FM(\033)889 2600 y FQ(\000)964 2560 y FL(cos\()p FM(\033)r(=)p FL(2\))p 964 2581 V 1021 2629 a(sin)g FM(\033)793 2492 y FO(:)j(:)g(:)1637 2488 y Fo(r)p 1637 2489 125 4 v 100 w(r)141 b(r)p 1928 2489 V 99 w(r)1528 2600 y FQ(\000)1653 2567 y FL(1)p 1602 2581 134 4 v 1602 2629 a(sin)12 b FM(\033)2036 2567 y FL(1)p 1985 2581 V 1985 2629 a(sin)f FM(\033)1789 2492 y FO(:)j(:)g(:)243 2703 y FP(In)34 b(what)g(follo)n(ws)g(w)n(e)g(study) g(irreducible)g(represen)n(tations)e(in)i(eac)n(h)g(of)118 2803 y(the)28 b(subspaces)f FO(H)710 2815 y FM(i)737 2803 y FP(,)h FO(i)23 b FP(=)f(1,)28 b(2,)f(3.)243 2902 y(1.)55 b(If)35 b FO(J)499 2914 y FL(1)536 2902 y FP(,)g FO(J)640 2914 y FL(2)712 2902 y FP(is)f(an)f(irreducible)h(represen)n (tation)e(acting)h(in)h FO(H)2317 2914 y FL(3)2355 2902 y FP(,)h(then)118 3002 y FO(\033)s FP(\()p FO(J)246 3014 y FL(1)284 3002 y FP(\))29 b FQ(\032)f FB(O)p FP(\()p FQ(f)p FP(sin)13 b FO(x\033)t(=)p FP(sin)h FO(\033)s FQ(g)p FP(\))28 b(=)g FQ(f)p FP(sin\(\()p FO(x)19 b FP(+)f FO(k)s FP(\))p FO(\033)s FP(\))q FO(=)p FP(sin)c FO(\033)32 b FQ(j)c FP(0)g FQ(\024)g FO(k)k FQ(\024)c FO(s)20 b FQ(\000)h FP(1)p FQ(g)p FP(;)118 3101 y(moreo)n(v)n(er,)37 b FQ(\006)p FP(1)p FO(=)p FP(sin)12 b FO(\033)t FP(,)39 b FQ(\006)p FP(cos)o(\()p FO(\033)s(=)p FP(2\))p FO(=)p FP(sin)13 b FO(\033)52 b(=)-51 b FQ(2)39 b FB(O)p FP(\()p FQ(f)p FP(sin)13 b FO(x\033)t(=)p FP(sin)g FO(\033)t FQ(g)p FP(\).)65 b(Denote)118 3201 y(the)27 b(pro)5 b(jection)26 b(on)n(to)g(the)i(eigenspace)d(of)i(the)g(op)r(erator)f FO(J)2005 3213 y FL(1)2069 3201 y FP(corresp)r(onding)118 3301 y(to)i FO(\025)268 3313 y FM(k)332 3301 y FP(=)23 b(sin)o(\(\()p FO(x)d FP(+)e FO(k)s FP(\))p FO(\033)s FP(\))q FO(=)p FP(sin)13 b FO(\033)31 b FP(b)n(y)c FO(P)1299 3313 y FM(k)1341 3301 y FP(.)243 3400 y(It)g(follo)n(ws)g(from)g (condition)g(\(2.40\))f(that)i FO(J)1630 3412 y FL(2)1667 3400 y FO(P)1720 3412 y FL(0)1758 3400 y FO(H)1827 3412 y FL(3)1887 3400 y FQ(\032)23 b FO(P)2028 3412 y FL(1)2065 3400 y FO(H)2134 3412 y FL(3)2189 3400 y FQ(\010)18 b FO(P)2325 3412 y FM(s)p FN(\000)p FL(1)2445 3400 y FO(H)2514 3412 y FL(3)2552 3400 y FP(,)118 3500 y FO(J)164 3512 y FL(2)201 3500 y FO(P)254 3512 y FM(s)p FN(\000)p FL(1)375 3500 y FO(H)444 3512 y FL(3)504 3500 y FQ(\032)23 b FO(P)645 3512 y FL(1)683 3500 y FO(H)752 3512 y FL(3)807 3500 y FQ(\010)18 b FO(P)943 3512 y FM(s)p FN(\000)p FL(2)1064 3500 y FO(H)1133 3512 y FL(3)1170 3500 y FP(,)28 b(and)f FO(J)1428 3512 y FL(2)1466 3500 y FO(P)1519 3512 y FM(k)1560 3500 y FO(H)1629 3512 y FL(3)1689 3500 y FQ(\032)c FO(P)1830 3512 y FM(k)q FL(+1)1955 3500 y FO(H)2024 3512 y FL(3)2079 3500 y FQ(\010)18 b FO(P)2215 3512 y FM(k)q FN(\000)p FL(1)2342 3500 y FO(H)2411 3512 y FL(3)2475 3500 y FP(for)118 3600 y FO(k)33 b FQ(6)p FP(=)c(0)p FO(;)14 b(s)20 b FQ(\000)h FP(1.)48 b(Th)n(us)31 b(the)h(op)r(erator)e FO(J)1371 3612 y FL(2)1440 3600 y FP(can)h(b)r(e)h(represen)n(ted)e(in)i(the)g (form)118 3699 y FO(J)164 3711 y FL(2)230 3699 y FP(=)27 b FO(X)g FP(+)20 b FO(X)579 3669 y FN(\003)617 3699 y FP(,)31 b(where)f FO(X)35 b FP(=)1110 3637 y Fz(P)1198 3657 y FM(s)p FN(\000)p FL(2)1198 3724 y FM(k)q FL(=0)1337 3699 y FO(P)1390 3711 y FM(k)q FL(+1)1515 3699 y FO(J)1561 3711 y FL(2)1598 3699 y FO(P)1651 3711 y FM(k)1713 3699 y FP(+)20 b FO(P)1851 3711 y FL(0)1888 3699 y FO(J)1934 3711 y FL(2)1972 3699 y FO(P)2025 3711 y FM(s)p FN(\000)p FL(1)2145 3699 y FP(.)47 b(Moreo)n(v)n(er,)118 3799 y(the)28 b(pair)f(\()p FO(J)510 3811 y FL(1)548 3799 y FO(;)14 b(J)631 3811 y FL(2)668 3799 y FP(\))28 b(is)g(irreducible)f(if)i(and)e (only)h(if)g(the)g(triple)g(\()p FO(J)2163 3811 y FL(1)2201 3799 y FO(;)14 b(X)r(;)g(X)2422 3769 y FN(\003)2459 3799 y FP(\))28 b(is)118 3898 y(irreducible.)36 b(One)26 b(can)g(easily)g (sho)n(w)g(that)h FO(J)1544 3910 y FL(1)1581 3898 y FP(,)g FO(J)1677 3910 y FL(2)1741 3898 y FP(satisfy)f(\(2.36\))g(i\013)h FO(X)7 b FP(,)26 b FO(X)2537 3868 y FN(\003)118 3998 y FP(are)h(additionally)g(connected)g(b)n(y)g(the)h(relations:)785 4147 y FO(\013)838 4159 y FM(k)879 4147 y FO(X)955 4113 y FN(\003)992 4147 y FO(X)7 b(P)1121 4159 y FM(k)1180 4147 y FP(+)18 b FO(\014)1310 4159 y FM(k)1351 4147 y FO(X)7 b(X)1503 4113 y FN(\003)1540 4147 y FO(P)1593 4159 y FM(k)1657 4147 y FP(=)22 b FO(\015)1787 4159 y FM(k)1828 4147 y FO(I)7 b(;)469 b FP(\(2.41\))p eop %%Page: 145 149 145 148 bop 118 100 a FK(2.3.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-deforemd)f FO(U)9 b FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(R)p FP(\)\))631 b(145)118 333 y(where)33 b FO(\013)417 345 y FM(k)492 333 y FP(=)g FQ(\000)p FP(2)14 b(cos)n(\(\()p FO(x)24 b FP(+)e FO(k)k FP(+)c(1\))p FO(\033)s FP(\),)36 b FO(\014)1461 345 y FM(k)1535 333 y FP(=)e(2)14 b(cos)n(\(\()p FO(x)24 b FP(+)e FO(k)j FQ(\000)e FP(1\))p FO(\033)s FP(\),)36 b FO(\015)2436 345 y FM(k)2510 333 y FP(=)118 432 y(sin\(\()p FO(x)20 b FP(+)e FO(k)s FP(\))p FO(\033)s FP(\))p FO(=)p FP(sin)c FO(\033)s FP(.)37 b(Since)26 b FQ(f\006)p FP(1)p FO(=)p FP(sin)12 b FO(\033)s FQ(g)26 b FP(do)r(es)g(not)g(b)r(elong)g(to)g(the)g(tra)5 b(jec-)118 532 y(tory)-7 b(,)27 b FO(\013)365 544 y FM(k)406 532 y FO(\014)453 544 y FM(k)517 532 y FQ(6)p FP(=)c(0.)243 632 y(Let)28 b(0)23 b FQ(\024)g FO(k)k FQ(\024)c FO(s)c FQ(\000)f FP(1)28 b(b)r(e)g(the)h(smallest)e(n)n(um)n(b)r(er)h(suc)n(h) g(that)g FO(\025)2209 644 y FM(k)2274 632 y FQ(2)c FO(\033)s FP(\()p FO(J)2481 644 y FL(1)2519 632 y FP(\).)118 731 y(Set)31 b FO(C)323 743 y FM(k)393 731 y FP(=)c FO(X)561 701 y FN(\003)599 731 y FO(X)7 b(P)728 743 y FM(k)768 731 y FP(,)32 b(and)f(denote)f(the)h(resolution)f(of)h(the)g(iden)n (tit)n(y)f(for)h FO(C)2534 743 y FM(k)118 831 y FP(b)n(y)j FO(E)301 843 y FM(C)349 852 y Fw(k)389 831 y FP(\()p FQ(\001)p FP(\).)57 b(F)-7 b(rom)34 b(\(2.41\))f(it)i(follo)n(ws)e (that)h([)p FO(A;)14 b(X)1780 801 y FM(s)1815 831 y FP(])34 b(=)f(0,)j([)p FO(X)2170 801 y FN(\003)2207 831 y FO(;)14 b(X)2320 801 y FM(s)2355 831 y FP(])34 b(=)f(0,)118 930 y(whic)n(h)38 b(yields)g FO(X)686 900 y FM(s)760 930 y FP(=)i FO(cI)45 b FP(if)38 b(\()p FO(J)1146 942 y FL(1)1184 930 y FO(;)14 b(X)r(;)g(X)1405 900 y FN(\003)1442 930 y FP(\))38 b(is)g(irreducible.)67 b(F)-7 b(rom)37 b(this)h(w)n(e)118 1030 y(conclude)32 b(that)g FQ(\010)713 995 y FM(s)p FN(\000)p FL(1)713 1055 y FM(l)p FL(=)p FM(k)833 1030 y FO(X)909 1000 y FM(l)p FN(\000)p FM(k)1022 1030 y FO(E)1083 1042 y FM(C)1131 1051 y Fw(k)1171 1030 y FP(\(\001\))14 b FO(W)45 b FP(is)31 b(in)n(v)-5 b(arian)n(t)31 b(with)h(resp)r(ect)g (to)f FO(J)2514 1042 y FL(1)2552 1030 y FP(,)118 1130 y FO(X)7 b FP(,)29 b FO(X)322 1100 y FN(\003)389 1130 y FP(for)g(an)n(y)g(\001)d FQ(2)h FA(B)p FP(\()p FJ(R)q FP(\))35 b(and)30 b(a)f(subspace)f FO(W)42 b FP(suc)n(h)29 b(that)h FO(C)2214 1142 y FM(k)2255 1130 y FO(W)38 b FQ(\032)26 b FO(W)12 b FP(.)118 1229 y(Hence,)37 b(if)f(\()p FO(J)559 1241 y FL(1)596 1229 y FO(;)14 b(X)r(;)g(X)817 1199 y FN(\003)854 1229 y FP(\))35 b(is)g(irreducible,)i(then)e(\001)g (is)g(concen)n(trated)f(in)h(one)118 1329 y(p)r(oin)n(t,)k(and)e(w)n(e) f(can)g(c)n(ho)r(ose)f(a)i(basis)e(consisting)h(of)h(eigen)n(v)n (ectors)d(of)i FO(J)2514 1341 y FL(1)2552 1329 y FP(,)118 1429 y(namely)-7 b(,)23 b FQ(f)p FO(e)499 1441 y FM(\025)538 1450 y Fw(k)578 1429 y FO(;)14 b(X)7 b(e)730 1441 y FM(\025)769 1450 y Fw(k)809 1429 y FO(=)p FQ(jj)p FO(X)g(e)1012 1441 y FM(\025)1051 1450 y Fw(k)1090 1429 y FQ(jj)p FO(;)14 b(:)g(:)g(:)28 b(;)14 b(X)1411 1398 y FM(s)p FN(\000)p FM(k)q FN(\000)p FL(1)1619 1429 y FO(e)1658 1441 y FM(\025)1697 1450 y Fw(k)1738 1429 y FO(=)p FQ(jj)p FO(X)1902 1398 y FM(s)p FN(\000)p FM(k)q FN(\000)p FL(1)2110 1429 y FO(e)2149 1441 y FM(\025)2188 1450 y Fw(k)2228 1429 y FQ(jjg)p FP(,)23 b(where)118 1528 y FO(e)157 1540 y FM(\025)196 1549 y Fw(k)267 1528 y FP(is)30 b(an)g(eigen)n(v)n(ector)e(of)j FO(C)1062 1540 y FM(k)1133 1528 y FP(whic)n(h)g(exists)f(due)g(to)g (the)h(last)f(argumen)n(ts.)118 1628 y(Let)e FO(X)343 1598 y FN(\003)380 1628 y FO(X)7 b(e)495 1640 y FM(\025)534 1649 y Fw(k)597 1628 y FQ(\021)23 b FO(C)744 1640 y FM(k)785 1628 y FO(e)824 1640 y FM(\025)863 1649 y Fw(k)926 1628 y FP(=)g FO(b)14 b(e)1103 1640 y FM(\025)1142 1649 y Fw(k)1182 1628 y FP(,)27 b FO(b)c FQ(2)h FJ(R)1424 1640 y FL(+)1485 1628 y FP(.)37 b(Using)27 b(\(2.41\))f(one)h(can)g(get)g (the)118 1727 y(action)21 b(of)h(the)g(op)r(erators)d FO(J)994 1739 y FL(1)1031 1727 y FP(,)k FO(X)7 b FP(,)22 b FO(X)1274 1697 y FN(\003)1312 1727 y FP(,)h(and)e FO(J)1559 1739 y FL(2)1618 1727 y FP(on)g(the)h(basis.)34 b(In)22 b(particular,)118 1827 y(if)g FO(\033)s FP(\()p FO(J)316 1839 y FL(1)354 1827 y FP(\))h(=)g FQ(f)p FO(\025)587 1839 y FL(0)624 1827 y FO(;)14 b(\025)709 1839 y FL(1)746 1827 y FO(;)g(:)g(:)g(:)g(;)g(\025)979 1839 y FM(s)p FN(\000)p FL(1)1100 1827 y FQ(g)p FP(,)22 b(w)n(e)e(will)i(ha)n(v)n(e)d (represen)n(tations)g(of)i(series)f(1,)118 1927 y(otherwise,)27 b(represen)n(tations)e(from)j(series)e(9.)243 2026 y(2.)62 b(If)37 b FO(J)508 2038 y FL(1)545 2026 y FP(,)i FO(J)653 2038 y FL(2)727 2026 y FP(is)d(an)g(irreducible)g(represen)n(tation)e (acting)i(in)h FO(H)2347 2038 y FL(1)2384 2026 y FP(,)h FO(H)2514 2038 y FL(2)2552 2026 y FP(,)118 2126 y(then)28 b(it)h(is)e(not)h(necessary)e(for)h FO(\033)s FP(\()p FO(J)1245 2138 y FL(1)1283 2126 y FP(\))h(to)g(b)r(e)g(simple.)38 b(First)27 b(let)h(us)g(consider)118 2226 y(the)g(case)f(where)g FO(J)723 2238 y FL(1)788 2226 y FP(is)g(concen)n(trated)g(on)g(the)h (tra)5 b(jectory)26 b(of)i(the)g(form)1139 2378 y Fo(r)p 1139 2380 4 4 v 1134 2375 V 1130 2370 V 1126 2365 V 1122 2361 V 1119 2356 V 1116 2352 V 1113 2348 V 1111 2344 V 1109 2340 V 1108 2336 V 1107 2332 V 1106 2329 V 1106 2325 V 1106 2322 V 1106 2319 V 1107 2316 V 1108 2313 V 1110 2310 V 1112 2307 V 1114 2305 V 1114 2305 V 1116 2303 V 1119 2300 V 1121 2299 V 1124 2297 V 1126 2296 V 1129 2294 V 1131 2294 V 1134 2293 V 1136 2293 V 1139 2292 V 1141 2293 V 1144 2293 V 1146 2294 V 1149 2294 V 1151 2296 V 1154 2297 V 1156 2299 V 1159 2300 V 1161 2303 V 1164 2305 V 1139 2380 V 1144 2375 V 1148 2370 V 1152 2365 V 1156 2361 V 1159 2356 V 1162 2352 V 1165 2348 V 1167 2344 V 1169 2340 V 1170 2336 V 1171 2332 V 1172 2329 V 1172 2325 V 1172 2322 V 1172 2319 V 1171 2316 V 1170 2313 V 1168 2310 V 1166 2307 V 1164 2305 V 1139 2380 125 4 v 99 w(r)141 b(r)p 1429 2380 V 100 w(r)p 1554 2380 4 4 v 1549 2375 V 1545 2370 V 1541 2365 V 1537 2361 V 1534 2356 V 1531 2352 V 1528 2348 V 1526 2344 V 1524 2340 V 1523 2336 V 1522 2332 V 1521 2329 V 1521 2325 V 1521 2322 V 1521 2319 V 1522 2316 V 1523 2313 V 1525 2310 V 1527 2307 V 1529 2305 V 1529 2305 V 1532 2303 V 1534 2300 V 1537 2299 V 1539 2297 V 1542 2296 V 1544 2294 V 1547 2294 V 1549 2293 V 1552 2293 V 1554 2292 V 1557 2293 V 1559 2293 V 1561 2294 V 1564 2294 V 1566 2296 V 1569 2297 V 1571 2299 V 1574 2300 V 1576 2303 V 1579 2305 V 1554 2380 V 1559 2375 V 1563 2370 V 1567 2365 V 1571 2361 V 1574 2356 V 1577 2352 V 1580 2348 V 1582 2344 V 1584 2340 V 1585 2336 V 1586 2332 V 1587 2329 V 1587 2325 V 1587 2322 V 1587 2319 V 1586 2316 V 1585 2313 V 1583 2310 V 1581 2307 V 1579 2305 V 1124 2478 a FM(\025)1163 2486 y Fy(1)1535 2478 y FM(\025)1574 2486 y Fw(n)1291 2382 y FO(:)14 b(:)g(:)2363 2396 y FP(\(2.42\))118 2640 y(where)27 b FO(\025)406 2652 y FL(1)467 2640 y FP(=)c FQ(\000)630 2600 y FL(cos)o(\()p FM(\033)r(=)p FL(2\))p 629 2621 249 4 v 686 2669 a(sin)12 b FM(\033)888 2640 y FP(,)27 b FO(\025)986 2652 y FM(n)1055 2640 y FP(=)1153 2600 y FL(cos)o(\()p FM(\033)r(=)p FL(2\))p 1153 2621 V 1210 2669 a(sin)11 b FM(\033)1411 2640 y FP(,)28 b(and)f FO(\033)s FP(\()p FO(J)1751 2652 y FL(1)1789 2640 y FP(\))c(=)g FQ(f)p FO(\025)2022 2652 y FL(1)2059 2640 y FO(;)14 b(:)g(:)g(:)g(;)g(\025)2292 2652 y FM(n)2337 2640 y FQ(g)p FP(.)243 2740 y(Denote)35 b(the)h(pro)5 b(jection)34 b(on)n(to)h(the)h(eigenspace)e(of)h FO(J)1996 2752 y FL(1)2069 2740 y FP(corresp)r(onding)118 2840 y(to)h(the)h(eigen)n(v)-5 b(alue)35 b FO(\025)833 2852 y FM(k)912 2840 y FP(=)i FQ(\000)p FP(cos)o(\(\(2)p FO(k)21 b FQ(\000)d FP(1\))p FO(\033)s(=)p FP(2)p FO(=)p FP(sin)13 b FO(\033)40 b FP(b)n(y)c FO(P)2072 2852 y FM(k)2113 2840 y FP(.)63 b(As)36 b(b)r(efore,)118 2939 y(the)31 b(op)r(erator)f FO(J)649 2951 y FL(2)717 2939 y FP(can)g(b)r(e)i (represen)n(ted)d(in)j(the)f(form)f FO(J)1925 2951 y FL(2)1991 2939 y FP(=)f FO(X)d FP(+)21 b FO(X)2342 2909 y FN(\003)2400 2939 y FP(+)f FO(Y)f FP(,)118 3039 y(where)34 b FO(X)42 b FP(=)576 2977 y Fz(P)663 2997 y FM(n)p FN(\000)p FL(1)663 3064 y FM(k)q FL(=1)807 3039 y FO(P)860 3051 y FM(k)q FL(+1)986 3039 y FO(J)1032 3051 y FL(2)1069 3039 y FO(P)1122 3051 y FM(k)1163 3039 y FP(,)37 b FO(Y)54 b FP(=)34 b FO(P)1477 3051 y FL(1)1515 3039 y FO(J)1561 3051 y FL(2)1598 3039 y FO(P)1651 3051 y FL(1)1712 3039 y FP(+)23 b FO(P)1853 3051 y FM(n)1898 3039 y FO(J)1944 3051 y FL(2)1982 3039 y FO(P)2035 3051 y FM(n)2080 3039 y FP(,)37 b(and)e FO(Y)53 b FQ(6)p FP(=)35 b(0,)118 3139 y FO(X)7 b(P)247 3151 y FM(k)322 3139 y FQ(6)p FP(=)33 b(0,)j(for)e FO(k)j(<)c(n)p FP(.)57 b(Moreo)n(v)n(er,)33 b(\()p FO(J)1437 3151 y FL(1)1475 3139 y FO(;)14 b(J)1558 3151 y FL(2)1595 3139 y FP(\))35 b(is)f(irreducible)f(if)i(and)f(only) 118 3238 y(if)29 b(the)g(family)g(\()p FO(J)672 3250 y FL(1)709 3238 y FO(;)14 b(X)r(;)g(X)930 3208 y FN(\003)967 3238 y FO(;)g(Y)19 b FP(\))29 b(is)f(irreducible.)40 b(Clearly)27 b FO(J)1998 3250 y FL(1)2035 3238 y FP(,)i FO(J)2133 3250 y FL(2)2199 3238 y FP(satisfy)f(the)118 3338 y(relation)f FO(Q)p FP(\()p 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FM(k)932 3608 y FO(X)1001 3620 y FM(k)q FN(\000)p FL(1)1126 3608 y FO(X)1202 3573 y FN(\003)1195 3628 y FM(k)q FN(\000)p FL(1)1344 3608 y FP(=)23 b FO(\015)1475 3620 y FM(k)1516 3608 y FO(I)7 b(;)180 b(k)25 b FQ(6)p FP(=)e(1)p FO(;)41 b(k)26 b FQ(6)p FP(=)d FO(n;)631 3743 y(\013)684 3755 y FL(1)721 3743 y FO(X)797 3708 y FN(\003)790 3763 y FL(1)835 3743 y FO(X)904 3755 y FL(1)959 3743 y FP(+)18 b FO(\014)1089 3755 y FL(1)1127 3743 y FO(Y)1193 3708 y FL(2)1231 3743 y FO(P)1284 3755 y FL(1)1344 3743 y FP(=)23 b FO(\015)1475 3755 y FL(1)1512 3743 y FO(I)7 b(;)429 3877 y(\014)476 3889 y FM(n)521 3877 y FO(X)590 3889 y FM(n)p FN(\000)p FL(1)720 3877 y FO(X)796 3843 y FN(\003)789 3898 y FM(n)p FN(\000)p FL(1)937 3877 y FP(+)18 b FO(\013)1073 3889 y FM(n)1119 3877 y FO(Y)1185 3843 y FL(2)1223 3877 y FO(P)1276 3889 y FM(n)1344 3877 y FP(=)23 b FO(\015)1475 3889 y FM(n)1520 3877 y FO(I)7 b(;)777 b FP(\(2.43\))118 4048 y(where)21 b FO(X)421 4060 y FM(k)485 4048 y FP(=)h FO(X)7 b(P)701 4060 y FM(k)742 4048 y FP(,)23 b FO(\013)841 4060 y FM(k)905 4048 y FP(=)f FQ(\000)p FP(2)14 b(sin)o(\(\(2)p FO(k)9 b FP(+)d(1\))p FO(\033)s(=)p FP(2\),)22 b FO(\014)1775 4060 y FM(k)1839 4048 y FP(=)g(2)14 b(sin\(\(2)p FO(k)9 b FQ(\000)d FP(3\))p FO(\033)s(=)p FP(2\),)118 4147 y FO(\015)161 4159 y FM(k)225 4147 y FP(=)23 b FQ(\000)p FP(cos)n(\(\(2)p FO(k)f FQ(\000)c FP(1\))p FO(\033)s(=)p FP(2\))o FO(=)p FP(sin)c FO(\033)s FP(.)p eop %%Page: 146 150 146 149 bop 118 100 a FP(146)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)243 333 y FP(Set)38 b FO(D)465 345 y FL(0)542 333 y FP(=)i(\()p FO(X)755 303 y FN(\003)793 333 y FP(\))825 303 y FM(n)p FN(\000)p FL(1)955 333 y FO(X)1031 303 y FM(n)p FN(\000)p FL(1)1161 333 y FO(P)1214 345 y FL(1)1251 333 y FP(.)68 b(Then)38 b FO(P)1622 345 y FL(1)1660 333 y FO(H)47 b FQ(\032)39 b FP(\(k)n(er)13 b FO(D)2106 345 y FL(0)2143 333 y FP(\))2175 303 y FN(?)2232 333 y FP(,)40 b(b)r(ecause)118 432 y(if)e(this)f(w)n(ere)g(not)g(the)h(case,)h(w)n(e) d(w)n(ould)h(conclude)g(that)h FO(W)51 b FP(=)39 b(k)n(er)12 b FO(D)2448 444 y FL(0)2510 432 y FQ(\010)118 532 y FO(X)20 b FP(k)n(er)13 b FO(D)401 544 y FL(0)464 532 y FQ(\010)26 b FO(:)14 b(:)g(:)25 b FQ(\010)h FO(X)844 502 y FM(n)p FN(\000)p FL(2)987 532 y FP(k)n(er)13 b FO(D)1181 544 y FL(0)1257 532 y FP(is)39 b(in)n(v)-5 b(arian)n(t)37 b(with)j(resp)r(ect)f(to)f FO(J)2366 544 y FL(1)2404 532 y FP(,)j FO(J)2514 544 y FL(2)2552 532 y FP(,)118 632 y(whic)n(h)28 b(con)n(tradicts)e(the)i(fact)g(that)g FO(\033)s FP(\()p FO(J)1398 644 y FL(1)1436 632 y FP(\))23 b(=)g FQ(f)p FO(\025)1669 644 y FL(1)1706 632 y FO(;)14 b(:)g(:)g(:)27 b(;)14 b(\025)1952 644 y FM(n)1998 632 y FQ(g)p FP(.)243 731 y(Let)23 b FO(X)29 b FP(=)23 b FO(U)639 661 y FQ(p)p 708 661 189 4 v 70 x FO(X)784 707 y FN(\003)821 731 y FO(X)29 b FP(b)r(e)24 b(the)f(p)r(olar)f(decomp)r (osition)g(of)h(the)g(op)r(erator)e FO(X)7 b FP(.)118 831 y(Put)22 b FO(Y)323 843 y FL(1)384 831 y FP(=)h FO(Y)18 b(P)591 843 y FL(1)629 831 y FP(,)23 b FO(Y)723 843 y FL(2)784 831 y FP(=)f(\()p FO(U)969 801 y FN(\003)1008 831 y FP(\))1040 801 y FM(n)p FN(\000)p FL(1)1170 831 y FO(Y)d(U)1303 801 y FM(n)p FN(\000)p FL(1)1433 831 y FO(P)1486 843 y FL(1)1523 831 y FP(.)35 b(Let)23 b(us)f(pro)n(v)n(e)e (that)j(\()p FO(J)2297 843 y FL(1)2335 831 y FP(,)g FO(J)2427 843 y FL(2)2464 831 y FP(\))g(is)118 930 y(irreducible)f(if)i(and)e (only)h(if)g(the)g(pair)g(\()p FO(Y)1386 942 y FL(1)1423 930 y FP(,)h FO(Y)1518 942 y FL(2)1556 930 y FP(\))f(is)g(irreducible.) 34 b(In)23 b(fact,)h(if)g FO(W)118 1030 y FP(is)e(in)n(v)-5 b(arian)n(t)22 b(with)h(resp)r(ect)f(to)g FO(Y)1147 1042 y FL(1)1185 1030 y FP(,)h FO(Y)1279 1042 y FL(2)1317 1030 y FP(,)g(then)g FO(W)1637 1000 y FN(0)1683 1030 y FP(=)g FO(W)d FQ(\010)8 b FO(U)h(W)20 b FQ(\010)8 b FO(:)14 b(:)g(:)f(U)2355 1000 y FM(n)p FN(\000)p FL(1)2485 1030 y FO(W)118 1130 y FP(is)25 b(in)n(v)-5 b(arian)n(t)24 b(with)i(resp)r(ect)e(to)h FO(J)1158 1142 y FL(1)1196 1130 y FP(,)g FO(X)7 b FP(,)25 b FO(X)1444 1100 y FN(\003)1482 1130 y FP(,)g FO(Y)19 b FP(,)26 b(and)f(hence)g(with)g(resp)r(ect)g(to) 118 1229 y FO(J)164 1241 y FL(1)201 1229 y FP(,)j FO(J)298 1241 y FL(2)335 1229 y FP(:)321 1397 y FO(X)7 b(U)463 1363 y FM(k)503 1397 y FO(W)35 b FP(=)22 b FO(U)769 1323 y FQ(p)p 838 1323 V 74 x FO(X)914 1373 y FN(\003)952 1397 y FO(X)6 b(U)1093 1363 y FM(k)1134 1397 y FO(W)616 1554 y FP(=)22 b FO(U)769 1519 y FM(k)q FL(+1)894 1487 y Fz(\000)932 1554 y FR(F)992 1519 y FL(\()p FM(k)q FL(\))1085 1487 y Fz(\000)1123 1554 y FP(\()p FO(\033)g FQ(\000)c FO(\031)s FP(\))p FO(=)p FP(2)p FO(;)c(X)1586 1519 y FN(\003)1623 1554 y FO(X)7 b FP(\))1731 1487 y Fz(\001)1769 1587 y FL(2)1806 1487 y Fz(\001)1844 1504 y FL(1)p FM(=)p FL(2)1948 1554 y FO(W)35 b FQ(\032)23 b FO(U)2215 1519 y FM(k)q FL(+1)2340 1554 y FO(W)n(;)240 1692 y(Y)c(U)373 1658 y FM(n)p FN(\000)p FL(1)503 1692 y FO(W)35 b FP(=)22 b FO(U)769 1658 y FM(n)p FN(\000)p FL(1)899 1692 y FP(\()p FO(U)997 1658 y FN(\003)1036 1692 y FP(\))1068 1658 y FM(n)p FN(\000)p FL(1)1198 1692 y FO(Y)d(U)1331 1658 y FM(n)p FN(\000)p FL(1)1461 1692 y FO(W)35 b FQ(\032)22 b FO(U)1727 1658 y FM(n)p FN(\000)p FL(1)1857 1692 y FO(Y)1905 1704 y FL(1)1943 1692 y FO(W)35 b FQ(\032)22 b FO(U)2209 1658 y FM(n)p FN(\000)p FL(1)2339 1692 y FO(W)n(;)118 1860 y FP(where)309 2027 y FR(F)p FP(\()p FO(x;)14 b(y)s FP(\))24 b(=)e(\()p FO(F)757 2039 y FL(1)795 2027 y FP(\()p FO(x;)14 b(y)s FP(\))p FO(;)g(F)1077 2039 y FL(2)1115 2027 y FP(\()p FO(x;)g(y)s FP(\)\))585 2211 y(=)672 2094 y Fz(\022)733 2211 y FO(x)19 b FP(+)f(1)p FO(;)1071 2155 y(y)f FP(cos)c FO(x\033)p 971 2192 482 4 v 971 2268 a FP(cos)o(\(\()p FO(x)19 b FP(+)f(2\))p FO(\033)s FP(\))1480 2211 y FQ(\000)1696 2155 y FP(sin\(\()p FO(x)h FP(+)f(1\))p FO(\033)s FP(\))p 1573 2192 717 4 v 1573 2268 a(2)c(sin)f FO(\033)18 b FP(cos)o(\(\()p FO(x)h FP(+)f(2\))p FO(\033)s FP(\))2300 2094 y Fz(\023)2361 2211 y FO(:)118 2424 y FP(Here)31 b(w)n(e)g(use)g(the)h(fact)f(that)h FO(U)9 b(U)1221 2394 y FN(\003)1290 2424 y FP(is)31 b(the)g(pro)5 b(jection)31 b(on)f(the)i(orthogonal)118 2524 y(complemen)n(t)c(to)f(k) n(er)o(\()p FO(A)19 b FP(+)f(\(cos\()p FO(\033)s(=)p FP(2\))p FO(=)c FP(sin)f FO(\033)s FP(\))p FO(I)7 b FP(\).)38 b(Moreo)n(v)n(er,)25 b(\(2.43\))h(giv)n(es)335 2691 y FO(Q)401 2703 y FL(2)438 2691 y FP(\()p FO(Y)518 2703 y FL(2)556 2691 y FP(\))d(=)g(\()p FO(U)797 2657 y FN(\003)835 2691 y FP(\))867 2657 y FM(n)p FN(\000)p FL(1)998 2691 y FO(X)7 b(X)1150 2657 y FN(\003)1186 2691 y FO(U)1252 2657 y FM(n)p FN(\000)p FL(1)1382 2691 y FO(P)1435 2703 y FL(1)1496 2691 y FP(=)23 b(\()p FO(U)1682 2657 y FN(\003)1720 2691 y FP(\))1752 2657 y FM(n)p FN(\000)p FL(2)1882 2691 y FO(X)1958 2657 y FN(\003)1996 2691 y FO(X)7 b(U)2138 2657 y FM(n)p FN(\000)p FL(2)2267 2691 y FO(P)2320 2703 y FL(1)611 2832 y FP(=)699 2765 y Fz(\000)737 2832 y FR(F)797 2798 y FL(\()p FM(n)p FN(\000)p FL(2\))979 2832 y FP(\(\()p FO(\033)23 b 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FO(\025)1594 4171 y FM(n)1639 4159 y FO(I)1682 3841 y Fz(1)1682 3987 y(C)1682 4040 y(A)1769 4032 y FO(;)p eop %%Page: 147 151 147 150 bop 118 100 a FK(2.3.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-deforemd)f FO(U)9 b FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(R)p FP(\)\))631 b(147)447 525 y FO(J)493 537 y FL(2)554 525 y FP(=)641 258 y Fz(0)641 405 y(B)641 454 y(B)641 504 y(B)641 554 y(B)641 607 y(@)793 319 y FO(Y)841 331 y FL(1)1040 319 y FO(\026)1090 331 y FL(1)1128 319 y FP(\()p FO(\025)p FP(\))p FO(I)714 474 y(\026)764 486 y FL(1)801 474 y FP(\()p FO(\025)p FP(\))p FO(I)192 b FP(0)1491 416 y(.)1523 441 y(.)1556 466 y(.)1118 571 y(.)1150 596 y(.)1183 621 y(.)1491 571 y(.)1523 596 y(.)1556 621 y(.)1786 629 y FO(\026)1836 641 y FM(n)p FN(\000)p FL(1)1966 629 y FP(\()p FO(\025)p FP(\))p FO(I)1367 729 y(\026)1417 741 y FM(n)p FN(\000)p FL(1)1547 729 y FP(\()p FO(\025)p FP(\))p FO(I)216 b(Y)1959 741 y FL(2)2122 258 y Fz(1)2122 405 y(C)2122 454 y(C)2122 504 y(C)2122 554 y(C)2122 607 y(A)2209 525 y FO(;)118 879 y FP(where)159 1086 y FO(\026)209 1098 y FM(m)272 1086 y FP(\()p FO(\025)p FP(\))24 b(=)495 969 y Fz(\022)556 1086 y FR(F)616 1052 y FM(m)p FN(\000)p FL(1)765 969 y Fz(\022)836 1030 y FO(\033)e FQ(\000)c FO(\031)p 836 1067 203 4 v 916 1143 a FP(2)1048 1086 y FO(;)c FQ(\000)1171 1030 y FP(sin\()p FO(\033)s(=)p FP(2\))p 1160 1067 324 4 v 1160 1143 a(sin)f(3)p FO(\033)s(=)p FP(2\))1506 1086 y FO(\025)1554 1052 y FL(2)1610 1086 y FP(+)1837 1030 y(cos)o(\()p FO(\033)s(=)p FP(2\))p 1703 1067 577 4 v 1703 1143 a(2)h(sin)o(\(3)p FO(\033)s(=)p FP(2\))g(sin)g FO(\033)2290 969 y Fz(\023\023)2412 986 y FL(1)p FM(=)p FL(2)2412 1169 y(2)408 1346 y FP(=)495 1228 y Fz(\022)770 1289 y FP(sin)872 1254 y FL(2)909 1289 y FP(\()p FO(m\033)s FP(\))19 b FQ(\000)f FP(4)p FO(\025)1288 1259 y FL(2)1339 1289 y FP(sin)1441 1254 y FL(2)1478 1289 y FP(\()p FO(\033)s(=)p FP(2\))c(sin)1792 1254 y FL(2)1843 1289 y FO(\033)p 566 1326 1531 4 v 566 1409 a FP(4)g(sin)724 1374 y FL(2)775 1409 y FO(\033)j FP(sin\(\(2)p FO(m)h FQ(\000)g FP(1\))p FO(\033)s(=)p FP(2\))c(sin\(\(2)p FO(m)k FP(+)g(1\))p FO(\033)s(=)p FP(2\))2107 1228 y Fz(\023)2168 1246 y FL(1)p FM(=)p FL(2)2272 1346 y FO(;)717 1532 y(m)23 b FP(=)f(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n)19 b FQ(\000)f FP(1)p FO(;)336 1714 y(\025)24 b FQ(2)486 1622 y Fz(n)541 1714 y FO(x)g FQ(2)f FJ(R)773 1619 y Fz(\014)773 1668 y(\014)773 1718 y(\014)824 1622 y(\020)874 1714 y FR(F)934 1680 y FL(\()p FM(m)p FN(\000)p FL(1\))1134 1622 y Fz(\020)1193 1658 y FO(\033)f FQ(\000)c FO(\031)p 1193 1695 203 4 v 1274 1771 a FP(2)1405 1714 y FO(;)c FQ(\000)1538 1658 y FP(sin\()p FO(\033)s(=)p FP(2\))p 1517 1695 342 4 v 1517 1771 a(sin\(3)p FO(\033)s(=)p FP(2\))1868 1714 y FO(x)1915 1680 y FL(2)901 1945 y FP(+)1128 1889 y(cos)o(\()p FO(\033)s(=)p FP(2\))p 994 1926 577 4 v 994 2002 a(2)g(sin\(3)p FO(\033)s(=)p FP(2\))g(sin)f FO(\033)1581 1853 y Fz(\021\021)1680 2003 y FL(2)1741 1945 y FO(>)22 b FP(0)p FO(;)27 b FP(1)c FQ(\024)g FO(m)g FQ(\024)f FO(n)c FQ(\000)g FP(1)2449 1828 y Fz(\033)2511 1945 y FO(:)243 2149 y FP(An)n(y)26 b(irreducible)h(pair)f(\()p FO(Y)1080 2161 y FL(1)1117 2149 y FO(;)14 b(Y)1202 2161 y FL(2)1240 2149 y FP(\))27 b(is)g(unitarily)f(equiv)-5 b(alen)n(t)27 b(to)f(one)h(of)f(the)118 2249 y(follo)n(wing:)243 2349 y(a\))d(one-dimensional:)34 b FO(Y)1028 2361 y FL(1)1088 2349 y FP(=)23 b(\()p FQ(\000)p FP(1\))1347 2319 y FM(i)1374 2349 y FO(\025)p FP(,)i FO(Y)1518 2361 y FL(2)1579 2349 y FP(=)e(\()p FQ(\000)p FP(1\))1838 2319 y FM(j)1872 2349 y FO(\025)p FP(,)i FO(\025)f(>)e FP(0,)j FO(i;)14 b(j)27 b FP(=)c(0)p FO(;)14 b FP(1;)243 2448 y(b\))28 b(t)n(w)n(o-dimensional:)529 2653 y FO(Y)577 2665 y FL(1)638 2653 y FP(=)726 2535 y Fz(\022)787 2602 y FO(\025)119 b FP(0)790 2702 y(0)86 b FQ(\000)p FO(\025)1031 2535 y Fz(\023)1106 2653 y FO(;)97 b(Y)1274 2665 y FL(2)1334 2653 y FP(=)23 b FO(\025)1484 2535 y Fz(\022)1545 2602 y FP(cos)13 b FO(')128 b FP(sin)13 b FO(')1550 2702 y FP(sin)h FO(')88 b FQ(\000)14 b FP(cos)e FO(')2066 2535 y Fz(\023)2140 2653 y FO(;)118 2857 y FP(with)28 b FO(\025)c(>)e FP(0,)28 b FO(')23 b FQ(2)g FP(\(0)p FO(;)14 b(\031)s FP(\).)243 2956 y(Hence,)23 b(all)e(irreducible)g(represen)n(tations)e FO(J)1637 2968 y FL(1)1674 2956 y FP(,)k FO(J)1766 2968 y FL(2)1825 2956 y FP(ha)n(v)n(e)d(dimensions)h FO(n)h FP(or)118 3056 y(2)p FO(n)p FP(;)f(moreo)n(v)n(er,)d(t)n(w)n(o)g(suc)n(h)h(pairs) e(\()p FO(J)1227 3068 y FL(1)1265 3056 y FO(;)d(J)1348 3068 y FL(2)1385 3056 y FP(\),)21 b(\()p FO(J)1547 3026 y FN(0)1539 3077 y FL(1)1577 3056 y FO(;)14 b(J)1668 3026 y FN(0)1660 3077 y FL(2)1697 3056 y FP(\))19 b(are)f(unitarily)g (equiv)-5 b(alen)n(t)118 3156 y(if)28 b(and)f(only)g(if)h(the)g (corresp)r(onding)e(pairs)g(\()p FO(Y)1573 3168 y FL(1)1611 3156 y FO(;)14 b(Y)1696 3168 y FL(2)1733 3156 y FP(\),)28 b(\()p FO(Y)1915 3126 y FN(0)1896 3176 y FL(1)1938 3156 y FO(;)14 b(Y)2042 3126 y FN(0)2023 3176 y FL(2)2065 3156 y FP(\))28 b(are)e(unitarily)118 3255 y(equiv)-5 b(alen)n(t.)37 b(Th)n(us)27 b(w)n(e)g(will)h(get)f(the)h(represen)n (tations)e(of)i(series)e(2)h(and)h(3)f(of)118 3355 y(the)h(theorem.)243 3455 y(If)j(\000)381 3467 y FL(0)446 3455 y FP(=)d(\000)g Fs(\026)654 3470 y FM(\033)r FL(\()p FM(J)757 3478 y Fy(1)789 3470 y FL(\))850 3455 y FP(is)j(a)f(prop)r(er)g(subgraph)g(of) g(\(2.42\),)h(then)g(argumen)n(ts)118 3554 y(similar)g(to)g(those)g(in) h(1\))f(giv)n(e)f(that)i FO(\033)s FP(\()p FO(J)1413 3566 y FL(1)1451 3554 y FP(\))g(is)f(simple)g(if)h(the)g(pair)f FO(J)2312 3566 y FL(1)2349 3554 y FP(,)h FO(J)2450 3566 y FL(2)2519 3554 y FP(is)118 3654 y(irreducible.)49 b(Using)31 b(\(2.43\),)h(one)g(can)f(easily)g(describ)r(e)h(all)f(unitarily)g (non-)118 3753 y(equiv)-5 b(alen)n(t)40 b(irreducible)g(represen)n (tations)e(connected)i(with)h(the)g(supp)r(ort)118 3853 y(\000)170 3865 y FL(0)207 3853 y FP(.)243 3953 y(3.)36 b(No)n(w)27 b(let)h(\000)23 b Fs(\026)763 3968 y FM(\033)r FL(\()p FM(J)866 3976 y Fy(1)898 3968 y FL(\))956 3953 y FP(b)r(e)28 b(the)g(follo)n(wing)f(graph)1139 4089 y Fo(r)p 1139 4091 125 4 v 99 w(r)141 b(r)p 1429 4091 V 100 w(r)1002 4202 y FQ(\000)1127 4169 y FL(1)p 1077 4183 134 4 v 1077 4230 a(sin)11 b FM(\033)1537 4169 y FL(1)p 1487 4183 V 1487 4230 a(sin)g FM(\033)1291 4093 y FO(:)j(:)g(:)2363 4107 y FP(\(2.44\))p eop %%Page: 148 152 148 151 bop 118 100 a FP(148)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(or)21 b(one)h(of)g(its)g(subgraphs.)34 b(Let)22 b FO(P)1180 345 y FM(k)1243 333 y FP(b)r(e)h(the)f(pro)5 b(jection)21 b(on)n(to)h(the)g(eigenspace)118 432 y(of)i FO(J)255 444 y FL(1)316 432 y FP(corresp)r(onding)e(to)i FO(\025)992 444 y FM(k)1056 432 y FP(=)f FQ(\000)14 b FP(cos)n(\(\()p FO(k)g FQ(\000)d FP(1\))p FO(\033)s FP(\))p FO(=)j FP(sin)g FO(\033)s FP(,)25 b FO(k)h FP(=)d(1,)g FO(:)14 b(:)g(:)28 b FP(,)d FO(n)11 b FP(+)g(1,)118 532 y FO(X)187 544 y FM(k)252 532 y FP(=)24 b FO(P)394 544 y FM(k)q FL(+1)519 532 y FO(J)565 544 y FL(2)603 532 y FO(P)656 544 y FM(k)697 532 y FP(,)k FO(X)j FP(=)937 470 y Fz(P)1025 490 y FM(n)1025 557 y(k)q FL(=1)1163 532 y FO(X)1232 544 y FM(k)1273 532 y FP(.)39 b(Using)29 b(the)f(same)g(argumen)n(ts)f(w)n(e)h(can)118 632 y(conclude)h(that)g FO(J)688 644 y FL(2)750 632 y FP(=)24 b FO(X)i FP(+)18 b FO(X)1093 601 y FN(\003)1131 632 y FP(,)29 b(and)g(that)f(the)i(op)r (erators)c FO(X)2108 644 y FM(k)2149 632 y FP(,)j FO(X)2277 601 y FN(\003)2270 655 y FM(k)2343 632 y FP(satisfy)118 731 y(the)f(follo)n(wing)f(relations:)851 893 y FO(\013)904 905 y FM(k)945 893 y FO(X)1021 859 y FN(\003)1014 914 y FM(k)1059 893 y FO(X)1128 905 y FM(k)1187 893 y FP(+)18 b FO(\014)1317 905 y FM(k)1358 893 y FO(X)1427 905 y FM(k)q FN(\000)p FL(1)1552 893 y FO(X)1628 859 y FN(\003)1621 914 y FM(k)q FN(\000)p FL(1)1770 893 y FP(=)23 b FO(\015)1901 905 y FM(k)1941 893 y FO(I)7 b(;)356 b FP(\(2.45\))118 1056 y(where)42 b FO(\013)426 1068 y FM(k)514 1056 y FP(=)47 b FQ(\000)p FP(2)14 b(sin)o(\()p FO(k)s(\033)s FP(\),)47 b FO(\014)1125 1068 y FM(k)1213 1056 y FP(=)g(2)14 b(sin)o(\(\()p FO(k)32 b FQ(\000)27 b FP(2\))p FO(\033)s FP(\),)47 b FO(\015)1982 1068 y FM(k)2070 1056 y FP(=)g FQ(\000)14 b FP(cos)o(\(\()p FO(k)31 b FQ(\000)118 1155 y FP(1\))p FO(\033)s FP(\))p FO(=)14 b FP(sin)g FO(\033)s FP(;)29 b(in)g(particular,)f FO(\013)1107 1167 y FM(n)1177 1155 y FP(=)c(0,)k FO(\014)1406 1167 y FL(2)1468 1155 y FP(=)c(0,)29 b FO(\013)1704 1167 y FM(k)1745 1155 y FO(\014)1792 1167 y FM(m)1879 1155 y FQ(6)p FP(=)c(0,)j FO(k)g FQ(6)p FP(=)c FO(n)p FP(,)k FO(m)d FQ(6)p FP(=)f(2.)118 1255 y(Moreo)n(v)n(er,)40 b(the)f(pair)g(\()p FO(J)936 1267 y FL(1)974 1255 y FO(;)14 b(J)1057 1267 y FL(2)1094 1255 y FP(\))39 b(is)h(irreducible)e(if)i(and)f(only)g(if)h(the)g (triple)118 1354 y(\()p FO(J)196 1366 y FL(1)234 1354 y FO(;)14 b(X)r(;)g(X)455 1324 y FN(\003)492 1354 y FP(\))35 b(is)g(irreducible.)57 b(If)36 b FQ(f)p FP(1)p FO(=)p FP(sin)12 b FO(\033)s FQ(g)35 b FP(or)f FQ(f\000)p FP(1)p FO(=)p FP(sin)12 b FO(\033)s FQ(g)34 b FP(do)r(es)h(not)g(b)r(e-)118 1454 y(long)27 b(to)g FO(\033)s FP(\()p FO(J)527 1466 y FL(1)565 1454 y FP(\),)h(then)h FO(X)7 b FP(,)27 b FO(X)1040 1424 y FN(\003)1105 1454 y FP(satisfy)g(the)h(relation:)475 1616 y FO(X)551 1582 y FN(\003)588 1616 y FO(X)h FP(=)23 b FO(F)827 1628 y FL(1)865 1616 y FP(\()p FO(X)7 b(X)1049 1582 y FN(\003)1086 1616 y FO(;)14 b(A)p FP(\))83 b(or)f FO(X)7 b(X)1609 1582 y FN(\003)1669 1616 y FP(=)23 b FO(F)1810 1628 y FL(2)1847 1616 y FP(\()p FO(X)1955 1582 y FN(\003)1993 1616 y FO(X)r(;)14 b(A)p FP(\))p FO(;)118 1778 y FP(resp)r(ectiv)n(ely)-7 b(,)27 b(where)716 2011 y FO(F)769 2023 y FL(1)806 2011 y FP(\()p FO(\026;)14 b(\025)973 2023 y FM(k)1015 2011 y FP(\))23 b(=)1158 1869 y Fz(\()1235 1916 y FM(\015)1270 1925 y Fw(k)1306 1916 y FN(\000)p FM(\014)1396 1925 y Fw(k)1432 1916 y FM(\026)p 1235 1934 238 4 v 1314 1982 a(\013)1357 1991 y Fw(k)1482 1953 y FO(;)83 b(k)26 b FQ(6)p FP(=)d FO(n;)1225 2074 y FP(0)p FO(;)298 b FP(otherwise)o FO(;)716 2293 y(F)769 2305 y FL(2)806 2293 y FP(\()p FO(\026;)14 b(\025)973 2305 y FM(k)1015 2293 y FP(\))23 b(=)1158 2151 y Fz(\()1235 2197 y FM(\015)1270 2206 y Fw(k)1306 2197 y FN(\000)p FM(\013)1401 2206 y Fw(k)1437 2197 y FM(\026)p 1235 2215 243 4 v 1319 2263 a(\014)1357 2272 y Fw(k)1488 2234 y FO(;)83 b(k)26 b FQ(6)p FP(=)c(2)p FO(;)1225 2358 y FP(0)p FO(;)304 b FP(otherwise)o FO(:)118 2525 y FP(Hence,)38 b(an)n(y)c(irreducible)h(represen)n(tation)f(can)h(b)r(e)h(realized)e (in)i(the)g(space)118 2625 y FO(H)30 b FP(=)23 b FO(l)330 2637 y FL(2)367 2625 y FP(\(\001\))28 b(b)n(y)f(the)h(form)n(ulae:)794 2787 y FO(J)840 2799 y FL(1)877 2787 y FO(e)916 2799 y FM(k)980 2787 y FP(=)22 b FO(\025)1115 2799 y FM(k)1157 2787 y FO(e)1196 2799 y FM(k)1236 2787 y FO(;)97 b(X)7 b(e)1471 2799 y FM(k)1534 2787 y FP(=)23 b FO(\026)1672 2799 y FM(k)1712 2787 y FO(e)1751 2799 y FM(k)q FL(+1)1876 2787 y FO(;)118 2949 y FP(where)29 b(\001)e(=)f FQ(f)p FO(\025)637 2961 y FM(p)676 2949 y FO(;)14 b(\025)761 2961 y FM(p)p FL(+1)883 2949 y FO(;)g(:)g(:)g(:)28 b(;)14 b(\025)1130 2961 y FM(m)1220 2949 y FQ(j)26 b FP(0)g FQ(\024)h FO(p;)14 b(m)26 b FQ(\024)g FO(n)p FQ(g)p FP(,)k(and)f (either)h(1)p FO(=)p FP(sin)12 b FO(\033)40 b(=)-52 b FQ(2)118 3049 y FP(\001)33 b(or)f FQ(\000)p FP(1)p FO(=)p FP(sin)13 b FO(\033)44 b(=)-51 b FQ(2)32 b FP(\001,)i(and)f FO(\013)1106 3061 y FM(k)1147 3049 y FO(\026)1197 3061 y FM(k)1260 3049 y FP(+)21 b FO(\014)1393 3061 y FM(k)1434 3049 y FO(\026)1484 3061 y FM(k)q FN(\000)p FL(1)1642 3049 y FP(=)31 b FO(\015)1781 3061 y FM(k)1855 3049 y FP(suc)n(h)i(that)g FO(\026)2283 3061 y FM(k)2356 3049 y FO(>)e FP(0)h(if)118 3148 y FO(\025)166 3160 y FM(k)207 3148 y FO(;)14 b(\025)292 3160 y FM(k)q FL(+1)441 3148 y FQ(2)23 b FP(\001)28 b(and)f FO(\026)827 3160 y FM(k)891 3148 y FP(=)c(0)k(if)h FO(\025)1172 3160 y FM(k)1246 3148 y FO(=)-51 b FQ(2)23 b FP(\001)28 b(or)f FO(\025)1562 3160 y FM(k)q FL(+1)1719 3148 y FO(=)-51 b FQ(2)23 b FP(\001.)243 3248 y(If)h FQ(f\006)p FP(1)p FO(=)p FP(sin)12 b FO(\033)s FQ(g)23 b(2)g FO(\033)s FP(\()p FO(J)948 3260 y FL(1)986 3248 y FP(\),)j(then)e(the)h(problem)e(of)h(describing) g(irreducible)118 3348 y(represen)n(tations)42 b(\()p FO(J)788 3360 y FL(1)825 3348 y FP(,)48 b FO(J)942 3360 y FL(2)979 3348 y FP(\))d(is)e(reduced)g(to)h(that)g(of)g(pairs)e(of)i (op)r(erators)118 3447 y(satisfying)d(some)f(quadratic)h(relations.)76 b(In)42 b(order)e(to)h(sho)n(w)g(that,)k(con-)118 3547 y(sider)37 b(the)h(follo)n(wing)f(op)r(erators)f(in)i(the)g(subspace)f FO(P)1889 3559 y FL(2)1926 3547 y FO(H)7 b FP(:)57 b FO(D)2151 3559 y FL(1)2228 3547 y FP(=)39 b FO(X)2401 3559 y FL(1)2438 3547 y FO(X)2514 3517 y FN(\003)2507 3568 y FL(1)2552 3547 y FP(,)118 3647 y FO(D)187 3659 y FL(2)262 3647 y FP(=)f(\()p FO(X)466 3659 y FM(n)525 3647 y FO(:)14 b(:)g(:)g(X)705 3659 y FL(2)742 3647 y FP(\))774 3616 y FN(\003)813 3647 y FO(X)882 3659 y FM(n)940 3647 y FO(:)g(:)g(:)g(X)1120 3659 y FL(2)1157 3647 y FP(.)64 b(F)-7 b(rom)37 b(\(2.45\))f(w)n(e)g(ha)n(v)n(e)g FO(X)2129 3616 y FN(\003)2122 3667 y FL(1)2166 3647 y FO(X)2235 3659 y FL(1)2310 3647 y FP(=)2427 3610 y FM(\015)2462 3618 y Fy(1)p 2423 3628 76 4 v 2423 3675 a FM(\013)2466 3683 y Fy(1)2509 3647 y FO(I)7 b FP(,)118 3771 y FO(X)187 3783 y FM(n)246 3771 y FO(:)14 b(:)g(:)g(X)426 3783 y FL(2)463 3771 y FP(\()p FO(X)564 3783 y FM(n)623 3771 y FO(:)g(:)g(:)f(X)802 3783 y FL(2)839 3771 y FP(\))871 3741 y FN(\003)954 3771 y FP(=)44 b FO(\026I)7 b FP(,)43 b(where)d FO(\026)k FP(=)1678 3709 y Fz(Q)1756 3729 y FM(n)p FN(\000)p FL(3)1756 3796 y FM(k)q FL(=0)1886 3771 y FP(\()p FR(F)1978 3741 y FM(k)2020 3771 y FP(\()p FO(\025)2100 3783 y FM(n)2146 3771 y FO(;)2194 3734 y FM(\015)2229 3742 y Fw(n)p 2192 3752 79 4 v 2192 3800 a FM(\014)2230 3808 y Fw(n)2281 3771 y FP(\)\))2345 3783 y FL(2)2394 3730 y FM(\015)2429 3738 y Fw(n)p Fy(+1)p 2393 3752 150 4 v 2393 3800 a FM(\014)2431 3808 y Fw(n)p Fy(+1)2552 3771 y FP(,)118 3885 y FR(F)p FP(\()p FO(\025)258 3897 y FM(k)q FL(+1)384 3885 y FO(;)14 b(x)p FP(\))44 b(=)f(\()p FO(\025)732 3897 y FM(k)773 3885 y FO(;)14 b FP(\()p FO(\015)885 3897 y FM(k)953 3885 y FQ(\000)26 b FO(\013)1097 3897 y FM(k)1138 3885 y FO(x)p FP(\))p FO(=\014)1306 3897 y FM(k)1347 3885 y FP(\).)73 b(One)40 b(can)f(c)n(hec)n(k)g(that)h FO(\015)2298 3897 y FL(1)2335 3885 y FO(=\013)2430 3897 y FL(1)2510 3885 y FP(=)118 3985 y(1)p FO(=)p FP(\(2)14 b(sin)391 3950 y FL(2)442 3985 y FO(\033)s FP(\).)37 b(Hence)474 4147 y FO(D)543 4159 y FL(1)580 4147 y FP(\()p FO(D)681 4159 y FL(1)736 4147 y FQ(\000)18 b FP(1)p FO(=)p FP(\(2)c(sin)1092 4112 y FL(2)1143 4147 y FO(\033)s FP(\))p FO(I)7 b FP(\))24 b(=)f(0)p FO(;)96 b(D)1642 4159 y FL(2)1679 4147 y FP(\()p FO(D)1780 4159 y FL(2)1836 4147 y FQ(\000)18 b FO(\026I)7 b FP(\))23 b(=)g(0)p FO(:)p eop %%Page: 149 153 149 152 bop 118 100 a FK(2.3.)36 b(Represen)n(tations)26 b(of)i FO(q)s FK(-deforemd)f FO(U)9 b FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(R)p FP(\)\))631 b(149)118 333 y(Let)34 b FO(H)342 345 y FL(0)412 333 y FP(b)r(e)g(a)f(subspace)f(of)h FO(H)40 b FP(whic)n(h)34 b(is)f(in)n(v)-5 b(arian)n(t)32 b(with)i(resp)r(ect)f(to)g FO(D)2515 345 y FL(1)2552 333 y FP(,)118 432 y FO(D)187 444 y FL(2)224 432 y FP(.)k(Then)391 627 y FO(X)467 593 y FN(\003)460 648 y FL(1)504 627 y FO(H)573 639 y FL(0)629 627 y FQ(\010)18 b FO(H)781 639 y FL(0)837 627 y FQ(\010)g FO(X)989 639 y FL(2)1026 627 y FO(H)1095 639 y FL(0)1150 627 y FQ(\010)g FO(X)1302 639 y FL(3)1339 627 y FO(X)1408 639 y FL(2)1445 627 y FO(H)1514 639 y FL(0)1570 627 y FQ(\010)g FO(:)c(:)g(:)k FQ(\010)g FO(X)1920 639 y FM(n)1979 627 y FO(:)c(:)g(:)g(X)2159 639 y FL(2)2196 627 y FO(H)2265 639 y FL(0)118 822 y FP(is)27 b(in)n(v)-5 b(arian)n(t)26 b(with)h(resp)r(ect)g(to)g FO(J)1168 834 y FL(1)1205 822 y FP(,)g FO(X)7 b FP(,)27 b FO(X)1457 792 y FN(\003)1494 822 y FP(.)37 b(In)27 b(fact,)g(using)g(\(2.45\))f(one)g(can)118 922 y(easily)35 b(pro)n(v)n(e)f(that)i FO(X)851 892 y FN(\003)844 946 y FM(k)888 922 y FO(X)957 934 y FM(k)1012 922 y FO(:)14 b(:)g(:)f(X)1191 934 y FL(2)1265 922 y FP(=)36 b FO(\026)1416 934 y FM(k)1457 922 y FO(X)1526 934 y FM(k)q FN(\000)p FL(1)1665 922 y FO(:)14 b(:)g(:)g(X)1845 934 y FL(2)1882 922 y FP(,)37 b(1)f FO(<)g(k)k(<)c(n)f FP(\(here)118 1022 y FO(\026)168 1034 y FM(k)232 1022 y FP(=)23 b(1)p FO(=)p FP(4)14 b(sin)560 987 y FL(2)611 1022 y FO(\033)s FP(\))28 b(.)37 b(Moreo)n(v)n(er,)801 1217 y FO(X)870 1229 y FM(n)p FN(\000)p FL(1)1013 1217 y FO(:)14 b(:)g(:)g(X)1193 1229 y FL(2)1230 1217 y FO(X)1306 1182 y FN(\003)1299 1237 y FL(2)1357 1217 y FO(:)g(:)g(:)g(X)1544 1182 y FN(\003)1537 1237 y FM(n)p FN(\000)p FL(1)1690 1217 y FP(=)23 b FO(\025I)7 b(;)118 1427 y FP(where)31 b FO(\025)e FP(=)533 1365 y Fz(Q)611 1385 y FM(n)p FN(\000)p FL(3)611 1452 y FM(k)q FL(=0)741 1427 y FP(\()p FR(F)833 1397 y FL(\()p FM(k)q FL(\))926 1427 y FP(\()p FO(\025)1006 1439 y FM(n)1052 1427 y FO(;)14 b(\015)1132 1439 y FM(n)1177 1427 y FO(=\014)1266 1439 y FM(n)1311 1427 y FP(\)\))1375 1439 y FL(2)1413 1427 y FP(.)48 b(Assuming)31 b(that)g FO(\025)f FP(=)e(0,)k(w)n(e)f(will)118 1527 y(ha)n(v)n(e)37 b(that)h FO(X)586 1497 y FN(\003)579 1547 y FL(3)637 1527 y FO(:)14 b(:)g(:)f(X)823 1497 y FN(\003)816 1547 y FM(n)p FN(\000)p FL(1)946 1527 y FO(P)999 1539 y FM(n)1045 1527 y FO(H)32 b FQ(\010)25 b FO(:)14 b(:)g(:)f(P)1399 1539 y FM(n)1445 1527 y FO(H)32 b FQ(\010)24 b FO(X)1704 1539 y FM(n)1749 1527 y FO(P)1802 1539 y FM(n)1848 1527 y FO(H)44 b FP(is)38 b(in)n(v)-5 b(arian)n(t)36 b(with)118 1626 y(resp)r(ect)i(to)g FO(J)571 1638 y FL(1)608 1626 y FP(,)j FO(J)718 1638 y FL(2)755 1626 y FP(,)g(hence)d FO(\033)s FP(\()p FO(J)1188 1638 y FL(1)1226 1626 y FP(\))j FQ(63)f(f)p FP(1)p FO(=)p FP(sin)13 b FO(\033)s FQ(g)38 b FP(if)g(the)h(pair)e(\()p FO(J)2264 1638 y FL(1)2302 1626 y FP(,)j FO(J)2411 1638 y FL(2)2449 1626 y FP(\))e(is)118 1726 y(irreducible,)27 b(whic)n(h)h(con)n(tradicts)e(the)i(assumption.) 36 b(Th)n(us)28 b FO(\025)23 b FQ(6)p FP(=)g(0,)633 1921 y FO(X)709 1887 y FN(\003)702 1942 y FM(n)747 1921 y FO(X)816 1933 y FM(n)875 1921 y FO(:)14 b(:)g(:)g(X)1055 1933 y FL(2)1092 1921 y FO(H)1161 1933 y FL(0)1221 1921 y FP(=)23 b FO(\025)1357 1887 y FN(\000)p FL(1)1446 1921 y FO(\025X)1570 1887 y FN(\003)1563 1942 y FM(n)1609 1921 y FO(X)1678 1933 y FM(n)1737 1921 y FO(:)14 b(:)g(:)f(X)1916 1933 y FL(2)1953 1921 y FO(H)2022 1933 y FL(0)486 2056 y FP(=)23 b FO(\025)622 2022 y FN(\000)p FL(1)711 2056 y FP(\()p FO(X)812 2068 y FM(n)p FN(\000)p FL(1)956 2056 y FO(:)14 b(:)g(:)g(X)1136 2068 y FL(2)1173 2056 y FO(X)1249 2022 y FN(\003)1242 2076 y FL(2)1300 2056 y FO(:)g(:)g(:)g(X)1487 2022 y FN(\003)1480 2076 y FM(n)p FN(\000)p FL(1)1610 2056 y FP(\))p FO(X)1718 2022 y FN(\003)1711 2076 y FM(n)1756 2056 y FO(X)1825 2068 y FM(n)1884 2056 y FO(:)g(:)g(:)g(X)2064 2068 y FL(2)2101 2056 y FO(H)2170 2068 y FL(0)1023 2180 y FQ(\032)23 b FO(X)1180 2192 y FM(n)p FN(\000)p FL(1)1324 2180 y FO(:)14 b(:)g(:)f(X)1503 2192 y FL(2)1540 2180 y FO(H)1609 2192 y FL(0)1647 2180 y FO(:)243 2382 y FP(An)n(y)37 b(irreducible)g(pair)g(\()p FO(D)1133 2394 y FL(1)1170 2382 y FO(;)14 b(D)1276 2394 y FL(2)1313 2382 y FP(\))38 b(is)f(one-)g(or)g(t)n(w)n(o-dimensional.)65 b(Let)118 2481 y(us)33 b(describ)r(e)g(the)g(corresp)r(onding)e(irreducible)h (pairs)g FO(J)1915 2493 y FL(1)1953 2481 y FP(,)i FO(J)2056 2493 y FL(2)2093 2481 y FP(.)53 b(Denote)33 b(the)118 2581 y(phase)h(of)g(the)h(op)r(erator)e FO(X)1018 2593 y FM(i)1059 2581 y FO(:)14 b(:)g(:)g(X)1239 2593 y FL(2)1310 2581 y FP(b)n(y)34 b FO(U)1489 2593 y FM(i)1517 2581 y FP(,)i(the)f(phase)f(of)g FO(X)2141 2551 y FN(\003)2134 2601 y FL(1)2213 2581 y FP(b)n(y)g FO(U)2401 2551 y FN(\003)2392 2601 y FL(1)2439 2581 y FP(.)57 b(If)118 2680 y(dim)14 b FO(H)339 2692 y FL(0)400 2680 y FQ(\024)22 b FP(2)27 b(and)g FO(e)756 2692 y FL(1)793 2680 y FP(,)g FO(e)882 2692 y FL(2)946 2680 y FP(is)g(an)g(orthonormal)e(basis)h(in)h(the)h (space)e FO(H)2351 2692 y FL(0)2415 2680 y FP(suc)n(h)118 2780 y(that)i FO(e)337 2792 y FL(1)397 2780 y FQ(2)23 b FP(\(k)n(er)13 b FO(D)701 2792 y FL(2)738 2780 y FP(\))770 2750 y FN(?)827 2780 y FP(,)27 b(then)325 3025 y FO(D)394 3037 y FL(1)454 3025 y FP(=)660 2969 y(1)p 552 3006 259 4 v 552 3088 a(2)14 b(sin)709 3053 y FL(2)760 3088 y FO(\033)834 2908 y Fz(\022)895 2974 y FP(1)k(+)g(cos)13 b FO(')160 b FP(sin)13 b FO(')971 3074 y FP(sin)h FO(')160 b FP(1)17 b FQ(\000)i FP(cos)12 b FO(')1623 2908 y Fz(\023)1698 3025 y FO(;)97 b(D)1887 3037 y FL(2)1947 3025 y FP(=)2034 2908 y Fz(\022)2095 2974 y FO(\026)84 b FP(0)2100 3074 y(0)j(0)2270 2908 y Fz(\023)2345 3025 y FO(;)118 3265 y(U)175 3277 y FM(i)202 3265 y FO(e)241 3277 y FL(1)278 3265 y FP(,)47 b FO(U)405 3277 y FM(i)433 3265 y FO(e)472 3277 y FL(2)552 3265 y FP(is)c(an)f(orthonormal)f(basis)i(in)g FO(X)1673 3277 y FM(i)1714 3265 y FO(:)14 b(:)g(:)g(X)1894 3277 y FL(2)1931 3265 y FO(H)2000 3277 y FL(0)2037 3265 y FP(,)47 b(2)i FQ(\024)f FO(i)h(<)f(n)p FP(,)118 3365 y(and)27 b(the)h(v)n(ectors)e FO(U)761 3377 y FM(n)806 3365 y FO(e)845 3377 y FL(1)910 3365 y FP(and)h FO(U)1128 3377 y FL(1)1165 3365 y FP(\(cos\()p FO('=)p FP(2\))p FO(e)1550 3377 y FL(1)1605 3365 y FP(+)18 b(sin\()p FO('=)p FP(2\))p FO(e)2031 3377 y FL(2)2067 3365 y FP(\))28 b(generate)e(the) 118 3465 y(spaces)20 b FO(X)435 3477 y FM(n)494 3465 y FO(:)14 b(:)g(:)g(X)674 3477 y FL(2)711 3465 y FO(H)780 3477 y FL(0)838 3465 y FP(and)21 b FO(X)1069 3434 y FN(\003)1062 3485 y FL(1)1107 3465 y FO(H)1176 3477 y FL(0)1213 3465 y FP(,)h(resp)r(ectiv)n(ely)-7 b(.)34 b(Note)22 b(that)f(cos)o(\()p FO('=)p FP(2\))14 b FO(e)2468 3477 y FL(1)2510 3465 y FP(+)118 3564 y(sin\()p FO('=)p FP(2\))g FO(e)475 3576 y FL(2)542 3564 y FQ(2)31 b FP(\(k)n(er)13 b FO(D)854 3576 y FL(1)891 3564 y FP(\))923 3534 y FN(?)979 3564 y FP(.)51 b(T)-7 b(o)31 b(describ)r(e)h(the)g(action)g(of)g(the)g(op)r (erator)f FO(J)2538 3576 y FL(2)118 3664 y FP(on)22 b(this)h(basis,)g (it)h(is)e(su\016cien)n(t)h(to)f(kno)n(w)g(it)h(for)g(the)g(op)r (erators)d FO(X)2187 3676 y FM(i)2215 3664 y FP(.)35 b(W)-7 b(e)23 b(ha)n(v)n(e)345 3934 y FO(X)414 3946 y FM(i)p FL(+1)526 3934 y FO(U)583 3946 y FM(i)610 3934 y FO(e)649 3946 y FM(k)713 3934 y FP(=)800 3842 y Fz(\020)850 3830 y FM(i)p FN(\000)p FL(1)851 3855 y Fz(Y)851 4034 y FM(l)p FL(=2)972 3934 y FO(\026)1022 3946 y FM(l)1048 3842 y Fz(\021)1097 3859 y FN(\000)p FL(1)p FM(=)p FL(2)1254 3934 y FO(X)1323 3946 y FM(i)p FL(+1)1448 3934 y FO(:)14 b(:)g(:)g(X)1628 3946 y FL(2)1665 3934 y FO(e)1704 3946 y FM(k)1767 3934 y FP(=)1855 3878 y FQ(p)p 1924 3878 78 4 v 56 x FO(\026)1974 3946 y FM(i)2015 3934 y FO(U)2072 3946 y FM(i)p FL(+1)2184 3934 y FO(e)2223 3946 y FM(k)713 4147 y FP(=)22 b(\(2)14 b(sin)g FO(\033)s FP(\))1086 4113 y FN(\000)p FL(1)1176 4147 y FO(U)1233 4159 y FM(i)p FL(+1)1344 4147 y FO(e)1383 4159 y FM(k)1423 4147 y FO(;)180 b(i)23 b(<)g(n)18 b FQ(\000)g FP(1;)p eop %%Page: 150 154 150 153 bop 118 100 a FP(150)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)309 399 y FO(X)378 411 y FM(n)423 399 y FO(U)480 411 y FM(n)p FN(\000)p FL(1)610 399 y FO(e)649 411 y FM(k)713 399 y FP(=)800 307 y Fz(\020)850 295 y FM(n)p FN(\000)p FL(1)860 320 y Fz(Y)860 499 y FM(l)p FL(=2)990 399 y FO(\026)1040 411 y FM(l)1065 307 y Fz(\021)1115 324 y FN(\000)p FL(1)p FM(=)p FL(2)1271 399 y FO(X)1340 411 y FM(n)1399 399 y FO(:)14 b(:)g(:)g(X)1579 411 y FL(2)1629 399 y FO(e)1668 411 y FM(k)713 686 y FP(=)800 544 y Fz(\()867 629 y FP(0)p FO(;)679 b(k)26 b FP(=)c(2)p FO(;)867 749 y FP(\()899 680 y FQ(p)p 969 680 42 4 v 969 749 a FP(2)13 b(sin)h FO(\033)s FP(\))1222 719 y FL(1)p FM(=)p FL(2)1327 749 y FO(U)1384 761 y FM(n)1429 749 y FO(e)1468 761 y FL(1)1505 749 y FO(;)83 b(k)26 b FP(=)c(1)p FO(;)358 893 y(X)427 905 y FL(1)464 893 y FO(U)530 858 y FN(\003)521 913 y FL(1)567 893 y FP(\(cos\()p FO('=)p FP(2\))14 b FO(e)966 905 y FL(1)1021 893 y FP(+)k(sin\()p FO('=)p FP(2\))c FO(e)1461 905 y FL(2)1498 893 y FP(\))713 1035 y(=)800 962 y FQ(p)p 869 962 V 73 x FP(2)g(sin)g FO(\033)s(X)1160 1047 y FL(1)1197 1035 y FO(X)1273 1001 y FN(\003)1266 1056 y FL(1)1311 1035 y FP(\(cos)o(\()p FO('=)p FP(2\))g FO(e)1709 1047 y FL(1)1764 1035 y FP(+)k(sin\()p FO('=)p FP(2\))c FO(e)2204 1047 y FL(2)2241 1035 y FP(\))23 b(=)713 1177 y(=)f(\()832 1105 y FQ(p)p 902 1105 V 902 1177 a FP(2)13 b(sin)h FO(\033)s FP(\))1155 1143 y FN(\000)p FL(1)1245 1177 y FP(\(cos)o(\()p FO('=)p FP(2\))g FO(e)1643 1189 y FL(1)1698 1177 y FP(+)k(sin\()p FO('=)p FP(2\))c FO(e)2138 1189 y FL(2)2175 1177 y FP(\))p FO(:)243 1348 y FP(Th)n(us)33 b(w)n(e)h(will)g(get)g(represen)n(tations)e(from)i (series)f(5.)55 b(If)35 b(dim)14 b FO(H)2341 1360 y FL(0)2412 1348 y FP(=)33 b(1,)118 1447 y(then)f FO(D)380 1459 y FL(1)446 1447 y FQ(6)p FP(=)d(0)i(and)g FO(D)847 1459 y FL(2)914 1447 y FQ(6)p FP(=)e(0,)j(since)f(otherwise)f FO(\033)s FP(\()p FO(J)1813 1459 y FL(1)1851 1447 y FP(\))g FQ(6)p FP(=)f FQ(f)p FO(\025)2097 1459 y FL(1)2134 1447 y FO(;)14 b(:)g(:)g(:)28 b(;)14 b(\025)2381 1459 y FM(n)p FL(+1)2510 1447 y FQ(g)p FP(.)118 1547 y(If)39 b FJ(C)15 b FO(e)305 1559 y FL(1)389 1547 y FP(=)40 b FO(H)563 1559 y FL(0)601 1547 y FP(,)h(then)e(using)f(the)g(same)g(argumen)n(ts) f(one)h(can)g(sho)n(w)g(that)118 1646 y FQ(f)p FO(U)226 1616 y FN(\003)217 1667 y FL(1)264 1646 y FO(e)303 1658 y FL(1)339 1646 y FO(;)14 b(e)415 1658 y FL(1)452 1646 y FO(;)g(U)546 1658 y FL(1)583 1646 y FO(e)622 1658 y FL(1)659 1646 y FO(;)g(:)g(:)g(:)g(U)864 1658 y FM(n)908 1646 y FO(e)947 1658 y FL(1)984 1646 y FQ(g)32 b FP(is)f(an)g (orthogonal)f(basis)g(in)i FO(H)7 b FP(,)33 b(and)e(the)h(cor-)118 1746 y(resp)r(onding)27 b(irreducible)g(represen)n(tations)e(\()p FO(J)1605 1758 y FL(1)1643 1746 y FO(;)14 b(J)1726 1758 y FL(2)1763 1746 y FP(\))28 b(form)f(series)g(6.)243 1846 y(4.)36 b(Supp)r(ose)28 b(that)g(the)g(supp)r(ort)f(of)h FO(J)1441 1858 y FL(2)1506 1846 y FP(is)1139 1998 y Fo(r)p 1139 2000 4 4 v 1134 1995 V 1130 1990 V 1126 1985 V 1122 1981 V 1119 1976 V 1116 1972 V 1113 1968 V 1111 1964 V 1109 1960 V 1108 1956 V 1107 1952 V 1106 1949 V 1106 1945 V 1106 1942 V 1106 1939 V 1107 1936 V 1108 1933 V 1110 1930 V 1112 1927 V 1114 1925 V 1114 1925 V 1116 1923 V 1119 1920 V 1121 1919 V 1124 1917 V 1126 1916 V 1129 1914 V 1131 1914 V 1134 1913 V 1136 1913 V 1139 1912 V 1141 1913 V 1144 1913 V 1146 1914 V 1149 1914 V 1151 1916 V 1154 1917 V 1156 1919 V 1159 1920 V 1161 1923 V 1164 1925 V 1139 2000 V 1144 1995 V 1148 1990 V 1152 1985 V 1156 1981 V 1159 1976 V 1162 1972 V 1165 1968 V 1167 1964 V 1169 1960 V 1170 1956 V 1171 1952 V 1172 1949 V 1172 1945 V 1172 1942 V 1172 1939 V 1171 1936 V 1170 1933 V 1168 1930 V 1166 1927 V 1164 1925 V 1139 2000 125 4 v 99 w(r)141 b(r)p 1429 2000 V 100 w(r)1117 2098 y FM(\025)1156 2106 y Fy(1)1510 2098 y FM(\025)1549 2109 y Fy(\()p Fw(n)p Fy(+1\))p Fw(=)p Fy(2)1291 2002 y FO(:)14 b(:)g(:)2363 2016 y FP(\(2.46\))118 2239 y(where)35 b FO(\025)414 2251 y FL(1)487 2239 y FP(=)h FQ(\006)p FP(\(cos)13 b FO(\033)s(=)p FP(2)o(\))p FO(=)p FP(sin)h FO(\033)s FP(,)37 b FO(\025)1291 2254 y FL(\()p FM(n)p FL(+1\))p FM(=)p FL(2)1576 2239 y FP(=)e(1)p FO(=)p FP(sin)13 b FO(\033)s FP(.)60 b(Let)36 b(us)f(consider)118 2338 y(the)f(case)e FO(\025)496 2350 y FL(1)566 2338 y FP(=)h(\(cos)13 b FO(\033)s(=)p FP(2)o(\))p FO(=)p FP(sin)h FO(\033)37 b FP(\(for)c FO(\025)1441 2350 y FL(1)1511 2338 y FP(=)f FQ(\000)p FP(\(cos)13 b FO(\033)s(=)p FP(2)o(\))p FO(=)p FP(sin)h FO(\033)37 b FP(the)c(pro)r(of)118 2438 y(is)c(the)h(same\).)42 b(The)29 b(op)r(erator)f FO(J)1181 2450 y FL(2)1248 2438 y FP(can)h(b)r(e)g(represen)n(ted)g(in) g(the)h(form)f FO(J)2447 2450 y FL(2)2510 2438 y FP(=)118 2554 y FO(X)g FP(+)22 b FO(X)379 2523 y FN(\003)439 2554 y FP(+)h FO(Y)18 b FP(,)36 b(where)d FO(X)40 b FP(=)1105 2491 y Fz(P)1193 2510 y FL(\()p FM(n)p FN(\000)p FL(1\))p FM(=)p FL(2)1193 2579 y FM(k)q FL(=1)1456 2554 y FO(P)1509 2566 y FM(k)q FL(+1)1634 2554 y FO(J)1680 2566 y FL(2)1717 2554 y FO(P)1770 2566 y FM(k)1811 2554 y FP(,)c FO(Y)52 b FP(=)33 b FO(P)2121 2566 y FL(1)2159 2554 y FO(J)2205 2566 y FL(2)2242 2554 y FO(P)2295 2566 y FL(1)2332 2554 y FP(,)j FO(P)2444 2566 y FM(k)2519 2554 y FP(is)118 2653 y(the)22 b(pro)5 b(jection)21 b(on)h(the)g(eigenspace)e(whic)n(h)i (corresp)r(onds)e(to)i(the)g(eigen)n(v)-5 b(alue)118 2753 y FO(\025)166 2765 y FM(k)230 2753 y FP(=)23 b(\(cos\(\(2)p FO(k)e FQ(\000)d FP(1\))p FO(\033)s(=)p FP(2\))o(\))p FO(=)p FP(sin)c FO(\033)s FP(.)37 b(Let)28 b FO(X)1472 2765 y FM(k)1536 2753 y FP(=)22 b FO(X)7 b(P)1752 2765 y FM(k)1793 2753 y FP(.)37 b(Then)561 2923 y FO(\013)614 2935 y FM(k)655 2923 y FO(X)731 2889 y FN(\003)724 2943 y FM(k)768 2923 y FO(X)837 2935 y FM(k)896 2923 y FP(+)18 b FO(\014)1026 2935 y FM(k)1067 2923 y FO(X)1136 2935 y FM(k)q FN(\000)p FL(1)1262 2923 y FO(X)1338 2889 y FN(\003)1331 2943 y FM(k)q FN(\000)p FL(1)1480 2923 y FP(=)k FO(\015)1610 2935 y FM(k)1651 2923 y FO(I)7 b(;)180 b(k)26 b FQ(6)p FP(=)d(1)p FO(;)767 3058 y(\013)820 3070 y FL(1)857 3058 y FO(X)933 3024 y FN(\003)926 3078 y FL(1)970 3058 y FO(X)1039 3070 y FL(1)1095 3058 y FP(+)18 b FO(\014)1225 3070 y FL(1)1262 3058 y FO(Y)1329 3024 y FL(2)1366 3058 y FO(P)1419 3070 y FL(1)1480 3058 y FP(=)k FO(\015)1610 3070 y FL(1)1647 3058 y FO(I)7 b(;)650 b FP(\(2.47\))118 3228 y(where)39 b FO(\013)423 3240 y FM(k)507 3228 y FP(=)k FQ(\000)p FP(2)14 b(sin)o(\(\(2)p FO(k)30 b FP(+)c(1\))p FO(\033)s(=)p FP(2\),)42 b FO(\014)1459 3240 y FM(k)1543 3228 y FP(=)h(2)14 b(sin)o(\(\(2)p FO(k)30 b FQ(\000)c FP(3\))p FO(\033)s(=)p FP(2\),)42 b FO(\015)2426 3240 y FM(k)2510 3228 y FP(=)118 3328 y FQ(\000)14 b FP(cos)o(\(\(2)p FO(k)24 b FQ(\000)e FP(1\))p FO(\033)s(=)p FP(2\))p FO(=)14 b FP(sin)f FO(\033)s FP(;)35 b(in)f(particular,)e FO(\013)1655 3340 y FM(k)1728 3328 y FQ(6)p FP(=)f(0)h(for)h FO(k)h FQ(6)p FP(=)d(\()p FO(n)22 b FQ(\000)g FP(1\))p FO(=)p FP(2,)118 3427 y FO(\013)171 3442 y FL(\()p FM(n)p FN(\000)p FL(1\))p FM(=)p FL(2)450 3427 y FP(=)29 b(0.)49 b(As)31 b(ab)r(o)n(v)n(e,)g(the)h(problem)f(of)h(describing)f (irreducible)g(rep-)118 3527 y(resen)n(tations)39 b(\()p FO(J)669 3539 y FL(1)706 3527 y FO(;)14 b(J)789 3539 y FL(2)827 3527 y FP(\))40 b(reduces)g(to)g(that)g(of)h(irreducible)e (pairs)g(\()p FO(D)2362 3539 y FL(1)2400 3527 y FO(;)14 b(D)2506 3539 y FL(2)2543 3527 y FP(\))118 3626 y(satisfying)27 b(some)g(quadratic)g(relation.)35 b(Here)432 3797 y FO(D)501 3809 y FL(1)561 3797 y FP(=)23 b FO(Y)5 b(;)96 b(D)890 3809 y FL(2)950 3797 y FP(=)23 b(\()p FO(X)1139 3812 y FL(\()p FM(n)p FN(\000)p FL(1\))p FM(=)p FL(2)1402 3797 y FO(:)14 b(:)g(:)g(X)1582 3809 y FL(1)1619 3797 y FP(\))1651 3762 y FN(\003)1689 3797 y FO(X)1758 3812 y FL(\()p FM(n)p FN(\000)p FL(1\))p FM(=)p FL(2)2021 3797 y FO(:)g(:)g(:)g(X)2201 3809 y FL(1)2238 3797 y FO(;)118 3977 y FP(and)28 b FO(D)349 3989 y FL(2)386 3977 y FP(\()p FO(D)487 3989 y FL(2)542 3977 y FQ(\000)18 b FO(\026I)7 b FP(\))24 b(=)e(0,)28 b FO(D)1025 3947 y FL(2)1023 3998 y(1)1085 3977 y FP(=)23 b(1)p FO(=)p FP(\(4)14 b(sin)1445 3942 y FL(2)1496 3977 y FO(\033)t FP(\))p FO(I)7 b FP(,)28 b FO(\026)f FP(is)h(de\014ned)g(b)n(y)633 4147 y FO(X)702 4162 y FL(\()p FM(n)p FN(\000)p FL(1\))p FM(=)p FL(2)965 4147 y FO(:)14 b(:)g(:)g(X)1145 4159 y FL(1)1182 4147 y FP(\()p FO(X)1283 4162 y FL(\()p FM(n)p FN(\000)p FL(1\))p FM(=)p FL(2)1546 4147 y FO(:)g(:)g(:)f(X)1725 4159 y FL(1)1762 4147 y FP(\))1794 4113 y FN(\003)1856 4147 y FP(=)23 b FO(\026I)7 b(;)p eop %%Page: 151 155 151 154 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(151)118 333 y(whic)n(h)28 b(can)f(easily)g(b)r(e)h(obtained)f(from)g(\(2.47\).)243 440 y(If)38 b FO(H)405 452 y FL(0)481 440 y FP(is)g(in)n(v)-5 b(arian)n(t)37 b(with)i(resp)r(ect)f(to)g FO(D)1610 452 y FL(1)1647 440 y FP(,)j FO(D)1780 452 y FL(2)1817 440 y FP(,)g(then)e FO(H)2150 452 y FL(0)2213 440 y FQ(\010)25 b FO(X)7 b(H)2448 452 y FL(0)2510 440 y FQ(\010)118 539 y FO(:)14 b(:)g(:)26 b FQ(\010)f FO(X)407 509 y FL(\()p FM(n)p FN(\000)p FL(1\))p FM(=)p FL(2)656 539 y FO(H)725 551 y FL(0)801 539 y FP(is)39 b(in)n(v)-5 b(arian)n(t)38 b(with)h(resp)r(ect)g(to)g FO(J)1911 551 y FL(1)1948 539 y FP(,)j FO(X)7 b FP(,)41 b FO(X)2229 509 y FN(\003)2266 539 y FP(.)71 b(More-)118 639 y(o)n(v)n(er,)23 b(the)i(dimensions)f(of) g(the)h(irreducible)f(represen)n(tations)e(are)h(\()p FO(n)12 b FP(+)g(1\))p FO(=)p FP(2.)118 739 y(This)26 b(follo)n(ws)g(b)n(y)g(the)h(same)e(metho)r(d)i(as)f(in)g(the)h (previous)e(case)h(and)g(w)n(e)g(get)118 838 y(represen)n(tations)g(6,) h(7.)243 945 y(If)44 b(\()p FO(J)420 957 y FL(1)458 945 y FO(;)14 b(J)541 957 y FL(2)578 945 y FP(\))45 b(is)f(irreducible)f (and)i FO(\033)s FP(\()p FO(J)1489 957 y FL(1)1527 945 y FP(\))51 b FQ(\032)f(f)p FO(\025)1815 957 y FL(1)1852 945 y FO(;)14 b(:)g(:)g(:)28 b(;)14 b(\025)2099 960 y FL(\()p FM(n)p FL(+1\))p FM(=)p FL(2)2347 945 y FQ(g)44 b FP(is)g(a)118 1045 y(prop)r(er)f(subset)g(then,)48 b(as)43 b(b)r(efore,)k(w)n(e)c(conclude)g(that)h FO(\033)s FP(\()p FO(J)2131 1057 y FL(1)2169 1045 y FP(\))f(is)h(simple)118 1145 y(and)34 b(the)g(irreducible)g(represen)n(tation)e(is)i(realized)f (in)h FO(l)1931 1157 y FL(2)1968 1145 y FP(\(\001\),)j(where)c(\001)h (=)118 1244 y FQ(f)p FO(\025)208 1256 y FM(p)246 1244 y FO(;)14 b(\025)331 1256 y FM(p)p FL(+1)454 1244 y FO(;)g(:)g(:)g(:)g (;)g(\025)687 1256 y FM(m)750 1244 y FQ(g)p FP(,)24 b(for)f(some)h(1)e FQ(\024)h FO(p;)14 b(m)22 b FQ(\024)h FP(\()p FO(n)11 b FP(+)g(1\))p FO(=)p FP(2)21 b(and)j(either)f FO(\025)2366 1256 y FL(1)2436 1244 y FO(=)-51 b FQ(2)24 b FP(\001)118 1344 y(or)j FO(\025)268 1359 y FL(\()p FM(n)p FL(+1\))p FM(=)p FL(2)549 1344 y FO(=)-51 b FQ(2)23 b FP(\001.)37 b(W)-7 b(e)28 b(obtain)f(series)g(8)g(if)h FO(\025)1564 1356 y FL(1)1625 1344 y FQ(2)23 b FP(\001)28 b(or)f(9)g(if)h FO(\025)2095 1356 y FL(1)2165 1344 y FO(=)-51 b FQ(2)23 b FP(\001.)p 2514 1344 4 57 v 2518 1291 50 4 v 2518 1344 V 2567 1344 4 57 v 118 1570 a FR(4.)37 b FP(Represen)n(tations)26 b(of)34 b FO(U)992 1582 y FM(q)1029 1570 y FP(\()p FO(so)p FP(\(3\)\),)28 b(where)g FO(q)j FP(is)c(not)h(a)f(ro)r(ot)g(of)h(unit)n (y)-7 b(.)38 b(Let)118 1670 y FO(q)j FP(=)c FO(e)337 1640 y FM(i\033)404 1670 y FP(,)i(where)d FO(\033)50 b(=)-51 b FQ(2)37 b FO(\031)s FJ(Q)6 b FP(.)69 b(Analysis)36 b(similar)f(to)h(that)h(as)e(in)i(the)f(pro)r(of)118 1770 y(of)e(Theorem)g(35)f(sho)n(ws)h(that)g(if)h(the)g(pair)e(\()p FO(J)1614 1782 y FL(1)1652 1770 y FO(;)14 b(J)1735 1782 y FL(2)1772 1770 y FP(\))35 b(de\014nes)f(a)g(non-trivial)118 1869 y(irreducible)39 b(represen)n(tation,)j(then)e FO(\033)s FP(\()p FO(J)1452 1881 y FL(1)1490 1869 y FP(\))k FQ(2)h FO(S)1717 1881 y FL(1)1797 1869 y FP(=)f([)p FQ(\000)p FP(1)o FO(=)p FP(sin)13 b FO(\033)t(;)h FP(1)p FO(=)p FP(sin)e FO(\033)t FP(].)118 1969 y(Moreo)n(v)n(er,)27 b(the)i(op)r(erator)e FO(J)1034 1981 y FL(2)1101 1969 y FP(is)i(concen)n(trated)e(on)i(\000)c(=)g FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))26 b FQ(j)f FO(t)2278 1939 y FL(2)2335 1969 y FQ(\000)19 b FP(\()p FO(q)j FP(+)118 2068 y FO(q)158 2038 y FN(\000)p FL(1)247 2068 y FP(\))14 b FO(ts)21 b FP(+)e FO(s)506 2038 y FL(2)571 2068 y FP(=)27 b(0)p FQ(g)p FP(.)44 b(An)n(y)30 b(tra)5 b(jectory)29 b(of)h(a)g(p)r(oin)n(t) g FO(\025)e FP(=)f(sin)14 b FO(x\033)s(=)g FP(sin)g FO(\033)31 b FQ(2)d FO(S)2538 2080 y FL(1)118 2168 y FP(with)22 b(resp)r(ect)g(to)g(\000)g(is)f(of)h(the)h(form)e FB(O)p FP(\()p FQ(f)p FO(\025)p FQ(g)p FP(\))i(=)g FQ(f)p FP(sin)o(\()p FO(x)7 b FP(+)g FO(k)s FP(\))p FO(\033)s(=)14 b FP(sin)g FO(\033)27 b FQ(j)c FO(k)j FQ(2)d FJ(Z)o FQ(g)118 2268 y FP(whic)n(h)32 b(is)g(clearly)g(dense)g(in)g FO(S)1102 2280 y FL(1)1139 2268 y FP(,)i(and)e(there)g(is)g(no)g(measurable)f (section)h(of)118 2367 y(\()p FO(S)201 2379 y FL(1)239 2367 y FO(;)14 b FP(\000\).)35 b(In)25 b(this)f(case)f(there)h(exist)g (irreducible)f(represen)n(tations)f(whic)n(h)i(are)118 2467 y(not)29 b(concen)n(trated)f(on)g(a)h(tra)5 b(jectory)-7 b(.)39 b(The)29 b(description)f(of)h(suc)n(h)f(represen-)118 2567 y(tations)21 b(is)f(problematic.)34 b(The)21 b(description)g(of)g (irreducible)f(represen)n(tations)118 2666 y(related)j(to)f(tra)5 b(jectories)22 b(can)g(b)r(e)i(obtained)e(using)h(the)h(same)e(tec)n (hnique)h(\(see)118 2766 y([246)o(])k(for)h(the)f(concrete)g(form)n (ulae\).)118 3047 y FH(2.4)112 b(Man)m(y-dimensional)38 b(dynamical)e(systems)118 3243 y FP(In)41 b(this)h(section)e(w)n(e)h (study)g(represen)n(tations)e(of)i(relations)f(with)i(sev)n(eral)118 3343 y(generators)23 b(using)h(the)h(man)n(y-dimensional)f(dynamical)g (system)h(approac)n(h.)243 3450 y(W)-7 b(e)32 b(start)g(with)h(some)e (examples.)51 b(Firstly)-7 b(,)33 b(in)f(Section)h(2.4.1)e(w)n(e)h (con-)118 3550 y(sider)20 b(the)g(so-called)f(\\direct)g(pro)r(ducts")g (of)h(one-dimensional)f(relations)g(and)118 3649 y(sho)n(w)26 b(ho)n(w)h(to)g(apply)g(results)f(of)h(Section)h(2.1)e(to)h(classify)f (their)i(irreducible)118 3749 y(represen)n(tations.)71 b(In)40 b(Section)g(2.4.2)e(w)n(e)h(study)h(the)g(more)f(complicated) 118 3848 y(\\triangular")24 b(case)h(and)h(apply)g(the)g(inductiv)n(e)h (algorithm)e(for)g(the)i(descrip-)118 3948 y(tion)32 b(of)g(irreducible)g(represen)n(tations)e(to)i(the)g(t)n(wisted)h(CCR)f (and)g(t)n(wisted)118 4048 y(CAR)e(algebras.)42 b(Then)30 b(\(see)g(Section)g(2.4.3\))f(w)n(e)g(consider)g(families)h(of)f(op-) 118 4147 y(erators)h(satisfying)h(a)h(general)e(class)h(of)h(relations) f(whose)g(represen)n(tations)p eop %%Page: 152 156 152 155 bop 118 100 a FP(152)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(can)34 b(b)r(e)g(describ)r(ed)f(in)h(terms)g (of)g(orbits)f(of)h(some)f(man)n(y-dimensional)f(dy-)118 432 y(namical)37 b(system.)64 b(The)37 b(results)g(obtained)f(are)g (illustrated)h(with)h(sev)n(eral)118 532 y(examples.)61 b(Namely)-7 b(,)37 b(w)n(e)f(study)g(irreducible)f(represen)n(tations)e (of)j(a)f(non-)118 632 y(standard)h(quan)n(tum)h(sphere)g(\(see)g (2.4.4\),)h(Heisen)n(b)r(erg)e(relations)g(for)g(the)118 731 y(quan)n(tum)e FO(E)536 743 y FL(2)607 731 y FP(group)e(\(see)67 b(2.4.5\),)34 b(and)g(a)f(wide)g(class)g(of)h(Wic)n(k)f(algebras)118 831 y(con)n(taining)27 b FO(q)559 843 y FM(ij)617 831 y FP(-CCR,)h FO(\026)p FP(-CCR)f(and)h(other)f(algebras)e(\(see)j (2.4.6\).)243 930 y(Let)e(us)g(note)f(that)i(all)e(examples)g(presen)n (ted)h(b)r(elo)n(w)f(are)g(Wic)n(k)h(algebras)118 1030 y(with)e(a)f(braided)g(op)r(erator)e FO(T)34 b FP(and)24 b(additional)e(relations)g(that)i(reduce)f(these)118 1130 y(algebras)i(to)i(the)g(form,)g(to)g(whic)n(h)g(dynamical)g (formalism)f(can)g(b)r(e)i(applied.)118 1229 y(F)-7 b(or)36 b(more)f(details)h(w)n(e)g(address)f(the)i(reader)e(to)h(Section)g (2.4.6)f(where)h(w)n(e)118 1329 y(giv)n(e)27 b(all)g(necessary)f (de\014nitions)i(and)f(facts)h(concerning)e(Wic)n(k)h(algebras.)118 1543 y FR(2.4.1)94 b(\\Direct)22 b(pro)s(ducts")f(of)h(one-dimensional) 17 b(dynamical)k(sys-)410 1643 y(tems)118 1796 y(1.)48 b FP(By)31 b(the)g(\\direct)g(pro)r(duct")g(of)g(one-dimensional)f (systems)h(w)n(e)g(mean)g(a)118 1896 y(dynamical)21 b(system)h(on)g FJ(R)945 1866 y FM(d)1011 1896 y FP(de\014ned)h(b)n(y)e(the)i(family)e (of)h(mappings)g FO(F)2297 1908 y FM(i)2325 1896 y FP(,)h FO(i)g FP(=)f(1,)118 1996 y FO(:)14 b(:)g(:)28 b FP(,)c FO(d)f FP(with)g(the)h(prop)r(ert)n(y)e(that)h(the)g FO(i)p FP(-th)g(mapping)g(only)f(c)n(hanges)g(the)h FO(i)p FP(-th)118 2095 y(co)r(ordinate,)k(i.e.,)352 2271 y FO(F)405 2283 y FM(i)433 2271 y FP(\()462 2249 y FO(~)465 2271 y(\025)p FP(\))d(=)e(\()p FO(\025)736 2283 y FL(1)774 2271 y FO(;)14 b(:)g(:)g(:)g(;)g(\025)1007 2283 y FM(i)p FN(\000)p FL(1)1120 2271 y FO(;)g(f)1198 2283 y FM(i)1225 2271 y FP(\()p FO(\025)1305 2283 y FM(i)1333 2271 y FP(\))p FO(;)g(\025)1450 2283 y FM(i)p FL(+1)1563 2271 y FO(;)g(:)g(:)g(:)f(;)h (\025)1795 2283 y FM(d)1834 2271 y FP(\))p FO(;)2067 2249 y(~)2070 2271 y(\025)23 b FQ(2)h FJ(R)2274 2236 y FM(d)2318 2271 y FO(:)118 2446 y FR(2.)55 b FP(Let)34 b(us)g(giv)n(e)f(an)h(example)g(of)g(a)f FQ(\003)p FP(-algebra)e(suc)n (h)j(that)g(its)h(irreducible)118 2546 y(represen)n(tations)k(can)g(b)r (e)i(classi\014ed)f(using)g(the)h(direct)f(pro)r(duct)g(of)h(one-)118 2645 y(dimensional)33 b(systems.)54 b(Namely)-7 b(,)35 b(w)n(e)e(consider)f(the)i(direct)f(pro)r(duct)g(of)h FO(d)118 2745 y FP(one-dimensional)e FO(q)s FP(-CCR,)h(whic)n(h)h(is)f (a)g(particular)f(case)g(of)i(the)f(so-called)118 2844 y FO(q)155 2856 y FM(ij)214 2844 y FP(-CCR)27 b(algebra)f(\(see)h (Section)h(2.4.6\).)36 b(Consider)26 b(the)i FQ(\003)p FP(-algebra:)363 3020 y FJ(C)417 2952 y Fz(\012)462 3020 y FO(x)509 3032 y FM(i)537 3020 y FO(;)14 b(x)621 2985 y FN(\003)621 3040 y FM(i)683 3020 y FQ(j)23 b FO(x)776 2985 y FN(\003)776 3040 y FM(i)815 3020 y FO(x)862 3032 y FM(i)913 3020 y FP(=)f(1)c(+)g FO(q)s(x)1230 3032 y FM(i)1259 3020 y FO(x)1306 2985 y FN(\003)1306 3040 y FM(i)1344 3020 y FO(;)743 3153 y(x)790 3119 y FN(\003)790 3174 y FM(i)829 3153 y FO(x)876 3165 y FM(j)934 3153 y FP(=)23 b FO(x)1069 3165 y FM(j)1104 3153 y FO(x)1151 3119 y FN(\003)1151 3174 y FM(i)1190 3153 y FO(;)k(i;)14 b(j)28 b FP(=)23 b(1)p FO(;)14 b(::;)g(d;)27 b FP(0)c FO(<)f(q)k(<)d FP(1)2056 3086 y Fz(\013)2095 3153 y FO(:)245 b FP(\(2.48\))118 3329 y(The)31 b(classi\014cation)e(of)i(irreducible)f (represen)n(tations)f(is)i(based)f(on)h(the)g(fol-)118 3428 y(lo)n(wing)i(prop)r(osition)g(whic)n(h)h(sho)n(ws)f(that)h(there) g(are)f(additional)g(relations)118 3528 y(b)r(et)n(w)n(een)25 b(generators)e(whic)n(h)i(hold)g(automatically)f(in)i(an)n(y)e (irreducible)h(rep-)118 3628 y(resen)n(tation)h(b)n(y)i(b)r(ounded)g (op)r(erators.)118 3788 y FR(Prop)s(osition)42 b(49.)k FC(L)l(et)39 b FO(\031)s FP(\()p FQ(\001)p FP(\))h FC(b)l(e)f(a)g(b)l (ounde)l(d)h(r)l(epr)l(esentation)f(of)57 b FP(\(2.48\))o FC(.)118 3887 y(Then)31 b FO(\031)s FP(\()p FO(x)464 3899 y FM(j)500 3887 y FO(x)547 3899 y FM(i)593 3887 y FQ(\000)18 b FO(x)723 3899 y FM(i)751 3887 y FO(x)798 3899 y FM(j)834 3887 y FP(\))23 b(=)g(0)p FC(.)118 4048 y(Pr)l(o)l(of.)43 b FP(A)22 b(simple)f(calculation)f(sho)n(ws)g(that)h FO(X)1601 4060 y FM(ij)1683 4048 y FP(=)h FO(\031)s FP(\()p FO(x)1899 4060 y FM(i)1928 4048 y FO(x)1975 4060 y FM(j)2016 4048 y FQ(\000)5 b FO(x)2133 4060 y FM(j)2168 4048 y FO(x)2215 4060 y FM(i)2243 4048 y FP(\))22 b(satis\014es)118 4147 y(the)31 b(relation)f FO(X)648 4117 y FN(\003)641 4169 y FM(ij)700 4147 y FO(X)769 4159 y FM(ij)856 4147 y FP(=)e FO(q)989 4117 y FL(2)1026 4147 y FO(X)1095 4159 y FM(ij)1153 4147 y FO(X)1229 4117 y FN(\003)1222 4169 y FM(ij)1281 4147 y FP(,)k(whic)n(h)f(implies)g(that)g FO(X)2114 4159 y FM(ij)2201 4147 y FP(=)d(0)j(\(since)p eop %%Page: 153 157 153 156 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(153)118 333 y(the)27 b(relation)f FO(x)611 303 y FN(\003)649 333 y FO(x)e FP(=)e FO(q)s(xx)941 303 y FN(\003)1007 333 y FP(do)r(es)k(not)h(ha)n(v)n(e)e(an)n(y)h (non-trivial)g(b)r(ounded)h(rep-)118 432 y(resen)n(tation,)g(see)g (Section)g(1.4.2\).)p 2514 432 4 57 v 2518 380 50 4 v 2518 432 V 2567 432 4 57 v 243 598 a(Let)k(us)h(note)f(that)h(the)g (elemen)n(ts)f FO(x)1415 610 y FM(j)1451 598 y FO(x)1498 610 y FM(i)1547 598 y FQ(\000)20 b FO(x)1679 610 y FM(i)1708 598 y FO(x)1755 610 y FM(j)1822 598 y FP(generate)30 b(a)h(quadratic)118 697 y(Wic)n(k)25 b(ideal)f(\(see)h(Section)f (2.4.6\).)35 b(So,)25 b(to)g(study)f(the)h(irreducible)f(represen-)118 797 y(tations,)30 b(one)f(has)g(to)h(consider)e(a)i(family)f(of)h(b)r (ounded)g(op)r(erators)e FO(X)2358 809 y FM(i)2385 797 y FP(,)i FO(X)2514 767 y FN(\003)2507 818 y FM(i)2552 797 y FP(,)118 896 y FO(i)23 b FP(=)f(1,)28 b FO(:)14 b(:)g(:)27 b FP(,)h FO(d)p FP(,)g(satisfying)f(the)h(relations)224 1075 y FO(X)300 1041 y FN(\003)293 1096 y FM(i)338 1075 y FO(X)407 1087 y FM(i)457 1075 y FP(=)23 b(1)18 b(+)g FO(q)s(X)797 1087 y FM(i)824 1075 y FO(X)900 1041 y FN(\003)893 1096 y FM(i)938 1075 y FO(;)96 b(X)1133 1041 y FN(\003)1126 1096 y FM(i)1171 1075 y FO(X)1240 1087 y FM(j)1298 1075 y FP(=)22 b FO(X)1454 1087 y FM(j)1489 1075 y FO(X)1565 1041 y FN(\003)1558 1096 y FM(i)1603 1075 y FO(;)97 b(X)1792 1087 y FM(i)1819 1075 y FO(X)1888 1087 y FM(j)1946 1075 y FP(=)22 b FO(X)2102 1087 y FM(j)2137 1075 y FO(X)2206 1087 y FM(i)2234 1075 y FO(:)106 b FP(\(2.49\))118 1254 y(It)27 b(is)f(easy)f(to)h(see)g(that)h(this)f(algebra)f(is)h (generated)f(b)n(y)h FO(d)g FP(comm)n(uting)g(alge-)118 1354 y(bras)652 1533 y FA(A)712 1545 y FM(i)762 1533 y FP(=)d FJ(C)904 1466 y Fz(\012)949 1533 y FO(X)1018 1545 y FM(i)1046 1533 y FO(;)14 b(X)1159 1499 y FN(\003)1152 1553 y FM(i)1219 1533 y FQ(j)23 b FO(X)1341 1499 y FN(\003)1334 1553 y FM(i)1379 1533 y FO(X)1448 1545 y FM(i)1498 1533 y FP(=)g(1)18 b(+)g FO(q)s(X)1838 1545 y FM(i)1865 1533 y FO(X)1941 1499 y FN(\003)1934 1553 y FM(i)1979 1466 y Fz(\013)2018 1533 y FO(:)243 1712 y FP(Let)28 b(us)h(no)n(w)f (consider)g(the)h(p)r(olar)f(decomp)r(osition)g FO(X)1983 1682 y FN(\003)1976 1733 y FM(i)2045 1712 y FP(=)d FO(U)2192 1724 y FM(i)2219 1712 y FO(C)2278 1724 y FM(i)2306 1712 y FP(.)40 b(Using)118 1811 y(relations)27 b(\(2.49\))f(one)i(can)f (rewrite)g(the)h(system)f(in)h(an)f(equiv)-5 b(alen)n(t)27 b(form,)621 1990 y FO(C)686 1956 y FL(2)680 2011 y FM(i)723 1990 y FO(U)789 1956 y FN(\003)780 2011 y FM(i)850 1990 y FP(=)c FO(U)1004 1956 y FN(\003)995 2011 y FM(i)1042 1990 y FP(\(1)18 b(+)g FO(q)s(C)1322 1956 y FL(2)1316 2011 y FM(i)1360 1990 y FP(\))p FO(;)97 b(C)1577 1956 y FL(2)1571 2011 y FM(i)1614 1990 y FO(U)1680 1956 y FN(\003)1671 2011 y FM(j)1741 1990 y FP(=)23 b FO(U)1895 1956 y FN(\003)1886 2011 y FM(j)1933 1990 y FO(C)1998 1956 y FL(2)1992 2011 y FM(i)2035 1990 y FO(;)807 2125 y(C)866 2137 y FM(i)894 2125 y FO(C)953 2137 y FM(j)1011 2125 y FP(=)g FO(C)1158 2137 y FM(j)1193 2125 y FO(C)1252 2137 y FM(i)1280 2125 y FO(;)97 b(U)1457 2137 y FM(j)1492 2125 y FO(U)1549 2137 y FM(i)1599 2125 y FP(=)23 b FO(U)1744 2137 y FM(i)1771 2125 y FO(U)1828 2137 y FM(j)1863 2125 y FO(:)477 b FP(\(2.50\))118 2304 y(As)28 b(in)h(the)g(one-dimensional) e(case,)g(the)i(op)r(erator)e FO(U)1838 2274 y FN(\003)1829 2326 y FM(i)1904 2304 y FP(determines)h(the)h(ac-)118 2404 y(tion)g(on)f(the)h(sp)r(ectrum)g(of)g FO(C)1075 2373 y FL(2)1069 2425 y FM(i)1113 2404 y FP(,)g FO(x)1212 2416 y FM(i)1265 2404 y FQ(7!)c FP(1)19 b(+)f FO(q)s(x)1604 2416 y FM(i)1632 2404 y FP(.)41 b(The)29 b(action)f(of)h FO(U)2280 2373 y FN(\003)2271 2425 y FM(j)2318 2404 y FP(,)g FO(i)24 b FQ(6)p FP(=)h FO(j)5 b FP(,)118 2512 y(on)27 b FO(\033)s FP(\()p FO(C)380 2482 y FL(2)374 2534 y FM(i)419 2512 y FP(\))h(is)f(iden)n(tical,)h(b)r(ecause)f(of)g (the)h(relation)f FO(C)1834 2482 y FL(2)1828 2534 y FM(i)1871 2512 y FO(U)1928 2524 y FM(j)1986 2512 y FP(=)c FO(U)2131 2524 y FM(j)2165 2512 y FO(C)2230 2482 y FL(2)2224 2534 y FM(i)2268 2512 y FP(.)118 2660 y FR(3.)33 b FP(No)n(w)19 b(w)n(e)g(are)f(able)h(to)g(giv)n(e)f(a)h(classi\014cation)e(of)i(b)r (ounded)h(represen)n(tations)118 2760 y(up)28 b(to)g(unitary)f(equiv)-5 b(alence)27 b(using)g(the)h(one-dimensional)e(tec)n(hnique.)243 2860 y(T)-7 b(o)41 b(classify)h(b)r(ounded)h(irreducible)e(represen)n (tations,)j(one)e(m)n(ust)g(de-)118 2959 y(scrib)r(e)35 b(b)r(ounded)g(orbits)g(of)g(the)g(dynamical)g(system)g(on)f FJ(R)2072 2971 y FL(+)2169 2959 y FP(determined)118 3059 y(b)n(y)27 b(the)h(mapping)g FO(f)9 b FP(\()p FO(\025)p FP(\))23 b(=)g(1)18 b(+)g FO(q)s(\025)p FP(.)37 b(It)28 b(has)f(only)h(t)n(w)n(o)e(orbits,)220 3222 y(1.)41 b FQ(f)p FO(f)418 3192 y FM(n)462 3222 y FP(\(0\))p FO(;)h(n)23 b FQ(\025)f FP(0)p FQ(g)p FP(,)27 b(the)h(\\F)-7 b(o)r(c)n(k")26 b(orbit;)220 3386 y(2.)41 b(the)28 b(\014xed)f(p)r(oin)n(t)h FQ(f)p FP(1)p FO(=)p FP(\(1)17 b FQ(\000)h FO(q)s FP(\))p FQ(g)p FP(.)118 3550 y(It)29 b(is)g(easy)e(to)i(see)f(that)h(the)g(sp)r (ectral)f(pro)5 b(jection)28 b FO(E)1810 3568 y FM(C)1862 3548 y Fy(2)1858 3586 y(1)1899 3550 y FP(\()p FB(O)p FP(\),)h(where)f FB(O)h FP(is)f(an)118 3649 y(orbit)35 b(of)h(the)g(one-dimensional)e(dynamical)h(system)g(\()p FO(f)t(;)14 b FJ(R)2058 3661 y FL(+)2119 3649 y FP(\),)38 b(comm)n(utes)118 3749 y(with)e(all)e(op)r(erators)f(of)i(the)h (represen)n(tation.)57 b(Consequen)n(tly)-7 b(,)36 b(for)f(an)f(the)118 3848 y(irreducible)d(represen)n(tation,)h(the)g(sp)r(ectrum)g(of)g(the) h(op)r(erator)d FO(C)2270 3818 y FL(2)2264 3869 y(1)2339 3848 y FP(lies)i(on)118 3948 y(a)k(single)g(orbit)g(of)g(the)h (dynamical)e(system.)63 b(Moreo)n(v)n(er,)36 b(since)g FO(U)2321 3918 y FN(\003)2312 3969 y FL(1)2395 3948 y FP(is)g(an)118 4048 y(isometry)-7 b(,)21 b(the)g(sp)r(ectrum)f (coincides)g(with)h(one)f(of)g(the)g(t)n(w)n(o)g(orbits)f(presen)n(ted) 118 4147 y(ab)r(o)n(v)n(e.)p eop %%Page: 154 158 154 157 bop 118 100 a FP(154)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)243 333 y FP(Let)37 b(us)h(consider)e(the)i(case)e(where)h FO(\033)s FP(\()p FO(C)1586 303 y FL(2)1580 353 y(1)1625 333 y FP(\))i(=)g FQ(f)p FO(f)1892 303 y FM(n)1937 333 y FP(\(0\))p FO(;)28 b(n)39 b FQ(\025)g FP(0)p FQ(g)p FP(.)65 b(W)-7 b(e)118 432 y(ha)n(v)n(e:)203 771 y FO(U)269 737 y FN(\003)260 792 y FL(1)330 771 y FP(=)417 530 y Fz(0)417 676 y(B)417 726 y(B)417 775 y(B)417 829 y(@)490 593 y FP(0)490 693 y(1)82 b(0)614 793 y(1)111 b(0)744 889 y(.)776 914 y(.)808 940 y(.)923 889 y(.)956 914 y(.)988 940 y(.)1016 530 y Fz(1)1016 676 y(C)1016 726 y(C)1016 775 y(C)1016 829 y(A)1107 771 y FQ(\012)18 b FO(I)7 b(;)97 b(C)1418 737 y FL(2)1412 792 y(1)1478 771 y FP(=)1566 579 y Fz(0)1566 726 y(B)1566 779 y(@)1639 643 y FO(f)9 b FP(\(0\))1877 743 y FO(f)1927 713 y FL(2)1964 743 y FP(\(0\))2158 840 y(.)2190 864 y(.)2222 890 y(.)2250 579 y Fz(1)2250 726 y(C)2250 779 y(A)2341 771 y FQ(\012)18 b FO(I)7 b(:)118 1126 y FP(It)28 b(is)g(easy)e(to)i(deduce)f(from)h (the)g(comm)n(utation)f(relations)f(that)623 1309 y FO(C)688 1275 y FL(2)682 1330 y FM(i)749 1309 y FP(=)c(1)c FQ(\012)998 1288 y FP(^)979 1309 y FO(C)1044 1275 y FL(2)1038 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FO(\033)s FP(\()p FO(C)1468 1761 y FL(2)1462 1811 y(1)1506 1791 y FP(\))d(=)e FQ(f)p FP(1)p FO(=)p FP(\(1)17 b FQ(\000)h FO(q)s FP(\))p FQ(g)p FP(.)37 b(In)28 b(this)f(case,)617 1973 y FO(C)682 1939 y FL(2)676 1994 y(1)743 1973 y FP(=)22 b(\(1)d FQ(\000)f FO(q)s FP(\))1078 1939 y FN(\000)p FL(1)1167 1973 y FO(I)7 b(;)97 b(U)1387 1985 y FL(1)1447 1973 y FP(=)23 b FO(\025I)7 b(;)180 b FQ(j)p FO(\025)p FQ(j)24 b FP(=)e(1)118 2156 y(\(since)g FO(U)405 2168 y FL(1)464 2156 y FP(comm)n(utes)f(with)i(all)e(op)r (erators)f(of)i(the)g(represen)n(tation\).)34 b(There-)118 2256 y(fore,)e(in)f(this)h(case)e(w)n(e)h(deal)g(with)h(the)g(\\direct) e(pro)r(duct")h(of)g FO(d)21 b FQ(\000)g FP(1)31 b(copies)118 2355 y(of)c FO(q)s FP(-CCR)g(algebras.)34 b(Let)28 b(us)e(in)n(tro)r (duce,)h(for)g(con)n(v)n(enience,)f(the)h(op)r(erators)118 2455 y(on)g FO(l)258 2467 y FL(2)295 2455 y FP(\()p FJ(N)5 b FP(\))34 b(giv)n(en)26 b(b)n(y)376 2815 y FO(S)i FP(=)542 2573 y Fz(0)542 2719 y(B)542 2769 y(B)542 2819 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FO(k)j FQ(\024)d FO(d)p FQ(g)p FC(.)52 b(T)-6 b(o)35 b(e)l(ach)g(such)g(subset,)g(asso)l (ciate)g(the)g(fol)t(lowing)i(irr)l(e-)118 3635 y(ducible)31 b(r)l(epr)l(esentation)6 b FP(:)583 3829 y FO(C)648 3795 y FL(2)642 3850 y FM(i)709 3829 y FP(=)858 3751 y Fz(O)796 3929 y FM(j)s(<i;j)s FN(62)p FL(\010)1059 3829 y FO(I)26 b FQ(\012)18 b FO(C)24 b FQ(\012)1432 3751 y Fz(O)1370 3929 y FM(j)s(>i;j)s FN(62)p FL(\010)1633 3829 y FO(I)7 b(;)184 b(i)22 b FQ(62)i FP(\010)p FO(;)587 4069 y(U)653 4035 y FN(\003)644 4090 y FM(i)714 4069 y FP(=)864 3990 y Fz(O)802 4169 y FM(j)s(<i;j)s FN(62)p FL(\010)1065 4069 y FO(I)h FQ(\012)18 b FO(S)23 b FQ(\012)1428 3990 y Fz(O)1366 4169 y FM(j)s(>i;j)s FN(62)p FL(\010)1629 4069 y FO(I)7 b(;)184 b(i)22 b FQ(62)i FP(\010)p FO(;)p eop %%Page: 155 159 155 158 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(155)622 333 y FO(U)679 345 y FM(i)730 333 y FP(=)22 b FO(\025)865 345 y FM(i)893 333 y FO(I)7 b(;)99 b(C)1123 298 y FL(2)1117 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FM(j)959 4135 y FP(=)22 b FO(\026x)1143 4147 y FM(j)1179 4135 y FO(x)1226 4101 y FN(\003)1226 4155 y FM(i)1264 4135 y FO(;)28 b(\026)23 b FQ(2)h FP([0)p FO(;)14 b FP(1])p FO(;)26 b(i)d FQ(6)p FP(=)g FO(j)1862 4043 y Fz(E)1912 4135 y FO(:)428 b FP(\(2.51\))p eop %%Page: 156 160 156 159 bop 118 100 a FP(156)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)243 333 y FP(W)-7 b(e)40 b(sho)n(w)f(that,)k(as)c(in)h (the)g(previous)f(example,)j(the)e(additional)g(re-)118 432 y(lations)c FO(x)442 444 y FM(i)470 432 y FO(x)517 444 y FM(j)591 432 y FP(=)i FO(\026x)791 444 y FM(j)827 432 y FO(x)874 444 y FM(i)902 432 y FP(,)h FO(i)f(>)g(j)5 b FP(,)39 b(hold)e(automatically)e(for)h(an)n(y)g(b)r(ounded)118 532 y(represen)n(tation)26 b(of)i(the)g FO(\026)p FP(-CCR)f(algebra.) 118 689 y FR(Prop)s(osition)33 b(51.)41 b FC(L)l(et)32 b FO(\031)s FP(\()p FQ(\001)p FP(\))g FC(b)l(e)g(an)f(irr)l(e)l (ducible)i(b)l(ounde)l(d)f(r)l(epr)l(esentation)118 789 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1676 y FL(2)1996 1710 y FP(\))2081 1632 y Fz(X)2042 1808 y FM(r)r(<s<j)2255 1710 y FO(X)2324 1722 y FM(sr)2391 1710 y FO(X)2467 1676 y FN(\003)2460 1731 y FM(sr)2528 1710 y FO(:)118 1976 y FP(This)37 b(expression)e(sho) n(ws)h(that)h FO(X)1239 1946 y FN(\003)1232 1998 y FM(ij)1290 1976 y FO(X)1359 1988 y FM(ij)1454 1976 y FP(consists)f(of)h(the)g (term)g FO(\026)2283 1946 y FL(6)2320 1976 y FO(X)2389 1988 y FM(ij)2448 1976 y FO(X)2524 1946 y FN(\003)2517 1998 y FM(ij)118 2076 y FP(and)k(the)h(ones)f(that)h(ha)n(v)n(e)e(an)h (index)h(less)f(than)g(\()p FO(i;)14 b(j)5 b FP(\))42 b(with)g(resp)r(ect)f(to)118 2175 y(the)27 b(lexicographic)e(ordering.) 35 b(Then)26 b(the)h(induction)g(on)g(the)g(lexicographic)118 2275 y(ordering)17 b(and)h(the)h(fact)g(that)g(the)g(relation)f FO(x)1526 2245 y FN(\003)1564 2275 y FO(x)24 b FP(=)e FO(q)s(xx)1856 2245 y FN(\003)1914 2275 y FP(do)r(es)c(not)h(ha)n(v)n (e)e(non-)118 2375 y(trivial)24 b(b)r(ounded)h(represen)n(tations)d (\014nishes)i(the)h(pro)r(of)f(of)g(the)h(prop)r(osition.)p 2514 2474 4 57 v 2518 2422 50 4 v 2518 2474 V 2567 2474 4 57 v 243 2638 a(This)43 b(means)h(that,)k(as)43 b(far)g(as)g(b)r (ounded)h(represen)n(tations)e(are)h(con-)118 2738 y(cerned,)37 b(no)f(information)f(is)g(lost)h(if)g(the)g FO(\026)p FP(-CCR)g(relations)e(are)h(replaced)118 2837 y(with)28 b(the)g(complete)g(t)n(wisted)f(CCR)h(relations:)339 3021 y FO(x)386 2987 y FN(\003)386 3042 y FM(i)424 3021 y FO(x)471 3033 y FM(i)523 3021 y FP(=)22 b(1)c(+)g FO(\026)803 2987 y FL(2)840 3021 y FO(x)887 3033 y FM(i)916 3021 y FO(x)963 2987 y FN(\003)963 3042 y FM(i)1020 3021 y FQ(\000)g FP(\(1)g FQ(\000)g FO(\026)1328 2987 y FL(2)1365 3021 y FP(\))1411 2942 y Fz(X)1418 3119 y FM(j)s(<i)1545 3021 y FO(x)1592 3033 y FM(j)1627 3021 y FO(x)1674 2987 y FN(\003)1674 3042 y FM(j)1713 3021 y FO(;)180 b(i)23 b FP(=)f(1)p FO(;)14 b(:)g(:)g(:)f(;)h(d;)331 3242 y(x)378 3208 y FN(\003)378 3262 y FM(i)417 3242 y FO(x)464 3254 y FM(j)523 3242 y 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b(+)i FO(\026)1091 3895 y FL(2)1128 3930 y FO(C)1193 3895 y FL(2)1187 3950 y FM(i)1249 3930 y FP(+)1332 3851 y Fz(X)1336 4030 y FM(k)q(<i)1466 3930 y FO(C)1531 3895 y FL(2)1525 3950 y FM(k)1568 3838 y Fz(\021)1618 3930 y FO(;)180 b(i)22 b FP(=)h(1)p FO(;)14 b(:)g(:)g(:)f(;)h(d;)427 4147 y(C)492 4113 y FL(2)486 4168 y FM(i)529 4147 y FO(U)595 4113 y FN(\003)586 4168 y FM(j)656 4147 y FP(=)23 b FO(U)810 4113 y FN(\003)801 4168 y FM(j)848 4147 y FO(C)913 4113 y FL(2)907 4168 y FM(i)950 4147 y FO(;)180 b(i)23 b(<)g(j;)p eop %%Page: 157 161 157 160 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(157)427 333 y FO(C)492 298 y FL(2)486 353 y FM(i)529 333 y FO(U)595 298 y FN(\003)586 353 y FM(j)656 333 y FP(=)23 b FO(\026)794 298 y FL(2)831 333 y FO(U)897 298 y FN(\003)888 353 y FM(j)935 333 y FO(C)1000 298 y FL(2)994 353 y FM(i)1038 333 y FO(;)180 b(i)22 b(>)h(j:)926 b FP(\(2.52\))118 520 y(It)25 b(follo)n(ws)e(from)h (these)g(relations)g(that)g(all)g FO(U)1556 532 y FM(i)1608 520 y FP(are)f(co-isometries.)34 b(It)25 b(is)f(also)118 620 y(easy)j(to)h(see)g(that)g(the)g(isometry)f FO(U)1267 590 y FN(\003)1258 642 y FM(i)1333 620 y FP(acts)h(non-trivially)f(on)g (the)i(sp)r(ectrum)118 720 y(of)k FO(C)283 690 y FL(2)277 741 y FM(j)321 720 y FP(,)27 b FO(i)c(<)f(j)5 b FP(,)28 b(and)e(the)i(corresp)r(onding)d(dynamical)h(system)h(is)g(triangular.) 243 822 y(Let)33 b(us)g(classify)g(the)h(irreducible)f(represen)n (tations)e(of)i(these)h(op)r(erator)118 921 y(relations.)k(F)-7 b(or)28 b(an)g(irreducible)g(represen)n(tation,)f(the)i(sp)r(ectrum)f (of)h FO(C)2403 891 y FL(2)2397 942 y(1)2469 921 y FP(co-)118 1021 y(incides)f(with)g(one)f(of)h(the)g(t)n(w)n(o)e(orbits:)220 1195 y(1.)41 b FO(\033)s FP(\()p FO(C)473 1165 y FL(2)467 1215 y(1)511 1195 y FP(\))23 b(=)g FQ(f)p FO(f)746 1165 y FM(n)790 1195 y FP(\(0\))28 b FO(;)41 b(n)23 b(>)g FP(0)p FQ(g)p FP(;)220 1371 y(2.)41 b FO(\033)s FP(\()p FO(C)473 1341 y FL(2)467 1392 y(1)511 1371 y FP(\))23 b(=)g FQ(f)p FP(1)p FO(=)p FP(\(1)17 b FQ(\000)h FO(\026)1004 1341 y FL(2)1041 1371 y FP(\))p FQ(g)p FP(.)118 1545 y(Here)25 b FO(f)9 b FP(\()p FO(x)p FP(\))24 b(=)f(1)14 b(+)g FO(\026)770 1515 y FL(2)807 1545 y FO(x)p FP(,)26 b FO(x)e FQ(2)f FJ(R)p FP(.)43 b(First,)25 b(consider)g(the)h(case)f (where)g FO(\033)s FP(\()p FO(C)2417 1515 y FL(2)2411 1565 y(1)2455 1545 y FP(\))e(=)118 1644 y FQ(f)p FO(f)210 1614 y FM(n)254 1644 y FP(\(0\))p FO(;)28 b(n)f(>)f FP(0)p FQ(g)p FP(.)43 b(In)30 b(this)g(case)f(the)i(op)r(erators)c FO(C)1757 1614 y FL(2)1751 1665 y(1)1825 1644 y FP(and)j FO(U)2055 1614 y FN(\003)2046 1665 y FL(1)2122 1644 y FP(are)f(unitarily)118 1744 y(equiv)-5 b(alen)n(t)27 b(to)h(the)g(follo)n(wing)e(op)r(erators:)196 2104 y FO(C)261 2070 y FL(2)255 2125 y(1)321 2104 y FP(=)409 1912 y Fz(0)409 2058 y(B)409 2112 y(@)482 1976 y FO(f)9 b FP(\(0\))720 2076 y FO(f)770 2045 y FL(2)807 2076 y FP(\(0\))1001 2172 y(.)1033 2197 y(.)1065 2223 y(.)1093 1912 y Fz(1)1093 2058 y(C)1093 2112 y(A)1184 2104 y FQ(\012)18 b FO(I)7 b(;)97 b(U)1496 2070 y FN(\003)1487 2125 y FL(1)1557 2104 y FP(=)1645 1862 y Fz(0)1645 2009 y(B)1645 2058 y(B)1645 2108 y(B)1645 2161 y(@)1717 1926 y FP(0)1717 2026 y(1)83 b(0)1842 2125 y(1)110 b(0)1971 2222 y(.)2003 2247 y(.)2036 2272 y(.)2151 2222 y(.)2183 2247 y(.)2215 2272 y(.)2243 1862 y Fz(1)2243 2009 y(C)2243 2058 y(C)2243 2108 y(C)2243 2161 y(A)2334 2104 y FQ(\012)18 b FO(I)7 b(;)169 2591 y(C)234 2557 y FL(2)228 2612 y FM(i)294 2591 y FP(=)382 2349 y Fz(0)382 2496 y(B)382 2545 y(B)382 2595 y(B)382 2648 y(@)454 2413 y FP(1)579 2513 y FO(\026)629 2483 y FL(2)749 2612 y FO(\026)799 2582 y FL(4)924 2709 y FP(.)956 2734 y(.)989 2759 y(.)1016 2349 y Fz(1)1016 2496 y(C)1016 2545 y(C)1016 2595 y(C)1016 2648 y(A)1108 2591 y FQ(\012)1209 2570 y FP(^)1191 2591 y FO(C)1256 2557 y FL(2)1250 2612 y FM(i)1293 2591 y FO(;)97 b(U)1479 2557 y FN(\003)1470 2612 y FM(i)1540 2591 y FP(=)23 b FO(I)i FQ(\012)1786 2570 y FP(^)1772 2591 y FO(U)1838 2557 y FN(\003)1829 2612 y FM(i)1876 2591 y FO(;)180 b(i)23 b FP(=)f(2)p FO(;)14 b(:)g(:)g(:)f(;)h(d;)118 2972 y FP(where)26 b(the)h(families)f FQ(f)861 2951 y FP(^)842 2972 y FO(C)907 2942 y FL(2)901 2994 y FM(i)945 2972 y FQ(g)p FP(,)g FQ(f)1092 2951 y FP(^)1078 2972 y FO(U)1135 2984 y FM(i)1162 2972 y FQ(g)g FP(satisfy)g(the)h (relations)e(\(2.52\))h(with)h FO(d)16 b FQ(\000)g FP(1)118 3072 y(generators.)243 3174 y(Consider)28 b(no)n(w)i(the)g(case)f FO(\033)s FP(\()p FO(C)1237 3144 y FL(2)1231 3195 y(1)1275 3174 y FP(\))e(=)f FQ(f)p FP(1)p FO(=)p FP(\(1)17 b FQ(\000)h FO(\026)1775 3144 y FL(2)1812 3174 y FP(\))p FQ(g)p FP(.)44 b(Here)29 b FO(U)2208 3186 y FL(1)2275 3174 y FP(is)g(a)h(uni-)118 3274 y(tary)g(op)r(erator,)g(and)h(it)g(is)g(easy)f(to)h(deduce)g(from) g FO(C)1835 3244 y FL(2)1829 3295 y FM(i)1872 3274 y FO(U)1938 3244 y FN(\003)1929 3295 y FM(i)2005 3274 y FP(=)d FO(\026)2148 3244 y FL(2)2185 3274 y FO(U)2251 3244 y FN(\003)2242 3295 y FM(i)2289 3274 y FO(C)2354 3244 y FL(2)2348 3295 y FM(i)2423 3274 y FP(that)118 3373 y FO(C)183 3343 y FL(2)177 3395 y FM(i)244 3373 y FP(=)22 b(0)g(for)f(a)h(b)r(ounded)g(represen)n(tation.)33 b(This)22 b(follo)n(ws)f(from)g(the)i(fact)f(that)118 3473 y(if)35 b FO(U)258 3485 y FL(1)322 3473 y FP(is)28 b(unitary)-7 b(,)27 b(then)h FO(\033)s FP(\()p FO(C)1051 3443 y FL(2)1045 3495 y FM(i)1089 3473 y FP(\))g(is)g(in)n(v)-5 b(arian)n(t)26 b(under)h(m)n(ultiplication)h(b)n(y)f FO(\026)2514 3443 y FL(2)2552 3473 y FP(,)118 3573 y FO(\026)168 3542 y FN(\000)p FL(2)257 3573 y FP(.)51 b(Consequen)n(tly)-7 b(,)33 b(the)g(sp)r(ectrum)g(of)f FO(C)1548 3542 y FL(2)1542 3594 y FM(i)1585 3573 y FP(,)i FO(i)d FP(=)f(2,)i FO(:)14 b(:)g(:)28 b FP(,)33 b FO(d)p FP(,)h(is)e(b)r(ounded)118 3672 y(if)f(and)f(only)g(if)g FO(\033)s FP(\()p FO(C)771 3642 y FL(2)765 3694 y FM(i)810 3672 y FP(\))d(=)g FQ(f)p FP(0)p FQ(g)i FP(and,)i(since)f(k)n(er)12 b FO(U)1691 3684 y FM(i)1746 3672 y FP(=)27 b(k)n(er)13 b FO(C)2022 3684 y FM(i)2050 3672 y FP(,)31 b(w)n(e)f(ha)n(v)n(e)f (that)118 3772 y FO(U)175 3784 y FM(i)225 3772 y FP(=)23 b(0.)36 b(Then,)26 b(the)f(op)r(erator)e FO(U)1181 3784 y FL(1)1244 3772 y FP(is)i(in)g(the)g(cen)n(ter)g(of)g(the)h(represen)n (tation,)118 3871 y(and)i FO(U)346 3841 y FN(\003)337 3892 y FL(1)407 3871 y FP(=)22 b FO(\025I)7 b FP(,)28 b FQ(j)p FO(\025)p FQ(j)c FP(=)f(1.)243 3974 y(T)-7 b(o)27 b(obtain)g(the)h(\014nal)g(result,)f(w)n(e)g(com)n(bine)g(these)h(t)n (w)n(o)f(cases.)118 4147 y FR(Prop)s(osition)45 b(52.)j FC(Fix)41 b(a)g(numb)l(er)g FO(i)p FC(,)j FP(1)f FQ(\024)g FO(i)h FQ(\024)f FO(d)p FC(,)i(and)c(c)l(onsider)i(the)p eop %%Page: 158 162 158 161 bop 118 100 a FP(158)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(fol)t(lowing)32 b(r)l(epr)l(esentation)e(of)h (the)f FO(\026)p FC(-CCR)g(algebr)l(a)6 b FP(:)571 578 y FO(C)636 544 y FL(2)630 599 y FM(j)697 578 y FP(=)789 471 y FM(j)s FN(\000)p FL(1)784 500 y Fz(O)787 678 y FM(k)q FL(=1)924 578 y FO(d)p FP(\()p FO(\026)p FP(\))19 b FQ(\012)f FO(d)p FP(\()p FO(f)9 b FP(\))19 b FQ(\012)1489 475 y FM(i)p FN(\000)p FL(1)1480 500 y Fz(O)1442 678 y FM(k)q FL(=)p FM(j)s FL(+1)1658 578 y FO(I)7 b(;)184 b(j)28 b(<)23 b(i;)570 886 y(U)636 851 y FN(\003)627 906 y FM(j)697 886 y FP(=)789 778 y FM(j)s FN(\000)p FL(1)784 807 y Fz(O)787 985 y FM(k)q FL(=1)924 886 y FO(I)i FQ(\012)18 b FO(S)23 b FQ(\012)1272 782 y FM(i)p FN(\000)p FL(1)1264 807 y Fz(O)1225 985 y FM(k)q FL(=)p FM(j)s FL(+1)1442 886 y FO(I)7 b(;)183 b(j)28 b(<)23 b(i;)571 1189 y(C)636 1155 y FL(2)630 1209 y FM(i)697 1189 y FP(=)889 1133 y(1)p 794 1170 231 4 v 794 1246 a(1)18 b FQ(\000)g FO(\026)987 1222 y FL(2)1057 1085 y FM(i)p FN(\000)p FL(1)1048 1110 y Fz(O)1051 1289 y FM(k)q FL(=1)1188 1189 y FO(d)p FP(\()p FO(\026)p FP(\))p FO(;)589 1400 y(U)646 1412 y FM(i)697 1400 y FP(=)k FO(\025)832 1412 y FM(i)861 1400 y FO(I)7 b(;)98 b(C)1084 1412 y FM(j)1143 1400 y FP(=)22 b FO(U)1287 1412 y FM(j)1345 1400 y FP(=)h(0)p FO(;)183 b(j)28 b(>)22 b(i;)118 1566 y FC(wher)l(e)253 1889 y FO(d)p FP(\()p FO(\026)p FP(\))i(=)521 1648 y Fz(0)521 1794 y(B)521 1844 y(B)521 1894 y(B)521 1947 y(@)594 1711 y FP(1)718 1811 y FO(\026)768 1781 y FL(2)889 1911 y FO(\026)939 1881 y FL(4)1064 2008 y FC(.)1098 2032 y(.)1133 2058 y(.)1163 1648 y Fz(1)1163 1794 y(C)1163 1844 y(C)1163 1894 y(C)1163 1947 y(A)1250 1889 y FO(;)98 b(d)p FP(\()p FO(f)9 b FP(\))24 b(=)1639 1698 y Fz(0)1639 1844 y(B)1639 1897 y(@)1712 1761 y FO(f)9 b FP(\(0\))1951 1861 y FO(f)2001 1831 y FL(2)2037 1861 y FP(\(0\))2231 1958 y FC(.)2266 1983 y(.)2301 2008 y(.)2331 1698 y Fz(1)2331 1844 y(C)2331 1897 y(A)2417 1889 y FO(;)118 2233 y(S)35 b FC(is)30 b(the)g(unilater)l(al)g(shift)g(in)g FO(l)1112 2245 y FL(2)1149 2233 y FP(\()p FJ(N)t FP(\))q FC(,)36 b(and)30 b FO(\025)1538 2245 y FM(i)1589 2233 y FQ(2)24 b FJ(C)15 b FC(,)36 b FQ(j)p FO(\025)1854 2245 y FM(i)1882 2233 y FQ(j)23 b FP(=)g(1)p FC(.)243 2333 y(Then)34 b(al)t(l)i(irr)l(e)l(ducible)f(r)l(epr)l(esentations)g(of)g (the)f FO(\026)p FC(-CCR)g(algebr)l(a)i(c)l(oin-)118 2432 y(cide)d(with)g(the)f(ones)g(c)l(onstructe)l(d)f(for)i(some)f FO(i)p FC(.)45 b(Mor)l(e)l(over,)34 b(two)e(r)l(epr)l(esen-)118 2532 y(tations)37 b(ar)l(e)f(e)l(quivalent)h(if)h(and)f(only)g(if)g (they)g(c)l(orr)l(esp)l(ond)g(to)g(the)f(same)h FO(i)118 2632 y FC(and)30 b FO(\025)327 2644 y FM(i)355 2632 y FC(.)118 2785 y FR(3.)36 b FO(\026)p FP(-CAR)26 b(algebra.)34 b(No)n(w)26 b(consider)e(the)j(an)n(ticomm)n(utativ)n(e)d(case.)36 b(By)25 b(the)118 2884 y FO(\026)p FP(-CAR)j(algebra)e(w)n(e)h(call)g (the)h(follo)n(wing)f FQ(\003)p FP(-algebra:)266 3071 y FJ(C)320 2979 y Fz(D)377 3071 y FO(x)424 3083 y FM(i)452 3071 y FO(;)14 b(x)536 3037 y FN(\003)536 3092 y FM(i)597 3071 y FQ(j)24 b FO(x)691 3037 y FN(\003)691 3092 y FM(i)729 3071 y FO(x)776 3083 y FM(i)827 3071 y FP(=)f(1)18 b FQ(\000)g FO(x)1105 3083 y FM(i)1133 3071 y FO(x)1180 3037 y FN(\003)1180 3092 y FM(i)1237 3071 y FQ(\000)g FP(\(1)g FQ(\000)g FO(\026)1545 3037 y FL(2)1583 3071 y FP(\))1629 2992 y Fz(X)1636 3169 y FM(j)s(<i)1763 3071 y FO(x)1810 3083 y FM(j)1845 3071 y FO(x)1892 3037 y FN(\003)1892 3092 y FM(j)1931 3071 y FO(;)27 b(i)c FP(=)g(1)p FO(;)14 b(:)g(:)g(:)f(;)h(d;)804 3317 y(x)851 3283 y FN(\003)851 3338 y FM(i)890 3317 y FO(x)937 3329 y FM(j)995 3317 y FP(=)23 b FQ(\000)p FO(\026x)1245 3329 y FM(j)1280 3317 y FO(x)1327 3283 y FN(\003)1327 3338 y FM(i)1366 3317 y FO(;)k(i)c FQ(6)p FP(=)g FO(j)1595 3225 y Fz(E)1645 3317 y FO(:)118 3509 y FC(R)l(emark)39 b FP(38)p FC(.)j FP(Let)28 b(us)g(note)g(that)h(an)n(y)e(represen)n(tation)g FO(\031)s FP(\()p FQ(\001)p FP(\))i(of)f(the)h FO(\026)p FP(-CAR)118 3609 y(algebra)d(is)h(b)r(ounded,)h(since)g FO(\031)s FP(\()p FO(x)1187 3578 y FN(\003)1187 3630 y FM(i)1226 3609 y FO(x)1273 3621 y FM(i)1301 3609 y FP(\))c FQ(\025)e FP(0)27 b(for)g(an)n(y)g FO(i)p FP(,)h(and,)f (therefore,)633 3793 y FR(1)18 b FQ(\000)g FO(\031)s FP(\()p FO(x)911 3805 y FM(i)940 3793 y FO(x)987 3758 y FN(\003)987 3813 y FM(i)1026 3793 y FP(\))g FQ(\000)g FP(\(1)h FQ(\000)f FO(\026)1385 3758 y FL(2)1422 3793 y FP(\))1468 3714 y Fz(X)1475 3891 y FM(j)s(<i)1602 3793 y FO(\031)s FP(\()p FO(x)1731 3805 y FM(j)1767 3793 y FO(x)1814 3758 y FN(\003)1814 3813 y FM(j)1852 3793 y FP(\))24 b FQ(\025)e FP(0)p FO(;)118 4048 y FP(whic)n(h)33 b(implies)g(that)g FQ(k)p FO(\031)s FP(\()p FO(x)1004 4060 y FM(i)1032 4048 y FO(x)1079 4018 y FN(\003)1079 4069 y FM(i)1118 4048 y FP(\))p FQ(k)e(\024)g FP(1.)52 b(Then)33 b FO(\031)s FP(\()p FO(x)1787 4060 y FM(i)1816 4048 y FP(\))g(is)f(b)r(ounded)h(for)g(an)n(y)118 4147 y FO(i)23 b FP(=)f(1,)28 b FO(:)14 b(:)g(:)27 b FP(,)h FO(d)p FP(.)p eop %%Page: 159 163 159 162 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(159)243 333 y(T)-7 b(o)31 b(reduce)h(the)g FO(\026)p FP(-CAR)g(algebra)e(to)i(a)f(dynamical)g (form,)i(w)n(e)e(ha)n(v)n(e)g(to)118 432 y(\014nd)j(additional)e (relations)g(for)h(the)g(generators)e FO(x)1766 444 y FM(i)1794 432 y FP(,)j FO(x)1898 444 y FM(j)1967 432 y FP(so)e(that)i(these)f(re-)118 532 y(lations)f(w)n(ould)h(b)r(e)g (compatible)g(with)g(the)g(de\014ning)g(relations.)52 b(F)-7 b(or)32 b(Wic)n(k)118 632 y(algebras)e(suc)n(h)i(relations)f (are)h(describ)r(ed)g(b)n(y)g(Wic)n(k)g(ideals,)h(in)f(particular,)118 731 y(quadratic)40 b(Wic)n(k)h(ideals)g(\(see)g(Section)h(2.4.6\).)77 b(The)41 b(largest)f(quadratic)118 831 y(Wic)n(k)27 b(ideal)g(of)f(the) i FO(\026)p FP(-CAR)f(algebra)e(is)i(generated)e(b)n(y)i(the)g(follo)n (wing)f(fam-)118 930 y(ily)i(of)f(elemen)n(ts:)318 1093 y FO(x)365 1105 y FM(i)393 1093 y FO(x)440 1105 y FM(j)494 1093 y FP(+)18 b FO(\026x)674 1105 y FM(j)709 1093 y FO(x)756 1105 y FM(i)784 1093 y FO(;)97 b FP(1)23 b FQ(\024)f FO(j)28 b(<)23 b(i)g FQ(\024)f FO(d;)97 b FP(and)83 b FO(x)1772 1059 y FL(2)1772 1114 y FM(i)1810 1093 y FO(;)97 b(i)22 b FP(=)h(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(d:)118 1256 y FP(According)24 b(to)h(the)h(previous)e(example,)h(w)n(e)g(ha)n (v)n(e)f(to)h(sho)n(w)f(that)i(the)f(gener-)118 1355 y(ators)31 b(of)h(this)h(ideal)f(are)f(annihilated)h(in)h(an)n(y)e (irreducible)h(represen)n(tation)118 1455 y(of)23 b(the)h FO(\026)p FP(-CAR)f(algebra.)34 b(Ho)n(w)n(ev)n(er,)22 b(this)i(is)f(not)g(true)g(for)g(the)h FO(\026)p FP(-CAR)f(alge-)118 1555 y(bra.)42 b(The)30 b(largest)e(quadratic)g(ideal)i(is)f(v)n(ery)f (large.)42 b(So,)29 b(consider)g(another)118 1654 y(quadratic)e(Wic)n (k)g(ideal,)318 1795 y(^)316 1817 y FB(I)352 1829 y FL(2)412 1817 y FP(=)499 1750 y Fz(\012)539 1817 y FO(x)586 1829 y FM(i)614 1817 y FO(x)661 1829 y FM(j)715 1817 y FP(+)18 b FO(\026x)895 1829 y FM(j)930 1817 y FO(x)977 1829 y FM(i)1005 1817 y FO(;)28 b FP(1)23 b FQ(\024)f FO(j)28 b(<)23 b(i)f FQ(\024)h FO(d;)14 b FP(;)g FO(x)1661 1783 y FL(2)1661 1838 y FM(i)1699 1817 y FO(;)28 b(i)22 b FP(=)h(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(d)k FQ(\000)g FP(1)2315 1750 y Fz(\013)2354 1817 y FO(:)118 1980 y FR(Theorem)26 b(36.)37 b FC(F)-6 b(or)26 b(any)g(irr)l(e)l(ducible)h(r) l(epr)l(esentation)g FO(\031)s FP(\()p FQ(\001)p FP(\))f FC(of)h(the)f FO(\026)p FC(-CAR)118 2089 y(algebr)l(a)31 b FO(\031)s FP(\()483 2066 y(^)481 2089 y FB(I)517 2101 y FL(2)555 2089 y FP(\))23 b(=)g FQ(f)p FP(0)p FQ(g)28 b FC(holds.)118 2239 y(Pr)l(o)l(of.)43 b FP(Let)28 b(us)f(denote)g FO(X)969 2251 y FM(i)1019 2239 y FP(=)c FO(\031)s FP(\()p FO(x)1236 2251 y FM(i)1265 2239 y FP(\),)k FO(A)d FP(=)e FO(\031)s FP(\()p FO(x)1649 2209 y FL(2)1649 2260 y(1)1688 2239 y FP(\),)27 b FO(B)h FP(=)22 b FO(\031)s FP(\()p FO(x)2077 2251 y FL(2)2115 2239 y FO(x)2162 2251 y FL(1)2218 2239 y FP(+)17 b FO(\026x)2397 2251 y FL(1)2435 2239 y FO(x)2482 2251 y FL(2)2519 2239 y FP(\).)118 2339 y(No)n(w)31 b(w)n(e)f(pro)n(v)n(e)g(that)h FO(A)e FP(=)g FO(B)k FP(=)28 b(0.)47 b(It)32 b(is)f(easy)f(to)h(see)g(from)f(the)i(de\014ning)118 2438 y(relations)27 b(that)547 2601 y FO(A)609 2567 y FN(\003)648 2601 y FO(A)c FP(=)g FO(AA)945 2567 y FN(\003)983 2601 y FO(;)97 b(A)1165 2567 y FN(\003)1204 2601 y FO(X)1273 2613 y FM(k)1337 2601 y FP(=)22 b FO(\026)1474 2567 y FL(2)1511 2601 y FO(X)1580 2613 y FM(k)1621 2601 y FO(A)1683 2567 y FN(\003)1721 2601 y FO(;)180 b(k)26 b(>)d FP(1)p FO(:)217 b FP(\(2.53\))118 2764 y(Since)29 b FO(A)h FP(is)e(a)h(normal) f(op)r(erator,)f(w)n(e)i(can)g(use)f(the)i(F)-7 b(uglede{Putnam)28 b(the-)118 2863 y(orem)f(and)g(obtain)h(the)g(follo)n(wing)e (relations:)864 3026 y FO(AX)995 3038 y FM(k)1059 3026 y FP(=)c FO(\026)1196 2992 y FL(2)1233 3026 y FO(X)1302 3038 y FM(k)1343 3026 y FO(A;)180 b(k)26 b(>)d FP(1)p FO(:)118 3189 y FP(\(It)29 b(is)e(ob)n(vious)g(that)h FO(AX)936 3201 y FL(1)997 3189 y FP(=)23 b FO(X)1154 3201 y FL(1)1191 3189 y FO(A)p FP(\).)39 b(Th)n(us,)28 b(in)g(an)n(y)f(irreducible)g(represen-)118 3288 y(tation,)34 b(either)f FO(A)g FP(=)e(0)i(or)f(k)n(er)13 b FO(A)32 b FP(=)g FQ(f)p FP(0)p FQ(g)f FP(\(b)r(ecause)i(k)n(er)13 b FO(A)33 b FP(is)g(an)g(in)n(v)-5 b(arian)n(t)118 3388 y(subspace\).)37 b(Let)27 b(k)n(er)13 b FO(A)23 b FP(=)g FQ(f)p FP(0)p FQ(g)p FP(.)35 b(Then)28 b(w)n(e)g(ha)n(v)n(e:)630 3551 y FO(B)697 3517 y FN(\003)735 3551 y FO(B)f FP(=)c FO(\026)963 3517 y FL(2)1000 3551 y FO(B)t(B)1134 3517 y FN(\003)1191 3551 y FP(+)18 b(\(1)g FQ(\000)g FO(\026)1499 3517 y FL(4)1537 3551 y FP(\)\(1)g(+)g FO(\026)1794 3517 y FL(2)1831 3551 y FP(\))p FO(AA)1987 3517 y FN(\003)2026 3551 y FO(;)780 3686 y(A)842 3651 y FN(\003)880 3686 y FO(B)28 b FP(=)22 b FO(\026)1108 3651 y FL(2)1146 3686 y FO(B)t(A)1275 3651 y FN(\003)1313 3686 y FO(;)97 b(AB)28 b FP(=)22 b FO(\026)1723 3651 y FL(2)1760 3686 y FO(B)t(A:)451 b FP(\(2.54\))118 3848 y(It)38 b(follo)n(ws)f(from)h(the)g(condition)f (k)n(er)13 b FO(A)40 b FP(=)g FQ(f)p FP(0)p FQ(g)d FP(that)h FO(AA)2044 3818 y FN(\003)2122 3848 y FO(>)i FP(0.)67 b(Equa-)118 3948 y(tion)26 b(\(2.4.2\))f(implies)h(that)g FO(B)1073 3918 y FN(\003)1111 3948 y FO(B)h(>)c FP(0.)36 b(Let)25 b(us)h(consider)f(the)h(p)r(olar)f(decom-)118 4048 y(p)r(osition)33 b FO(B)507 4018 y FN(\003)577 4048 y FP(=)e FO(W)12 b(T)g FP(,)34 b(k)n(er)12 b FO(W)44 b FP(=)32 b(k)n(er)12 b FO(T)g FP(,)34 b FO(T)43 b FQ(\025)31 b FP(0,)j FO(T)1814 4018 y FL(2)1882 4048 y FP(=)d FO(B)t(B)2112 4018 y FN(\003)2151 4048 y FP(,)j(and,)g(since)118 4147 y FO(B)185 4117 y FN(\003)223 4147 y FO(B)28 b(>)22 b FP(0,)27 b(w)n(e)h(ha)n(v)n(e)e(that)i FO(W)40 b FP(is)27 b(a)g(co-isometry)-7 b(.)p eop %%Page: 160 164 160 163 bop 118 100 a FP(160)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)243 333 y FP(No)n(w)g(w)n(e)g(can)g(rewrite)g(relations)f (\(2.4.2\))h(in)h(an)f(equiv)-5 b(alen)n(t)28 b(form:)539 506 y FO(T)600 472 y FL(2)636 506 y FO(W)726 472 y FN(\003)787 506 y FP(=)23 b FO(W)965 472 y FN(\003)1003 439 y Fz(\000)1041 506 y FO(\026)1091 472 y FL(2)1128 506 y FO(T)1189 472 y FL(2)1244 506 y FP(+)18 b(\(1)g FQ(\000)g FO(\026)1552 472 y FL(4)1589 506 y FP(\)\(1)h(+)f FO(\026)1847 472 y FL(2)1884 506 y FP(\))p FO(AA)2040 472 y FN(\003)2079 439 y Fz(\001)2117 506 y FO(;)808 641 y(AW)960 606 y FN(\003)1021 641 y FP(=)23 b FO(\026)1159 606 y FL(2)1196 641 y FO(W)1286 606 y FN(\003)1324 641 y FO(A;)98 b(AT)34 b FP(=)23 b FO(T)12 b(A:)477 b FP(\(2.55\))118 814 y(These)31 b(equalities)g(and)g(relations)e(\(2.53\))i(imply)g(that)h(the)f(sp)r (ectrum)g(of)h FO(A)118 913 y FP(coincides)21 b(with)h(the)g(set)f FQ(f)p FO(\025\026)1046 883 y FL(2)p FM(n)1125 913 y FO(x)1172 925 y FL(1)1209 913 y FO(;)28 b(n)23 b FQ(2)g FJ(N)10 b FQ([)d(f)p FP(0)p FQ(gg)25 b FP(for)c(some)g FO(\025)j FQ(2)f FJ(C)15 b FP(,)29 b FQ(j)p FO(\025)p FQ(j)24 b FP(=)e(1,)118 1013 y FO(x)165 1025 y FL(1)228 1013 y FO(>)k FP(0.)41 b(Since)29 b FO(W)733 983 y FN(\003)800 1013 y FP(is)g(an)g(isometry)-7 b(,)29 b(the)h(eigen)n(v)-5 b(alues)28 b(of)h(the)g(op)r(erator)f FO(A)118 1113 y FP(ha)n(v)n(e)f(the)i(same)e(m)n(ultiplicit)n(y)-7 b(,)29 b(and)f(c)n(ho)r(osing)f(the)h(corresp)r(onding)e(basis)i(in)118 1212 y(the)e(represen)n(tation)d(space)h(w)n(e)h(can)g(write)g(the)g (op)r(erators)f FO(A)h FP(and)g FO(W)37 b FP(in)25 b(the)118 1312 y(follo)n(wing)i(form:)314 1608 y FO(A)c FP(=)487 1416 y Fz(0)487 1562 y(B)487 1615 y(@)560 1480 y FO(\025x)655 1492 y FL(1)693 1480 y FO(I)819 1579 y(\025x)914 1591 y FL(1)952 1579 y FO(\026)1002 1549 y FL(2)1039 1579 y FO(I)1170 1676 y FP(.)1202 1701 y(.)1235 1726 y(.)1262 1416 y Fz(1)1262 1562 y(C)1262 1615 y(A)1349 1608 y FO(;)97 b(W)35 b FP(=)1669 1416 y Fz(0)1669 1562 y(B)1669 1615 y(@)1742 1480 y FP(0)82 b FO(I)1867 1579 y FP(0)110 b FO(I)1997 1676 y FP(.)2029 1701 y(.)2062 1726 y(.)2177 1676 y(.)2209 1701 y(.)2242 1726 y(.)2269 1416 y Fz(1)2269 1562 y(C)2269 1615 y(A)2356 1608 y FO(:)118 1904 y FP(The)28 b(condition)f FO(T)12 b(A)23 b FP(=)f FO(AT)39 b FP(then)28 b(giv)n(es)f(that)868 2249 y FO(T)34 b FP(=)1039 2008 y Fz(0)1039 2154 y(B)1039 2204 y(B)1039 2253 y(B)1039 2307 y(@)1112 2071 y FO(T)1161 2083 y FL(0)1281 2171 y FO(T)1330 2183 y FL(1)1449 2271 y FO(T)1498 2283 y FL(2)1623 2367 y FP(.)1655 2392 y(.)1687 2418 y(.)1715 2008 y Fz(1)1715 2154 y(C)1715 2204 y(C)1715 2253 y(C)1715 2307 y(A)1801 2249 y FO(:)118 2595 y FP(Since)e(k)n(er)13 b FO(W)41 b FP(=)30 b(k)n(er)12 b FO(T)g FP(,)32 b(w)n(e)f(ha)n(v)n(e)g (that)h FO(T)1473 2607 y FL(0)1539 2595 y FP(=)d(0.)49 b(Moreo)n(v)n(er,)30 b(it)i(is)f(easy)g(to)118 2695 y(obtain)c(from)h (\(2.4.2\))f(that)516 2868 y FO(T)565 2880 y FM(n)633 2868 y FP(=)c FO(x)768 2880 y FL(1)819 2868 y FP(\(1)c(+)f FO(\026)1045 2833 y FL(2)1082 2868 y FP(\))c FO(\026)1178 2833 y FM(n)p FN(\000)p FL(1)1308 2868 y FP(\(1)k FQ(\000)g FO(\026)1533 2833 y FL(2)p FM(n)1612 2868 y FP(\))1644 2833 y FL(1)p FM(=)p FL(2)1748 2868 y FO(;)180 b(n)23 b FQ(\025)g FP(1)p FO(:)118 3041 y FP(Since)28 b FO(X)404 3053 y FL(1)441 3041 y FO(A)23 b FP(=)g FO(AX)745 3053 y FL(1)782 3041 y FP(,)28 b FO(X)909 3011 y FN(\003)902 3062 y FL(1)946 3041 y FO(A)c FP(=)e FO(AX)1257 3011 y FN(\003)1250 3062 y FL(1)1295 3041 y FP(,)28 b(w)n(e)f(get)943 3341 y FO(X)1012 3353 y FL(1)1072 3341 y FP(=)1159 3149 y Fz(0)1159 3296 y(B)1159 3349 y(@)1232 3213 y FO(b)1268 3225 y FL(1)1388 3313 y FO(b)1424 3325 y FL(2)1548 3409 y FP(.)1581 3434 y(.)1613 3460 y(.)1641 3149 y Fz(1)1641 3296 y(C)1641 3349 y(A)1727 3341 y FO(:)118 3637 y FP(Let)h(us)f(write) h(the)g(op)r(erator)e FO(B)h FP(=)22 b FO(T)12 b(W)1392 3607 y FN(\003)1457 3637 y FP(in)28 b(the)g(matrix)f(form,)859 3983 y FO(B)g FP(=)1037 3741 y Fz(0)1037 3887 y(B)1037 3937 y(B)1037 3987 y(B)1037 4040 y(@)1132 3805 y FP(0)1110 3904 y FO(T)1159 3916 y FL(1)1301 3904 y FP(0)1279 4004 y FO(T)1328 4016 y FL(2)1475 4004 y FP(0)1452 4101 y(.)1484 4126 y(.)1517 4151 y(.)1632 4101 y(.)1664 4126 y(.)1696 4151 y(.)1724 3741 y Fz(1)1724 3887 y(C)1724 3937 y(C)1724 3987 y(C)1724 4040 y(A)1811 3983 y FO(:)p eop %%Page: 161 165 161 164 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(161)118 333 y(The)27 b(condition)g FO(X)728 303 y FN(\003)721 353 y FL(1)765 333 y FO(B)g FP(=)c FO(\026B)t(X)1136 303 y FN(\003)1129 353 y FL(1)1200 333 y FP(expressed)j(in)h(terms)g(of)g(the)g(matrix)f(co)r(ef-)118 432 y(\014cien)n(ts)i(tak)n(es)e(the)i(follo)n(wing)f(form:)1031 613 y FO(b)1067 579 y FN(\003)1067 634 y FM(n)p FL(+1)1196 613 y FO(T)1245 625 y FM(n)1313 613 y FP(=)22 b FO(\026)14 b(T)1513 625 y FM(n)1558 613 y FO(b)1594 579 y FN(\003)1594 634 y FM(n)1639 613 y FO(:)118 794 y FP(Let)31 b(us)g(note)f(that)h FO(x)797 806 y FL(1)863 794 y FQ(6)p FP(=)d(0,)j(hence,)h FO(T)1359 806 y FM(n)1432 794 y FQ(6)p FP(=)c(0,)j(and)f(w)n(e)h(ha)n (v)n(e)e FO(b)2141 806 y FM(k)2210 794 y FP(=)f FO(\026)2353 764 y FM(k)q FN(\000)p FL(1)2479 794 y FO(b)2515 806 y FL(1)2552 794 y FP(.)118 894 y(F)-7 b(rom)27 b(the)h(relation)f FO(x)831 864 y FN(\003)831 914 y FL(1)870 894 y FO(x)917 906 y FL(1)977 894 y FP(=)c(1)18 b FQ(\000)g FO(x)1255 906 y FL(1)1293 894 y FO(x)1340 864 y FN(\003)1340 914 y FL(1)1406 894 y FP(it)28 b(follo)n(ws)e(that)641 1075 y FO(b)677 1040 y FN(\003)677 1095 y FL(1)714 1075 y FO(b)750 1087 y FL(1)810 1075 y FP(=)d(1)18 b FQ(\000)g FO(b)1077 1087 y FL(1)1114 1075 y FO(b)1150 1040 y FN(\003)1150 1095 y FL(1)1188 1075 y FO(;)97 b(\026)1358 1040 y FL(2)1395 1075 y FO(b)1431 1087 y FL(1)1468 1075 y FO(b)1504 1040 y FN(\003)1504 1095 y FL(1)1565 1075 y FP(=)22 b(1)c FQ(\000)g FO(\026)1845 1040 y FL(2)1883 1075 y FO(b)1919 1087 y FL(1)1955 1075 y FO(b)1991 1040 y FN(\003)1991 1095 y FL(1)2029 1075 y FO(:)118 1256 y FP(These)31 b(equalities)h(are) e(compatible)i(if)g(and)f(only)h(if)g FO(\026)1867 1225 y FL(2)1934 1256 y FP(=)d(1,)k(whic)n(h)e(is)h(im-)118 1355 y(p)r(ossible.)67 b(Hence,)40 b FO(x)822 1367 y FL(1)900 1355 y FP(=)f(0,)h FO(A)g FP(=)f(0,)h(and)e(relation)e (\(2.4.2\))h(yields)h(that)118 1455 y FO(B)185 1425 y FN(\003)223 1455 y FO(B)28 b FP(=)22 b FO(\026)451 1425 y FL(2)489 1455 y FO(B)t(B)623 1425 y FN(\003)661 1455 y FP(.)37 b(Since)28 b FO(B)k FP(is)27 b(a)g(b)r(ounded)h(op)r(erator,) e FO(B)h FP(=)c(0.)243 1554 y(Let)37 b(us)g(denote)g FO(B)857 1566 y FM(k)937 1554 y FP(=)h FO(\031)s FP(\()p FO(x)1169 1566 y FM(k)1211 1554 y FO(x)1258 1566 y FL(1)1321 1554 y FP(+)24 b FO(\026x)1507 1566 y FL(1)1545 1554 y FO(x)1592 1566 y FM(k)1633 1554 y FP(\),)40 b(2)e FO(<)h(k)j FQ(\024)c FO(d)p FP(.)66 b(F)-7 b(rom)36 b(the)118 1654 y(de\014ning)28 b(relations)e(w)n(e)h(obtain:)148 1847 y FO(B)215 1813 y FN(\003)211 1868 y FM(k)253 1847 y FO(B)316 1859 y FM(k)380 1847 y FP(=)c FO(\026)518 1813 y FL(2)555 1847 y FO(B)618 1859 y FM(k)659 1847 y FO(B)726 1813 y FN(\003)722 1868 y FM(k)782 1847 y FP(+)18 b FO(\026)915 1813 y FL(2)953 1847 y FP(\(1)g FQ(\000)g FO(\026)1178 1813 y FL(2)1215 1847 y FP(\))1300 1768 y Fz(X)1261 1947 y FL(1)p FM(<i<k)1472 1847 y FO(B)1535 1859 y FM(i)1563 1847 y FO(B)1630 1813 y FN(\003)1626 1868 y FM(i)1687 1847 y FP(+)g(\(1)g(+)g FO(\026)1995 1813 y FL(2)2032 1847 y FP(\)\(1)h FQ(\000)f FO(\026)2290 1813 y FL(4)2327 1847 y FP(\))p FO(AA)2483 1813 y FN(\003)2522 1847 y FO(:)118 2107 y FP(It)k(is)g(easy)f(to)g(see)g(that)h(the)h(induction)f (in)g FO(d)g FP(giv)n(es)e(that)i FO(B)1945 2119 y FM(k)2009 2107 y FP(=)h(0,)f FO(k)k FP(=)d(3,)e FO(:)14 b(:)g(:)28 b FP(,)118 2206 y FO(d)p FP(.)37 b(Th)n(us)25 b(the)i(op)r(erators)d FO(X)1006 2218 y FM(i)1059 2206 y FP(can)i(b)r(e)g(decomp)r(osed)f(in)h (the)h(tensor)e(pro)r(duct,)357 2433 y FO(X)426 2445 y FL(1)486 2433 y FP(=)573 2315 y Fz(\022)634 2382 y FP(0)83 b(0)634 2482 y(1)g(0)800 2315 y Fz(\023)880 2433 y FQ(\012)18 b FO(I)7 b(;)97 b(X)1195 2445 y FM(k)1259 2433 y FP(=)1346 2315 y Fz(\022)1407 2382 y FP(1)120 b(0)1407 2482 y(0)83 b FQ(\000)p FO(\026)1647 2315 y Fz(\023)1726 2433 y FQ(\012)1831 2412 y FP(~)1809 2433 y FO(A)1871 2445 y FM(k)1912 2433 y FO(;)180 b(k)26 b(>)d FP(1)p FO(;)118 2677 y FP(where)d FQ(f)416 2656 y FP(~)393 2677 y FO(X)462 2689 y FM(k)502 2677 y FO(;)34 b(k)26 b(>)d FP(1)p 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y(is\014ed)k(b)n(y)f(the)h(op)r(erators)e FO(X)1026 444 y FM(j)1060 432 y FP(,)i FO(j)g FP(=)23 b(1,)k FO(:)14 b(:)g(:)28 b FP(,)f FO(n)p FP(,)h(of)f(the)h(follo)n(wing)f(form:)775 614 y FO(X)851 580 y FN(\003)844 635 y FM(j)888 614 y FO(X)957 626 y FM(j)1015 614 y FP(=)22 b FO(F)1155 626 y FM(j)1191 614 y FP(\()p FO(X)1292 626 y FL(1)1329 614 y FO(X)1405 580 y FN(\003)1398 635 y FL(1)1443 614 y FO(;)14 b(:)g(:)g(:)f(;)h(X)1696 626 y FM(n)1741 614 y FO(X)1817 580 y FN(\003)1810 635 y FM(n)1855 614 y FP(\))p FO(;)769 749 y(X)845 715 y FN(\003)838 769 y FM(j)882 749 y FO(X)951 761 y FM(k)1015 749 y FP(=)22 b FO(\026)1152 761 y FM(j)s(k)1238 749 y FO(X)1307 761 y FM(k)1348 749 y FO(X)1424 715 y FN(\003)1417 769 y FM(j)1461 749 y FO(;)778 884 y(X)847 896 y FM(j)882 884 y FO(X)951 896 y FM(k)1015 884 y FP(=)g FO(\025)1150 896 y FM(j)s(k)1236 884 y FO(X)1305 896 y FM(k)1346 884 y FO(X)1415 896 y FM(j)1450 884 y FO(;)890 b FP(\(2.56\))118 1066 y(where)37 b FO(F)421 1078 y FL(1)459 1066 y 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FO(:)14 b(:)g(:)27 b FP(,)h FO(n)p FP(,)g FO(j)g FQ(6)p FP(=)22 b FO(k)s FP(.)243 4048 y(In)30 b(what)h(follo)n(ws,)f(w)n(e)g (will)h(assume)f(that)h(the)g FO(n)p FP(-dimensional)e(dynam-)118 4147 y(ical)34 b(system)g(generated)e(b)n(y)i(the)g(mappings)g FR(F)1655 4159 y FL(1)1692 4147 y FP(\()p FQ(\001)p FP(\),)h FO(:)14 b(:)g(:)28 b FP(,)35 b FR(F)2080 4159 y FM(n)2126 4147 y FP(\()p FQ(\001)p FP(\))f(p)r(ossesses)p eop %%Page: 164 168 164 167 bop 118 100 a FP(164)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(a)i(measurable)e(section,)i(a)f(measurable)g (set)h(whic)n(h)f(meets)h(eac)n(h)f(orbit)h(at)f(a)118 432 y(single)k(p)r(oin)n(t.)53 b(In)33 b(this)h(case,)f(for)f(an)n(y)g (irreducible)h(represen)n(tation)e(of)h(the)118 532 y(relations,)39 b(the)f(sp)r(ectral)g(measure)e(of)i(the)g(comm)n(uting)g(family)g FO(C)2329 502 y FL(2)2323 553 y(1)2366 532 y FP(,)g FO(:)14 b(:)g(:)28 b FP(,)118 632 y FO(C)183 601 y FL(2)177 652 y FM(n)252 632 y FP(is)h(concen)n(trated)f(on)h(\(a)g(subset)g(of)6 b(\))30 b(a)f(single)f(orbit,)h(and)g(w)n(e)g(can)g(clas-)118 731 y(sify)f(all)f(irreducible)f(represen)n(tations)g(up)h(to)h (unitary)f(equiv)-5 b(alence.)36 b(In)27 b(the)118 831 y(case)j(of)h(more)f(complicated)g(dynamical)g(systems)h(without)g(a)f (measurable)118 930 y(section,)d(non-trivial)f(ergo)r(dic)g(measures)g (can)h(arise,)f(whic)n(h)i(giv)n(es)e(rise)g(to)h(a)118 1030 y(m)n(uc)n(h)20 b(more)f(complicated)g(structure)g(of)h(represen)n (tations,)g(including)g(factor)118 1130 y(represen)n(tations)29 b(not)i(of)h(t)n(yp)r(e)f(I)g(etc.)48 b(Notice)31 b(that)h(the)f (linear)g(dynamical)118 1229 y(system)c(of)h(the)g(form)f(\(2.58\))g (alw)n(a)n(ys)f(p)r(ossesses)g(a)h(measurable)f(section.)243 1330 y(W)-7 b(e)28 b(pro)r(ceed)e(with)i(a)g(more)e(detailed)i(study)g (of)f(the)h(irreducible)f(collec-)118 1429 y(tions)34 b FO(X)397 1441 y FM(j)432 1429 y FP(,)i FO(j)k FP(=)34 b(1,)g FO(:)14 b(:)g(:)27 b FP(,)36 b FO(n)p FP(,)h(satisfying)d (\(2.56\),)i(whic)n(h)e(corresp)r(ond)f(to)h(an)118 1529 y(orbit)h(\012.)59 b(Denote)35 b(b)n(y)g(\001)g(the)g(supp)r(ort)g(of)g (the)g(sp)r(ectral)g(measure)f(of)g(the)118 1628 y(comm)n(uting)i (family)g FO(C)885 1598 y FL(2)879 1650 y FM(j)922 1628 y FP(,)j FO(j)j FP(=)36 b(1,)g FO(:)14 b(:)g(:)27 b FP(,)39 b FO(n)p FP(.)62 b(It)36 b(is)g(a)g(general)e(fact)i(that)h(in)118 1728 y(the)28 b(basis)f(of)g(eigen)n(v)n(ectors)e(of)i(the)h(comm)n (uting)f(family)-7 b(,)28 b(the)g(op)r(erators)d FO(X)2540 1740 y FM(j)118 1828 y FP(act)33 b(as)g(w)n(eigh)n(ted)g(shift)h(op)r (erators)d([202)n(],)k(but)f(w)n(e)f(need)h(to)f(tak)n(e)g(in)n(to)g (ac-)118 1927 y(coun)n(t)e(that)h FO(C)593 1939 y FM(j)658 1927 y FQ(\025)e FP(0,)i(and)f(that)h FO(U)1256 1939 y FM(j)1291 1927 y FO(U)1357 1897 y FN(\003)1348 1949 y FM(j)1426 1927 y FP(is)g(a)f(pro)5 b(jection)30 b(on)i(the)g(co-k)n (ernel)118 2039 y(\(k)n(er)13 b FO(C)340 2009 y FL(2)334 2061 y FM(j)378 2039 y FP(\))410 2009 y FN(?)466 2039 y FP(,)28 b FO(j)g FP(=)22 b(1,)28 b FO(:)14 b(:)g(:)27 b FP(,)h FO(n)p FP(.)118 2207 y FR(Lemma)i(11.)40 b FC(F)-6 b(or)30 b(any)g FO(\025)24 b FP(=)e(\()p FO(\025)1173 2219 y FL(1)1211 2207 y FO(;)14 b(:)g(:)g(:)g(;)g(\025)1444 2219 y FM(n)1489 2207 y FP(\))24 b FQ(2)f FP(\001)p FC(,)31 b(we)f(have)243 2307 y FP(i\))g FO(\025)376 2319 y FM(j)434 2307 y FQ(\025)23 b FP(0)p FC(,)29 b FO(j)f FP(=)23 b(1)p FC(,)30 b FO(:)14 b(:)g(:)27 b FC(,)j FO(n)p FP(;)243 2407 y(ii\))g FC(either)g FR(F)647 2419 y FM(j)682 2407 y FP(\()p FO(\025)p FP(\))24 b FQ(2)g FP(\001)p FC(,)30 b(or)g FP(\()p FR(F)1220 2419 y FM(j)1256 2407 y FP(\()p FO(\025)p FP(\)\))1400 2419 y FM(j)1459 2407 y FP(=)23 b(0;)243 2508 y(iii\))30 b FC(similarly,)i(either)e FR(F)1038 2472 y FN(\000)p FL(1)1038 2531 y FM(j)1128 2508 y FP(\()p FO(\025)p FP(\))24 b FQ(2)f FP(\001)p FC(,)31 b(or)f FO(\025)1622 2520 y FM(j)1680 2508 y FP(=)23 b(0)p FC(.)118 2686 y(Pr)l(o)l(of.)43 b FP(i\))28 b(Indeed,)g(since)g FO(C)1023 2656 y FL(2)1017 2708 y FM(j)1083 2686 y FQ(\025)23 b FP(0,)k(w)n(e)g(ha)n(v)n(e)g FO(\025)1625 2698 y FM(j)1683 2686 y FQ(\025)c FP(0,)k FO(j)h FP(=)23 b(1,)k FO(:)14 b(:)g(:)27 b FP(,)h FO(n)p FP(.)243 2787 y(ii\))c(If)g FR(F)484 2799 y FM(j)519 2787 y FP(\()p FO(\025)p FP(\))34 b FO(=)-52 b FQ(2)24 b FP(\001,)h(then)f FO(U)1093 2799 y FM(j)1128 2787 y FO(e)1167 2799 y FM(\025)1233 2787 y FP(=)e(0,)j(where)e FO(e)1685 2799 y FM(\025)1752 2787 y FP(is)h(the)g(basis)f(eigen)n(v)n(ector)118 2886 y(of)38 b(the)h(comm)n(uting)e(family)h(corresp)r(onding)f(to)h(the)g(join)n(t) g(eigen)n(v)-5 b(alue)37 b FO(\025)p FP(.)118 2986 y(Then)28 b(w)n(e)f(also)g(ha)n(v)n(e)f FO(U)872 2998 y FM(j)907 2986 y FO(U)973 2956 y FN(\003)964 3008 y FM(j)1011 2986 y FO(e)1050 3001 y Fn(F)1097 3009 y Fw(j)1127 3001 y FL(\()p FM(\025)p FL(\))1246 2986 y FP(=)c(0)28 b(and)779 3183 y FO(C)844 3149 y FL(2)838 3204 y FM(j)882 3183 y FO(e)921 3198 y Fn(F)968 3206 y Fw(j)999 3198 y FL(\()p FM(\025)p FL(\))1117 3183 y FP(=)23 b(\()p FR(F)1297 3195 y FM(j)1332 3183 y FP(\()p FO(\025)p FP(\)\))1476 3195 y FM(j)1526 3183 y FO(e)1565 3198 y Fn(F)1612 3206 y Fw(j)1643 3198 y FL(\()p FM(\025)p FL(\))1761 3183 y FP(=)g(0)p FO(;)118 3367 y FP(whic)n(h)28 b(implies)f(that)h FR(F)877 3379 y FM(j)912 3367 y FP(\()p FO(\025)p FP(\)\))1056 3379 y FM(j)1116 3367 y FP(=)22 b(0.)243 3467 y(iii\))k(Similarly)-7 b(,)26 b(if)g FR(F)872 3432 y FN(\000)p FL(1)872 3490 y FM(j)961 3467 y FP(\()p FO(\025)p FP(\))34 b FO(=)-52 b FQ(2)24 b FP(\001,)i(then)g FO(U)1547 3437 y FN(\003)1538 3489 y FM(j)1599 3467 y FO(e)1638 3479 y FM(\025)1704 3467 y FP(=)d(0.)36 b(Then)26 b FO(U)2165 3479 y FM(j)2200 3467 y FO(U)2266 3437 y FN(\003)2257 3489 y FM(j)2317 3467 y FO(e)2356 3479 y FM(\025)2423 3467 y FP(=)c(0,)118 3578 y(and)28 b FO(C)345 3548 y FL(2)339 3599 y FM(j)396 3578 y FO(e)435 3590 y FM(\025)501 3578 y FP(=)23 b FO(\025)637 3590 y FM(j)672 3578 y FO(e)711 3590 y FM(\025)777 3578 y FP(=)g(0,)k(whic)n(h)h(giv)n(es)e FO(\025)1446 3590 y FM(j)1505 3578 y FP(=)c(0.)p 2514 3578 4 57 v 2518 3525 50 4 v 2518 3578 V 2567 3578 4 57 v 118 3749 a FR(Corollary)29 b(5.)37 b FC(If)27 b(for)g(some)f FO(\025)e FP(=)e(\()p FO(\025)1312 3761 y FL(1)1350 3749 y FO(;)14 b(:)g(:)g(:)g(;)g(\025) 1583 3761 y FM(n)1629 3749 y FP(\))23 b FQ(2)g FP(\012)p FC(,)28 b(we)e(have)h FO(\025)2229 3761 y FM(j)2288 3749 y FO(>)c FP(0)i FC(and)118 3848 y FP(\()p FR(F)210 3860 y FM(j)245 3848 y FP(\()p FO(\025)p FP(\)\))389 3860 y FM(j)459 3848 y FO(<)33 b FP(0)p FC(,)k(then)e FO(\025)43 b(=)-51 b FQ(2)33 b FP(\001)p FC(.)56 b(This)37 b(c)l(ondition)f (implies)h(that)e(irr)l(e)l(ducible)118 3948 y(r)l(epr)l(esentations)23 b(c)l(orr)l(esp)l(ond)h(only)g(to)f(the)g(orbits)h(such)f(that)g FO(\025)2110 3960 y FM(j)2168 3948 y FO(>)g FP(0)f FC(implies)118 4048 y FP(\()p FR(F)210 4060 y FM(j)245 4048 y FP(\()p FO(\025)p FP(\)\))389 4060 y FM(j)450 4048 y FQ(\025)h FP(0)p FC(,)31 b FP(\()p FR(F)728 4012 y FN(\000)p FL(1)728 4071 y FM(j)817 4048 y FP(\()p FO(\025)p FP(\)\))961 4060 y FM(j)1021 4048 y FQ(\025)24 b FP(0)p FC(.)39 b(Notic)l(e)31 b(also)g(that)38 b FP(\(2.58\))30 b FC(implies)i(that)e(it)118 4147 y(fol)t(lows)i(fr)l(om)e FO(\025)638 4159 y FM(j)697 4147 y FO(>)22 b FP(0)30 b FC(that)f FP(\()p FR(F)1117 4159 y FM(k)1158 4147 y FP(\()p FO(\025)p FP(\)\))1302 4159 y FM(j)1362 4147 y FO(>)22 b FP(0)30 b FC(for)g FO(k)c FQ(6)p FP(=)d FO(j)5 b FC(.)p eop %%Page: 165 169 165 168 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(165)243 333 y(Consider)25 b(p)r(ossible)h(t)n(yp)r(es)h(of)f(orbits)g(and)h(describ)r(e)f(the)h (corresp)r(onding)118 432 y(irreducible)g(represen)n(tations)f(of)34 b(\(2.56\))o(.)118 584 y FR(Theorem)e(37.)41 b FC(A)n(ny)30 b(irr)l(e)l(ducible)i(r)l(epr)l(esentation)f(c)l(an)g(b)l(e)g(r)l(e)l (alize)l(d)h(in)f(the)118 683 y(sp)l(ac)l(e)f FO(l)359 695 y FL(2)396 683 y FP(\(\001\))p FC(.)40 b(F)-6 b(or)29 b(any)38 b FO(l)24 b FP(=)f(1)p FC(,)29 b FO(:)14 b(:)g(:)28 b FC(,)i FO(n)f FC(one)h(of)g(the)g(fol)t(lowing)i(c)l(ases)d(holds)7 b FP(:)243 783 y FO(a)p FP(\))p FC(.)57 b(The)37 b(mapping)45 b FR(F)986 795 y FM(l)1012 783 y FP(\()p FQ(\001)p FP(\))36 b FC(p)l(ossesses)h(a)f(stationary)h(p)l(oint)g FO(\025)d FQ(2)h FP(\001)h(\()p FC(in)118 883 y(this)24 b(c)l(ase)h(al)t(l)g (other)f(p)l(oints)g(ar)l(e)h(also)f(stationary)7 b FP(\))p FC(.)39 b(If)24 b FO(\025)1898 895 y FM(l)1947 883 y FP(=)f(0)p FC(,)i(then)e FO(X)2374 895 y FM(l)2423 883 y FP(=)f(0;)118 982 y FC(otherwise,)32 b(the)e(op)l(er)l(ator)g FO(X)1045 994 y FM(l)1100 982 y FC(has)h(the)f(form)1063 1147 y FO(X)1132 1159 y FM(l)1158 1147 y FO(e)1197 1159 y FM(\025)1263 1147 y FP(=)22 b FO(\014)1397 1159 y FM(l)1437 1147 y FO(\025)1485 1159 y FM(l)1524 1147 y FO(e)1563 1159 y FM(\025)1607 1147 y FO(;)118 1311 y FC(wher)l(e)31 b FO(\014)400 1323 y FM(l)455 1311 y FC(is)f(a)g(p)l(ar)l(ameter)g (with)h(the)f(absolute)g(value)g(e)l(qual)g(to)g(one)6 b FP(;)243 1411 y FO(b)p FP(\))p FC(.)52 b(The)35 b(mapping)h FR(F)962 1423 y FM(l)987 1411 y FP(\()p FQ(\001)p FP(\))f FC(do)l(es)g(not)f(have)i(stationary)f(p)l(oints.)52 b(In)34 b(this)118 1510 y(c)l(ase)c(the)g(op)l(er)l(ator)h FO(X)830 1522 y FM(l)885 1510 y FC(has)f(the)g(form)986 1675 y FO(X)1055 1687 y FM(l)1080 1675 y FO(e)1119 1687 y FM(\025)1185 1675 y FP(=)23 b FO(F)1326 1687 y FM(l)1352 1675 y FP(\()p FO(\025)p FP(\))14 b FO(e)1517 1690 y Fn(F)1564 1699 y Fw(l)1589 1690 y FL(\()p FM(\025)p FL(\))1684 1675 y FO(:)656 b FP(\(2.60\))118 1839 y FC(The)37 b(kernel)e(of)i(the) e(op)l(er)l(ator)i FO(X)1195 1851 y FM(l)1255 1839 y FC(is)f(gener)l(ate)l(d)g(by)g(ve)l(ctors)g FO(e)2162 1851 y FM(\025)2240 1839 y FC(such)g(that)118 1939 y FO(F)171 1951 y FM(l)197 1939 y FP(\()p FO(\025)p FP(\))41 b(=)e(0)p FC(;)k(the)d(kernel)f(of)h FO(X)1151 1908 y FN(\003)1144 1962 y FM(l)1227 1939 y FC(is)f(gener)l(ate)l(d)h(by)f(ve) l(ctors)g FO(e)2147 1951 y FM(\025)2229 1939 y FC(for)h(which)118 2038 y FO(\025)166 2050 y FM(l)215 2038 y FP(=)23 b(0)p FC(.)118 2190 y(Pr)l(o)l(of.)43 b FP(The)d(pro)r(of)f(is)h(essen)n (tially)f(based)g(on)g(the)h(follo)n(wing)f(statemen)n(ts)118 2289 y(from)27 b([202)o(].)118 2439 y FR(Theorem)44 b(38.)j FC(L)l(et)40 b(the)g(dynamic)l(al)i(system)e(on)h FJ(R)1878 2409 y FM(n)1969 2439 y FC(gener)l(ate)l(d)g(by)g(the)118 2538 y(mappings)e FR(F)555 2550 y FM(l)581 2538 y FC(,)h FO(l)f FP(=)f(1)p FC(,)f FO(:)14 b(:)g(:)28 b FC(,)40 b FO(n)p FC(,)g(p)l(ossess)f(a)f(me)l(asur)l(able)g(se)l(ction.)64 b(Then,)118 2638 y(for)38 b(e)l(ach)h(irr)l(e)l(ducible)g(c)l(ol)t(le)l (ction)g(of)f(op)l(er)l(ators)g FO(C)1771 2650 y FM(j)1807 2638 y FC(,)h FO(U)1928 2650 y FM(j)1963 2638 y FC(,)h FO(j)i FP(=)37 b(1)p FC(,)g FO(:)14 b(:)g(:)28 b FC(,)40 b FO(n)p FC(,)118 2738 y(satisfying)f FP(\(2.57\))o FC(,)30 b(the)g(fol)t(lowing)i(holds.)243 2837 y(i.)46 b(Ther)l(e)34 b(exists)e(a)g(unique)g(orbit)h FP(\012)f FC(of)h(the)g(dynamic)l(al)h (system)e(of)h(ful)t(l)118 2937 y(sp)l(e)l(ctr)l(al)42 b(me)l(asur)l(e)g(of)h(the)f(c)l(ommuting)g(c)l(ol)t(le)l(ction)h FO(C)1895 2949 y FM(j)1931 2937 y FC(,)i FO(j)50 b FP(=)45 b(1)p FC(,)d FO(:)14 b(:)g(:)28 b FC(,)46 b FO(n)p FC(,)118 3036 y FO(E)5 b FP(\(\012\))24 b(=)e(1;)243 3136 y FC(ii.)48 b(If)j FP(k)n(er)13 b FO(U)656 3148 y FM(l)710 3136 y FP(=)28 b FQ(f)p FP(0)p FQ(g)p FC(,)k(then)h(the)g(sp)l(e)l(ctr)l(al)g (me)l(asur)l(e)f(is)i(quasi-invariant)118 3236 y(with)43 b(r)l(esp)l(e)l(ct)e(to)h(the)g(mapping)h FR(F)1262 3248 y FM(l)1288 3236 y FP(\()p FQ(\001)p FP(\);)48 b FC(in)42 b(the)g(c)l(ase)h(of)f(unitary)g FO(U)2371 3248 y FM(l)2396 3236 y FC(,)k(the)118 3335 y(me)l(asur)l(e)30 b(is)g(also)g (quasi-invariant)h(with)g(r)l(esp)l(e)l(ct)e(to)h FR(F)1883 3300 y FN(\000)p FL(1)1883 3360 y FM(l)1972 3335 y FP(\()p FQ(\001)p FP(\);)243 3435 y FC(iii.)43 b(The)32 b(joint)f(sp)l(e)l (ctrum)g(of)g(the)g(c)l(ommuting)g(family)h FO(C)2081 3447 y FM(j)2117 3435 y FC(,)f FO(j)f FP(=)25 b(1)p FC(,)31 b FO(:)14 b(:)g(:)27 b FC(,)118 3535 y FO(n)p FC(,)j(is)g(simple.)118 3684 y FR(Theorem)45 b(39.)j FC(The)42 b(irr)l(e)l(ducible)g(c)l(ol)t (le)l(ction)g FO(C)1741 3696 y FM(j)1777 3684 y FC(,)i FO(U)1903 3696 y FM(j)1938 3684 y FC(,)g FO(j)49 b FP(=)43 b(1)p FC(,)e FO(:)14 b(:)g(:)27 b FC(,)45 b FO(n)p FC(,)118 3784 y(satisfying)37 b FP(\(2.57\))27 b FC(acts)h(in)g(the)h(sp)l(ac)l (e)f FO(l)1379 3796 y FL(2)1416 3784 y FP(\(\001\))p FC(,)i(wher)l(e)f FP(\001)23 b FQ(\032)g FP(\012)28 b FC(is)g(a)g(subset)g(of)118 3883 y(some)e(orbit)f FP(\012)g(\()p FC(for)i(unitary)e FO(U)1109 3895 y FM(l)1134 3883 y FC(,)h FO(l)f FP(=)e(1)p FC(,)i FO(:)14 b(:)g(:)27 b FC(,)g FO(n)p FC(,)f FP(\001)d(=)g(\012\))p FC(,)j(by)g(the)f(fol)t (lowing)118 3983 y(formulae)710 4147 y FO(C)769 4159 y FM(l)795 4147 y FO(e)834 4159 y FM(\025)900 4147 y FP(=)e FO(x)1035 4159 y FM(k)1076 4147 y FO(e)1115 4159 y FM(\025)1158 4147 y FO(;)99 b(U)1337 4159 y FM(l)1362 4147 y FO(e)1401 4159 y FM(\025)1467 4147 y FP(=)23 b FO(u)1603 4159 y FM(l)1628 4147 y FP(\()p FO(\025)p FP(\))14 b FO(e)1793 4162 y Fn(F)1840 4171 y Fw(l)1865 4162 y FL(\()p FM(\025)p FL(\))1960 4147 y FO(;)p eop %%Page: 166 170 166 169 bop 118 100 a FP(166)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(wher)l(e)38 b FO(u)408 345 y FM(l)433 333 y FP(\()p FO(\025)p FP(\))g FC(ar)l(e)f(c)l(onstants)f (which)j(determine)e(the)g(action)h(of)55 b FO(U)2324 345 y FM(l)2350 333 y FC(.)60 b(The)118 432 y(subset)29 b FP(\001)h FC(satis\014es)g(the)g(fol)t(lowing)i(\\b)l(oundary)e(c)l (onditions")6 b FP(:)826 622 y FO(u)874 634 y FM(l)899 622 y FP(\()p FO(\025)p FP(\))24 b(=)f(0)p FO(;)98 b FQ(8)14 b FO(\025)22 b FQ(2)i FP(\001)9 b(:)28 b FR(F)1685 634 y FM(l)1710 622 y FP(\()p FO(\025)p FP(\))34 b FO(=)-52 b FQ(2)24 b FP(\001)p FO(;)612 758 y(u)660 770 y FM(l)685 758 y FP(\()p FR(F)777 723 y FN(\000)p FL(1)777 783 y FM(l)867 758 y FP(\()p FO(\025)p FP(\)\))g(=)f(0)p FO(;)98 b FQ(8)14 b FO(\025)22 b FQ(2)i FP(\001)9 b(:)28 b FR(F)1685 723 y FN(\000)p FL(1)1685 783 y FM(l)1774 758 y FP(\()p FO(\025)p FP(\))33 b FO(=)-51 b FQ(2)24 b FP(\001)p FO(;)295 b FP(\(2.61\))118 949 y FC(and)32 b(is)g(\\c)l(onne)l(cte)l(d")f(in)g (the)h(fol)t(lowing)i(sense)6 b FP(:)41 b FO(u)1741 961 y FM(l)1766 949 y FP(\()p FO(\025)p FP(\))28 b FQ(6)p FP(=)d(0)31 b FC(for)i(al)t(l)f FO(\025)27 b FQ(2)f FP(\001)p FC(,)118 1048 y(such)k(that)g FR(F)537 1060 y FM(l)562 1048 y FP(\()p FO(\025)p FP(\))24 b FQ(2)g FP(\001)p FC(,)30 b FO(l)25 b FP(=)d(1)p FC(,)30 b FO(:)14 b(:)g(:)27 b FC(,)k FO(n)p FC(.)243 1224 y FP(Let)26 b FO(\025)g FP(b)r(e)g(a)f(stationary)g(p)r(oin)n(t)h(of)f(the)i(mapping)e FR(F)1884 1236 y FM(l)1910 1224 y FP(\()p FQ(\001)p FP(\).)37 b(If)26 b FO(\025)2186 1236 y FM(l)2235 1224 y FP(=)d(0,)i(then)118 1323 y(for)e(all)g(p)r(oin)n(ts)g FO(\025)h FQ(2)f FP(\001,)h(the)g (comm)n(utation)f(of)g FR(F)1653 1335 y FM(l)1679 1323 y FP(\()p FQ(\001)p FP(\))g(and)h FR(F)2007 1335 y FM(k)2048 1323 y FP(\()p FQ(\001)p FP(\))f(also)g(implies)118 1423 y(that)28 b FO(\025)346 1435 y FM(l)396 1423 y FP(=)23 b(0.)38 b(Then)28 b FO(X)873 1435 y FM(l)922 1423 y FP(=)c(0.)38 b(If)28 b FO(\025)1245 1435 y FM(l)1295 1423 y FQ(6)p FP(=)23 b(0,)28 b(then)g(also)f FO(\025)1880 1435 y FM(l)1930 1423 y FQ(6)p FP(=)c(0)28 b(for)f(all)h FO(\025)c FQ(2)h FP(\001.)118 1523 y(In)34 b(this)h(case,)g(the)f(op)r(erator)e FO(U)1151 1535 y FM(l)1211 1523 y FP(comm)n(utes)h(with)i(all)e(the)i (op)r(erators)d FO(X)2517 1535 y FM(j)2552 1523 y FP(,)118 1622 y FO(X)194 1592 y FN(\003)187 1644 y FM(j)232 1622 y FP(,)27 b(and,)h(therefore,)f(is)g(a)g(m)n(ultiple)h(of)g(the)g(iden) n(tit)n(y)-7 b(.)243 1726 y(In)37 b(the)g(case)e(where)i(the)g(mapping) f FR(F)1504 1738 y FM(l)1530 1726 y FP(\()p FQ(\001)p FP(\))h(do)r(es)f(not)h(ha)n(v)n(e)e(stationary)118 1825 y(p)r(oin)n(ts,)30 b(the)f(op)r(erator)f FO(U)931 1837 y FM(l)986 1825 y FP(is)h(unitarily)g(equiv)-5 b(alen)n(t)29 b(to)g(the)h(shift)g(op)r(erator;)118 1925 y(taking)c(in)n(to)g(accoun) n(t)g(that)h FO(X)1094 1937 y FM(l)1119 1925 y FO(X)1195 1895 y FN(\003)1188 1948 y FM(l)1256 1925 y FP(=)22 b FO(C)1408 1895 y FL(2)1402 1948 y FM(l)1446 1925 y FP(,)27 b(w)n(e)f(get)g(the)h(needed)g(form)n(ula)e(for)118 2024 y FO(X)187 2036 y FM(l)212 2024 y FP(.)p 2514 2024 4 57 v 2518 1972 50 4 v 2518 2024 V 2567 2024 4 57 v 118 2260 a FR(2.4.4)94 b(Represen)m(tations)40 b(of)i(the)f(non-standard)h (real)g(quan)m(tum)410 2360 y(sphere)118 2520 y(1.)h FP(The)30 b(algebra)e(of)i(functions)h(on)e(the)i(non-standard)d (three-dimensional)118 2620 y(real)23 b(quan)n(tum)h(sphere)f(\(see)g ([185)o(]\))h(is)g(an)f(asso)r(ciativ)n(e)f FQ(\003)p FP(-algebra)f(generated)118 2719 y(b)n(y)27 b(the)h(elemen)n(ts)g FO(x)p FP(,)g FO(y)s FP(,)f FO(u)p FP(,)h FO(v)s FP(,)g FO(c)p FP(,)f(and)h FO(d)g FP(satisfying)f(the)h(relations:)483 2909 y FO(ux)23 b FP(=)g FO(q)s(xu;)97 b(v)s(x)23 b FP(=)g FO(q)s(xv)s(;)98 b(y)s(u)22 b FP(=)g FO(q)s(uy)s(;)97 b(y)s(v)26 b FP(=)c FO(q)s(v)s(y)s(;)306 3044 y(v)s(u)c FQ(\000)g FO(uv)26 b FP(=)d(\()p FO(q)f FQ(\000)c FO(q)914 3010 y FN(\000)p FL(1)1003 3044 y FP(\))c FO(d;)97 b(xy)22 b FQ(\000)c FO(q)1445 3010 y FN(\000)p FL(1)1534 3044 y FO(uv)26 b FP(=)c FO(y)s(x)d FQ(\000)f FO(q)s(v)s(u)23 b FP(=)f FO(c)d FP(+)f FO(d;)311 3179 y(dx)24 b FP(=)f FO(q)553 3145 y FL(2)590 3179 y FO(xd;)98 b(dv)26 b FP(=)d FO(q)1038 3145 y FL(2)1075 3179 y FO(v)s(d;)97 b(ud)23 b FP(=)g FO(q)1523 3145 y FL(2)1560 3179 y FO(du;)97 b(y)s(d)23 b FP(=)f FO(q)2008 3145 y FL(2)2046 3179 y FO(dy)s(;)207 b FP(\(2.62\))118 3369 y(with)37 b FO(c)g FP(lying)f(in)h(the)g(cen)n(ter,)h(and)e(the)h(in)n(v)n(olution)f (de\014ned)h(b)n(y)f FO(x)2329 3339 y FN(\003)2406 3369 y FP(=)h FO(y)s FP(,)118 3469 y FO(u)166 3439 y FN(\003)229 3469 y FP(=)25 b FQ(\000)p FO(q)424 3439 y FN(\000)p FL(1)512 3469 y FO(v)s FP(,)30 b FO(c)644 3439 y FN(\003)707 3469 y FP(=)25 b FO(c)p FP(,)k FO(d)928 3439 y FN(\003)991 3469 y FP(=)c FO(d)p FP(.)41 b(F)-7 b(or)28 b(the)h(generators)e FO(x)p FP(,)i FO(u)p FP(,)g FO(c)p FP(,)g FO(d)p FP(,)h(the)f(rela-)118 3568 y(tions)24 b(\(2.62\))f(ha)n(v)n(e)f(the)i(form)g(\(2.58\))o(,)g (and)g(are)f(equiv)-5 b(alen)n(t)23 b(to)h(the)g(follo)n(wing)118 3668 y(relations)889 3858 y FO(ux)f FP(=)g FO(q)s(xu;)97 b(u)1398 3824 y FN(\003)1435 3858 y FO(x)24 b FP(=)f FO(q)s(xu)1729 3824 y FN(\003)1767 3858 y FO(;)190 3993 y(u)238 3959 y FN(\003)276 3993 y FO(u)f FP(=)h FO(q)474 3959 y FN(\000)p FL(2)563 3993 y FO(uu)659 3959 y FN(\003)715 3993 y FQ(\000)18 b FP(\(1)g FQ(\000)g FO(q)1013 3959 y FN(\000)p FL(2)1102 3993 y FP(\)\()p FO(xx)1260 3959 y FN(\003)1318 3993 y FQ(\000)g FO(c)p FP(\))p FO(;)98 b(x)1637 3959 y FN(\003)1675 3993 y FO(x)24 b FP(=)e FO(q)1873 3959 y FL(2)1911 3993 y FO(xx)2005 3959 y FN(\003)2062 3993 y FP(+)c(\(1)g FQ(\000)g FO(q)2360 3959 y FL(2)2398 3993 y FP(\))p FO(c;)1005 4117 y(d)24 b FP(=)e FO(xx)1253 4083 y FN(\003)1311 4117 y FP(+)c FO(uu)1490 4083 y FN(\003)1546 4117 y FQ(\000)g FO(c:)675 b FP(\(2.63\))p eop %%Page: 167 171 167 170 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(167)118 333 y FR(2.)45 b FP(The)30 b(corresp)r(onding)f(dynamical)h(system)g(on)g FJ(R)1800 303 y FL(2)1874 333 y FP(is)g(generated)f(b)n(y)h(the)118 432 y(mappings)531 620 y FR(F)591 632 y FL(1)628 620 y FP(\()p FO(\025)708 632 y FL(1)746 620 y FO(;)14 b(\025)831 632 y FL(2)868 620 y FP(\))24 b(=)e(\()p FO(q)1083 586 y FL(2)1121 620 y FO(\025)1169 632 y FL(1)1225 620 y FP(+)c(\(1)g FQ(\000)g FO(q)1523 586 y FL(2)1561 620 y FP(\))p FO(c;)c(q)1706 586 y FL(2)1743 620 y FO(\025)1791 632 y FL(2)1829 620 y FP(\))p FO(;)531 755 y FR(F)591 767 y FL(2)628 755 y FP(\()p FO(\025)708 767 y FL(1)746 755 y FO(;)g(\025)831 767 y FL(2)868 755 y FP(\))24 b(=)e(\()p FO(\025)1091 767 y FL(1)1129 755 y FO(;)14 b(q)1206 721 y FN(\000)p FL(2)1295 755 y FO(\025)1343 767 y FL(2)1400 755 y FQ(\000)k FP(\(1)g FQ(\000)g FO(q)1698 721 y FN(\000)p FL(2)1787 755 y FP(\)\()p FO(\025)1899 767 y FL(1)1956 755 y FQ(\000)g FO(c)p FP(\)\))p FO(;)118 943 y FP(whic)n(h)28 b(satisfy)-7 b(,)27 b(as)g(one)g(can)g(see,)h(the)g(conditions)f (\(2.59\))o(.)243 1046 y(An)n(y)g(orbit)g(of)h(the)g(dynamical)f (system)g(consists)g(of)h(the)g(p)r(oin)n(ts)218 1234 y FO(\025)266 1199 y FL(\()p FM(k)q(l)p FL(\))404 1234 y FP(=)23 b FR(F)552 1199 y FM(k)552 1254 y FL(1)593 1234 y FP(\()p FR(F)685 1199 y FM(l)685 1254 y FL(2)722 1234 y FP(\()p FO(\025)p FP(\)\))404 1375 y(=)g(\()p FO(q)564 1340 y FL(2)p FM(k)638 1375 y FO(\025)686 1387 y FL(1)742 1375 y FP(+)18 b(\(1)g FQ(\000)g FO(q)1040 1340 y FL(2)p FM(k)1114 1375 y FP(\))c FO(c;)g(q)1273 1340 y FL(2\()p FM(k)q FN(\000)p FM(l)p FL(\))1472 1375 y FO(\025)1520 1387 y FL(2)1576 1375 y FQ(\000)k FO(q)1699 1340 y FL(2)p FM(k)1773 1375 y FP(\(1)h FQ(\000)f FO(q)1989 1340 y FN(\000)p FL(2)p FM(l)2099 1375 y FP(\)\()p FO(c)h FQ(\000)f FO(\025)2349 1387 y FL(1)2387 1375 y FP(\)\))p FO(;)118 1572 y FP(where)24 b FO(\025)f FP(=)g(\()p FO(\025)594 1584 y FL(1)632 1572 y FO(;)14 b(\025)717 1584 y FL(2)754 1572 y FP(\),)26 b FR(F)895 1542 y FM(k)895 1595 y(l)936 1572 y FP(\()p FQ(\001)p FP(\))f(is)f(the)g FO(k)s FP(-th)h(iterations) e(of)h(the)h(mapping)f FR(F)2439 1584 y FM(l)2464 1572 y FP(\()p FQ(\001)p FP(\).)243 1674 y(The)c(mapping)g FR(F)800 1686 y FL(1)838 1674 y FP(\()p FQ(\001)p FP(\))h(has)f(a)g (single)g(stationary)f(p)r(oin)n(t)h(\()p FO(c;)14 b FP(0\),)22 b(Stationary)118 1774 y(p)r(oin)n(ts)30 b(of)g(the)h (mapping)f FR(F)1017 1786 y FL(2)1054 1774 y FP(\()p FQ(\001)p FP(\))h(ha)n(v)n(e)e(the)i(co)r(ordinates)d(\()p FO(\025;)14 b(c)21 b FQ(\000)f FO(\025)p FP(\).)45 b(There)118 1873 y(are)20 b(no)h(p)r(erio)r(dic)g(p)r(oin)n(ts)h(of)f FR(F)1063 1885 y FL(1)1100 1873 y FP(\()p FQ(\001)p FP(\),)i FR(F)1293 1885 y FL(2)1331 1873 y FP(\()p FQ(\001)p FP(\),)g(apart)d (from)h(the)h(stationary)d(ones.)118 2031 y FR(3.)54 b FP(The)34 b(follo)n(wing)e(is)i(a)f(list)h(of)f(orbits,)i(the)f (corresp)r(onding)d(sets)j(\001,)h(and)118 2131 y(the)28 b(corresp)r(onding)e(irreducible)h(represen)n(tations.)243 2233 y(1\))33 b(A)i(single)e(stationary)f(p)r(oin)n(t)i(\()p FO(c;)14 b FP(0\).)56 b(F)-7 b(or)33 b FO(c)h FP(=)f(0,)i(this)f(orbit) g(corre-)118 2333 y(sp)r(onds)k(to)g(the)h(trivial)e(represen)n(tation) g FO(X)47 b FP(=)40 b FO(U)49 b FP(=)40 b(0,)h(and)d(for)f FO(c)k(>)f FP(0,)118 2432 y(to)30 b(the)h(family)f(of)g (one-dimensional)f(irreducible)h(represen)n(tations)e FO(U)36 b FP(=)27 b(0,)118 2532 y FO(X)34 b FP(=)27 b FO(\013)14 b(c)p FP(,)31 b(where)f FQ(j)p FO(\013)p FQ(j)e FP(=)f(1.)45 b(Therefore,)29 b(the)i(set)f(of)h(one-dimensional)e(rep-) 118 2632 y(resen)n(tations)d(is)i(parametrized)e(b)n(y)h(p)r(oin)n(ts)h (of)f(the)h(cone.)243 2734 y(2\))38 b(If)g FO(c)i(>)g FP(0,)g(then)f(there)f(exists)f(a)h(unique)g(orbit)g(that)g(is)g(in)n (v)-5 b(arian)n(t)118 2834 y(with)39 b(resp)r(ect)g(to)f(the)h(mapping) g FR(F)1293 2846 y FL(2)1330 2834 y FP(\()p FQ(\001)p FP(\))g(and)g(satis\014es)f(the)h(conditions)f(of)118 2933 y(Lemma)25 b(11.)35 b(Namely)-7 b(,)25 b(this)g(orbit)f(con)n (tains)g(the)h(p)r(oin)n(t)g FO(\025)f FP(=)e(\(0)p FO(;)14 b(c)p FP(\).)36 b(The)25 b(set)118 3033 y(\001)g(consists)f(of)h(the)g (p)r(oin)n(ts)g FO(\025)1042 3003 y FL(\()p FM(k)q FL(\))1158 3033 y FP(=)e(\(\(1)13 b FQ(\000)g FO(q)1483 3003 y FL(2)p FM(k)1557 3033 y FP(\))p FO(c;)h(q)1702 3003 y FL(2)p FM(k)1776 3033 y FO(c)p FP(\),)26 b(and)e(the)i(irreducible)118 3132 y(represen)n(tation)e(corresp)r(onding)g(to)i(this)g(orbit)g(is)g (realized)f(on)g(the)i(space)e FO(l)2538 3144 y FL(2)118 3232 y FP(b)n(y)i(the)h(form)n(ulae:)508 3420 y FO(X)7 b(e)623 3432 y FM(k)686 3420 y FP(=)22 b(\(\(1)d FQ(\000)f FO(q)1021 3386 y FL(2)p FM(k)1095 3420 y FP(\))c FO(c)p FP(\))1209 3386 y FL(1)p FM(=)p FL(2)1314 3420 y FO(e)1353 3432 y FM(k)q FL(+1)1477 3420 y FO(;)517 3558 y(U)9 b(e)622 3570 y FM(k)686 3558 y FP(=)22 b FO(\013)15 b(q)881 3524 y FM(k)q FN(\000)p FL(1)1006 3494 y FQ(p)p 1076 3494 36 4 v 1076 3558 a FO(c)e(e)1164 3570 y FM(k)1205 3558 y FO(;)180 b FQ(j)p FO(\013)p FQ(j)23 b FP(=)g(1)p FO(;)41 b(k)26 b FP(=)d(1)p FO(;)14 b FP(2)p FO(;)g(:)g(:)g(:)26 b(:)118 3746 y FP(The)h(represen)n(tations)e(of)i(this)h(series)e(are)g (de\014ned)i(b)n(y)e(the)i(parameters)d FO(c)e(>)118 3846 y FP(0,)k FO(\013)d FQ(2)f FO(S)421 3816 y FL(1)458 3846 y FP(.)243 3948 y(3\))30 b(If)g FO(c)e(>)f FP(0,)j(then)h(the)g (orbits)e(that)i(con)n(tain)e(the)i(p)r(oin)n(ts)f(\()p FO(c;)14 b(y)s FP(\),)31 b FO(y)f(>)d FP(0)118 4048 y(lie)j(in)g(the)g (\014rst)g(quadran)n(t.)42 b(They)30 b(consist)f(of)h(the)g(p)r(oin)n (ts)g(\()p FO(c;)14 b(q)2174 4018 y FL(2)p FM(n)2252 4048 y FP(\),)31 b FO(n)26 b FQ(2)h FJ(Z)p FP(,)118 4147 y(and)j(the)h(set)g(of)f(these)h(orbits)e(is)i(naturally)e (parametrized)g(b)n(y)h(p)r(oin)n(ts)h(of)f(a)p eop %%Page: 168 172 168 171 bop 118 100 a FP(168)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(circle)d FO(S)388 303 y FL(1)425 333 y FP(.)36 b(The)24 b(corresp)r(onding)f(irreducible)h(represen)n (tations)e(are)i(realized)118 432 y(on)j FO(l)258 444 y FL(2)295 432 y FP(\()p FJ(Z)p FP(\))22 b(b)n(y)27 b(the)h(form)n (ulae:)509 616 y FO(X)7 b(e)624 628 y FM(k)687 616 y FP(=)775 552 y FQ(p)p 844 552 36 4 v 64 x FO(c)14 b(e)933 628 y FM(k)q FL(+1)1057 616 y FO(;)97 b(U)9 b(e)1282 628 y FM(k)1346 616 y FP(=)22 b FO(\025)14 b(q)1535 582 y FM(k)1590 616 y FO(e)1629 628 y FM(k)q FN(\000)p FL(1)1755 616 y FO(;)180 b(k)26 b FQ(2)d FJ(Z)o FO(:)118 800 y FP(The)31 b(parameters)e FO(\025)g FQ(2)g FP(\()p FO(q)958 770 y FL(2)996 800 y FO(;)14 b FP(1],)31 b FO(c)e FQ(\025)f FP(0,)k(determine)f(the)g(set)g(of)g(represen)n(ta-)118 900 y(tions)d(of)f(this)h(series.)243 1000 y(4\))20 b(Consider)f(no)n (w)g(the)h(orbits)g(con)n(taining)f(the)h(p)r(oin)n(ts)g(\(0)p FO(;)14 b(y)s FP(\),)21 b FO(y)26 b(>)d(c)g(>)f 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4147 y(and)37 b(in)n(v)n(estigated)f(in)h([282)o(,)g(306)o(,)g(305)o(].)65 b(The)38 b(algebra)d(of)i(\\functions)g(on)p eop %%Page: 170 174 170 173 bop 118 100 a FP(170)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FO(E)179 345 y FM(q)216 333 y FP(\(2\)")37 b(is)g(an)g(algebra)e(generated)h(b)n(y)h(the)h(elemen)n (ts)f FO(v)k FP(and)c FO(n)p FP(,)i FO(v)i FP(b)r(eing)118 432 y(unitary)27 b(and)h FO(n)f FP(b)r(eing)h(normal,)f(whic)n(h)g (satisfy)g(the)h(relation)962 604 y FO(v)s(n)23 b FP(=)g FO(q)s(nv)s(;)180 b(q)26 b(>)d FP(0)p FO(;)645 b FP(\(2.64\))118 777 y(sub)5 b(ject)39 b(to)g(an)f(additional)g(condition:)60 b(the)39 b(sp)r(ectrum)g(of)f(the)i(op)r(erator)118 876 y FQ(j)p FO(n)p FQ(j)34 b FP(lies)g(in)h FQ(f)p FO(q)584 846 y FM(n)629 876 y FO(;)27 b(n)34 b FQ(2)h FJ(Z)o FQ(g)p FP(.)51 b(Although)34 b(this)g(condition)g(is)g(rather)g(natural,)118 976 y(w)n(e)f(will)h(not)f(assume)g(it)h(in)g(our)f(study)g(of)h(the)g 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Fw(N)5 b Fy(+)p Fw(I)p 1915 2604 122 3 v 1961 2637 a Fy(2)2050 2651 y FO(v)s(:)247 b FP(\(2.66\))118 2824 y FR(3.)34 b FP(In)23 b(the)f(sequel)g(w)n(e)f(consider)g(a)h FQ(\003)p FP(-algebra)d FB(A)j FP(generated)f(b)n(y)h(the)g(elemen)n (ts)118 2923 y FO(v)s FP(,)30 b FO(n)p FP(,)g FO(N)9 b FP(,)30 b(and)f FO(b)g FP(suc)n(h)g(that)g FO(v)k FP(is)c(unitary)-7 b(,)30 b FO(N)38 b FP(is)29 b(self-adjoin)n(t,)g FO(n)h FP(and)f FO(b)g FP(are)118 3023 y(normal,)e(and)g(the)h(generators)d (satisfy)j(relations)e(\(2.64\))o(,)i(\(2.65\))o(,)g(\(2.66\))o(.)243 3122 y(Instead)j(of)g FO(b)g FP(and)g FO(N)9 b FP(,)33 b(in)n(tro)r(duce)e(the)h(new)f(generators,)f FO(d)g FP(=)f FO(bv)2372 3092 y FN(\003)2441 3122 y FP(and)118 3222 y FO(M)j FP(=)22 b(\(1)d FQ(\000)f FO(q)534 3192 y FL(2)571 3222 y FP(\))c FO(q)657 3192 y FN(\000)p FL(\()p FM(N)6 b FL(+)p FM(I)f FL(\))p FM(=)p FL(2)976 3222 y FP(.)37 b(Then)28 b(the)g(relations)e(are:)284 3394 y FO(v)s(n)d FP(=)g FO(q)s(nv)s(;)97 b(nn)841 3360 y 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b(mind)h(that)f(the)h(op)r(erator)d FO(M)43 b FP(m)n(ust)34 b(b)r(e)g(p)r(ositiv)n(e)g(for)g(0)f FO(<)g(q)k(<)c FP(1)h(and)118 4048 y(negativ)n(e)e(for)g FO(q)i(>)e FP(1.)51 b(W)-7 b(e)34 b(will)e(consider)g(the)h(case)f(0)f FO(<)g(q)k(<)c FP(1)h(\(the)i(case)118 4147 y FO(q)26 b(>)d FP(1)k(is)h(quite)f(similar\).)p eop %%Page: 171 175 171 174 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(171)118 333 y FR(Lemma)32 b(12.)42 b FC(Supp)l(ose)33 b(we)f(have)h(a)f FQ(\003)p FC(-r)l(epr)l(esentation)g(of)g(the)h(Heisenb)l(er)l(g)118 432 y(r)l(elations)j FP(\()p FC(gener)l(al)t(ly)29 b(sp)l(e)l(aking,)h (by)e(unb)l(ounde)l(d)g(op)l(er)l(ators)7 b FP(\))29 b FC(and)f(ther)l(e)g(ex-)118 532 y(ists)i(a)g(ve)l(ctor)g FO(f)i FQ(2)23 b FO(H)36 b FC(such)30 b(that)g FO(f)i FQ(2)23 b FP(k)n(er)13 b FO(n)29 b FC(and)963 715 y FO(nd)-14 b(f)27 b FQ(\000)18 b FO(q)1233 681 y FL(3)p FM(=)p FL(2)1338 715 y FO(dnf)31 b FP(=)23 b FO(M)9 b(f)118 898 y FP(\()p FC(it)35 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FP(=)23 b FO(n)1173 2932 y FN(\003)1211 2966 y FO(n;)97 b(y)s(M)31 b FP(=)23 b FO(M)9 b(y)s(;)96 b FP([)p FO(y)s(;)14 b(y)2026 2932 y FN(\003)2063 2966 y FP(])23 b(=)g(0)p FO(;)493 3107 y(v)s(M)31 b FP(=)23 b FO(q)776 3073 y FL(1)p FM(=)p FL(2)880 3107 y FO(M)9 b(v)s(;)97 b(v)s(y)26 b FP(=)d FO(q)s(y)s(v)s(;)97 b(nM)31 b FP(=)23 b FO(q)1868 3073 y FN(\000)p FL(1)p FM(=)p FL(2)2024 3107 y FO(M)9 b(n;)899 3232 y(n)949 3197 y FN(\003)987 3232 y FO(y)26 b FP(=)c FO(q)s(y)s(n)1275 3197 y FN(\003)1313 3232 y FO(;)97 b(ny)26 b FP(=)c FO(q)s(y)s(n:)569 b FP(\(2.67\))243 3415 y(F)-7 b(rom)27 b(no)n(w)g(on,)g(w)n(e)g(deal)h (with)g FQ(\003)p FP(-represen)n(tations)d(of)i(the)h(algebra)e FB(A)p FP(.)118 3582 y FR(Prop)s(osition)e(53.)35 b FC(Ther)l(e)26 b(ar)l(e)e(no)h(r)l(epr)l(esentations)g(of)43 b FP(\(2.67\))23 b FC(by)i(b)l(ounde)l(d)118 3682 y(op)l(er)l(ators.)118 3848 y(Pr)l(o)l(of.)43 b FP(Indeed,)26 b(since)g FO(M)9 b(u)22 b FP(=)h FO(q)1159 3818 y FN(\000)p FL(1)p FM(=)p FL(2)1315 3848 y FO(uM)34 b FP(and)25 b FO(u)g FP(is)g(unitary)-7 b(,)26 b(the)g(sp)r(ectrum)118 3948 y(of)k FO(M)39 b FP(is)30 b(in)n(v)-5 b(arian)n(t)29 b(under)h(m)n(ultiplication)h(b)n (y)f FO(q)1704 3918 y FN(\000)p FL(1)p FM(=)p FL(2)1860 3948 y FP(.)45 b(But)30 b(since)g FO(M)36 b(>)27 b FP(0,)118 4048 y(the)k(sp)r(ectrum)g(of)g FO(M)40 b FP(do)r(es)30 b(not)h(con)n(tain)f(zero)g(and,)i(th)n(us,)g(is)e(un)n(b)r(ounded.)p 2514 4147 4 57 v 2518 4095 50 4 v 2518 4147 V 2567 4147 4 57 v eop %%Page: 172 176 172 175 bop 118 100 a FP(172)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(R)l(emark)44 b FP(41)p FC(.)h FP(As)33 b(one)g(can)g(see,)i(relations)d(\(2.67\))h(ha)n(v)n(e)f(the)i (form)f(\(2.57\))o(.)118 432 y(This)e(fact)h(enables)f(one)g(to)g (select)g(the)h(class)f(of)g(\\go)r(o)r(d")f(un)n(b)r(ounded)i(rep-)118 532 y(resen)n(tations,)g(whic)n(h)f(can)h(b)r(e)g(con)n(tin)n(ued)g(to) f(represen)n(tations)f(of)i(the)g(cor-)118 632 y(resp)r(onding)i(b)r (ounded)g(op)r(erators)f(\(see)h(Section)g(2.4.3\),)h(and)f(to)h(list)f (irre-)118 731 y(ducible)28 b(represen)n(tations)e(from)h(this)h (class.)243 868 y(In)n(tro)r(duce)f(the)h(follo)n(wing)e(op)r(erators)g (on)h(the)h(space)f FO(l)1985 880 y FL(2)2022 868 y FP(\()p FJ(Z)p FP(\):)223 1054 y FO(S)5 b(e)318 1066 y FM(k)381 1054 y FP(=)23 b FO(e)508 1066 y FM(k)q FL(+1)633 1054 y FO(;)97 b(T)12 b(e)853 1066 y FM(k)915 1054 y FP(=)23 b FO(k)s(e)1088 1066 y FM(k)1128 1054 y FO(;)97 b(Qe)1353 1066 y FM(k)1416 1054 y FP(=)23 b FO(q)1544 1020 y FM(k)q(=)p FL(2)1652 1054 y FO(e)1691 1066 y FM(k)1755 1054 y FP(=)f FO(e)1881 1020 y FM(T)7 b(=)p FL(2)1998 1054 y FO(e)2037 1066 y FM(k)2244 1054 y FO(k)26 b FQ(2)d FJ(Z)p FO(:)118 1243 y FR(Theorem)28 b(40.)38 b FC(A)n(l)t(l)28 b(irr)l(e)l(ducible)h FQ(\003)p FC(-r)l(epr)l(esentations)e(of)h(the)g(algebr)l(a)g FB(A)p FC(,)h(up)118 1342 y(to)h(unitary)g(e)l(quivalenc)l(e,)h(ar)l (e:)243 1471 y FO(a)p FP(\))f FC(r)l(epr)l(esentations)f(on)h FO(l)1067 1483 y FL(2)1104 1471 y FP(\()p FJ(Z)p FP(\))13 b FQ(\012)18 b FO(l)1351 1483 y FL(2)1388 1471 y FP(\()p FJ(Z)o FP(\):)470 1657 y FO(n)23 b FP(=)g FO(\025)14 b(S)23 b FQ(\012)18 b FO(Q)916 1623 y FL(2)953 1657 y FO(;)99 b(v)26 b FP(=)d FO(S)1285 1623 y FN(\003)1341 1657 y FQ(\012)18 b FO(S)1480 1623 y FN(\003)1518 1657 y FO(;)99 b(N)32 b FP(=)22 b FO(\013)d FQ(\000)f FO(T)30 b FQ(\012)18 b FO(I)7 b(;)735 1798 y(b)22 b FP(=)h FO(q)921 1764 y FN(\000)p FM(\013=)p FL(2)p FN(\000)p FL(1)1172 1798 y FO(\025)1220 1764 y FN(\000)p FL(1)1310 1798 y FO(Q)p FP(\()p FO(S)1464 1764 y FN(\003)1502 1798 y FP(\))1534 1764 y FL(2)1590 1798 y FQ(\012)18 b FO(Q)1739 1764 y FN(\000)p FL(2)1827 1798 y FO(S)1883 1764 y FN(\003)1921 1798 y FP(;)243 1986 y FO(b)p FP(\))29 b FC(r)l(epr)l(esentations)h(on) g FO(l)1059 1998 y FL(2)1096 1986 y FP(\()p FJ(Z)p FP(\))12 b FQ(\012)18 b FO(l)1342 1998 y FL(2)1379 1986 y FP(\()p FJ(Z)p FP(\))13 b FQ(\012)18 b FO(l)1626 1998 y 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FM(n)788 3572 y FP(=)22 b FO(S)h FQ(\012)18 b FO(I)7 b(;)97 b(u)1243 3584 y FM(y)1306 3572 y FP(=)22 b(0)p FO(;)97 b(v)26 b FP(=)d FO(S)1765 3537 y FN(\003)1821 3572 y FQ(\012)18 b FO(S)1960 3537 y FN(\003)1998 3572 y FO(:)118 3758 y FP(Then,)28 b(since)686 3945 y FO(b)23 b FP(=)f FO(M)922 3910 y FN(\000)p FL(1)1011 3945 y FO(n)1061 3910 y FN(\000)p FL(1)1150 3945 y FO(y)s(v)f FP(+)d(\(1)g FQ(\000)h FO(q)1554 3910 y FL(2)1591 3945 y FP(\))1623 3910 y FN(\000)p FL(1)1712 3945 y FO(n)1762 3910 y FN(\000)p FL(1)1851 3945 y FO(M)9 b(v)s(:)118 4131 y FP(w)n(e)26 b(get)h(the)g(necessary)d(expressions)h(b)n(y)h(using)h(the)f (expression)g(for)g FO(M)9 b FP(.)p 2514 4131 4 57 v 2518 4078 50 4 v 2518 4131 V 2567 4131 4 57 v eop %%Page: 173 177 173 176 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(173)118 333 y FR(2.4.6)94 b(Wic)m(k)32 b(algebras)g(related)g(to)f(dynamical)g(systems)118 486 y(1.)52 b FP(In)33 b(this)g(section)f(w)n(e)h(consider)e(some)h 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y FM(i)2044 4048 y FP(,)k FO(a)2152 4018 y FN(\003)2152 4069 y FM(i)2190 4048 y FP(,)h(where)c(in)118 4147 y(eac)n(h)33 b(monomial,)h(the)f(v)-5 b(ariables)32 b FO(a)1266 4159 y FM(i)1327 4147 y FP(are)h(placed)g(to) g(the)g(left)h(of)g FO(a)2287 4117 y FN(\003)2287 4169 y FM(j)2325 4147 y FP(.)54 b(Suc)n(h)p eop %%Page: 174 178 174 177 bop 118 100 a FP(174)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(monomials)19 b(are)g(called)g(Wic)n(k)h (ordered)f(monomials,)h(and)g(they)g(form)f(a)h(basis)118 432 y(in)28 b FB(W)p FP(.)243 532 y(When)33 b(studying)g(the)g(prop)r (erties)f(of)g FB(W)p FP(,)j(one)d(can)h(\014nd)g(the)g(follo)n(wing) 118 632 y(op)r(erators)26 b(useful:)247 797 y FO(T)20 b FP(:)28 b FB(H)18 b FQ(\012)g FB(H)23 b FQ(\000)-49 b(!)23 b FB(H)18 b FQ(\012)g FB(H)q FO(;)96 b(T)12 b(e)1234 809 y FM(k)1292 797 y FQ(\012)18 b FO(e)1414 809 y FM(l)1462 797 y FP(=)1550 718 y Fz(X)1573 895 y FM(i;j)1683 797 y FO(T)1744 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b(dynamical)h(systems)796 b FP(175)243 333 y(In)30 b(the)h(case)e (where)h FO(d)d FP(=)g(2,)k(condition)f(1)g(is)g(called)g(the)g (\\linear",)g(con-)118 432 y(dition)e(2)f(is)h(called)f(\\quadratic".) 118 585 y FR(3.)36 b FP(Let)28 b(us)g(consider)e(the)i(follo)n(wing)f (class)f(of)i(Wic)n(k)f(algebras:)348 847 y FO(x)395 813 y FN(\003)395 867 y FM(i)434 847 y FO(x)481 859 y FM(i)532 847 y FP(=)c(1)18 b(+)805 743 y FM(d)763 768 y Fz(X)765 945 y FM(j)s FL(=1)896 847 y FO(\013)949 859 y FM(ij)1008 847 y FO(x)1055 859 y FM(j)1090 847 y FO(x)1137 813 y FN(\003)1137 867 y FM(j)1176 847 y FO(;)97 b(x)1343 813 y FN(\003)1343 867 y FM(i)1381 847 y FO(x)1428 859 y FM(j)1487 847 y FP(=)23 b FO(\025)1623 859 y FM(ij)1682 847 y FO(q)1719 859 y FM(ij)1777 847 y FO(x)1824 859 y FM(j)1859 847 y FO(x)1906 813 y FN(\003)1906 867 y FM(i)1945 847 y FO(;)180 b(i)23 b FQ(6)p FP(=)f FO(j;)118 1132 y FP(0)31 b FO(<)g(\013)340 1144 y FM(ii)422 1132 y FO(<)g FP(1,)i FO(q)653 1144 y FM(ij)743 1132 y FP(=)e 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1133 y FR(Prop)s(osition)25 b(57.)36 b FC(L)l(et)25 b FO(\013)g FC(b)l(e)h(a)f(c)l(anonic)l(al)i(solution.) 37 b(Then)26 b(it)g(is)f(inde)l(c)l(om-)118 1233 y(p)l(osable)31 b(if)g(and)f(only)h(if)f FO(\013)958 1245 y FL(21)1052 1233 y FP(=)22 b FQ(\001)14 b(\001)g(\001)23 b FP(=)g FO(\013)1400 1245 y FM(d)p FL(1)1495 1233 y FP(=)g FO(\013)1636 1245 y FL(1)1691 1233 y FQ(\000)18 b FP(1)p FC(.)243 1391 y FP(Let)28 b FO(\013)g FP(b)r(e)g(a)f(canonical)g(solution,)g FO(A)d FP(=)f(\()p FO(\013)1616 1403 y FM(ij)1675 1391 y FP(\).)37 b(It)28 b(follo)n(ws)f(from)h(Prop)r(o-)118 1490 y(sition)34 b(57)g(and)g(Remark)g(44,)h(that)g(w)n(e)f(can)g(supp) r(ose)g(that,)i(for)e(an)n(y)g(\014xed)118 1590 y FO(j)5 b FP(,)40 b(all)e(non-zero)e FO(\013)746 1602 y FM(ij)805 1590 y FP(,)k FO(i)g(>)f(j)5 b FP(,)41 b(are)36 b(placed)i(b)r(efore)f (all)h(zero)r(es.)66 b(Consider)114 1681 y FO(~)118 1703 y(k)46 b FP(=)c(\()p FO(k)389 1715 y FL(1)426 1703 y FO(;)14 b(:)g(:)g(:)28 b(;)14 b(k)668 1715 y FM(d)p FN(\000)p FL(1)792 1703 y FP(\),)42 b(where)d FO(i)j FQ(\024)h FO(k)1363 1715 y FM(i)1433 1703 y FQ(\024)f FO(d)e FP(are)e(natural)h (n)n(um)n(b)r(ers)g(con-)118 1803 y(structed)e(as)e(follo)n(ws:)54 b(if,)39 b(for)d(a)h(\014xed)f FO(j)42 b FP(and)36 b(all)g FO(i)i(>)f(j)5 b FP(,)39 b FO(\013)2111 1815 y FM(ij)2207 1803 y FP(=)f(0,)g(then)118 1902 y FO(k)161 1914 y FM(j)219 1902 y FP(=)23 b(0;)i(otherwise)f FO(k)807 1914 y FM(j)867 1902 y FP(is)h(the)g(greatest)f(n)n(um)n(b)r(er)g FO(l)i FP(suc)n(h)f(that)g FO(\013)2165 1914 y FM(lj)2244 1902 y FP(=)e FO(\013)2385 1914 y FM(j)2433 1902 y FQ(\000)13 b FP(1.)118 2016 y(The)28 b(c)n(haracteristic)d(prop)r(ert)n(y)i(of) 1229 1994 y FO(~)1233 2016 y(k)k FP(is)c(the)h(follo)n(wing.)118 2173 y FR(Prop)s(osition)i(58.)41 b FC(If)48 b FO(i)22 b(>)h(j)35 b FC(and)30 b FO(i)22 b FQ(\024)h FO(k)1454 2185 y FM(j)1489 2173 y FC(,)30 b(then)g FO(k)1772 2185 y FM(i)1823 2173 y FQ(\024)23 b FO(k)1954 2185 y FM(j)1989 2173 y FC(.)243 2345 y FP(Con)n(v)n(ersely)-7 b(,)20 b(let)783 2323 y FO(~)788 2345 y(k)k FP(b)r(e)d(a)g(v)n(ector)f(with)i (the)f(c)n(haracteristic)f(prop)r(ert)n(y)-7 b(,)21 b(and)118 2444 y FO(A)33 b FP(=)g(\()p FO(\013)396 2456 y FM(ij)455 2444 y FP(\))h(b)r(e)g(a)f(matrix)g(suc)n(h)g(that)h FO(\013)1423 2456 y FM(ii)1508 2444 y FP(=)e FO(\013)1658 2456 y FM(i)1686 2444 y FP(,)j FO(\013)1797 2456 y FM(ij)1889 2444 y FP(=)e(0,)h FO(i)f(<)g(j)5 b FP(.)55 b(Then,)118 2544 y FO(\013)171 2556 y FM(ij)268 2544 y FP(=)38 b(0)f FO(;)14 b FQ(8)p FO(i)37 b(>)h(j)k FP(if)c FO(k)908 2556 y FM(j)982 2544 y FP(=)g FO(j)5 b FP(;)41 b(otherwise)36 b FO(\013)1620 2556 y FM(lj)1716 2544 y FP(=)i FO(\013)1872 2556 y FM(j)1931 2544 y FQ(\000)25 b FP(1,)39 b FO(j)k(<)38 b(l)i FQ(\024)f FO(k)2517 2556 y FM(j)2552 2544 y FP(,)118 2644 y FO(\013)171 2656 y FM(lj)267 2644 y FP(=)f(0,)i FO(l)g(>)f(k)687 2656 y FM(j)722 2644 y FP(.)66 b(Then)37 b(it)h(is)f(easy)g(to)g(v)n(erify)f(that)i FO(A)f FP(is)g(a)g(matrix)g (of)118 2757 y(a)c(canonical)g(solution.)55 b(W)-7 b(e)34 b(will)g(denote)f(suc)n(h)h(a)f(matrix)g(b)n(y)h FO(A)p FP(\()2271 2735 y FO(~)2275 2757 y(k)s FP(\).)56 b(The)118 2856 y(follo)n(wing)27 b(statemen)n(t)g(has)g(b)r(een)h(pro)n(v)n(ed.) 118 3014 y FR(Theorem)c(41.)35 b FC(L)l(et)23 b FO(\013)i FC(b)l(e)f(a)g(solution)h(of)g(system)30 b FP(\(2.70\))o FC(.)37 b(Then)25 b(ther)l(e)f(exist)118 3124 y FO(\031)i FQ(2)e FO(S)321 3136 y FM(d)382 3124 y FC(and)541 3102 y FO(~)545 3124 y(k)i FC(having)e(the)f(char)l(acteristic)i(pr)l(op)l (erty)f(so)f(that)g FP(\()p FO(\013)2175 3094 y FM(\031)2175 3146 y(ij)2234 3124 y FP(\))g(=)g FO(A)p FP(\()2467 3102 y FO(~)2471 3124 y(k)s FP(\))p FC(.)118 3247 y(Conversely,)37 b(for)d(any)867 3225 y FO(~)871 3247 y(k)i FC(with)e(char)l(acteristic) h(pr)l(op)l(erty,)h FO(A)30 b FP(=)g FO(A)p FP(\()2285 3225 y FO(~)2289 3247 y(k)s FP(\))k FC(gives)118 3346 y(a)c(solution.)118 3504 y FR(5.)50 b FP(No)n(w)32 b(w)n(e)f(describ)r (e)h(irreducible)g(represen)n(tations)e(of)i(the)g(algebras)e(ob-)118 3604 y(tained.)81 b(Let)43 b FO(A)48 b FP(=)f FO(A)p FP(\()924 3582 y FO(~)928 3604 y(k)t FP(\),)f FB(U)i FP(=)f FB(U)p FP(\()p FO(A;)14 b FP(\003\).)82 b(Then)43 b FB(U)g FP(has)f(the)g(largest)118 3703 y(quadratic)27 b(ideal)g(generated)f(b)n(y)711 3875 y FO(X)780 3887 y FM(ij)862 3875 y FP(=)c FO(x)996 3887 y FM(j)1032 3875 y FO(x)1079 3887 y FM(i)1125 3875 y FQ(\000)c FO(\025)1256 3887 y FM(ij)1315 3875 y FO(q)1352 3887 y FM(ij)1424 3875 y FO(x)1471 3887 y FM(i)1500 3875 y FO(x)1547 3887 y FM(j)1582 3875 y FO(;)180 b(i)23 b(<)f(j:)118 4048 y FR(Prop)s(osition)32 b(59.)42 b FC(L)l(et)31 b FO(\031)s FP(\()p FQ(\001)p FP(\))h FC(b)l(e)f(a)h(b)l(ounde)l(d)f(r)l(epr)l (esentation)h(of)50 b FB(U)p FP(\()p FO(A;)14 b FP(\003\))p FC(.)118 4147 y(Then)31 b FO(\031)s FP(\()p FO(X)486 4159 y FM(ij)544 4147 y FP(\))24 b(=)e(0)p FC(.)p eop %%Page: 178 182 178 181 bop 118 100 a FP(178)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(R)l(emark)34 b FP(45)p FC(.)j FP(The)22 b(pro)r(of)g(basically)g(coincides)g(with)h(the)g(pro)r(of)e (of)i(the)g(anal-)118 432 y(ogous)g(fact)j(for)e(the)h(t)n(wisted)g (comm)n(utation)g(relations)e(\(see)i(Section)50 b(2.4.2\))118 532 y(giv)n(en)27 b(b)n(y)g(\003)c(=)g(1,)k FO(A)c FP(=)g FO(A)p FP(\()p FO(d;)14 b(:)g(:)g(:)g(;)g(d)p FP(\),)28 b(and)g FO(\013)1547 544 y FM(j)1605 532 y FP(=)22 b FO(\026)1742 502 y FL(2)1780 532 y FP(.)243 672 y(This)c(means)g(that)h (in)g(order)e(to)h(describ)r(e)h(irreducible)e(represen)n(tations)g(of) 118 772 y FB(U)p FP(,)27 b(it)f(is)g(necessary)f(to)g(describ)r(e)h (families)g(of)g(op)r(erators)e FQ(f)p FO(X)2047 784 y FM(i)2074 772 y FO(;)k(i)22 b FP(=)h(1)p FO(;)14 b(:)g(:)g(:)f(;)h(d) p FQ(g)118 872 y FP(suc)n(h)27 b(that)507 1062 y FO(X)583 1028 y FN(\003)576 1083 y FM(i)620 1062 y FO(X)689 1074 y FM(i)740 1062 y FP(=)22 b(1)c(+)g FO(\013)1023 1074 y FM(i)1051 1062 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FO(;)83 b(i)23 b(<)f(j;)28 b(k)1348 1578 y FM(i)1399 1566 y FQ(\025)23 b FO(j;)894 1685 y FP(1)p FO(;)122 b FP(otherwise)o FO(:)2363 1622 y FP(\(2.71\))118 1887 y(Let)25 b FO(X)340 1857 y FN(\003)333 1909 y FM(i)401 1887 y FP(=)d FO(U)545 1899 y FM(i)573 1887 y FO(C)632 1899 y FM(i)685 1887 y FP(b)r(e)j(the)h(p)r(olar)e(decomp)r(osition.)35 b(Then)25 b(system)g(\(2.71\))f(can)118 1987 y(b)r(e)k(rewritten)f(in)h (an)g(equiv)-5 b(alen)n(t)27 b(form:)640 2177 y FR(C)p FO(U)775 2143 y FN(\003)766 2198 y FM(i)836 2177 y FP(=)22 b FO(U)989 2143 y FN(\003)980 2198 y FM(i)1027 2177 y FR(F)1087 2189 y FM(i)1115 2177 y FP(\()p FR(C)p FP(\))p FO(;)98 b FR(C)23 b FP(=)f(\()p FO(C)1645 2143 y FL(2)1639 2198 y(1)1683 2177 y FO(;)14 b(:)g(:)g(:)27 b(;)14 b(C)1946 2143 y FL(2)1940 2198 y FM(d)1984 2177 y FP(\))p FO(;)230 2315 y FP([)p FO(C)312 2327 y FM(i)340 2315 y FO(;)g(C)436 2327 y FM(j)471 2315 y FP(])23 b(=)g(0)p FO(;)96 b(U)823 2327 y FM(i)850 2315 y FO(U)907 2327 y FM(j)965 2315 y FP(=)p 1053 2247 49 4 v 23 w FO(\025)1101 2327 y FM(ij)1160 2315 y FO(U)1217 2327 y FM(j)1251 2315 y FO(U)1308 2327 y FM(i)1336 2315 y FO(;)h(U)1513 2327 y FM(i)1540 2315 y FO(U)1606 2281 y FN(\003)1597 2335 y FM(j)1667 2315 y FP(=)23 b FO(\025)1803 2327 y FM(ij)1861 2315 y FO(U)1927 2281 y FN(\003)1918 2335 y FM(j)1965 2315 y FO(U)2022 2327 y FM(i)2050 2315 y FO(;)180 b(i)22 b(<)h(j;)1093 2450 y FR(F)1153 2462 y FM(i)1180 2450 y FP(\()p FO(x)1259 2462 y FL(1)1297 2450 y FO(;)14 b(:)g(:)g(:)g(;)g(x)1529 2462 y FM(d)1568 2450 y FP(\))393 2584 y(=)481 2517 y Fz(\000)519 2584 y FO(x)566 2596 y FL(1)604 2584 y FO(;)g(:)g(:)g(:)f (;)h(x)835 2596 y FM(i)p FN(\000)p FL(1)948 2584 y FO(;)g(F)1038 2596 y FM(i)1066 2584 y FP(\()p FO(x)1145 2596 y FL(1)1183 2584 y FO(;)g(:)g(:)g(:)g(;)g(x)1415 2596 y FM(d)1454 2584 y FP(\))p FO(;)g(q)1563 2550 y FL(2)1560 2605 y FM(ii)p FL(+1)1695 2584 y FO(x)1742 2596 y FM(i)p FL(+1)1855 2584 y FO(;)g(:)g(:)g(:)f(;)h(q)2079 2550 y FL(2)2076 2605 y FM(id)2138 2584 y 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y(satisfy)j(the)i(relations)d FO(U)930 3557 y FM(i)958 3545 y FO(U)1015 3557 y FM(j)1081 3545 y FP(=)g FO(\025)1225 3557 y FM(ij)1284 3545 y FO(U)1341 3557 y FM(j)1376 3545 y FO(U)1433 3557 y FM(i)1460 3545 y FP(,)j FO(i)d(<)h(j)5 b FP(.)52 b(First,)34 b(in)n(tro)r(duce)e(some)118 3645 y(notations:)k(let)28 b FO(D)r FP(\()p FO(\026)p FP(\))g(denote)g(an)f (op)r(erator)f(in)i FO(l)1688 3657 y FL(2)1725 3645 y FP(\()p FO(N)9 b FP(\))28 b(giv)n(en)f(b)n(y)803 3835 y FO(D)r FP(\()p FO(\026)p FP(\))p FO(e)1027 3847 y FM(n)1095 3835 y FP(=)c FO(\026)1233 3801 y FM(n)p FN(\000)p FL(1)1363 3835 y FO(e)1402 3847 y FM(n)1447 3835 y FO(;)180 b(n)23 b FQ(2)g FJ(N)t FO(;)795 4043 y(D)r FP(\()p FO(j;)14 b(k)1012 4055 y FM(i)1040 4043 y FP(\))23 b(=)1183 3902 y Fz(\()1250 3987 y FP(1)p FO(;)265 b(j)28 b(>)22 b(k)1772 3999 y FM(i)1800 3987 y FO(;)1250 4106 y(D)r FP(\()p FO(\013)1406 4118 y FM(j)1441 4106 y FP(\))p FO(;)84 b(j)28 b FQ(\024)22 b FO(k)1772 4118 y FM(i)1800 4106 y FO(;)p eop %%Page: 179 183 179 182 bop 118 100 a FK(2.4.)36 b(Man)n(y-dimensional)26 b(dynamical)h(systems)796 b FP(179)118 333 y(and)23 b(let)g FO(S)k FP(stand)c(for)f(the)h(unilateral)f(shift.)36 b(Then)23 b FO(D)r FP(\()p FO(f)1893 345 y FM(j)1928 333 y FP(\))14 b FO(e)2013 345 y FM(n)2081 333 y FP(=)22 b FO(f)2218 297 y FM(n)p FN(\000)p FL(1)2209 356 y FM(j)2348 333 y FP(\(0\))14 b FO(e)2507 345 y FM(n)2552 333 y FP(,)118 432 y(where)27 b FO(f)399 444 y FM(j)434 432 y FP(\()p FO(x)p FP(\))d(=)f(1)17 b(+)i FO(\013)853 444 y FM(j)888 432 y FO(x)p FP(,)28 b(and)f FO(f)1197 402 y FM(n)1242 432 y FP(\()p FQ(\001)p FP(\))h(denotes)g(the)g FO(n)p FP(-th)f(iteration)g(of)h FO(f)9 b FP(.)243 549 y(Let)27 b(1)c FQ(\024)g FO(i)573 561 y FL(1)633 549 y FO(<)f FQ(\001)14 b(\001)g(\001)23 b FO(<)g(i)957 561 y FM(l)1005 549 y FQ(\024)f FO(d)28 b FP(b)r(e)g(natural)f(n)n(um)n(b)r(ers)g(suc)n (h)h(that)709 766 y FO(k)752 778 y FM(i)775 786 y Fw(j)830 766 y FP(+)18 b(1)k FQ(\024)h FO(i)1094 778 y FM(j)s FL(+1)1213 766 y FO(;)179 b(j)28 b FP(=)23 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(l)20 b FQ(\000)e FP(1)p FO(:)118 1003 y FP(Fix)25 b(suc)n(h)g(a)g(family)-7 b(.)36 b(Denote)25 b(\010)e(=)1247 941 y Fz(S)1316 961 y FM(l)1316 1028 y(j)s FL(=1)1435 1003 y FQ(f)p FO(i)1506 1015 y FM(j)1554 1003 y FP(+)13 b(1)p FO(;)h(:)g(:)g(:)26 b(;)14 b(k)1914 1015 y FM(i)1937 1023 y Fw(j)1973 1003 y FQ(g)p FP(.)36 b(Construct)24 b(the)118 1103 y(follo)n(wing)j(irreducible)g(represen)n (tation)e(for)j(a)f(\014xed)g(family)h(of)g FQ(f)p FO(i)2239 1115 y FL(1)2289 1103 y FO(:)14 b(:)g(:)27 b(;)14 b(i)2479 1115 y FM(l)2504 1103 y FQ(g)p FP(:)216 1320 y FO(C)275 1332 y FM(j)334 1320 y FP(=)22 b FO(U)478 1332 y FM(j)536 1320 y FP(=)h(0)p FO(;)179 b FQ(8)p FO(j)27 b FQ(2)d FP(\010)p FO(;)208 1537 y(C)273 1503 y FL(2)267 1558 y FM(j)334 1537 y FP(=)485 1430 y FM(j)s FN(\000)p FL(1)480 1458 y Fz(O)421 1640 y FM(i)p FL(=1)p FM(;i)7 b(=)-41 b FN(2)q FL(\010)678 1537 y FO(D)r FP(\()p FO(j;)14 b(k)895 1549 y FM(i)924 1537 y FP(\))k FQ(\012)g FO(D)r FP(\()p FO(f)1201 1549 y FM(j)1236 1537 y FP(\))h FQ(\012)f FO(I)26 b FQ(\012)18 b(\001)c(\001)g(\001)k(\012)g FO(I)7 b(;)180 b(j)28 b FQ(6)p FP(=)22 b FO(i)2137 1549 y FM(k)2178 1537 y FO(;)207 1848 y(U)273 1813 y FN(\003)264 1868 y FM(j)334 1848 y FP(=)485 1740 y FM(j)s FN(\000)p FL(1)480 1769 y Fz(O)421 1947 y FM(i)p FL(=1)p FM(;i)p FN(62)p FL(\010)678 1848 y FO(D)r FP(\()p FO(\025)829 1860 y FM(ij)888 1848 y FP(\))d FQ(\012)f FO(S)23 b FQ(\012)18 b FO(I)26 b FQ(\012)18 b(\001)c(\001)g(\001)k(\012)g FO(I)7 b(;)180 b(j)28 b FQ(6)p FP(=)23 b FO(i)1947 1860 y FM(k)1987 1848 y FO(;)190 2087 y(U)256 2053 y FN(\003)247 2108 y FM(i)270 2117 y Fw(k)334 2087 y FP(=)494 2009 y Fz(O)421 2187 y FM(i<i)519 2196 y Fw(k)557 2187 y FM(;i)p FN(62)p FL(\010)706 2087 y FO(D)r FP(\()p FO(\025)857 2099 y FM(ii)903 2108 y Fw(k)945 2087 y FP(\))c FQ(\012)1151 2009 y Fz(O)1079 2187 y FM(i>i)1177 2196 y Fw(k)1214 2187 y FM(;i)p FN(62)p FL(\010)1363 2087 y FO(D)r FP(\()p 1466 2020 49 4 v FO(\025)1515 2099 y FM(ii)1561 2108 y Fw(k)1602 2087 y FP(\))g FQ(\012)1750 2066 y FP(^)1736 2087 y FO(U)1802 2053 y FN(\003)1793 2108 y FM(i)1816 2117 y Fw(k)1857 2087 y FO(;)180 b(k)25 b FP(=)e(1)p FO(;)14 b(:)g(:)g(:)f(;)h(l)r(;)187 2350 y(C)252 2315 y FL(2)246 2370 y FM(i)269 2379 y Fw(k)334 2350 y FP(=)541 2294 y(1)p 431 2331 261 4 v 431 2407 a(1)k FQ(\000)g FO(\013)627 2419 y FM(i)650 2428 y Fw(k)788 2271 y Fz(O)715 2450 y FM(i<i)813 2459 y Fw(k)850 2450 y FM(;i)p FN(62)p FL(\010)1000 2350 y FO(D)r FP(\()p FO(i;)c(k)1212 2362 y FM(i)1235 2371 y Fw(k)1276 2350 y FP(\))k FQ(\012)g FO(I)26 b FQ(\012)18 b(\001)c(\001)g(\001)k(\012)g FO(I)7 b(;)97 b(k)26 b FP(=)d(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(l)r(;)118 2674 y FP(where)30 b FQ(f)417 2653 y FP(^)403 2674 y FO(U)460 2686 y FM(i)483 2695 y Fw(k)523 2674 y FQ(g)g FP(is)g(an)g(irreducible)g(family)g(of)g(unitary)g(op)r (erators)f(satisfying)118 2782 y(the)f(relations)613 2761 y(^)598 2782 y FO(U)655 2794 y FM(i)697 2761 y FP(^)683 2782 y FO(U)740 2794 y FM(j)798 2782 y FP(=)22 b FO(\025)933 2794 y FM(ij)1006 2761 y FP(^)992 2782 y FO(U)1049 2794 y FM(j)1098 2761 y FP(^)1084 2782 y FO(U)1141 2794 y FM(j)1175 2782 y FP(.)118 3000 y FR(Theorem)37 b(42.)44 b FC(A)n(l)t(l)35 b(irr)l(e)l(ducible)h(r)l(epr)l(esentations)f(of)h (the)f(algebr)l(a)h FB(U)f FC(c)l(an)118 3099 y(b)l(e)i(obtaine)l(d)i (in)e(the)h(way)g(describ)l(e)l(d)h(ab)l(ove)6 b FP(;)42 b FC(mor)l(e)l(over,)f(two)c(r)l(epr)l(esenta-)118 3199 y(tions)g(ar)l(e)h(unitarily)g(e)l(quivalent)g(if)g(and)f(only)h(if)g (they)g(c)l(orr)l(esp)l(ond)g(to)g(the)118 3298 y(same)32 b(family)h FQ(f)p FO(i)657 3310 y FL(1)693 3298 y FO(;)14 b(:)g(:)g(:)g(;)g(i)907 3310 y FM(l)932 3298 y FQ(g)p FC(,)32 b(and)f(the)h(c)l(orr)l(esp)l(onding)h(unitary)e(families)i(ar) l(e)118 3398 y(unitarily)e(e)l(quivalent.)118 3616 y(R)l(emark)36 b FP(46)p FC(.)i FP(1.)d(If)25 b(at)f(least)g(one)g(of)g(the)h FO(\025)1476 3628 y FM(ij)1559 3616 y FP(is)f(not)g(a)g(ro)r(ot)f(of)i (1,)f(then)h(there)118 3715 y(exists)i(a)h(represen)n(tation)d(that)j (is)g(not)f(of)h(t)n(yp)r(e)g(one.)243 3832 y(2.)48 b(If)32 b(all)f FO(\025)610 3844 y FM(ij)701 3832 y FP(are)f(ro)r(ots)h(of)g (1,)h(then)g(the)g(problem)f(of)h(classi\014cation)e(of)118 3932 y(families)c FQ(f)p FO(U)518 3944 y FM(i)545 3932 y FQ(g)f FP(can)g(b)r(e)h(reduced)f(to)h(the)g(case)e(where)i FO(\025)1881 3892 y FM(q)1881 3955 y(ij)1963 3932 y FP(=)c(1;)k(here)f FO(q)h FP(=)d FO(p)2512 3902 y FM(m)118 4031 y FP(for)h(some)f(prime)i FO(p)p FP(.)35 b(In)25 b(this)f(case,)g(the)h(families)f FQ(f)p FO(U)1772 4043 y FM(i)1799 4031 y FQ(g)g FP(can)g(b)r(e)g (describ)r(ed)g(b)n(y)118 4131 y(a)j(simple)h(reduction)f(algorithm.)p eop %%Page: 180 184 180 183 bop 118 100 a FP(180)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FH(2.5)112 b(On)38 b(represen)m(tations)f(of)h (some)f(n)m(uclear)g(algebras)118 514 y FR(2.5.1)94 b(Comm)m(utativ)m (e)30 b(mo)s(dels)118 668 y FP(In)40 b(previous)f(sections,)j (considering)c(represen)n(tations)g(of)i(op)r(erator)e(rela-)118 767 y(tions,)h(w)n(e)e(ha)n(v)n(e)f(seen)h(that)g(the)h(complete)f (unitary)g(description)f(of)h(irre-)118 867 y(ducible)30 b(represen)n(tations)d(is)i(p)r(ossible)g(only)g(if)h(the)g(underlying) f(dynamical)118 967 y(system)g(has)g(a)g(measurable)f(section.)42 b(No)n(w)29 b(w)n(e)g(will)h(lo)r(ok)f(at)g(what)g(can)g(b)r(e)118 1066 y(said)24 b(ab)r(out)h(the)g(relation)f(in)h(the)g(case)f(where)g (the)h(dynamical)f(system)g(do)r(es)118 1166 y(not)f(necessarily)f(ha)n (v)n(e)g(a)h(measurable)f(section.)35 b(Of)23 b(course,)g(one)g(should) g(not)118 1265 y(exp)r(ect)h(a)g(complete)f(unitary)g(description;)i (ho)n(w)n(ev)n(er,)d(w)n(e)i(will)g(see)f(that)h(in)g(a)118 1365 y(certain)j(sense,)g(a)f(uniform)h(realization)f(\(comm)n(utativ)n (e)h(mo)r(del\))h(of)f(all)g(rep-)118 1465 y(resen)n(tations)32 b(as)h(op)r(erators)f(of)h(m)n(ultiplication)h(and)f(w)n(eigh)n(ted)g (shifts)h(can)118 1564 y(b)r(e)28 b(constructed.)243 1664 y(Instead)g(of)g(considering)g(the)h(relation)e(of)i(the)g(form)f FO(X)7 b(X)2106 1634 y FN(\003)2167 1664 y FP(=)25 b FO(F)12 b FP(\()p FO(X)2430 1634 y FN(\003)2467 1664 y FO(X)7 b FP(\))118 1764 y(or)33 b(its)h(m)n(ulti-dimensional)f(v)n (ersion,)h(w)n(e)f(will)h(consider)f(a)g(family)h(of)g(com-)118 1863 y(m)n(uting)40 b(self-adjoin)n(t)f(or)f(normal)h(op)r(erators)e FR(A)43 b FP(=)f(\()p FO(A)1955 1875 y FM(k)1997 1863 y FP(\),)h(and)c(a)g(family)118 1963 y FR(B)23 b FP(=)g(\()p FO(B)392 1975 y FM(j)427 1963 y FP(\))28 b(whic)n(h)f(satisfy)h(more)e (general)h(relations)1011 2144 y FO(A)1073 2156 y FM(k)1114 2144 y FO(B)1177 2156 y FM(j)1235 2144 y FP(=)22 b FO(B)1385 2156 y FM(j)1420 2144 y FO(F)1473 2156 y FM(k)q(j)1545 2144 y FP(\()p FR(A)p FP(\))118 2324 y(\(w)n(e)39 b(allo)n(w)f (families)h FR(A)g FP(and)f FR(B)h FP(to)g(ha)n(v)n(e)f(di\013eren)n(t) h(cardinalit)n(y)-7 b(,)40 b(and)f(no)118 2424 y(conditions)h(on)h(the) g(join)n(t)g(sp)r(ectrum)g(of)g FR(A)g FP(or)f(conditions)g(of)h(the)g (form)118 2524 y(k)n(er)13 b FO(A)305 2536 y FM(k)369 2524 y FP(=)23 b(k)n(er)12 b FO(B)644 2536 y FM(k)713 2524 y FP(are)27 b(assumed\).)243 2623 y(W)-7 b(e)25 b(start)f(with)i(the)f(case)f(of)h(a)f(unitary)g(op)r(erator)g FO(U)33 b FP(and)25 b(a)f(self-adjoin)n(t)118 2723 y(op)r(erator)d FO(A)i FP(satisfying)f(a)g(relation)g(of)h(the)g(form)f FO(AU)32 b FP(=)23 b FO(U)9 b(F)j FP(\()p FO(A)p FP(\),)24 b(where)e(the)118 2823 y(mapping)27 b FO(F)12 b FP(\()p FQ(\001)p FP(\))29 b(is)e(measurable)f(and)i(one-to-one)e(on)h(the)h (sp)r(ectrum)g(of)f FO(A)p FP(.)118 2987 y FR(Theorem)40 b(43.)46 b FC(L)l(et)37 b(a)g(self-adjoint)j(op)l(er)l(ator)e FO(A)g FC(and)g(a)g(unitary)f(op)l(er)l(a-)118 3087 y(tor)k FO(U)50 b FC(satisfy)42 b(the)f(r)l(elation)g FO(AU)53 b FP(=)43 b FO(U)9 b(F)j FP(\()p FO(A)p FP(\))41 b FC(with)h(a)f(me)l (asur)l(able)g(map-)118 3186 y(ping)35 b FO(F)12 b FP(\()p FQ(\001)p FP(\))35 b FC(which)h(is)f(one-to-one)f(on)h(the)f(sp)l(e)l (ctrum)g(of)h FO(A)p FC(.)53 b(The)35 b(sp)l(ac)l(e)g FO(H)118 3286 y FC(c)l(an)42 b(b)l(e)h(uniquely)f(de)l(c)l(omp)l(ose)l (d)i(into)e(a)h(dir)l(e)l(ct)f(sum,)j FO(H)53 b FP(=)45 b FO(H)2220 3298 y FL(1)2284 3286 y FQ(\010)28 b FO(H)2446 3298 y FL(2)2510 3286 y FQ(\010)118 3386 y(\001)14 b(\001)g(\001)h (\010)f FO(H)378 3398 y FN(1)448 3386 y FC(,)29 b(of)g(invariant)g (subsp)l(ac)l(es)7 b FP(;)28 b FC(e)l(ach)h FO(H)1605 3398 y FM(m)1696 3386 y FC(is)g(unitarily)f(e)l(quivalent)h(to)118 3485 y FQ(\010)183 3455 y FM(m)183 3509 y(k)q FL(=1)308 3485 y FO(L)365 3497 y FL(2)401 3485 y FP(\()p FJ(R)488 3455 y FL(1)531 3485 y FO(;)14 b(d\026)661 3497 y FM(m)724 3485 y FP(\))29 b FC(in)f(such)h(a)g(way)g(that)g FO(\026)1529 3497 y FM(m)1592 3485 y FP(\()p FQ(\001)p FP(\))g FC(is)g(a)g(pr)l(ob)l (ability)i FO(F)12 b FC(-quasi-)118 3585 y(invariant)32 b(me)l(asur)l(e,)f FO(m)24 b FP(=)h(1)p FC(,)30 b FO(:)14 b(:)g(:)28 b FC(,)j FQ(1)p FC(,)g(and)h(the)e(op)l(er)l(ators)i(in)f FO(H)2266 3597 y FM(m)2360 3585 y FC(act)f(as)118 3685 y(fol)t(lows)7 b FP(:)461 3865 y(\()p FO(Af)i FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(\025)14 b(f)9 b FP(\()p FO(\025)p FP(\))p FO(;)457 4006 y FP(\()p FO(U)g(f)g FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(u)909 4018 y FM(m)972 4006 y FP(\()p FO(\025)p FP(\))14 b FO(\032)1141 4018 y FM(m)1204 4006 y FP(\()p FO(\025)p FP(\))1316 3972 y FL(1)p FM(=)p FL(2)1436 4006 y FO(f)9 b FP(\()p FO(F)1583 3972 y FN(\000)p FL(1)1671 4006 y FP(\()p FO(\025)p FP(\)\))p FO(;)419 4147 y FP(\()p FO(U)517 4113 y FN(\003)555 4147 y FO(f)g FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(u)909 4113 y FN(\003)909 4168 y FM(m)972 4147 y FP(\()p FO(F)12 b FP(\()p FO(\025)p FP(\)\))i FO(\032)1270 4159 y FM(m)1334 4147 y FP(\()p FO(F)e FP(\()p FO(\025)p FP(\)\))1575 4113 y FN(\000)p FL(1)p FM(=)p FL(2)1746 4147 y FO(f)d FP(\()p FO(F)j FP(\()p FO(\025)p FP(\)\))p FO(;)316 b FP(\(2.72\))p eop %%Page: 181 185 181 184 bop 118 100 a FK(2.5.)36 b(Represen)n(tations)26 b(of)i(some)f(n)n(uclear)f(algebras)673 b FP(181)118 333 y FC(wher)l(e)37 b FO(f)9 b FP(\()p FQ(\001)p FP(\))36 b FC(is)g(a)h(ve)l(ctor)f(function)g(in)g FQ(\010)1463 303 y FM(m)1463 356 y(k)q FL(=1)1588 333 y FO(L)1645 345 y FL(2)1682 333 y FP(\()p FJ(R)1768 303 y FL(1)1811 333 y FO(;)14 b(d\026)1941 345 y FM(m)2004 333 y FP(\))p FC(,)39 b(the)d(functions)118 432 y FO(\032)161 444 y FM(m)224 432 y FP(\()p FO(\025)p FP(\))h(=)f FO(d\026)567 444 y FM(m)630 432 y FP(\()p FO(F)727 402 y FN(\000)p FL(1)817 432 y FP(\()p FO(\025)p FP(\)\))p FO(=d\026)p FP(\()p FO(\025)p FP(\))j FC(ar)l(e)e(the)g(R)l(adon{Niko)l(dym)i (derivatives,)118 532 y FO(u)166 544 y FM(m)229 532 y FP(\()p FQ(\001)p FP(\))g FC(ar)l(e)h(me)l(asur)l(able)f(mappings)i (taking)e(values)h(in)f(unitary)g(op)l(er)l(ators)118 632 y(on)30 b FJ(C)291 601 y FM(m)360 632 y FC(.)118 811 y(Pr)l(o)l(of.)43 b FP(The)28 b(decomp)r(osition)e(of)i(the)g (space)e FO(H)7 b FP(,)27 b(and)h(form)n(ula)e(for)h FO(A)h FP(follo)n(w)118 911 y(from)j(the)h(decomp)r(osition)f(of)h(the) f(self-adjoin)n(t)g(op)r(erator)f FO(A)i FP(with)g(resp)r(ect)118 1010 y(to)26 b(m)n(ultiplicit)n(y)h(of)f(its)h(sp)r(ectrum.)36 b(W)-7 b(e)27 b(need)f(to)g(sho)n(w)g(that)g(the)h(subspaces)118 1110 y FO(H)187 1122 y FM(m)278 1110 y FP(are)f(in)n(v)-5 b(arian)n(t)27 b(with)h(resp)r(ect)f(to)h FO(U)9 b FP(,)27 b FO(U)1522 1080 y FN(\003)1560 1110 y FP(.)243 1214 y(W)-7 b(e)40 b(shall)g(pro)n(v)n(e)f(that)h(eac)n(h)g(measure)f FO(\026)1620 1226 y FM(m)1724 1214 y FP(is)h(quasi-in)n(v)-5 b(arian)n(t.)73 b(Let)118 1314 y FO(h)166 1326 y FL(1)203 1314 y FP(,)36 b FO(:)14 b(:)g(:)28 b FP(,)37 b FO(h)495 1326 y FM(m)594 1314 y FP(b)r(e)f(an)g(orthonormal)d(basis)i(in)h FJ(C)1693 1283 y FM(m)1762 1314 y FP(.)61 b(F)-7 b(or)35 b(an)n(y)g(measurable)118 1413 y(\001)g FQ(\032)g FJ(R)p FP(,)42 b(consider)34 b(functions)h FO(f)1178 1425 y FM(k)1218 1413 y FP(\()p FO(\025)p FP(\))h(=)e FO(\037)1517 1425 y FL(\001)1576 1413 y FP(\()p FO(\025)p FP(\))p FO(h)1736 1425 y FM(k)1813 1413 y FQ(\032)g FO(H)1981 1425 y FM(m)2044 1413 y FP(,)j(\()p FO(\037)2188 1425 y FL(\001)2247 1413 y FP(\()p FQ(\001)p FP(\))e(is)f(the)118 1513 y(c)n(haracteristic)c(function)j(of)g(\001\))g(whic)n(h)f(are)f (ob)n(viously)g(orthogonal)f(in)j FO(H)7 b FP(.)118 1612 y(F)-7 b(rom)27 b(the)h(relation)f(that)h(connects)f FO(A)h FP(and)f FO(U)9 b FP(,)28 b(w)n(e)f(ha)n(v)n(e)f(that)365 1804 y FO(U)431 1770 y FN(\003)469 1804 y FO(f)510 1816 y FM(k)551 1804 y FP(\()p FO(\025)p FP(\))e(=)f FO(U)841 1770 y FN(\003)878 1804 y FO(E)939 1816 y FM(A)994 1804 y FP(\(\001\))p FO(f)1168 1816 y FM(k)1209 1804 y FP(\()p FO(\025)p FP(\))687 1939 y(=)g FO(E)5 b FP(\()p FO(F)938 1905 y FN(\000)p FL(1)1027 1939 y FP(\(\001\)\))p FO(U)1258 1905 y FN(\003)1297 1939 y FO(f)1338 1951 y FM(k)1378 1939 y FP(\()p FO(\025)p FP(\))24 b(=)f FO(\037)1654 1955 y FM(F)1705 1938 y Fx(\000)p Fy(1)1782 1955 y FL(\(\001\))1893 1939 y FP(\()p FO(\025)p FP(\))p FO(U)2071 1905 y FN(\003)2110 1939 y FO(f)2151 1951 y FM(k)2191 1939 y FP(\()p FO(\025)p FP(\))p FO(;)118 2130 y FP(and)29 b(all)g FO(U)464 2100 y FN(\003)502 2130 y FO(f)543 2142 y FM(k)583 2130 y FP(\()p FQ(\001)p FP(\),)h FO(k)e FP(=)d(1,)k FO(:)14 b(:)g(:)27 b FP(,)j FO(m)f FP(ha)n(v)n(e)f(supp)r(orts)g(in)h FO(F)1954 2100 y FN(\000)p FL(1)2043 2130 y FP(\(\001\).)42 b(Since)29 b(the)118 2230 y(op)r(erator)17 b FO(U)28 b FP(is)18 b(unitary)-7 b(,)20 b(the)f(latter)g(functions)g(again)e (are)h(orthonormal;)i(this)118 2330 y(implies)35 b(that)h(for)e(almost) h(all)g FO(\025)g FP(with)h(resp)r(ect)f(to)g(the)g(sp)r(ectral)g (measure)118 2429 y(of)j FO(A)p FP(,)j(the)e(v)n(ectors)d(\()p FO(U)893 2399 y FN(\003)931 2429 y FO(f)972 2441 y FM(k)1013 2429 y FP(\)\()p FO(\025)p FP(\),)42 b FO(k)h FP(=)d(1,)e FO(:)14 b(:)g(:)27 b FP(,)41 b FO(m)p FP(,)g(are)c(orthogonal,)h(and) 118 2529 y(therefore,)25 b(the)h(sp)r(ectral)e(m)n(ultiplicit)n(y)i(of) f FO(A)h FP(at)f(p)r(oin)n(ts)g(of)g FO(F)12 b FP(\(\001\))26 b(is)f(not)g(less)118 2629 y(than)k(at)h(p)r(oin)n(ts)f(of)g(\001.)42 b(Applying)30 b(the)f(same)g(argumen)n(ts)f(to)h(the)h(op)r(erator)118 2728 y FO(U)40 b FP(instead)31 b(of)g FO(U)669 2698 y FN(\003)707 2728 y FP(,)h(w)n(e)f(conclude)g(that)h(the)f(sp)r(ectral)g (m)n(ultiplicit)n(y)g(of)g FO(A)h FP(is)118 2828 y(in)n(v)-5 b(arian)n(t)26 b(with)h(resp)r(ect)f(to)h FO(F)12 b FP(\()p FQ(\001)p FP(\).)37 b(This)27 b(implies)f(that)h FO(H)1967 2840 y FM(m)2057 2828 y FP(is)g(an)f(in)n(v)-5 b(arian)n(t)118 2927 y(subspace,)27 b(and)g(that)h FO(\026)879 2939 y FM(k)948 2927 y FP(is)f(quasi-in)n(v)-5 b(arian)n(t.)243 3032 y(In)27 b(the)h(subspace)f FO(H)905 3044 y FM(m)968 3032 y FP(,)h(in)n(tro)r(duce)f(the)h(unitary)f(op)r(erator)593 3242 y(\()p FO(V)673 3254 y FM(k)715 3242 y FO(f)9 b FP(\)\()p FO(\025)p FP(\))24 b(=)1020 3174 y Fz(\000)1059 3242 y FO(d\026)p FP(\()p FO(F)12 b FP(\()p FO(\025)p FP(\)\))p FO(=d\026)p FP(\()p FO(\025)p FP(\))1640 3174 y Fz(\001)1680 3192 y FL(1)p FM(=)p FL(2)1784 3242 y FO(f)d FP(\()p FO(F)j FP(\()p FO(\025)p FP(\)\))p FO(:)118 3447 y FP(Then)45 b(the)g(op)r(erator)879 3426 y(~)865 3447 y FO(U)60 b FP(=)52 b FO(U)9 b(V)63 b FP(comm)n(utes)45 b(with)g FO(A)p FP(,)50 b(and,)f(therefore,)118 3547 y(\()165 3526 y(~)150 3547 y FO(U)9 b(f)g FP(\)\()p FO(\025)p FP(\))38 b(=)e FO(u)597 3559 y FM(m)659 3547 y FP(\()p FO(\025)p FP(\))14 b FO(f)9 b FP(\()p FO(\025)p FP(\).)63 b(F)-7 b(rom)35 b FO(U)45 b FP(=)1477 3526 y(~)1462 3547 y FO(U)9 b(V)1595 3517 y FN(\003)1633 3547 y FP(,)38 b(w)n(e)d(ha)n(v)n(e)g(the)h(represen)n(ta-)118 3647 y(tion)28 b(required.)p 2514 3647 4 57 v 2518 3594 50 4 v 2518 3647 V 2567 3647 4 57 v 243 3848 a(Instead)g(of)g(decomp)r (osing)g FO(H)35 b FP(in)n(to)29 b(a)f(direct)g(sum)h(with)g(resp)r (ect)f(to)g(the)118 3948 y(m)n(ultiplicit)n(y)22 b(of)g(the)h(sp)r (ectrum)f(of)g FO(A)p FP(,)h(one)e(can)h(decomp)r(ose)f FO(H)29 b FP(in)n(to)22 b(a)f(direct)118 4048 y(in)n(tegral)e(of)h (generalized)f(eigenspaces)g(of)h FO(A)p FP(.)35 b(Then)21 b(the)f(theorem)g(ab)r(o)n(v)n(e)f(can)118 4147 y(b)r(e)28 b(reform)n(ulated)e(as)h(follo)n(ws.)p eop %%Page: 182 186 182 185 bop 118 100 a FP(182)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FR(Theorem)i(44.)39 b FC(L)l(et)28 b FO(A)h FC(and)g FO(U)37 b FC(b)l(e)28 b(as)h(in)g(the)f(pr)l(evious)i (the)l(or)l(em.)38 b(Then)29 b FO(H)118 432 y FC(c)l(an)h(b)l(e)g(de)l (c)l(omp)l(ose)l(d)h(into)f(a)g(dir)l(e)l(ct)g(inte)l(gr)l(al,)999 669 y FO(H)g FP(=)1185 556 y Fz(Z)1268 577 y FN(\010)1232 745 y Fv(R)1279 728 y Fy(1)1338 669 y FO(H)1407 681 y FM(\025)1465 669 y FO(d\026)p FP(\()p FO(\025)p FP(\))p FO(;)118 900 y FC(wher)l(e)38 b(the)f(Bor)l(el)h(pr)l(ob)l(ability)h (me)l(asur)l(e)e FO(\026)g FC(is)h FO(F)12 b FP(\()p FQ(\001)p FP(\))p FC(-quasi-invariant,)40 b(and)118 1000 y FO(N)9 b FP(\()p FO(\025)p FP(\))27 b(=)e(dim)14 b FO(H)644 1012 y FM(\025)719 1000 y FC(is)31 b(invariant)h(with)g(r)l (esp)l(e)l(ct)f(to)g FO(F)12 b FP(\()p FQ(\001)p FP(\))p FC(,)33 b(so)f(that)f(the)g(op)l(er)l(a-)118 1099 y(tors)f(take)g(the)g (form)611 1283 y FP(\()p FO(Af)9 b FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(\025)14 b(f)9 b FP(\()p FO(\025)p FP(\))p FO(;)607 1424 y FP(\()p FO(U)g(f)g FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(u)p FP(\()p FO(\025)p FP(\))14 b FO(\032)p FP(\()p FO(\025)p FP(\))1340 1390 y FL(1)p FM(=)p FL(2)1459 1424 y FO(f)9 b FP(\()p FO(F)1606 1390 y FN(\000)p FL(1)1695 1424 y FP(\()p FO(\025)p FP(\)\))p FO(;)569 1565 y FP(\()p FO(U)667 1531 y FN(\003)705 1565 y FO(f)g FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(u)1059 1531 y FN(\003)1097 1565 y FP(\()p FO(F)12 b FP(\()p FO(\025)p FP(\)\))i FO(\032)p FP(\()p FO(F)e FP(\()p FO(\025)p FP(\)\))1636 1531 y FN(\000)p FL(1)p FM(=)p FL(2)1809 1565 y FO(f)d FP(\()p FO(F)j FP(\()p FO(\025)p FP(\)\))p FO(;)253 b FP(\(2.73\))118 1749 y FC(wher)l(e)34 b FO(\032)p FP(\()p FO(\025)p FP(\))d(=)f FO(d\026)p FP(\()p FO(F)827 1719 y FN(\000)p FL(1)916 1749 y FP(\()p FO(\025)p FP(\)\))p FO(=d\026)p FP(\()p FO(\025)p FP(\))36 b FC(is)e(the)g(R)l(adon{Niko)l(dym)h(derivative,) 118 1848 y FO(u)p FP(\()p FQ(\001)p FP(\))30 b FC(is)g(a)g(unitary)g (me)l(asur)l(able)g(op)l(er)l(ator-value)l(d)h(function.)243 2016 y FP(The)19 b(follo)n(wing)f(statemen)n(t)h(giv)n(es)f(conditions) g(of)h(irreducibilit)n(y)g(and)g(uni-)118 2116 y(tary)26 b(equiv)-5 b(alence)26 b(of)h(represen)n(tations)e(in)i(the)g(form)f (pro)n(vided)g(b)n(y)g(the)i(pre-)118 2216 y(vious)f(theorem.)118 2384 y FR(Prop)s(osition)44 b(60.)k FC(If)41 b(r)l(epr)l(esentation)47 b FP(\(2.73\))40 b FC(is)i(irr)l(e)l(ducible,)j(then)c(the)118 2483 y(me)l(asur)l(e)26 b FO(\026)p FP(\()p FQ(\001)p FP(\))h FC(is)f(er)l(go)l(dic,)j(and)d(the)g(dimension)i(function)e FO(N)9 b FP(\()p FQ(\001)p FP(\))26 b FC(is)h(c)l(onstant)118 2583 y FO(\026)p FC(-a.e.)49 b(If)33 b(the)g(me)l(asur)l(e)g FO(\026)p FP(\()p FQ(\001)p FP(\))h FC(is)f(er)l(go)l(dic,)i(and)f FP(dim)14 b FO(H)1892 2595 y FM(\025)1964 2583 y FP(=)29 b(1)j FO(\026)p FC(-a.e.,)j FP(\()p FC(i.e.,)118 2682 y FO(H)30 b FP(=)23 b FO(L)362 2694 y FL(2)398 2682 y FP(\()p FJ(R)q FO(;)14 b(d\026)p FP(\)\))p FC(,)36 b(then)30 b(the)g(r)l(epr)l(esentation)g(is)g(irr)l(e)l(ducible.)243 2783 y(Two)23 b(r)l(epr)l(esentations,)h(c)l(orr)l(esp)l(onding)f(to)f (the)h(me)l(asur)l(e)e FO(\026)p FC(,)j(multiplicity)118 2882 y(function)i FO(N)9 b FP(\()p FQ(\001)p FP(\))p FC(,)27 b(and)g(multiplier)g FO(u)p FP(\()p FQ(\001)p FP(\))p FC(,)g(and)f(c)l(orr)l(esp)l(onding)i(to)e(the)g(triple)33 b FP(~)-49 b FO(\026)p FC(,)142 2961 y FP(~)118 2982 y FO(N)9 b FP(\()p FQ(\001)p FP(\))p FC(,)29 b(and)k FP(~)-47 b FO(u)p FP(\()p FQ(\001)p FP(\))p FC(,)29 b(ar)l(e)f (unitarily)g(e)l(quivalent)g(if)g(and)g(only)g(if)h(the)f(me)l(asur)l (es)f FO(\026)118 3082 y FC(and)j FP(~)-48 b FO(\026)23 b FC(ar)l(e)h(e)l(quivalent)31 b FP(\()p FC(have)25 b(the)e(same)h(zer) l(o)g(sets)7 b FP(\))p FC(,)24 b(multiplicity)h(functions)118 3181 y(c)l(oincide)39 b FO(\026)p FC(-a.e.,)h(and)e(the)f(multipliers)h (ar)l(e)f(e)l(quivalent)g(in)g(the)g(fol)t(lowing)118 3281 y(sense)6 b FP(:)64 b FC(ther)l(e)42 b(exists)g(a)h(me)l(asur)l (able)g(unitary)f(op)l(er)l(ator-value)l(d)i(function)118 3380 y FO(v)s FP(\()p FQ(\001)p FP(\))31 b FC(such)e(that)399 3564 y FP(~)-47 b FO(u)o FP(\()p FO(\025)p FP(\))24 b(=)f FO(v)s FP(\()p FO(\025)p FP(\))14 b FO(u)p FP(\()p FO(\025)p FP(\))g FO(v)1051 3530 y FN(\003)1091 3564 y FP(\()p FO(F)1188 3530 y FN(\000)p FL(1)1277 3564 y FP(\()p FO(\025)p FP(\)\))p FO(;)185 b FC(for)31 b FO(\026)p FC(-almost)f(al)t(l)g FO(\025)p FC(.)118 3749 y(Pr)l(o)l(of.)43 b FP(If)26 b(the)g(measure)f(is)g(not)g(ergo)r(dic,)g(its)h(supp)r(ort)f(can)h(b)r (e)f(decomp)r(osed)118 3848 y(in)n(to)i(the)g(union)h(of)f(t)n(w)n(o)f (in)n(v)-5 b(arian)n(ts)26 b(subsets)h(of)g(p)r(ositiv)n(e)f(measure.) 36 b(Suc)n(h)27 b(a)118 3948 y(decomp)r(osition)i(giv)n(es)f(rise)g(to) h(a)g(decomp)r(osition)f(of)h FO(H)36 b FP(in)n(to)29 b(a)g(direct)g(sum)118 4048 y(of)35 b(in)n(v)-5 b(arian)n(t)34 b(subspaces.)59 b(Since)35 b FO(N)9 b FP(\()p FQ(\001)p FP(\))36 b(is)f FO(F)12 b FP(\()p FQ(\001)p FP(\)-in)n(v)-5 b(arian)n(t,)36 b(it)g(is)f(constan)n(t)118 4147 y FO(\026)p FP(-a.e.)p eop %%Page: 183 187 183 186 bop 118 100 a FK(2.5.)36 b(Represen)n(tations)26 b(of)i(some)f(n)n(uclear)f(algebras)673 b FP(183)243 333 y(Let)26 b FO(\026)h FP(b)r(e)g(ergo)r(dic)e(and)i FO(N)9 b FP(\()p FO(\025)p FP(\))23 b(=)g(1)j(for)g FO(\026)p FP(-almost)g(all)g FO(\025)p FP(.)37 b(An)n(y)26 b(b)r(ounded)118 432 y(self-adjoin)n(t)21 b(op)r(erator)f FO(C)28 b FP(that)22 b(comm)n(utes)f(with)h FO(A)g FP(is)f(the)h(op)r(erator)e(of)h(m)n(ul-) 118 532 y(tiplication)33 b(b)n(y)f(a)g(b)r(ounded)h(measurable)f (function)h FO(c)p FP(\()p FQ(\001)p FP(\).)52 b(If)33 b FO(C)39 b FP(comm)n(utes)118 632 y(with)h FO(U)9 b FP(,)43 b(then)d(this)g(function)h(is)e(in)n(v)-5 b(arian)n(t,)42 b(and)e(b)n(y)f(the)h(ergo)r(dicit)n(y)-7 b(,)42 b(is)118 731 y(constan)n(t)27 b FO(\026)p FP(-a.e.,)g(i.e.,)h(the)g(pair)f FO(A)p FP(,)h FO(U)36 b FP(is)28 b(irreducible.)243 831 y(No)n(w)i(consider)f(t)n(w)n(o)h(pairs)f FO(A)p FP(,)j FO(U)39 b FP(on)30 b FO(H)37 b FP(corresp)r(onding)29 b(to)h(a)g(triple)h FO(\026)p FP(,)118 930 y FO(N)9 b FP(\()p FQ(\001)p FP(\),)33 b FO(u)p FP(\()p FQ(\001)p FP(\),)f(and)714 909 y(~)692 930 y FO(A)p FP(,)824 909 y(~)809 930 y FO(U)40 b FP(on)1047 909 y(~)1025 930 y FO(H)e FP(corresp)r(onding)29 b(to)38 b(~)-49 b FO(\026)p FP(,)1903 909 y(~)1879 930 y FO(N)9 b FP(\()p FQ(\001)p FP(\),)38 b(~)-47 b FO(u)o FP(\()p FQ(\001)p FP(\).)48 b(If)32 b(these)118 1030 y(pairs)24 b(are)g(unitarily)h(equiv)-5 b(alen)n(t,)25 b(then)g(the)h(sp)r(ectral)e(measures)g(of)h(the)g (self-)118 1130 y(adjoin)n(t)32 b(op)r(erators)e FO(A)j FP(and)1059 1109 y(~)1037 1130 y FO(A)g FP(are)e(equiv)-5 b(alen)n(t,)33 b(and)f FO(N)9 b FP(\()p FQ(\001)p FP(\))31 b(=)2178 1109 y(~)2154 1130 y FO(N)9 b FP(\()p FQ(\001)p FP(\))33 b FO(\026)p FP(-a.e.;)118 1229 y(no)n(w)e(w)n(e)f(can)h (naturally)f(iden)n(tify)i FO(H)38 b FP(and)1536 1208 y(~)1515 1229 y FO(H)6 b FP(.)48 b(A)31 b(unitary)g(op)r(erator)e FO(V)50 b FP(on)118 1329 y FO(H)39 b FP(that)33 b(in)n(tert)n(wines)f (these)g(pairs)g(comm)n(utes)g(with)h FO(A)p FP(,)h(and)e(therefore,)h (is)118 1429 y(a)h(m)n(ultiplication)h(b)n(y)g(a)f(measurable)f (unitary)i(op)r(erator-v)-5 b(alued)32 b(function)118 1528 y FO(v)s FP(\()p FQ(\001)p FP(\).)46 b(It)30 b(directly)g(follo)n (ws)f(from)h(\(2.73\))g(that)g(the)h(relation)2080 1507 y(~)2066 1528 y FO(U)36 b FP(=)27 b FO(V)19 b(U)9 b(V)2451 1498 y FN(\003)2519 1528 y FP(is)118 1628 y(equiv)-5 b(alen)n(t)27 b(to)33 b(~)-47 b FO(u)o FP(\()p FO(\025)p FP(\))25 b(=)d FO(v)s FP(\()p FO(\025)p FP(\))14 b FO(u)p FP(\()p FO(\025)p FP(\))g FO(v)1271 1598 y FN(\003)1311 1628 y FP(\()p FO(F)1408 1598 y FN(\000)p FL(1)1497 1628 y FP(\()p FO(\025)p FP(\)\).)p 2514 1628 4 57 v 2518 1575 50 4 v 2518 1628 V 2567 1628 4 57 v 118 1791 a FC(R)l(emark)40 b FP(47)p FC(.)i FP(One)29 b(can)g(easily)g(see)g(that)g(the)h(same)f (statemen)n(t)g(holds)g(true)118 1891 y(for)f(a)g(\014nite)h(or)f(coun) n(table)f(comm)n(uting)h(family)h(of)f(self-adjoin)n(t)g(or)g(normal) 118 1990 y(op)r(erators)j FR(A)g FP(=)f(\()p FO(A)783 2002 y FM(k)825 1990 y FP(\))857 2002 y FM(k)q FN(2)p FM(X)1034 1990 y FP(whic)n(h)j(are)e(related)h(with)h(a)f(unitary)g(op) r(erator)118 2090 y FO(U)46 b FP(b)n(y)36 b(the)h(relations)e FO(A)905 2102 y FM(k)946 2090 y FO(U)47 b FP(=)38 b FO(U)9 b(F)1272 2102 y FM(k)1313 2090 y FP(\()p FR(A)p FP(\),)40 b FO(k)g FQ(2)f FO(X)7 b FP(.)63 b(Indeed,)39 b(w)n(e)e(actually)118 2189 y(use)28 b(the)g(prop)r(erties)e(of)i(the)g(join)n(t)g(resolution) e(of)i(the)g(iden)n(tit)n(y)g(of)f(this)h(fam-)118 2289 y(ily)-7 b(,)36 b FO(E)321 2301 y Fn(A)382 2289 y FP(\()p FQ(\001)p FP(\).)58 b(T)-7 b(o)34 b(pro)n(v)n(e)f(the)i(theorem,)g(w)n (e)f(use)g(the)h(relation)f FO(E)2216 2301 y Fn(A)2276 2289 y FP(\(\001\))p FO(U)44 b FP(=)118 2389 y FO(U)9 b(E)245 2401 y FM(A)299 2389 y FP(\()p FR(F)391 2359 y FN(\000)p FL(1)481 2389 y FP(\()p FR(A)p FP(\)\),)25 b(where)d FR(F)9 b FP(:)28 b FJ(R)1106 2359 y FM(X)1199 2389 y FQ(\000)-49 b(!)23 b FJ(R)1375 2359 y FM(X)1467 2389 y FP(is)g FR(F)p FP(\()p FQ(\001)p FP(\))h(=)f(\()p FO(F)1890 2401 y FM(k)1931 2389 y FP(\()p FQ(\001)p FP(\)\))2050 2401 y FM(k)q FN(2)p FM(X)2196 2389 y FP(.)35 b(The)23 b(only)118 2488 y(di\013erence)h(is)f(that)h(the)f(dynamical)g(system)h (acts)f(on)g FJ(R)1878 2458 y FM(X)1947 2488 y FP(,)h(and)g(the)f(sp)r (ectral)118 2588 y(measure)k FO(\026)g FP(should)h(b)r(e)g(considered)e (on)i(this)g(space.)243 2688 y(Moreo)n(v)n(er,)e(since)i(the)h(decomp)r (osition)f(of)h FO(H)36 b FP(is)28 b(constructed)g(from)g(the)118 2787 y(comm)n(utativ)n(e)h(family)g FR(A)p FP(,)h(the)f(comm)n(utativ)n (e)g(mo)r(del)g(can)g(b)r(e)g(constructed)118 2887 y(for)34 b(a)h(\014nite)g(or)f(coun)n(table)g(family)h(of)f(unitary)h(op)r (erators)e FR(U)i FP(=)g(\()p FO(U)2375 2899 y FM(j)2409 2887 y FP(\))2441 2899 y FM(j)s FN(2)p FM(Y)118 2986 y FP(whic)n(h)28 b(satisfy)f(the)h(relations)e FO(A)1157 2998 y FM(k)1198 2986 y FO(B)1261 2998 y FM(j)1319 2986 y FP(=)d FO(B)1470 2998 y FM(j)1505 2986 y FO(F)1558 2998 y FM(k)q(j)1630 2986 y FP(\()p FR(A)p FP(\),)29 b FO(k)c FQ(2)f FO(X)7 b FP(,)27 b FO(j)h FQ(2)23 b FO(Y)c FP(.)243 3086 y(T)-7 b(o)24 b(extend)g(this)h(theorem)e(to)h(the)h (case)e(of)i(an)f(arbitrary)e(\(uncoun)n(table\))118 3186 y(n)n(um)n(b)r(er)33 b(of)f(op)r(erators,)h(one)f(should)g (require)g(the)h(existence)g(of)g(a)f(n)n(uclear)118 3285 y(rigging)26 b(for)h(the)h(op)r(erators)e([28)o(,)i(192)n(].)243 3412 y(No)n(w)39 b(consider)g(a)g(family)h(of)g(comm)n(uting)g(b)r (ounded)g(self-adjoin)n(t)f(\(or)118 3512 y(normal\))28 b(op)r(erators)f(\()p FO(A)896 3524 y FM(k)938 3512 y FP(\),)i(where)g FO(k)f FQ(2)e FO(X)35 b FP(ranges)27 b(o)n(v)n(er)g(a)i(\014nite)g(or)f(coun)n(t-)118 3612 y(able)33 b(set,)h(and)e(a)h(family)g(of)g(b)r(ounded)g(op)r(erators)e (\()p FO(B)1870 3624 y FM(j)1905 3612 y FP(\),)j FO(j)j FQ(2)32 b FO(Y)19 b FP(,)34 b(whic)n(h)f(is)118 3711 y(connected)28 b(with)g(the)g(op)r(erators)d FO(A)1267 3723 y FM(k)1336 3711 y FP(b)n(y)i(the)h(follo)n(wing)f(relations)992 3879 y FO(A)1054 3891 y FM(k)1095 3879 y FO(B)1158 3891 y FM(j)1216 3879 y FP(=)c FO(B)1367 3891 y FM(j)1402 3879 y FO(F)1455 3891 y FM(k)q(j)1527 3879 y FP(\()p FR(A)p FP(\))p FO(;)677 b FP(\(2.74\))118 4048 y(where)33 b FO(F)417 4060 y FM(k)q(j)523 4048 y FP(is)g(a)g(measurable)g (function)h(of)f(the)h(comm)n(uting)f(family)h FR(A)f FP(=)118 4147 y(\()p FO(A)212 4159 y FM(k)254 4147 y FP(\),)g FO(k)g FQ(2)e FO(X)7 b FP(,)32 b FO(j)k FQ(2)30 b FO(Y)19 b FP(,)33 b(suc)n(h)f(that)g FR(F)1349 4159 y FM(j)1384 4147 y FP(\()p FQ(\001)p FP(\))f(=)f(\()p FO(F)1682 4159 y FM(k)q(j)1755 4147 y FP(\()p FQ(\001)p FP(\)\))1874 4159 y FM(k)q FN(2)p FM(X)2028 4147 y FP(:)f FJ(R)2134 4117 y FM(X)2234 4147 y FQ(\000)-49 b(!)31 b FJ(R)2418 4117 y FM(X)2519 4147 y FP(is)p eop %%Page: 184 188 184 187 bop 118 100 a FP(184)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(an)g(injectiv)n(e)h(on)g(the)f(join)n(t)h(sp) r(ectrum)g(of)g FR(A)f FP(measurable)g(mapping.)118 498 y FR(Theorem)46 b(45.)j FC(L)l(et)42 b(families)i FP(\()p FO(A)1283 510 y FM(k)1325 498 y FP(\))p FC(,)i FO(A)1490 510 y FM(k)1577 498 y FP(=)g FO(A)1750 467 y FN(\003)1750 521 y FM(k)1791 498 y FC(,)g FO(k)j FQ(2)d FO(X)7 b FC(,)45 b FP(\()p FO(B)2296 510 y FM(j)2331 498 y FP(\))p FC(,)i FO(j)j FQ(2)118 597 y FO(Y)19 b FC(,)41 b(satisfy)e(r)l(elations)46 b FP(\(2.74\))37 b FC(on)i(a)g(Hilb)l(ert)f(sp)l(ac)l(e)h FO(H)7 b FC(.)64 b(Then)39 b FO(H)45 b FC(c)l(an)39 b(b)l(e)118 697 y(de)l(c)l(omp)l(ose)l(d)31 b(into)f(the)g(dir)l(e)l(ct)g(inte)l (gr)l(al)664 931 y FO(H)f FP(=)850 818 y Fz(Z)933 839 y FN(\010)896 1007 y Fv(R)943 990 y Fw(X)1010 931 y FO(H)1079 943 y FM(\025)1136 931 y FO(d\026)p FP(\()p FO(\025)p FP(\))p FO(;)185 b(\025)23 b FP(=)g(\()p FO(\025)1788 943 y FM(k)1830 931 y FP(\))1862 943 y FM(k)q FN(2)p FM(X)2006 931 y FO(;)118 1154 y FC(and)30 b(the)g(op)l(er)l(ators)h (act)f(as)g(fol)t(lows)7 b FP(:)185 1336 y(\()p FO(A)279 1348 y FM(k)320 1336 y FO(f)i FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(\025)674 1348 y FM(k)729 1336 y FO(f)9 b FP(\()p FO(\025)p FP(\))p FO(;)190 1476 y FP(\()p FO(B)285 1488 y FM(j)320 1476 y FO(f)g FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(b)662 1488 y FM(j)696 1476 y FP(\()p FO(\025)p FP(\))14 b FO(\037)874 1504 y FL(\001)929 1476 y Fw(j)929 1522 y Fy(0)966 1476 y FP(\()p FO(\025)p FP(\))g FO(\032)1135 1488 y FM(j)1171 1476 y FP(\()p FO(\025)p FP(\))1283 1442 y FL(1)p FM(=)p FL(2)1402 1476 y FO(f)9 b FP(\()p FR(F)1544 1441 y FN(\000)p FL(1)1544 1500 y FM(j)1633 1476 y FP(\()p FO(\025)p FP(\)\))p FO(;)183 1633 y FP(\()p FO(B)282 1598 y FN(\003)278 1653 y FM(j)320 1633 y FO(f)g FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(b)662 1598 y FN(\003)662 1653 y FM(j)700 1633 y FP(\()p FR(F)792 1645 y FM(j)827 1633 y FP(\()p FO(\025)p FP(\)\))14 b FO(\037)1037 1660 y Fn(F)1084 1633 y Fx(\000)p Fy(1)1084 1679 y Fw(j)1162 1660 y FL(\(\001)1243 1632 y Fw(j)1243 1678 y Fy(0)1275 1660 y FL(\))1305 1633 y FP(\()p FO(\025)p FP(\))g FO(\032)1474 1645 y FM(j)1510 1633 y FP(\()p FR(F)1602 1645 y FM(j)1637 1633 y FP(\()p FO(\025)p FP(\)\))1781 1598 y FN(\000)p FL(1)p FM(=)p FL(2)1953 1633 y FO(f)9 b FP(\()p FR(F)2095 1645 y FM(j)2130 1633 y FP(\()p FO(\025)p FP(\)\))p FO(:)66 b FP(\(2.75\))118 1855 y FC(Her)l(e)35 b FP(\001)391 1815 y FM(j)391 1877 y FL(0)463 1855 y FC(is)g(the)g(set)f(de\014ne)l (d)h(by)g(the)g(pr)l(op)l(erty)h(that)f(for)g(any)g(me)l(asur)l(able) 118 1964 y FO(\016)f FQ(\032)c FP(\001)353 1924 y FM(j)353 1986 y FL(0)390 1964 y FC(,)36 b(we)e(have)h FO(B)836 1976 y FM(j)871 1964 y FO(E)932 1976 y Fn(A)993 1964 y FP(\()p FO(\016)s FP(\))c FQ(6)p FP(=)f(0;)36 b FC(the)e(me)l(asur)l (e)g FO(\037)1844 1991 y FL(\001)1899 1963 y Fw(j)1899 2009 y Fy(0)1935 1964 y FP(\()p FO(\025)p FP(\))14 b FO(d\026)p FP(\()p FR(F)2246 1928 y FN(\000)p FL(1)2246 1987 y FM(j)2336 1964 y FP(\()p FO(\025)p FP(\)\))35 b FC(is)118 2080 y(absolutely)d(c)l(ontinuous)e(with)i(r)l(esp)l(e)l (ct)e(to)h FO(d\026)p FP(\()p FO(\025)p FP(\))p FC(,)i FO(\032)1778 2092 y FM(j)1813 2080 y FP(\()p FQ(\001)p FP(\))f FC(is)f(the)g(c)l(orr)l(esp)l(ond-)118 2179 y(ing)26 b(R)l(adon{Niko)l(dym)h(derivative,)i FO(b)1305 2191 y FM(j)1340 2179 y FP(\()p FQ(\001)p FP(\))d FC(is)g(a)g(me)l(asur)l (able)g(op)l(er)l(ator-value)l(d)118 2290 y(function,)k FO(b)506 2302 y FM(j)541 2290 y FP(\()p FO(\025)p FP(\))24 b FQ(6)p FP(=)f(0)29 b FC(for)h FO(\026)p FC(-almost)g(al)t(l)h FO(\025)24 b FQ(2)f FP(\001)1651 2250 y FM(j)1651 2312 y FL(0)1689 2290 y FC(.)118 2455 y(Pr)l(o)l(of.)43 b FP(The)32 b(pro)r(of)g(of)f(the)i(theorem)e(in)h(a)g(general)e (situation)i(is)g(based)f(on)118 2554 y(the)24 b(theory)g(of)f(decomp)r (osition)h(with)g(resp)r(ect)g(to)g(generalized)e(eigen)n(v)n(ectors) 118 2654 y(of)28 b(families)f(of)h(self-adjoin)n(t)f(op)r(erators,)e (and)j(uses)f(tec)n(hniques)g(b)r(ey)n(ond)g(the)118 2753 y(scop)r(e)d(of)h(the)g(b)r(o)r(ok;)g(w)n(e)g(consider)e(only)i (the)g(case)e(where)i(the)g(comm)n(utativ)n(e)118 2853 y(family)k FR(A)g FP(has)f(a)g(simple)h(join)n(t)g(sp)r(ectrum)g(in)g (order)e(to)i(illustrate)f(the)h(idea)118 2953 y(of)f(the)g(pro)r(of.) 243 3052 y(In)j(this)h(case,)g(the)g(represen)n(tation)d(space)i(is)g FO(L)1783 3064 y FL(2)1820 3052 y FP(\()p FJ(R)1906 3022 y FM(X)1975 3052 y FO(;)14 b(d\026)p FP(\),)33 b(where)e FO(\026)h FP(is)118 3152 y(the)h(sp)r(ectral)f(measure)f(of)h(the)h (comm)n(utativ)n(e)f(family)-7 b(,)34 b(whic)n(h)e(acts)g(as)f(the)118 3252 y(op)r(erators)g(of)i(m)n(ultiplication,)i(\()p FO(A)1248 3264 y FM(k)1289 3252 y FO(f)9 b FP(\)\()p FO(\025)p FP(\))33 b(=)f FO(\025)1661 3264 y FM(k)1716 3252 y FO(f)9 b FP(\()p FO(\025)p FP(\),)35 b FO(k)g FQ(2)e FO(X)7 b FP(.)52 b(T)-7 b(ak)n(e)32 b(the)118 3351 y(v)n(ector)f FO(e)p FP(\()p FO(\025)p FP(\))g FQ(\021)f FP(1;)j(then)g(for)e(all)h(c)n(haracteristic)e(functions)i(of)g (measurable)118 3451 y(subsets)27 b(of)34 b FJ(R)560 3421 y FM(X)630 3451 y FP(,)27 b(w)n(e)h(ha)n(v)n(e)391 3632 y(\()p FO(U)480 3644 y FM(j)515 3632 y FO(\037)567 3644 y FM(\016)603 3632 y FP(\)\()p FO(\025)p FP(\))c(=)f(\()p FO(U)948 3644 y FM(j)983 3632 y FO(E)1044 3644 y Fn(A)1105 3632 y FP(\()p FO(\016)s FP(\))14 b FO(e)p FP(\)\()p FO(\025)p FP(\))771 3767 y(=)23 b FO(E)920 3779 y Fn(A)981 3767 y FP(\()p FR(F)1073 3779 y FM(j)1108 3767 y FP(\()p FO(\016)s FP(\)\)\()p FO(U)1333 3779 y FM(j)1369 3767 y FO(e)p FP(\)\()p FO(\025)p FP(\))h(=)f FO(e)1703 3779 y FM(j)1737 3767 y FP(\()p FO(\025)p FP(\))14 b FO(\037)1915 3779 y FM(\016)1952 3767 y FP(\()p FR(F)2044 3733 y FN(\000)p FL(1)2134 3767 y FP(\()p FO(\025)p FP(\)\))p FO(;)118 3948 y FP(where)33 b(w)n(e)g(write)g FO(e)749 3960 y FM(j)783 3948 y FP(\()p FO(\025)p FP(\))h(=)e(\()p FO(U)1115 3960 y FM(j)1150 3948 y FO(e)p FP(\)\()p FO(\025)p FP(\).)55 b(No)n(w)33 b(w)n(e)f(sho)n(w)h(that)g(the)h(measure)118 4048 y FO(\037)170 4060 y FL(\001)225 4068 y Fy(0)261 4048 y FP(\()p FO(\025)p FP(\))14 b FO(d\026)p FP(\()p FO(F)577 4018 y FN(\000)p FL(1)668 4048 y FP(\()p FO(\025)p FP(\)\))34 b(is)e(absolutely)g(con)n(tin)n(uous)g(with)h(resp)r(ect)g (to)g FO(d\026)p FP(\()p FO(\025)p FP(\).)118 4147 y(Indeed,)38 b(for)c(an)n(y)h(measurable)f FO(\016)k FP(w)n(e)d(ha)n(v)n(e)f FO(E)1627 4159 y Fn(A)1688 4147 y FP(\()p FO(\016)s FP(\))p FO(U)1849 4159 y FM(j)1920 4147 y FP(=)h FO(U)2077 4159 y FM(j)2112 4147 y FO(E)2173 4159 y Fn(A)2233 4147 y FP(\()p FR(F)2325 4112 y FN(\000)p FL(1)2325 4170 y FM(j)2415 4147 y FP(\()p FO(\016)s FP(\)\),)p eop %%Page: 185 189 185 188 bop 118 100 a FK(2.5.)36 b(Represen)n(tations)26 b(of)i(some)f(n)n(uclear)f(algebras)673 b FP(185)118 333 y(and)44 b(since)g FO(U)582 303 y FN(\003)573 354 y FM(j)620 333 y FO(U)677 345 y FM(j)762 333 y FP(=)50 b FO(P)930 345 y FM(j)1010 333 y FP(comm)n(utes)43 b(with)i FO(E)1683 345 y Fn(A)1743 333 y FP(\()p FR(F)1835 297 y FN(\000)p FL(1)1835 356 y FM(j)1925 333 y FP(\()p FO(\016)s FP(\)\),)k(w)n(e)44 b(see)g(that)118 432 y(\()p FO(P)203 444 y FM(j)239 432 y FO(f)9 b FP(\)\()p FO(\025)p FP(\))24 b(=)e FO(\037)596 459 y FL(\001)651 431 y Fw(j)651 477 y Fy(0)687 432 y FP(\()p FO(\025)p FP(\).)38 b(No)n(w)27 b(w)n(e)h(ha)n(v)n(e)804 619 y FO(P)857 631 y FM(j)893 619 y FO(E)954 631 y Fn(A)1014 619 y FP(\()p FR(F)1106 585 y FN(\000)p FL(1)1196 619 y FP(\()p FO(\016)s FP(\)\))c(=)f FO(U)1510 585 y FN(\003)1501 639 y FM(j)1547 619 y FO(E)1608 631 y Fn(A)1669 619 y FP(\()p FO(\016)s FP(\))p FO(U)1830 631 y FM(j)1865 619 y FO(;)118 780 y FP(whic)n(h)c(giv)n(es)f(the)h (absolute)f(con)n(tin)n(uit)n(y)h(required,)h(since)e FO(\026)p FP(\()p FQ(\001)p FP(\))24 b(=)f(\()p FO(E)2257 792 y Fn(A)2318 780 y FP(\()p FQ(\001)p FP(\))p FO(e;)14 b(e)p FP(\).)118 879 y(Notice)41 b(that)h(supp)14 b FO(e)810 891 y FM(j)844 879 y FP(\()p FQ(\001)p FP(\))47 b(=)e(\001)1157 839 y FM(j)1157 902 y FL(0)1195 879 y FP(;)j(then)42 b(w)n(e)e(can)h(rewrite)g(it)g(as)g FO(e)2317 891 y FM(j)2351 879 y FP(\()p FO(\025)p FP(\))47 b(=)118 979 y FO(u)166 991 y FM(j)201 979 y FP(\()p FO(\025)p FP(\))14 b FO(\032)370 991 y FM(j)405 979 y FP(\()p FO(\025)p FP(\))517 949 y FL(1)p FM(=)p FL(2)623 979 y FP(,)26 b(whic)n(h)h(pro)n(v)n(es)d(the) j(form)n(ula)e(for)h(c)n(haracteristic)f(functions)118 1079 y(of)j(measurable)e(sets,)h(and)h(therefore,)f(for)g(the)h(whole)f FO(H)7 b FP(.)p 2514 1079 4 57 v 2518 1026 50 4 v 2518 1079 V 2567 1079 4 57 v 118 1291 a FR(2.5.2)94 b(Cen)m(tered)32 b(op)s(erators)118 1444 y FP(Recall)26 b(that)h(a)f(b)r(ounded)h(op)r (erator)e FO(T)37 b FP(is)27 b(called)f(cen)n(tered)g(if)h(the)g(op)r (erators)118 1544 y FO(T)179 1513 y FM(k)219 1544 y FP(\()p FO(T)312 1513 y FN(\003)350 1544 y FP(\))382 1513 y FM(k)423 1544 y FP(,)g(\()p FO(T)566 1513 y FN(\003)604 1544 y FP(\))636 1513 y FM(k)677 1544 y FO(T)738 1513 y FM(k)806 1544 y FP(form)g(a)g(comm)n(utativ)n(e)g(family)-7 b(,)28 b(i.e.,)f(for)h(all)f FO(k)s(;)14 b(j)28 b FQ(2)23 b FJ(N)611 1705 y FP([)p FO(T)695 1670 y FM(k)735 1705 y FP(\()p FO(T)828 1670 y FN(\003)865 1705 y FP(\))897 1670 y FM(k)938 1705 y FO(;)14 b(T)1036 1670 y FM(j)1070 1705 y FP(\()p FO(T)1163 1670 y FN(\003)1201 1705 y FP(\))1233 1670 y FM(j)1268 1705 y FP(])23 b(=)g([)p FO(T)1486 1670 y FM(k)1526 1705 y FP(\()p FO(T)1619 1670 y FN(\003)1656 1705 y FP(\))1688 1670 y FM(k)1729 1705 y FO(;)14 b FP(\()p FO(T)1859 1670 y FN(\003)1897 1705 y FP(\))1929 1670 y FM(j)1964 1705 y FO(T)2025 1670 y FM(j)2059 1705 y FP(])875 1842 y(=)22 b([\()p FO(T)1078 1808 y FN(\003)1116 1842 y FP(\))1148 1808 y FM(k)1189 1842 y FO(T)1250 1808 y FM(k)1290 1842 y FO(;)14 b FP(\()p FO(T)1420 1808 y FN(\003)1457 1842 y FP(\))1489 1808 y FM(j)1525 1842 y FO(T)1586 1808 y FM(j)1620 1842 y FP(])23 b(=)g(0)p FO(:)544 b FP(\(2.76\))243 2003 y(In)24 b(this)h(section)f(w)n(e)f (study)i(b)r(ounded)g(cen)n(tered)e(op)r(erators.)34 b(W)-7 b(e)25 b(rewrite)118 2103 y(relations)31 b(\(2.76\))h(in)g(the)h (form)f(whic)n(h)g(enables)g(one)g(to)g(construct)g(a)g(com-)118 2202 y(m)n(utativ)n(e)e(mo)r(del)h(for)f(cen)n(tered)h(op)r(erators,)e (and)i(sho)n(w)e(that)i(the)g(problem)118 2302 y(of)f(the)g(unitary)g (classi\014cation)e(of)i(cen)n(tered)f(op)r(erators)f(is)i(not)g(wild)g (in)g(the)118 2402 y(sense)25 b(discussed)g(in)g(Chapter)g(3.)36 b(Similarly)24 b(to)i(Section)f(2.1.1,)f(w)n(e)h(describ)r(e)118 2501 y(cen)n(tered)30 b(op)r(erators)e(suc)n(h)i(that)g(k)n(er)13 b FO(T)38 b FQ(6)p FP(=)27 b FQ(f)p FP(0)p FQ(g)i FP(or)g(k)n(er)13 b FO(T)1942 2471 y FN(\003)2006 2501 y FQ(6)p FP(=)27 b FQ(f)p FP(0)p FQ(g)p FP(.)43 b(W)-7 b(e)31 b(also)118 2601 y(describ)r(e,)36 b(up)e(to)g(the)h(unitary)f(equiv)-5 b(alence,)35 b(all)f(\014nite-dimensional)g(irre-)118 2701 y(ducible)28 b(cen)n(tered)f(op)r(erators.)118 2841 y FR(1.)46 b FP(W)-7 b(e)31 b(will)g(rewrite)g(relations)e(\(2.76\))h (in)h(a)g(form)f(that)i(will)f(enable)f(us)h(to)118 2941 y(in)n(v)n(estigate)h(the)h(relations)f(using)h(the)g(formalism)f(of)h (dynamical)f(systems)118 3041 y(dev)n(elop)r(ed)f(ab)r(o)n(v)n(e)f(in)i (this)f(c)n(hapter.)47 b(W)-7 b(e)32 b(will)g(sho)n(w)e(that)i (irreducible)e(\(or)118 3140 y(factor\))h(represen)n(tations)e(of)i (the)g(relations)f(fall)h(in)n(to)g(t)n(w)n(o)g(cases:)42 b(the)32 b(case)118 3240 y(where)i(k)n(er)13 b FO(T)34 b FQ([)23 b FP(k)n(er)13 b FO(T)837 3210 y FN(\003)909 3240 y FP(=)34 b FQ(f)p FP(0)p FQ(g)f FP(and)h(degenerate)f(cases)g (\(similarly)h(to)g(the)118 3339 y(W)-7 b(old)28 b(decomp)r(osition)f (of)g(isometries\),)g(whic)n(h)h(are)f(studied)h(separately)-7 b(.)243 3439 y(Let)29 b FO(T)37 b FP(=)26 b FO(U)9 b(C)36 b FP(b)r(e)30 b(the)f(p)r(olar)g(decomp)r(osition)g(of)g(the)h(op)r (erator)e FO(T)12 b FP(.)41 b(W)-7 b(e)118 3539 y(also)28 b(in)n(tro)r(duce)g(the)h(comm)n(uting)g(self-adjoin)n(t)f(op)r (erators)f FO(A)2092 3551 y FM(k)2158 3539 y FP(=)d FO(T)2308 3509 y FM(k)2348 3539 y FP(\()p FO(T)2441 3509 y FN(\003)2479 3539 y FP(\))2511 3509 y FM(k)2552 3539 y FP(,)118 3638 y FO(B)181 3650 y FM(k)245 3638 y FP(=)f(\()p FO(T)426 3608 y FN(\003)463 3638 y FP(\))495 3608 y FM(k)536 3638 y FO(T)597 3608 y FM(k)637 3638 y FP(,)i FO(k)h FQ(\025)d FP(1,)i(and)f(denote)g(the)h(join)n(t)f(resolution)g(of)g(the)h(iden)n (tit)n(y)118 3738 y(of)j(the)g(comm)n(utativ)n(e)e(self-adjoin)n(t)i (family)f(\()p FO(A)1626 3750 y FM(k)1668 3738 y FO(;)14 b(B)1768 3750 y FM(k)1808 3738 y FP(\))1840 3750 y FM(k)q FN(2)p Fv(N)1996 3738 y FP(b)n(y)27 b FO(E)2172 3750 y Fn(A)p FM(;)p Fn(B)2306 3738 y FP(\()p FQ(\001)p FO(;)14 b FQ(\001)p FP(\).)118 3887 y FR(Prop)s(osition)32 b(61.)42 b FC(The)32 b(r)l(elations)39 b FP(\(2.76\))30 b FC(ar)l(e)i(e)l (quivalent)f(to)h(the)f(fol)t(low-)118 3986 y(ing)7 b FP(:)313 4147 y FO(E)374 4159 y Fn(A)p FM(;)p Fn(B)508 4147 y FP(\(\001\))p FO(U)32 b FP(=)23 b FO(U)9 b(E)c FP(\()p FO(F)1047 4113 y FN(\000)p FL(1)1136 4147 y FP(\(\001\)\))p FO(;)185 b FP(\001)23 b FQ(2)g FA(B)p FP(\()p FJ(R)1839 4113 y Fv(N)1909 4147 y FQ(\002)18 b FJ(R)2046 4113 y Fv(N)2099 4147 y FP(\))p FO(;)209 b FP(\(2.77\))p eop %%Page: 186 190 186 189 bop 118 100 a FP(186)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(wher)l(e)k(the)e(mapping)j FO(F)891 303 y FN(\000)p FL(1)980 333 y FP(\()p FQ(\001)p FP(\))e FC(is)g(de\014ne)l(d)g(by)201 531 y FO(F)266 497 y FN(\000)p FL(1)355 531 y FP(\()p FR(x)p FO(;)14 b FR(y)q FP(\))25 b(=)d FO(F)734 497 y FN(\000)p FL(1)824 531 y FP(\(\()p FO(x)935 543 y FL(1)973 531 y FO(;)14 b(x)1057 543 y FL(2)1094 531 y FO(;)g(:)g(:)g(:)g FP(\))p FO(;)g FP(\()p FO(y)1384 543 y FL(1)1421 531 y FO(;)g(y)1499 543 y FL(2)1536 531 y FO(;)g(:)g(:)g(:)g FP(\)\))477 739 y(=)565 598 y Fz(\()632 683 y FP(\(\()p FO(x)743 695 y FL(2)781 683 y FO(=x)870 695 y FL(1)907 683 y FO(;)g(x)991 695 y FL(3)1029 683 y FO(=x)1118 695 y FL(1)1155 683 y FO(;)g(:)g(:)g(:)g FP(\))p FO(;)g FP(\()p FO(x)1451 695 y FL(1)1489 683 y FO(;)g(y)1567 695 y FL(1)1604 683 y FO(x)1651 695 y FL(1)1688 683 y FO(;)g(y)1766 695 y FL(2)1803 683 y FO(x)1850 695 y FL(1)1888 683 y FO(;)g(:)g(:)g(:)g FP(\)\))p FO(;)85 b(x)2255 695 y FL(1)2316 683 y FQ(6)p FP(=)22 b(0)p FO(;)632 802 y FP(\(\(0)p FO(;)14 b FP(0)p FO(;)g(:)g(:)g(:)f FP(\))p FO(;)h FP(\(0)p FO(;)g FP(0)p FO(;)g(:)g(:)g(:)f FP(\)\))p FO(;)788 b(x)2255 814 y FL(1)2316 802 y FP(=)22 b(0)p FO(:)243 983 y FC(The)30 b(phase)i FO(U)38 b FC(of)31 b(the)e(op)l(er)l(ator)i FO(T)41 b FC(is)30 b(a)g(c)l(enter)l(e)l(d)g(op)l(er)l(ator.)118 1142 y(Pr)l(o)l(of.)43 b FP(F)-7 b(or)27 b(an)n(y)g FO(k)f FQ(2)d FJ(N)497 1315 y FO(A)559 1327 y FM(k)600 1315 y FO(U)9 b(C)29 b FP(=)23 b FO(T)903 1281 y FM(k)943 1315 y FP(\()p FO(T)1036 1281 y FN(\003)1073 1315 y FP(\))1105 1281 y FM(k)1147 1315 y FO(T)34 b FP(=)22 b FO(T)12 b(A)1440 1327 y FM(k)q FN(\000)p FL(1)1565 1315 y FO(B)1628 1327 y FL(1)1689 1315 y FP(=)22 b FO(U)9 b(C)d(A)1969 1327 y FM(k)q FN(\000)p FL(1)2095 1315 y FO(B)2158 1327 y FL(1)754 1440 y FP(=)23 b FO(U)9 b(A)970 1452 y FM(k)q FN(\000)p FL(1)1096 1440 y FO(B)1159 1452 y FL(1)1196 1440 y FO(C)q(:)118 1623 y FP(Denote)28 b(b)n(y)g FO(P)40 b FP(the)28 b(pro)5 b(jection)27 b(on)h(\(k)n(er)13 b FO(C)6 b FP(\))1520 1592 y FN(?)1577 1623 y FP(.)38 b(Then,)28 b(since)g(k)n(er)13 b FO(C)30 b FP(=)23 b(k)n(er)13 b FO(U)118 1722 y FP(and)30 b FO(U)9 b(P)39 b FP(=)27 b FO(U)9 b FP(,)30 b(w)n(e)g(ha)n(v)n(e)f FO(A)1032 1734 y FM(k)1073 1722 y FO(U)36 b FP(=)27 b FO(U)9 b(A)1386 1734 y FM(k)q FN(\000)p FL(1)1512 1722 y FO(B)1575 1734 y FL(1)1612 1722 y FP(.)44 b(Similarly)30 b(one)f(can)h(obtain)118 1822 y(the)e(equalities)f FO(A)690 1834 y FL(1)728 1822 y FO(U)32 b FP(=)22 b FO(U)9 b(B)1033 1834 y FL(1)1098 1822 y FP(and)27 b FO(B)1322 1834 y FM(k)1363 1822 y FO(A)1425 1834 y FL(1)1463 1822 y FO(U)k FP(=)23 b FO(U)9 b(B)1768 1834 y FM(k)q FL(+1)1893 1822 y FP(.)243 1921 y(Applying)28 b(similar)f(argumen)n(ts)f(as)i(in)g(Section)f(2.2.2)g (to)h FO(A)2133 1933 y FM(k)2174 1921 y FP(,)g FO(A)2287 1933 y FM(k)q FN(\000)p FL(1)2441 1921 y FP(and)118 2021 y FO(B)181 2033 y FL(1)218 2021 y FP(,)21 b(one)e(can)f(sho)n(w)g(that) h(for)g(an)n(y)f(Borel)g(set)h(\001)k FQ(\032)g FJ(R)1751 1991 y Fv(N)1822 2021 y FP(the)c(follo)n(wing)f(relation)118 2121 y(holds:)578 2294 y FO(E)639 2306 y FM(B)689 2314 y Fy(1)722 2306 y FM(;)p Fn(A)802 2294 y FP(\()p FJ(R)25 b FQ(\002)18 b FP(\001\))p FO(U)32 b FP(=)23 b FO(U)9 b(E)1401 2306 y FM(B)1451 2314 y Fy(1)1484 2306 y FM(;)p Fn(A)1564 2294 y FP(\()p FO(F)1661 2259 y FN(\000)p FL(1)1649 2316 y(1)1750 2294 y FP(\()p FJ(R)25 b FQ(\002)18 b FP(\001\)\))p FO(;)263 b FP(\(2.78\))118 2468 y(where)479 2641 y FO(F)544 2606 y FN(\000)p FL(1)532 2663 y(1)633 2641 y FP(\()p FO(y)706 2653 y FL(1)743 2641 y FO(;)14 b(x)827 2653 y FL(1)865 2641 y FO(;)g(x)949 2653 y FL(2)986 2641 y FO(;)g(x)1070 2653 y FL(3)1108 2641 y FO(;)g(:)g(:)g(:)g FP(\))23 b(=)g(\()p FO(x)1478 2653 y FL(1)1515 2641 y FO(;)14 b(x)1599 2653 y FL(2)1637 2641 y FO(=x)1726 2653 y FL(1)1763 2641 y FO(;)g(x)1847 2653 y FL(3)1885 2641 y FO(=x)1974 2653 y FL(1)2011 2641 y FO(;)g(:)g(:)g(:)g FP(\))p FO(:)118 2815 y FP(Note)19 b(that)f(the)h(righ)n(t-hand)e(side) i(of)25 b(\(2.78\))17 b(can)h(b)r(e)h(de\014ned)g(ev)n(en)f(for)g FO(x)2362 2827 y FL(1)2423 2815 y FP(=)k(0.)118 2914 y(Indeed,)29 b(since)f FO(E)677 2926 y FM(B)727 2934 y Fy(1)760 2926 y FM(;)p Fn(A)840 2914 y FP(\()p FQ(f)p FP(0)p FQ(g)18 b(\002)g FJ(R)25 b FQ(\002)19 b FJ(R)25 b FQ(\002)18 b(\001)c(\001)g(\001)g FP(\))28 b(is)h(a)f(pro)5 b(jection)27 b(on)h(k)n(er)13 b FO(B)2449 2926 y FL(1)2510 2914 y FP(=)118 3014 y(k)n(er)g FO(U)c FP(,)33 b(the)g(whole)f (expression)f(is)h(zero.)51 b(Th)n(us,)33 b(for)f(con)n(v)n(enience,)g (w)n(e)g(can)118 3114 y(set)691 3213 y FO(F)756 3178 y FN(\000)p FL(1)744 3235 y(1)846 3213 y FP(\()p FO(y)919 3225 y FL(1)956 3213 y FO(;)14 b FP(0)p FO(;)g(x)1119 3225 y FL(2)1156 3213 y FO(;)g(x)1240 3225 y FL(3)1277 3213 y FO(;)g(:)g(:)g(:)g FP(\))23 b(=)g(\(0)p FO(;)14 b FP(0)p FO(;)g FP(0)p FO(;)g(:)g(:)g(:)e FP(\))p FO(:)243 3357 y FP(In)41 b(a)f(similar)g(manner,)j(one)e(can)f(easily)g(deriv)n (e)g(from)g(the)h(equation)118 3457 y FO(B)181 3469 y FM(k)222 3457 y FO(A)284 3469 y FL(1)321 3457 y FO(U)32 b FP(=)23 b FO(U)9 b(B)627 3469 y FM(k)q FL(+1)779 3457 y FP(the)28 b(equalit)n(y)562 3630 y FO(E)623 3642 y FM(A)673 3650 y Fy(1)706 3642 y FM(;)p Fn(B)783 3630 y FP(\()p FJ(R)c FQ(\002)18 b FP(\001\))p FO(U)1143 3596 y FN(\003)1205 3630 y FP(=)k FO(U)1358 3596 y FN(\003)1396 3630 y FO(E)1457 3642 y FM(A)1507 3650 y Fy(1)1540 3642 y FM(;)p Fn(B)1617 3630 y FP(\()p FO(F)1714 3595 y FN(\000)p FL(1)1702 3653 y(2)1803 3630 y FP(\()p FJ(R)j FQ(\002)18 b FP(\001\)\))233 b(\(2.79\))118 3804 y(with)502 3904 y FO(F)567 3868 y FN(\000)p FL(1)555 3926 y(2)657 3904 y FP(\()p FO(x)736 3916 y FL(1)774 3904 y FO(;)14 b(y)852 3916 y FL(1)888 3904 y FO(;)g(y)966 3916 y FL(2)1003 3904 y FO(;)g(y)1081 3916 y FL(3)1118 3904 y FO(;)g(:)g(:)g(:)g FP(\))23 b(=)g(\()p FO(y)1482 3916 y FL(1)1519 3904 y FO(;)14 b(y)1597 3916 y FL(2)1634 3904 y FO(=y)1717 3916 y FL(1)1753 3904 y FO(;)g(y)1831 3916 y FL(3)1868 3904 y FO(=y)1951 3916 y FL(1)1987 3904 y FO(;)g(:)g(:)g(:)g FP(\))p FO(:)118 4048 y FP(P)n(assing)38 b(to)i(the)g(adjoin)n(t)f(op)r (erators)f(in)i(\(2.79\))f(and)h(com)n(bining)f(it)h(with)118 4147 y(\(2.78\))27 b(w)n(e)g(get)h(relations)e(\(2.77\))o(.)p eop %%Page: 187 191 187 190 bop 118 100 a FK(2.5.)36 b(Represen)n(tations)26 b(of)i(some)f(n)n(uclear)f(algebras)673 b FP(187)243 333 y(The)27 b(fact)h(that)g FO(U)36 b FP(is)28 b(cen)n(tered)f(follo)n (ws)f(directly)i(from)f(\(2.77\))o(.)243 434 y(The)i(pro)r(of)g(that)g (the)h(collection)f(of)g(non-negativ)n(e)e(self-adjoin)n(t)i(op)r(era-) 118 534 y(tors)23 b(\()p FO(A)375 546 y FM(k)416 534 y FO(;)14 b(B)516 546 y FM(k)557 534 y FP(\))24 b(and)f(the)h(cen)n (tered)e FO(U)33 b FP(\(whic)n(h)23 b(is)h(a)f(partial)f(isometry\))h (whic)n(h)118 633 y(satisfy)28 b(relations)e(\(2.77\))i(generates)e(a)i (cen)n(tered)f(op)r(erator)f(is)i(a)g(direct)g(cal-)118 733 y(culation.)p 2514 733 4 57 v 2518 680 50 4 v 2518 733 V 2567 733 4 57 v 243 913 a(It)d(is)f(a)g(standard)g(argumen)n(t)f (similar)h(to)h(the)g(one)f(in)h(Section)f(2.2.2)g(that)118 1012 y(allo)n(ws)18 b(one)h(to)h(pro)n(v)n(e)d(the)j(follo)n(wing)f (decomp)r(osition)g(of)g(cen)n(tered)g(op)r(erators)118 1112 y(similar)27 b(to)g(the)h(W)-7 b(old)28 b(decomp)r(osition)f(of)h (isometries.)118 1283 y FR(Prop)s(osition)40 b(62.)46 b FC(L)l(et)38 b FO(T)48 b FC(b)l(e)38 b(a)h(c)l(enter)l(e)l(d)e(op)l (er)l(ator)i(in)f(a)g(Hilb)l(ert)g(sp)l(ac)l(e)118 1383 y FO(H)7 b FC(.)50 b(The)34 b(sp)l(ac)l(e)g FO(H)40 b FC(c)l(an)34 b(b)l(e)f(de)l(c)l(omp)l(ose)l(d)i(into)f(the)f(dir)l(e)l (ct)h(sum)f(of)h(the)g(two)118 1482 y(invariant)g(with)g(r)l(esp)l(e)l (ct)f(to)h FO(T)12 b FC(,)33 b FO(T)1223 1452 y FN(\003)1294 1482 y FC(subsp)l(ac)l(es,)i FO(H)h FP(=)29 b FO(H)1959 1494 y FL(0)2017 1482 y FQ(\010)21 b FO(H)2172 1494 y FL(1)2242 1482 y FC(such)34 b(that)118 1582 y FP(k)n(er)13 b FO(T)29 b FQ([)19 b FP(k)n(er)13 b FO(T)581 1552 y FN(\003)641 1582 y FP(=)23 b FQ(f)p FP(0)p FQ(g)28 b FC(in)i FO(H)1054 1594 y FL(1)1091 1582 y FC(,)g(and)h FO(H)1377 1594 y FL(0)1444 1582 y FC(is)f(gener)l(ate)l(d)g(by)g FP(k)n(er)13 b FO(T)29 b FQ([)19 b FP(k)n(er)13 b FO(T)2472 1552 y FN(\003)2509 1582 y FC(.)243 1753 y FP(The)18 b(degenerate)f(represen)n(tations)g(can)h(b)r(e)h(completely)f(describ) r(ed)g(up)h(to)118 1853 y(a)f(unitary)g(equiv)-5 b(alence)18 b(\(see)h(b)r(elo)n(w\).)33 b(The)19 b(structure)f(of)g(represen)n (tations)f(in)118 1952 y FO(H)187 1964 y FL(1)244 1952 y FP(is)j(more)f(complicated;)j(ho)n(w)n(ev)n(er,)d(realization)f(of)i (cen)n(tered)f(op)r(erators)f(as)118 2052 y(\\op)r(erator-v)-5 b(alued)18 b(w)n(eigh)n(ted)i(shifts")g(follo)n(ws)g(from)g(Theorem)f (45)h(if)h(applied)118 2152 y(to)28 b(relations)e(\(2.77\))o(.)118 2323 y FR(Theorem)42 b(46.)k FC(L)l(et)39 b FO(T)49 b FC(b)l(e)39 b(a)g(c)l(enter)l(e)l(d)g(op)l(er)l(ator)h(with)f(zer)l(o)g (kernel)g(and)118 2422 y(dense)22 b(image.)37 b(Then)22 b(it)f(c)l(an)h(b)l(e)f(r)l(e)l(alize)l(d)i(in)e(the)h(sp)l(ac)l(e)g FO(L)1906 2434 y FL(2)1943 2422 y FP(\()p FJ(R)2029 2392 y FN(1)2029 2443 y FL(+)2106 2422 y FQ(\002)p FJ(R)2225 2392 y FN(1)2225 2443 y FL(+)2301 2422 y FO(;)14 b FB(H)q FO(;)g(d\026)p FP(\))118 2522 y FC(of)33 b(ve)l(ctor-value)l(d)f(squar) l(e-inte)l(gr)l(able)h(functions)e(having)j(their)e(values)g(in)g(a)118 2622 y(c)l(ertain)e(Hilb)l(ert)g(sp)l(ac)l(e)g FB(H)g FC(by)g(the)g(formula)240 2819 y FP(\()p FO(T)12 b(f)d FP(\)\()p FR(x)p FO(;)14 b FR(y)q FP(\))24 b(=)f FO(x)776 2776 y FL(1)p FM(=)p FL(2)776 2841 y(1)881 2819 y FO(u)p FP(\()p FR(x)p FO(;)14 b FR(y)q FP(\)\()p FO(d\026)p FP(\()p FO(F)e FP(\()p FR(x)p FO(;)i FR(y)q FP(\)\))p FO(=d\026)p FP(\()p FR(x)p FO(;)g FR(y)q FP(\)\))1956 2785 y FL(1)p FM(=)p FL(2)2065 2819 y FO(f)9 b FP(\))p FO(F)j FP(\()p FR(x)p FO(;)i FR(y)q FP(\))p FO(;)2363 2919 y FP(\(2.80\))118 3106 y FC(wher)l(e)39 b FO(F)12 b FP(\()p FQ(\001)p FO(;)i FQ(\001)p FP(\))39 b FC(is)f(intr)l(o)l(duc) l(e)l(d)g(ab)l(ove,)k FO(\026)p FP(\()p FQ(\001)p FO(;)14 b FQ(\001)p FP(\))39 b FC(is)g(a)f FO(F)12 b FP(\()p FQ(\001)p FO(;)i FQ(\001)p FP(\))p FC(-quasi-invariant)118 3205 y(pr)l(ob)l(ability)24 b(Bor)l(el)e(me)l(asur)l(e)f(and)h FO(u)p FP(\()p FQ(\001)p FO(;)14 b FQ(\001)p FP(\))22 b FC(is)g(a)g(unitary)f(me)l(asur)l(able)h(op)l(er)l(ator-)118 3305 y(value)l(d)31 b(function.)243 3406 y(Conversely,)44 b(any)c(c)l(ol)t(le)l(ction)h FB(H)q FC(,)h FO(\026)p FP(\()p FQ(\001)p FO(;)14 b FQ(\001)p FP(\))p FC(,)43 b(and)e FO(u)p FP(\()p FQ(\001)p FO(;)14 b FQ(\001)p FP(\))39 b FC(which)j(has)f(the)118 3506 y(mentione)l(d)d(pr)l(op)l (erties)h(gener)l(ates)f(a)g(c)l(enter)l(e)l(d)g(op)l(er)l(ator)h(by)f (the)g(formula)118 3606 y(ab)l(ove.)243 3777 y FP(Represen)n(tations)c (with)i(non-zero)e(k)n(ernel)h(can)g(b)r(e)i(completely)e(classi-)118 3876 y(\014ed.)118 4048 y FR(Theorem)43 b(47.)k FC(A)n(l)t(l)41 b(irr)l(e)l(ducible)g(r)l(epr)l(esentations)f(of)58 b FP(\(2.76\))39 b FC(for)i(which)118 4147 y FP(k)n(er)13 b FO(T)29 b FQ([)19 b FP(k)n(er)13 b FO(T)581 4117 y FN(\003)641 4147 y FQ(6)p FP(=)23 b FQ(f)p FP(0)p FQ(g)28 b FC(fal)t(l)j(into)f(the)g(fol)t(lowing)i(classes)7 b FP(:)p eop %%Page: 188 192 188 191 bop 118 100 a FP(188)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)191 333 y FP(\()p FO(i)p FP(\))42 b FC (\014nite-dimensional)30 b(r)l(epr)l(esentations)g(in)g FJ(C)1733 303 y FM(n)655 515 y FO(T)12 b(e)755 527 y FM(j)812 515 y FP(=)23 b FO(\025)948 527 y FM(j)983 515 y FO(e)1022 527 y FM(j)s FL(+1)1141 515 y FO(;)184 b(j)28 b FP(=)22 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n)19 b FQ(\000)f FP(1)p FO(;)43 b(\025)2031 527 y FM(j)2089 515 y FO(>)23 b FP(0)p FO(;)645 639 y(T)12 b(e)745 651 y FM(n)812 639 y FP(=)23 b(0;)162 855 y(\()p FO(ii)p FP(\))42 b FC (in\014nite-dimensional)31 b(in)e FO(l)1201 867 y FL(2)733 1037 y FO(T)12 b(e)833 1049 y FM(j)890 1037 y FP(=)22 b FO(\025)1025 1049 y FM(j)1061 1037 y FO(e)1100 1049 y FM(j)s FL(+1)1218 1037 y FO(;)184 b(j)28 b FP(=)23 b(1)p FO(;)14 b FP(2)p FO(;)g(:)g(:)g(:)e(;)44 b(\025)1957 1049 y FM(j)2015 1037 y FO(>)23 b FP(0;)134 1252 y(\()p FO(iii)p FP(\))41 b FC(in\014nite-dimensional)31 b(in)e FO(l)1201 1264 y FL(2)535 1435 y FO(T)12 b(e)635 1447 y FL(1)694 1435 y FP(=)23 b(0)p FO(;)98 b(T)12 b(e)1045 1447 y FM(j)1102 1435 y FP(=)23 b FO(\025)1238 1447 y FM(j)1273 1435 y FO(e)1312 1447 y FM(j)s FN(\000)p FL(1)1432 1435 y FO(;)183 b(j)28 b FP(=)23 b(2)p FO(;)14 b FP(3)p FO(;)g(:)g(:)g(:)f(;)27 b(\025)2154 1447 y FM(j)2213 1435 y FO(>)22 b FP(0)p FO(:)118 1650 y FC(Pr)l(o)l(of.)43 b FP(Indeed,)34 b(the)f(phase)f FO(U)41 b FP(of)32 b(the)h(op)r(erator) d FO(T)44 b FP(is)32 b(a)g(cen)n(tered)g(partial)118 1750 y(isometry)g(\(non-unitary\);)k(in)e(the)f(same)g(w)n(a)n(y)f(as)h (in)g(Theorem)g(19)f(w)n(e)h(get)118 1849 y(that)27 b(the)h(represen)n (tation)d(space)h(for)h(an)f(irreducible)h FO(T)38 b FP(is)26 b(the)i(same)e(as)g(for)118 1949 y(an)i(irreducible)g FO(U)9 b FP(.)40 b(Use)29 b(Theorem)f(18)g(to)g(represen)n(t)g(the)h (op)r(erator)d FO(U)9 b FP(;)29 b(the)118 2049 y(rest)e(of)h(the)g(pro) r(of)f(follo)n(ws)f(immediately)i(from)f(\(2.77\).)p 2514 2049 4 57 v 2518 1996 50 4 v 2518 2049 V 2567 2049 4 57 v 118 2215 a FR(3.)49 b FP(Finally)-7 b(,)33 b(w)n(e)f(giv)n(e)f (a)g(complete)h(list)g(of)g(\014nite-dimensional)g(irreducible)118 2314 y(cen)n(tered)39 b(op)r(erators.)72 b(W)-7 b(e)41 b(ha)n(v)n(e)d(already)h(seen)h(that)g(there)f(is)h(a)g(family)118 2414 y(of)d(\014nite-dimensional)f(represen)n(tations)f(\(Theorem)h (47\).)63 b(In)37 b(fact,)i(there)118 2513 y(are)32 b(some)g (irreducible)f(\014nite-dimensional)i(represen)n(tations)d(with)j(a)f (non-)118 2613 y(degenerate)26 b FO(T)12 b FP(.)118 2779 y FR(Theorem)37 b(48.)43 b FC(A)n(l)t(l)35 b(irr)l(e)l(ducible)h (\014nite-dimensional)g(r)l(epr)l(esentations)f(of)118 2878 y FP(\(2.76\))c FC(ar)l(e)i(either)f(the)h(\014nite-dimensional)g (r)l(epr)l(esentations)f(describ)l(e)l(d)h(by)118 2978 y(The)l(or)l(em)k FP(47)29 b FC(or)h(r)l(epr)l(esentations)g(of)h(the)f (form)6 b FP(:)315 3160 y FO(T)12 b(e)415 3172 y FM(j)472 3160 y FP(=)22 b FO(\025)607 3172 y FM(j)643 3160 y FO(e)682 3172 y FM(j)s FL(+1)800 3160 y FO(;)184 b(j)28 b FP(=)23 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n)k FQ(\000)g FP(1)p FO(;)305 3285 y(T)12 b(e)405 3297 y FM(n)472 3285 y FP(=)22 b FO(\013\025)660 3297 y FM(n)706 3285 y FO(e)745 3297 y FL(1)782 3285 y FO(;)184 b(\025)1037 3297 y FM(j)1095 3285 y FO(>)23 b FP(0)p FO(;)43 b(\025)1339 3297 y FM(j)1398 3285 y FQ(6)p FP(=)22 b FO(\025)1533 3297 y FM(k)1634 3285 y FC(for)60 b FO(j)28 b FQ(6)p FP(=)23 b FO(k)s(;)98 b FQ(j)p FO(\013)p FQ(j)24 b FP(=)f(1)p FO(:)118 3467 y FC(Pr)l(o)l(of.)43 b FP(Indeed,)36 b(it)d(is)h(easy)e(to)i(see)f (that)h(represen)n(tation)d(\(2.80\))i(is)h(\014nite-)118 3567 y(dimensional)f(only)g(if)h(the)g(measure)f FO(\026)p FP(\()p FQ(\001)p FP(\))h(is)f(concen)n(trated)g(on)g(a)g(p)r(erio)r (dic)118 3666 y(orbit)25 b(of)g FO(F)12 b FP(\()p FQ(\001)p FP(\).)37 b(Irreducibilit)n(y)25 b(in)g(this)h(case)e(implies)i(that)g (the)f(sp)r(ectrum)h(of)118 3766 y(the)j(op)r(erator)e FO(B)661 3778 y FL(1)726 3766 y FP(is)h(simple.)40 b(P)n(assing)26 b(to)j(a)e(unitarily)i(equiv)-5 b(alen)n(t)28 b(realiza-)118 3866 y(tion)g(w)n(e)f(get)g(the)h(necessary)e(form)n(ulae.)p 2514 3866 V 2518 3813 50 4 v 2518 3866 V 2567 3866 4 57 v 118 4031 a FR(Corollary)32 b(6.)39 b FC(A)n(ny)29 b(factor)g(gener)l(ate)l(d)g(by)g(a)g(c)l(enter)l(e)l(d)f(op)l(er)l (ator)i(is)f(hyp)l(er-)118 4131 y(\014nite.)p eop %%Page: 189 193 189 192 bop 118 100 a FK(2.5.)36 b(Represen)n(tations)26 b(of)i(some)f(n)n(uclear)f(algebras)673 b FP(189)118 333 y FC(Pr)l(o)l(of.)43 b FP(Indeed,)d(\(2.77\))c(imply)i(that)f(the)h (corresp)r(onding)d(v)n(on)i(Neumann)118 432 y(algebra)30 b(is)h(either)g(of)g(t)n(yp)r(e)g FO(I)7 b FP(,)32 b(if)g(the)g (represen)n(tation)d(is)i(degenerate,)g(or)f(a)118 532 y(crossed)f(pro)r(duct)i(of)f(a)h(comm)n(utativ)n(e)e(algebra)g(b)n(y)i (the)g(group)e FJ(Z)o FP(.)40 b(By)31 b([76)o(])118 632 y(w)n(e)c(get)h(the)g(assertion.)p 2514 632 4 57 v 2518 579 50 4 v 2518 632 V 2567 632 4 57 v 118 831 a FR(Corollary)36 b(7.)43 b FC(Sinc)l(e)32 b(for)h(a)g(p)l(air)g(of)g(self-adjoint)i(op)l (er)l(ators)e(ther)l(e)f(exists)118 931 y(a)i(non-hyp)l(er\014nite)f (factor)h(r)l(epr)l(esentation,)h(the)e(description)i(pr)l(oblem)f(for) 118 1030 y(c)l(enter)l(e)l(d)28 b(op)l(er)l(ators)h(by)g(the)f(pr)l (evious)h(statement)e(is)i(not)f(wild)h(in)g(the)f(sense)118 1130 y(discusse)l(d)j(in)f(Se)l(ction)f FP(3)p FO(:)p FP(1)p FC(.)118 1368 y FR(2.5.3)94 b(Represen)m(tations)31 b(of)g(Cun)m(tz)i(algebras)118 1529 y FP(W)-7 b(e)32 b(consider)e(represen)n(tations)f(of)i(the)h(Cun)n(tz)g(algebras)d FB(O)2053 1541 y FM(n)2098 1529 y FP(.)48 b(Recall)31 b(that)118 1629 y(the)g(Cun)n(tz)g(algebra)f(is)g(generated)g(b)n(y)h FO(n)g FP(isometries,)f FO(S)1943 1641 y FL(1)1981 1629 y FP(,)h FO(:)14 b(:)g(:)27 b FP(,)32 b FO(S)2265 1641 y FM(n)2310 1629 y FP(,)g(whic)n(h)118 1728 y(satisfy)27 b(the)h(follo)n(wing)f(relations)573 1981 y FO(S)629 1947 y FN(\003)624 2002 y FM(i)667 1981 y FO(S)718 1993 y FM(i)768 1981 y FP(=)c FO(I)7 b(;)180 b(i)23 b FP(=)f(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n;)1676 1878 y FM(n)1637 1903 y Fz(X)1643 2079 y FM(i)p FL(=1)1771 1981 y FO(S)1822 1993 y FM(i)1849 1981 y FO(S)1905 1947 y FN(\003)1900 2002 y FM(i)1966 1981 y FP(=)23 b FO(I)7 b(:)243 b FP(\(2.81\))118 2245 y(Notice)28 b(that)g(the)g(relations)e(imply)i(that)g FO(S)1508 2215 y FN(\003)1503 2267 y FM(i)1546 2245 y FO(S)1597 2257 y FM(j)1655 2245 y FP(=)22 b(0)27 b(for)g FO(i)c FQ(6)p FP(=)g FO(j)5 b FP(.)243 2349 y(Our)21 b(goal)g(is)g(to)h(construct)g(a)f(comm)n(utativ)n(e)g(mo)r(del)h(for)g (represen)n(tations)118 2449 y(of)30 b(the)g(Cun)n(tz)g(algebra)d(and)j (to)f(sho)n(w)g(ho)n(w)g(it)h(can)g(b)r(e)g(used)f(to)h(study)g(rep-) 118 2548 y(resen)n(tations)g(of)h(the)h(Cun)n(tz)g(algebra.)47 b(The)31 b(main)h(statemen)n(t)f(here)g(is)g(the)118 2648 y(follo)n(wing)c(theorem.)118 2826 y FR(Theorem)35 b(49.)42 b FC(F)-6 b(or)33 b(any)h(r)l(epr)l(esentation)f(of)h(the)f (Cuntz)f(algebr)l(a)j FB(O)2389 2838 y FM(n)2467 2826 y FC(the)118 2926 y(fol)t(lowing)d(form)f(for)f FO(S)854 2938 y FL(1)892 2926 y FC(,)g FO(:)14 b(:)g(:)27 b FC(,)j FO(S)1177 2938 y FM(n)1252 2926 y FC(holds)7 b FP(:)679 3170 y FO(H)29 b FP(=)865 3057 y Fz(Z)948 3078 y FN(\010)911 3246 y Fv(Z)956 3229 y Fx(1)956 3263 y Fw(n)1028 3170 y FO(H)1097 3182 y FM(x)1153 3170 y FO(d\026)p FP(\()p FO(x)p FP(\))p FO(;)143 3366 y FP(\()p FO(S)226 3378 y FM(i)254 3366 y FO(f)9 b FP(\)\()p FO(x)415 3378 y FL(1)453 3366 y FO(;)14 b(x)537 3378 y FL(2)574 3366 y FO(;)g(:)g(:)g(:)g FP(\))23 b(=)g FO(\016)902 3378 y FM(i)929 3366 y FP(\()p FO(x)1008 3378 y FL(1)1046 3366 y FP(\))14 b FO(U)1149 3378 y FM(i)1177 3366 y FP(\()p FO(x)1256 3378 y FL(2)1294 3366 y FO(;)g(x)1378 3378 y FL(3)1415 3366 y FO(;)g(:)g(:)g(:)g FP(\))858 3572 y FQ(\002)941 3455 y Fz(\022)1012 3516 y FO(d)p FP(\()p FO(\016)1124 3528 y FM(i)1152 3516 y FP(\()p FO(x)1231 3528 y FL(1)1269 3516 y FP(\))k FQ(\012)g FO(\026)p FP(\()p FO(x)1531 3528 y FL(2)1569 3516 y FO(;)c(x)1653 3528 y FL(3)1691 3516 y FO(;)g(:)g(:)g(:)g FP(\))p 1012 3553 860 4 v 1185 3629 a FO(d\026)p FP(\()p FO(x)1357 3641 y FL(1)1396 3629 y FO(;)g(x)1480 3641 y FL(2)1517 3629 y FO(;)g(:)g(:)g(:)g FP(\))1881 3455 y Fz(\023)1942 3472 y FL(1)p FM(=)p FL(2)2060 3572 y FO(f)9 b FP(\()p FO(x)2189 3584 y FL(2)2227 3572 y FO(;)14 b(x)2311 3584 y FL(3)2348 3572 y FO(;)g(:)g(:)g(:)g FP(\))p FO(;)128 3754 y FP(\()p FO(S)216 3720 y FN(\003)211 3774 y FM(i)254 3754 y FO(f)9 b FP(\)\()p FO(x)415 3766 y FL(1)453 3754 y FO(;)14 b(x)537 3766 y FL(2)574 3754 y FO(;)g(:)g(:)g(:)g FP(\))23 b(=)g FO(U)931 3720 y FN(\003)922 3774 y FM(i)969 3754 y FP(\()p FO(x)1048 3766 y FL(1)1086 3754 y FO(;)14 b(x)1170 3766 y FL(2)1207 3754 y FO(;)g(:)g(:)g(:)g FP(\))858 3960 y FQ(\002)941 3843 y Fz(\022)1012 3904 y FO(d\026)p FP(\()p FO(i;)g(x)1250 3916 y FL(1)1287 3904 y FO(;)g(x)1371 3916 y FL(2)1409 3904 y FO(;)g(:)g(:)g(:)g FP(\))p 1012 3941 578 4 v 1044 4017 a FO(d\026)p FP(\()p FO(x)1216 4029 y FL(1)1255 4017 y FO(;)g(x)1339 4029 y FL(2)1376 4017 y FO(;)g(:)g(:)g(:)g FP(\))1599 3843 y Fz(\023)1660 3861 y FL(1)p FM(=)p FL(2)1778 3960 y FO(f)9 b FP(\()p FO(i;)14 b(x)1973 3972 y FL(1)2010 3960 y FO(;)g(x)2094 3972 y FL(2)2132 3960 y FO(;)g(:)g(:)g(:)f FP(\))p FO(;)2363 4117 y FP(\(2.82\))p eop %%Page: 190 194 190 193 bop 118 100 a FP(190)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(Her)l(e,)h FO(\026)p FP(\()p FQ(\001)p FP(\))f FC(is)h(a)f(pr)l(ob)l(ability)i(me)l(asur)l(e)d (de\014ne)l(d)i(on)e(the)i(cylinder)g FO(\033)s FC(-algebr)l(a,)118 432 y(quasi-invariant)33 b(with)g(r)l(esp)l(e)l(ct)f(to)g(the)h(tr)l (ansformations)g FO(\026)p FP(\()p FO(x)2125 444 y FL(1)2163 432 y FO(;)14 b(x)2247 444 y FL(2)2284 432 y FO(;)g(:)g(:)g(:)g FP(\))28 b FQ(7!)118 532 y FO(\016)155 544 y FM(i)183 532 y FP(\()p FO(x)262 544 y FL(1)300 532 y FP(\))c FQ(\012)f FO(\026)p FP(\()p FO(x)573 544 y FL(2)611 532 y FO(;)14 b(x)695 544 y FL(3)733 532 y FO(;)g(:)g(:)g(:)f FP(\))p FC(,)40 b FO(i)d FP(=)f(1)p FC(,)j FP(2)p FC(,)e FO(:)14 b(:)g(:)51 b FP(;)41 b FO(H)1635 544 y FM(x)1714 532 y FC(is)d(a)f(me)l(asur)l(able)h(\014eld)g(of)118 632 y(Hilb)l(ert)33 b(sp)l(ac)l(es)g(such)f(that)h FO(d)p FP(\()p FO(x)p FP(\))c(=)e(dim)14 b FO(H)1507 644 y FM(x)1582 632 y FC(is)32 b(invariant)i FO(\026)p FC(-a.e.)47 b(with)33 b(r)l(e-)118 731 y(sp)l(e)l(ct)e(to)h(the)f(tr)l(ansformations)h FO(d)p FP(\()p FO(x)1286 743 y FL(1)1324 731 y FO(;)14 b(x)1408 743 y FL(2)1446 731 y FO(;)g(:)g(:)g(:)g FP(\))26 b FQ(7!)g FO(d)p FP(\()p FO(i;)14 b(x)1949 743 y FL(1)1987 731 y FO(;)g(x)2071 743 y FL(2)2108 731 y FO(;)g(:)g(:)g(:)g FP(\);)32 b FO(U)2400 743 y FL(1)2437 731 y FP(\()p FO(x)p FP(\))p FC(,)118 831 y FO(:)14 b(:)g(:)28 b FC(,)i FO(U)355 843 y FM(n)400 831 y FP(\()p FO(x)p FP(\))g FC(ar)l(e)g(me)l(asur)l (able)h(unitary)e(op)l(er)l(ator-value)l(d)j(functions.)243 930 y(The)e(er)l(go)l(dicity)g(of)g(the)f(sp)l(e)l(ctr)l(al)h(me)l (asur)l(e)e FO(\026)h FC(is)h(a)f(ne)l(c)l(essary)h(c)l(ondition)118 1030 y(of)41 b(the)f(irr)l(e)l(ducibility)h(of)g(the)f(r)l(epr)l (esentation)6 b FP(;)45 b FC(in)40 b(the)g(c)l(ase)g(of)h(a)f(simple) 118 1130 y(joint)33 b(sp)l(e)l(ctrum)k FP(\()p FC(if)c FO(H)859 1142 y FM(x)933 1130 y FC(ar)l(e)f(one-dimensional)h FO(\026)p FC(-a.e.)p FP(\))p FC(,)i(the)d(er)l(go)l(dicity)h(is)118 1229 y(also)e(su\016cient)e(for)i(the)f(irr)l(e)l(ducibility.)243 1329 y(Two)i(r)l(epr)l(esentations)g(of)g(the)g(form)39 b FP(\(2.82\))30 b FC(ar)l(e)i(unitary)g(e)l(quivalent)g(if)118 1429 y(and)e(only)h(if)18 b FP(:)243 1528 y FC(i.)39 b(the)30 b(sp)l(e)l(ctr)l(al)g(me)l(asur)l(es)f FO(\026)h FC(and)37 b FP(~)-49 b FO(\026)30 b FC(ar)l(e)g(e)l(quivalent)8 b FP(;)243 1628 y FC(ii.)46 b(the)32 b(multiplicity)h(functions)f FO(d)p FP(\()p FO(x)p FP(\))c(=)f(dim)14 b FO(H)1801 1640 y FM(x)1875 1628 y FC(and)2053 1606 y FP(~)2038 1628 y FO(d)p FP(\()p FO(x)p FP(\))28 b(=)f(dim)2486 1607 y(~)2464 1628 y FO(H)2533 1640 y FM(x)118 1727 y FC(c)l(oincide)32 b FO(\026)p FC(-a.e.)p FP(;)243 1827 y FC(iii.)86 b(the)45 b(c)l(ol)t(le)l(ctions)i(of)f(unitary)f(op)l(er)l (ator)h(functions)f FP(\()p FO(U)2226 1839 y FM(i)2254 1827 y FP(\()p FO(x)p FP(\)\))h FC(and)118 1927 y FP(\()165 1906 y(~)150 1927 y FO(U)207 1939 y FM(i)235 1927 y FP(\()p FO(x)p FP(\)\))41 b FC(ar)l(e)e(e)l(quivalent)h(in)g(the)g(fol)t (lowing)i(sense)6 b FP(:)58 b FC(ther)l(e)40 b(exists)f(a)h(me)l(a-)118 2026 y(sur)l(able)30 b(unitary)g(op)l(er)l(ator-value)l(d)h(function)f FO(V)19 b FP(\()p FO(x)p FP(\))30 b FC(such)g(that)201 2232 y FO(V)268 2197 y FN(\003)306 2232 y FP(\()p FO(x)385 2244 y FL(1)423 2232 y FO(;)14 b(x)507 2244 y FL(2)545 2232 y FO(;)g(:)g(:)g(:)f FP(\))h FO(U)804 2197 y FN(\003)795 2252 y FM(i)842 2232 y FP(\()p FO(x)921 2244 y FL(1)959 2232 y FO(;)g(x)1043 2244 y FL(2)1081 2232 y FO(;)g(:)g(:)g(:)g FP(\))g FO(V)19 b FP(\(1)p FO(;)14 b(x)1500 2244 y FL(1)1537 2232 y FO(;)g(x)1621 2244 y FL(2)1659 2232 y FO(;)g(:)g(:)g(:)f FP(\))1237 2371 y(=)1339 2350 y(~)1325 2371 y FO(U)1391 2337 y FN(\003)1382 2392 y FM(i)1429 2371 y FP(\()p FO(x)1508 2383 y FL(1)1545 2371 y FO(;)h(x)1629 2383 y FL(2)1667 2371 y FO(;)g(:)g(:)g(:)g FP(\))p FO(;)184 b(i)22 b FP(=)h(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n:)243 2551 y FC(Conversely,)26 b(a)d(family)i(c)l(onsisting)e(of)h(a)f(quasi-invariant)h(me)l(asur)l (e)f FO(\026)p FP(\()p FQ(\001)p FP(\))p FC(,)118 2651 y(a)g(dimension)g(function)f FO(d)p FP(\()p FQ(\001)p FP(\))p FC(,)j(and)e(a)g(c)l(ol)t(le)l(ction)g(of)g(unitary)f(op)l(er)l (ator-value)l(d)118 2751 y(functions)31 b FP(\()p FO(U)569 2763 y FM(i)597 2751 y FP(\()p FQ(\001)p FP(\)\))p FC(,)i(p)l (ossessing)f(the)g(liste)l(d)g(pr)l(op)l(erties,)h(determines)f(a)g(r)l (ep-)118 2850 y(r)l(esentation)e(of)g(the)g(Cuntz)f(algebr)l(a)i FB(O)1365 2862 y FM(n)1410 2850 y FC(.)118 3014 y(Pr)l(o)l(of.)43 b FP(First)28 b(w)n(e)g(in)n(tro)r(duce)g(some)g(notations.)38 b(F)-7 b(or)28 b(an)n(y)f(m)n(ulti-index)i FO(\013)24 b FP(=)118 3114 y(\()p FO(\013)203 3126 y FL(1)241 3114 y FO(;)14 b(:)g(:)g(:)f(;)h(\013)478 3126 y FM(s)514 3114 y FP(\),)35 b FO(\013)657 3126 y FM(j)724 3114 y FQ(2)e FJ(Z)873 3126 y FM(n)912 3114 y FP(,)i FO(s)d FQ(\025)g FP(0,)i(write)f FO(S)1506 3126 y FM(\013)1586 3114 y FP(=)f FO(S)1734 3126 y FM(\013)1777 3134 y Fy(1)1827 3114 y FO(:)14 b(:)g(:)g(S)1989 3126 y FM(\013)2032 3134 y Fw(s)2068 3114 y FP(,)35 b FO(P)2179 3126 y FM(\013)2258 3114 y FP(=)d FO(S)2406 3126 y FM(\013)2454 3114 y FO(S)2510 3084 y FN(\003)2505 3135 y FM(\013)2552 3114 y FP(.)118 3214 y(The)g(set)g(of)f(all)h(\014nite)g(m)n(ulti-indices)g FO(\013)g FP(will)g(b)r(e)h(denoted)e(b)n(y)h(\000.)49 b(The)32 b(fol-)118 3313 y(lo)n(wing)27 b(statemen)n(t)g(is)h(obtained) f(b)n(y)g(an)h(easy)e(direct)i(calculation.)118 3463 y FR(Prop)s(osition)g(63.)39 b FC(The)29 b(op)l(er)l(ators)f FO(P)1373 3475 y FM(\013)1421 3463 y FC(,)h(wher)l(e)f FO(\013)g FC(r)l(anges)h(over)f FP(\000)p FC(,)h(form)f(a)118 3562 y(c)l(ommuting)h(family)j(of)f(pr)l(oje)l(ctions.)39 b(A)n(lso,)368 3743 y FO(S)419 3755 y FM(i)446 3743 y FO(P)499 3755 y FM(\013)542 3763 y Fy(1)575 3755 y FM(:::)o(\013)677 3763 y Fw(s)737 3743 y FP(=)22 b FO(P)877 3755 y FM(i\013)943 3763 y Fy(1)977 3755 y FM(:::)o(\013)1079 3763 y Fw(s)1115 3743 y FO(S)1166 3755 y FM(i)1194 3743 y FO(;)98 b(S)1371 3708 y FN(\003)1366 3763 y FM(i)1409 3743 y FO(P)1462 3755 y FM(\013)1505 3763 y Fy(1)1538 3755 y FM(:::)o(\013)1640 3763 y Fw(s)1699 3743 y FP(=)23 b FO(\016)1824 3755 y FM(i\013)1890 3763 y Fy(1)1927 3743 y FO(P)1980 3755 y FM(\013)2023 3763 y Fy(2)2056 3755 y FM(:::)o(\013)2158 3763 y Fw(s)2195 3743 y FO(S)2251 3708 y FN(\003)2246 3763 y FM(i)2288 3743 y FO(;)608 3867 y(i;)14 b(\013)727 3879 y FL(1)764 3867 y FO(;)g(:)g(:)g(:)g(;)g(\013)1002 3879 y FM(s)1060 3867 y FP(=)23 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n;)183 b(s)23 b FP(=)g(1)p FO(;)14 b FP(2)p FO(;)g(:)g(:)g(:)26 b(:)279 b FP(\(2.83\))243 4048 y(According)25 b(to)h(Theorem)g(45,)g (relations)f(\(2.83\))h(imply)h(that)f(the)h(op)r(era-)118 4147 y(tors)i FO(S)338 4159 y FL(1)375 4147 y FP(,)h FO(:)14 b(:)g(:)28 b FP(,)j FO(S)658 4159 y FM(n)733 4147 y FP(act)e(as)h(w)n(eigh)n(ted)f(op)r(erator-v)-5 b(alued)28 b(shifts)i(on)f(the)i(join)n(t)p eop %%Page: 191 195 191 194 bop 118 100 a FK(2.5.)36 b(Represen)n(tations)26 b(of)i(some)f(n)n(uclear)f(algebras)673 b FP(191)118 333 y(sp)r(ectrum)31 b(of)f(the)h(comm)n(uting)f(family)h(\()p FO(P)1503 345 y FM(\013)1551 333 y FP(\))1583 345 y FM(\013)p FN(2)p FL(\000)1747 333 y FP(in)g(the)g(space)e(of)i(F)-7 b(ourier)118 432 y(images)37 b(relativ)n(e)h(to)g(this)h(comm)n(uting)f (family)-7 b(.)69 b(Our)38 b(further)g(task)g(is)g(to)118 532 y(describ)r(e)30 b(the)g(join)n(t)f(sp)r(ectrum)h(of)g(the)g(comm)n (uting)g(family)f(of)h(pro)5 b(jections)118 632 y(men)n(tioned)25 b(ab)r(o)n(v)n(e,)f(and)h(to)g(study)g(the)h(corresp)r(onding)d (dynamical)h(system)118 731 y(on)j(it.)243 831 y(According)19 b(to)i(the)g(sp)r(ectral)g(theorem)f(for)g(an)h(in\014nite)h(comm)n (uting)e(fam-)118 930 y(ily)34 b(of)h(b)r(ounded)f(self-adjoin)n(t)g (op)r(erators)e(\(see,)k(for)e(example,)h([243)o(])f(etc.\),)118 1030 y(for)27 b(an)n(y)g FO(\013)c FQ(2)h FP(\000)j(w)n(e)h(ha)n(v)n(e) 856 1244 y FO(P)909 1256 y FM(\013)980 1244 y FP(=)1067 1131 y Fz(Z)1113 1319 y FN(f)p FL(0)p FM(;)p FL(1)p FN(g)1267 1303 y Fy(\000)1324 1244 y FO(\025)p FP(\()p FO(\013)p FP(\))14 b FO(dE)5 b FP(\()p FO(\025)p FP(\()p FQ(\001)p FP(\)\))p FO(;)118 1488 y FP(where)36 b FQ(f)p FP(0)p FO(;)14 b FP(1)p FQ(g)572 1458 y FL(\000)652 1488 y FQ(3)37 b FO(\025)p FP(\()p FQ(\001)p FP(\))g(is)f(a)f(set)h(of)g(measurable)f (functions)h(on)g(\000)f(taking)118 1588 y(v)-5 b(alues)28 b(0)g(or)g(1,)g(and)h FO(E)5 b FP(\()p FQ(\001)p FP(\))29 b(is)f(the)h(join)n(t)g(resolution)e(of)i(the)g(iden)n(tit)n(y)f(for)g (the)118 1687 y(comm)n(uting)33 b(family)h(of)g(pro)5 b(jections)32 b(de\014ned)i(on)f(the)h(cylinder)g FO(\033)s FP(-algebra)118 1787 y(in)28 b FQ(f)p FP(0)p FO(;)14 b FP(1)p FQ(g)420 1757 y FL(\000)463 1787 y FP(.)243 1887 y(F)-7 b(or)27 b(an)n(y)f(\()p FO(i)609 1899 y FL(1)647 1887 y FO(;)14 b(:)g(:)g(:)f(;)h(i)860 1899 y FM(k)901 1887 y FP(\))23 b FQ(2)g FP(\000,)28 b(w)n(e)f(ha)n(v)n(e)312 2025 y FM(n)272 2050 y Fz(X)261 2226 y FM(m)p FL(=1)417 2129 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FP(=)23 b FO(P)2191 2417 y FM(i)2214 2425 y Fy(1)2247 2417 y FM(;:::)o(;i)2369 2426 y Fw(k)2409 2405 y FO(:)118 2657 y FP(Then,)h(according)d(to)h ([25)o(],)i(the)e(sp)r(ectral)g(measure)g(of)g(the)h(comm)n(uting)f (fam-)118 2756 y(ily)28 b(is)f(concen)n(trated)g(on)g(functions)h FO(\025)p FP(\()p FQ(\001)p FP(\))c FQ(2)f(f)p FP(0)p FO(;)14 b FP(1)p FQ(g)1726 2726 y FL(\000)1797 2756 y FP(suc)n(h)27 b(that)764 2895 y FM(n)725 2920 y Fz(X)713 3096 y FM(m)p FL(=1)870 2999 y FO(\025)p FP(\()p FO(i)979 3011 y FL(1)1017 2999 y FO(;)14 b(:)g(:)g(:)f(;)h(i)1230 3011 y FM(k)1271 2999 y FO(;)g(m)p FP(\))23 b(=)g FO(\025)p FP(\()p FO(i)1633 3011 y FL(1)1670 2999 y FO(;)14 b(:)g(:)g(:)g(;)g(i) 1884 3011 y FM(k)1924 2999 y FP(\))p FO(:)243 3251 y FP(Let)41 b FO(\025)p FP(\()p FQ(\001)p FP(\))g(b)r(e)h(suc)n(h)e(a)g (function.)77 b(Then)41 b FO(\025)p FP(\()p FJ(?)p FP(\))h(is)f(either) f(0)h(or)e(1.)76 b(If)118 3350 y FO(\025)p FP(\()p FJ(?)p FP(\))32 b(=)f(0,)j(then)f FO(\025)p FP(\(1\))22 b(+)f 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b FQ(2)h(f)p FP(1)p FO(;)14 b(:)g(:)g(:)e(;)i(n)p FQ(g)1349 3818 y FN(1)1445 3848 y FP(=)25 b FJ(Z)1596 3818 y FN(1)1596 3869 y FM(n)1690 3848 y FP(suc)n(h)k(that)h(for)e(an)n(y)h FO(k)g FP(=)c(1,)118 3948 y(2,)32 b FO(:)14 b(:)g(:)46 b FP(w)n(e)32 b(ha)n(v)n(e)f FO(\025)p FP(\()p FO(x)808 3960 y FL(1)847 3948 y FO(;)14 b(:)g(:)g(:)f(;)h(x)1078 3960 y FM(k)1120 3948 y FP(\))31 b(=)f(1)i(and)h(vice)f(v)n(ersa.)49 b(T)-7 b(o)32 b(a)g(cylinder)g(set)118 4048 y(in)e FJ(Z)278 4018 y FN(1)278 4068 y FM(n)372 4048 y FP(there)f(corresp)r(onds)f(a)h (cylinder)g(set)g(in)h FQ(f)p FP(0)p FO(;)14 b FP(1)p FQ(g)1868 4018 y FL(\000)1911 4048 y FP(,)30 b(whic)n(h)f(enables)g(us) 118 4147 y(to)d(de\014ne)g(the)h(image)e(of)h(the)g(pro)5 b(jection-v)-5 b(alued)25 b(measure)g FO(E)5 b FP(\()p FQ(\001)p FP(\))27 b(under)e(the)p eop %%Page: 192 196 192 195 bop 118 100 a FP(192)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(mapping)32 b FO(\036)9 b FP(:)30 b FJ(Z)636 303 y FN(1)636 353 y FM(n)732 333 y FQ(\000)-49 b(!)31 b(f)p FP(0)p FO(;)14 b FP(1)p FQ(g)1067 303 y FL(\000)1143 333 y FP(whic)n(h)32 b(w)n(as)f(just)i(de\014ned;)i(this)e (image)f(will)118 432 y(also)21 b(b)r(e)h(denoted)g(b)n(y)g FO(E)5 b FP(\()p FQ(\001)p FP(\))23 b(as)e(long)g(as)g(this)i(will)f (not)g(lead)f(to)h(an)n(y)f(confusion.)243 536 y(Therefore,)32 b(the)h(sp)r(ectral)f(decomp)r(osition)g(of)h(the)g(comm)n(uting)f (family)118 636 y(of)c(pro)5 b(jections)26 b FO(P)693 648 y FM(\013)741 636 y FP(,)h FO(\013)d FQ(2)f FP(\000,)28 b(tak)n(es)e(the)i(form)430 864 y FO(P)483 876 y FM(\013)554 864 y FP(=)641 751 y Fz(Z)687 940 y Fv(Z)732 923 y Fx(1)732 957 y Fw(n)805 864 y FO(\037)857 876 y FM(\013)p FN(\002)p Fv(Z)997 860 y Fx(1)997 893 y Fw(n)1055 864 y FP(\()p FO(x)p FP(\))14 b FO(dE)5 b FP(\()p FO(x)p FP(\))26 b(=)c FO(E)5 b FP(\()p FO(\013)1664 876 y FL(1)1702 864 y FO(;)14 b(:)g(:)g(:)g(;)g(\013)1940 876 y FM(k)1999 864 y FQ(\002)k FJ(Z)2143 830 y FN(1)2143 885 y FM(n)2208 864 y FP(\))p FO(:)243 1118 y FP(No)n(w,)27 b(Theorem)g(45)f(implies)i(the)g (statemen)n(t.)p 2514 1118 4 57 v 2518 1065 50 4 v 2518 1118 V 2567 1118 4 57 v 243 1317 a(The)34 b(established)g(theorem)f (pro)n(vides)g(a)h(w)n(a)n(y)f(to)h(construct)f(represen-)118 1417 y(tations)d(of)g(the)h(Cun)n(tz)f(algebra.)43 b(Indeed,)31 b(one)e(should)h(tak)n(e)g(an)g(appropri-)118 1516 y(ate)d(quasi-in)n (v)-5 b(arian)n(t)25 b(measure)h(on)g FJ(Z)1318 1486 y FN(1)1318 1537 y FM(n)1382 1516 y FP(,)h(c)n(ho)r(ose)f(an)h(in)n(v) -5 b(arian)n(t)25 b(m)n(ultiplicit)n(y)118 1616 y(function,)j(and)g(a)f (unitary)g(op)r(erator-v)-5 b(alued)26 b(function.)118 1757 y FC(R)l(emark)34 b FP(48)p FC(.)i FP(The)22 b(constructed)g(comm) n(utativ)n(e)f(mo)r(del)h(can)g(also)f(b)r(e)i(rewrit-)118 1857 y(ten)29 b(in)g(the)g(space)f(of)h(\(v)n(ector-v)-5 b(alued,)28 b(in)h(general\))f(functions)h(on)f(the)h(unit)118 1956 y(in)n(terv)-5 b(al)27 b(as)g(follo)n(ws.)243 2060 y(T)-7 b(o)27 b(eac)n(h)f(p)r(oin)n(t)h FO(x)h FP(of)f(the)h(unit)g(in) n(terv)-5 b(al)27 b([0)p FO(;)14 b FP(1\))26 b(w)n(e)h(put)h(in)n(to)f (corresp)r(on-)118 2160 y(dence)j(its)g FO(n)p FP(-ary)e(represen)n (tation)h FO(x)e FP(=)f(0)p FO(:x)1517 2172 y FL(1)1555 2160 y FO(x)1602 2172 y FL(2)1653 2160 y FO(:)14 b(:)g(:)g FP(.)44 b(This)30 b(corresp)r(ondence)118 2259 y(is)f(one-to-one)e (except)i(for)f(the)i(p)r(oin)n(ts)f(with)g(tails)g(consisting)f(of)h FO(n)p FP('s)f(\(they)118 2359 y(are)k(iden)n(ti\014ed)h(with)h(p)r (oin)n(ts)e(ha)n(ving)g(\014nite)h(decomp)r(osition\).)53 b(With)33 b(suc)n(h)118 2459 y(a)25 b(corresp)r(ondence,)f(cylinder)i (sets)f(in)h FJ(Z)1417 2428 y FN(1)1417 2479 y FM(n)1507 2459 y FP(corresp)r(ond)e(to)h(Borel)g(sets)g(from)118 2558 y([0)p FO(;)14 b FP(1\),)27 b(and)h(the)g(represen)n(tation)d (\(2.82\))i(tak)n(es)g(the)h(form)505 2791 y FO(H)i FP(=)692 2678 y Fz(Z)738 2867 y FL([0)p FM(;)p FL(1\))887 2791 y FO(H)956 2803 y FM(\025)1013 2791 y FO(dE)5 b FP(\()p FO(\025)p FP(\))p FO(;)275 2985 y FP(\()p FO(S)358 2997 y FM(i)386 2985 y FO(f)k FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(\037)744 3018 y FP([)777 2987 y Fw(i)p Fx(\000)p Fy(1)p 777 3000 96 3 v 806 3032 a Fw(n)882 3012 y FM(;)919 2990 y Fw(i)p 912 2999 37 3 v 912 3032 a(n)958 3018 y FP(\))995 2985 y(\()p FO(\025)p FP(\))14 b FO(U)1178 2997 y FM(i)1206 2985 y FP(\()p FO(n\025)19 b FQ(\000)f FP(\()p FO(i)g FQ(\000)g FP(1\)\))683 3216 y FQ(\002)766 3099 y Fz(\022)837 3160 y FO(d\026)p FP(\()p FO(n\025)h FQ(\000)f FP(\()p FO(i)g FQ(\000)g FP(1\)\))p 837 3197 594 4 v 1030 3273 a FO(d\026)p FP(\()p FO(\025)p FP(\))1440 3099 y Fz(\023)1502 3116 y FL(1)p FM(=)p FL(2)1606 3216 y FO(f)9 b FP(\()p FO(n\025)18 b FQ(\000)g FP(\()p FO(i)h FQ(\000)f FP(1\)\))p FO(;)260 3471 y FP(\()p FO(S)348 3437 y FN(\003)343 3492 y FM(i)386 3471 y FO(f)9 b FP(\)\()p FO(\025)p FP(\))24 b(=)f FO(U)758 3437 y FN(\003)749 3492 y FM(i)796 3471 y FP(\()p FO(\025)p FP(\))922 3354 y Fz(\022)994 3415 y FO(d\026)p FP(\(\()p FO(\025)c FP(+)f FO(i)g FQ(\000)g FP(1\))p FO(=n)p FP(\))p 994 3452 636 4 v 1208 3528 a FO(d\026)p FP(\()p FO(\025)p FP(\))1639 3354 y Fz(\023)1700 3371 y FL(1)p FM(=)p FL(2)1818 3471 y FO(f)9 b FP(\(\()p FO(\025)19 b FP(+)f FO(i)g FQ(\000)g FP(1\))p FO(=n)p FP(\))p FO(;)118 3707 y FP(whic)n(h)28 b(holds)g(for)f(all)h(represen)n(tations)e(that)i(do)g(not)g(con)n (tain)f(a)h(single)g(rep-)118 3807 y(resen)n(tation)35 b(related)h(to)g(the)h(orbit)f(of)g(the)h(p)r(oin)n(t)f(\()p FO(n;)14 b(n;)g(:)g(:)g(:)g FP(\))38 b FQ(2)g FJ(Z)2335 3777 y FN(1)2335 3827 y FM(n)2436 3807 y FP(\(see)118 3907 y(example)27 b(b)r(elo)n(w\).)118 4048 y FC(Example)42 b FP(18)p FC(.)i FP(\(Represen)n(tations)32 b(related)h(to)g(a)f (single)h(orbit\).)53 b(The)33 b(sim-)118 4147 y(plest)c(class)e(of)h (irreducible)g(represen)n(tations)e(can)i(b)r(e)g(obtained)g(as)g (follo)n(ws.)p eop %%Page: 193 197 193 196 bop 118 100 a FK(2.5.)36 b(Represen)n(tations)26 b(of)i(some)f(n)n(uclear)f(algebras)673 b FP(193)118 333 y(T)-7 b(ak)n(e)26 b(an)h(arbitrary)e(p)r(oin)n(t)j FO(x)23 b FP(=)g(\()p FO(x)1240 345 y FL(1)1278 333 y FO(;)14 b(x)1362 345 y FL(2)1399 333 y FO(;)g(:)g(:)g(:)g FP(\))23 b FQ(2)h FJ(Z)1742 303 y FN(1)1742 353 y FM(n)1806 333 y FP(.)37 b(The)28 b(simplest)f(quasi-)118 432 y(in)n(v)-5 b(arian)n(t)32 b(ergo)r(dic)f(measure)h(is)g(an)g(atomic)g(measure)g FO(\026)h FP(concen)n(trated)e(on)118 532 y(the)39 b(orbit)f(of)g(a)g (p)r(oin)n(t)g FO(x)h FP(with)g(resp)r(ect)f(to)g(the)h(action)f(giv)n (en)f(in)i(\(2.82\))o(.)118 632 y(The)32 b(orbit)g(consists)f(of)h(all) g(p)r(oin)n(ts)g(ha)n(ving)f(similar)h(tails,)h(i.e.,)g(the)g(p)r(oin)n (ts)118 731 y FO(y)50 b FP(=)d(\()p FO(y)394 743 y FL(1)432 731 y FO(;)14 b(y)510 743 y FL(2)546 731 y FO(;)g(:)g(:)g(:)g FP(\))43 b(suc)n(h)f(that)g(there)g(exist)g FO(l)49 b FQ(\025)e FP(1,)f FO(m)h FQ(2)h FJ(Z)36 b FP(suc)n(h)42 b(that)118 831 y FO(x)165 843 y FM(k)233 831 y FP(=)26 b FO(y)365 843 y FM(k)q FL(+)p FM(m)545 831 y FP(for)j(all)g FO(k)h(>)c(l)r FP(.)42 b(Denote)30 b(suc)n(h)f(an)g(orbit)h(b)n(y)f FO(O)2026 843 y FM(x)2068 831 y FP(.)43 b(In)30 b(this)g(case,)118 930 y(an)n(y)c(op)r(erator-v)-5 b(alued)25 b(function)j FO(U)9 b FP(\()p FO(x)p FP(\))28 b(is)f(equiv)-5 b(alen)n(t)26 b(to)h(the)h(iden)n(tit)n(y)-7 b(,)27 b(and)118 1030 y(the)h(irreducibilit)n(y)f(implies)h(that)g FO(d)p FP(\()p FO(x)p FP(\))c(=)f(1)k FO(\026)p FP(-a.e.)243 1130 y(No)n(w,)d FO(\016)s FP(-functions)h(concen)n(trated)e(at)i(p)r(oin)n(ts)f(of)g (the)h(orbit)f(form)h(a)f(basis)118 1229 y(of)k FO(H)7 b FP(,)27 b(and)h(the)g(represen)n(tation)d(acts)j(as)e(follo)n(ws:)399 1381 y FO(S)450 1393 y FM(i)491 1381 y FO(\016)528 1396 y FL(\()p FM(x)592 1404 y Fy(1)624 1396 y FM(;x)682 1404 y Fy(2)714 1396 y FM(;:::)10 b FL(\))857 1381 y FP(=)23 b FO(\016)982 1396 y FL(\()p FM(i;x)1089 1404 y Fy(1)1121 1396 y FM(;x)1179 1404 y Fy(2)1211 1396 y FM(;:::)10 b FL(\))1331 1381 y FO(;)384 1505 y(S)440 1471 y FN(\003)435 1526 y FM(i)491 1505 y 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FO(;)g(:)g(:)g(:)f FP(\))30 b(tak)n(en)f(to)g(construct)g(the)h(orbit)f(has)g(a)118 2254 y(constan)n(t)k(tail)h(of)g(the)g(form)g(\()p FO(:)14 b(:)g(:)g(;)g(k)s(;)g(k)s(;)g(k)s(;)g(:)g(:)g(:)e FP(\).)56 b(Then)34 b FO(O)2001 2266 y FM(x)2078 2254 y FP(is)f(exactly)g(the)118 2354 y(orbit)c(of)h(the)g(p)r(oin)n(t)g(\()p FO(k)s(;)14 b(k)s(;)g(k)s(;)g(:)g(:)g(:)f FP(\).)44 b(The)30 b(op)r(erator)e FO(S)1835 2366 y FM(k)1906 2354 y FP(has)h(a)g(unitary)g(part)118 2454 y(formed)c(b)n(y)f(its)h(restriction)f(to)h(the)g(subspace)f (spanned)g(b)n(y)h(the)g(eigen)n(v)n(ector)118 2553 y FO(\016)155 2568 y FL(\()p FM(k)q(;k)q(;:::)11 b FL(\))394 2553 y FP(,)28 b(while)g(the)g(others)f(are)f(m)n(ultiples)i(of)g(the)g (unilateral)f(shift.)243 2653 y(F)-7 b(or)27 b(suc)n(h)h(a)g(represen)n (tation,)e(its)i(restriction)f(to)h(UHF)2010 2665 y FM(n)2084 2653 y FP(is)g(irreducible,)118 2752 y(since)21 b(the)h(action)f(of)h (elemen)n(ts)f(of)g(the)h(form)f FO(S)1583 2764 y FM(i)1606 2772 y Fy(1)1657 2752 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b(\()p FO(x)400 3719 y FN(0)400 3769 y FL(1)438 3749 y FO(;)14 b(x)522 3719 y FN(0)522 3769 y FL(2)560 3749 y FO(;)g(:)g(:)g(:)g FP(\))34 b(b)r(elong)g(to)g(the)g(same)g (orbit)g(of)g(UHF)2007 3761 y FM(n)2087 3749 y FP(if)h(and)f(only)f(if) 118 3848 y FO(x)165 3860 y FM(i)216 3848 y FQ(6)p FP(=)23 b FO(x)351 3818 y FN(0)351 3870 y FM(i)407 3848 y FP(for)k(a)g (\014nite)i(n)n(um)n(b)r(er)e(of)h FO(i)p FP('s.)36 b(This)28 b(implies)g(that)g(the)g(corresp)r(ond-)118 3948 y(ing)33 b(represen)n(tation)e(of)j FB(O)970 3960 y FM(n)1048 3948 y FP(decomp)r(oses)e(in)n(to)h(an)g(in\014nite)h(direct)f(sum)g (of)118 4048 y(inequiv)-5 b(alen)n(t)36 b(irreducible)g(represen)n (tations)f(of)h(UHF)1876 4060 y FM(n)1959 4048 y FP(corresp)r(onding)e (to)118 4147 y(the)28 b(sub-orbits.)p eop %%Page: 194 198 194 197 bop 118 100 a FP(194)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FC(Example)41 b FP(19)p FC(.)i FP(Another)31 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FO(f)9 b FP(\)\()p FO(x)545 1775 y FL(1)583 1763 y FO(;)14 b(x)667 1775 y FL(2)704 1763 y FO(;)g(:)g(:)g(:)g FP(\))23 b(=)g FO(\013\016)1085 1775 y FM(j)1120 1763 y FP(\()p FO(x)1199 1775 y FL(1)1237 1763 y FP(\))p FO(n)1319 1728 y FL(1)p FM(=)p FL(2)1424 1763 y FO(f)9 b FP(\()p FO(x)1553 1775 y FL(2)1590 1763 y FO(;)14 b(x)1674 1775 y FL(3)1712 1763 y FO(;)g(:)g(:)g(:)f FP(\))p FO(;)159 1923 y FP(\()p FO(S)247 1880 y FL(\()p FM(\013)p FL(\))242 1947 y FM(j)346 1857 y FN(\003)384 1923 y FO(f)c FP(\)\()p FO(x)545 1935 y FL(1)583 1923 y FO(;)14 b(x)667 1935 y FL(2)704 1923 y FO(;)g(:)g(:)g(:)g FP(\))23 b(=)g FO(\013)1048 1889 y FN(\000)p FL(1)1138 1923 y FO(n)1188 1889 y FN(\000)p FL(1)p FM(=)p FL(2)1344 1923 y FO(f)9 b FP(\()p FO(j;)14 b(x)1544 1935 y FL(1)1581 1923 y FO(;)g(x)1665 1935 y FL(2)1703 1923 y FO(;)g(:)g(:)g(:)g FP(\))p FO(;)180 b(j)28 b FP(=)22 b(1)p FO(;)14 b(:)g(:)g(:)g(;)g(n:) 2363 2036 y FP(\(2.85\))118 2253 y(A)39 b(direct)g(computation)g(sho)n 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FP(\))g(is)f(indep)r(enden)n(t)h(of)g FO(x)1807 1787 y FL(1)1844 1775 y FP(.)35 b(But)19 b(then)h(the)g(righ) n(t-)118 1875 y(hand)25 b(side)g(of)32 b(\(2.86\))24 b(is)h(in)n(v)-5 b(arian)n(t)24 b(with)h(resp)r(ect)g(to)g FO(x)1864 1887 y FL(1)1902 1875 y FP(,)g(and)g(the)h(left-hand)118 1974 y(side)i(is)f(indep)r(enden)n(t)i(of)f FO(x)981 1986 y FL(1)1018 1974 y FP(,)g FO(x)1116 1986 y FL(2)1154 1974 y FP(,)g(etc.)37 b(Therefore,)27 b FO(v)s FP(\()p FO(x)p FP(\))i(is)e(indep)r(enden)n(t)i(of)118 2074 y(an)n(y)34 b(\014nite)h(n)n(um)n(b)r(er)f(of)g(v)-5 b(ariables;)37 b(since)d(a)g(pro)r(duct)g(measure)g(is)g(ergo)r(dic)118 2174 y(with)g(resp)r(ect)f(to)g(suc)n(h)g(transformations,)g(w)n(e)g (get)g(that)h FO(v)s FP(\()p FO(x)p FP(\))g(is)g(constan)n(t)118 2273 y FO(\026)p FP(-a.e.,)27 b(and)h(th)n(us,)f FO(\013)791 2285 y FL(1)852 2273 y FP(=)22 b FO(\013)992 2285 y FL(2)1030 2273 y FP(.)p 2514 2273 4 57 v 2518 2220 50 4 v 2518 2273 V 2567 2273 4 57 v 118 2435 a FC(Example)40 b FP(21)p 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FP(Cho)r(ose)i(a)i(v)n(ector)e(\012)d(=)g FO(e)1060 3996 y FL(0)1115 3984 y FQ(\012)18 b FP(1\()p FO(x)p FP(\).)37 b(Then)28 b(w)n(e)f(ha)n(v)n(e)599 4147 y(\()p FO(S)682 4159 y FM(j)717 4147 y FP(\012\)\()p FO(x)p FP(\))d(=)f FO(e)1071 4159 y FL(1)1126 4147 y FQ(\012)18 b FO(\016)1246 4159 y FM(i)1274 4147 y FP(\()p FO(x)1353 4159 y FL(1)1391 4147 y FP(\))1423 4083 y FQ(p)p 1492 4083 50 4 v 64 x FO(n)c FP(1\()p FO(x)1677 4159 y FL(2)1714 4147 y FO(;)g(x)1798 4159 y FL(3)1836 4147 y FO(;)g(:)g(:)g(:)g FP(\))p FO(;)p eop %%Page: 196 200 196 199 bop 118 100 a FP(196)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)591 333 y FP(\()p FO(S)679 298 y FN(\003)674 353 y FM(j)717 333 y FP(\012\)\()p FO(x)p FP(\))d(=)f FO(e)1071 345 y FN(\000)p FL(1)1178 333 y FQ(\012)18 b FP(\()1293 269 y FQ(p)p 1363 269 50 4 v 1363 333 a FO(n)p FP(\))1445 298 y FN(\000)p FL(1)1548 333 y FP(1\()p FO(j;)c(x)1740 345 y FL(1)1778 333 y FO(;)g(x)1862 345 y FL(2)1899 333 y FO(;)g(:)g(:)g(:)g FP(\))p FO(;)118 496 y FP(and)28 b(writing)g FO(S)618 508 y FM(\013)690 496 y FP(=)c FO(S)830 508 y FM(\013)873 516 y Fy(1)923 496 y FO(:)14 b(:)g(:)g(S)1085 508 y FM(\013)1128 516 y Fw(s)1193 496 y FP(for)27 b(a)h(m)n(ulti-index)h FO(\013)c FP(=)f(\()p FO(\013)2086 508 y FL(1)2123 496 y FO(;)14 b(:)g(:)g(:)g(;)g(\013)2361 508 y FM(s)2396 496 y FP(\),)29 b(w)n(e)118 596 y(get)133 759 y(\()p FO(S)216 771 y FM(\013)264 759 y FO(S)320 725 y FN(\003)315 779 y FM(\014)359 759 y FP(\012\)\()p FO(x)p FP(\))c(=)d FO(e)713 774 y FN(j)p FM(\013)p FN(j\000j)p FM(\014)s FN(j)950 759 y FQ(\012)c FO(\016)1070 771 y FM(\013)1113 779 y Fy(1)1150 759 y FP(\()p FO(x)1229 771 y FL(1)1267 759 y FP(\))c FO(:)g(:)g(:)f(\016) 1460 771 y FM(\013)1503 779 y Fw(s)1540 759 y FP(\()p FO(x)1619 771 y FM(s)1655 759 y FP(\))h FO(n)1751 725 y FL(\()p FN(j)p FM(\013)p FN(j\000j)p FM(\014)s FN(j)p FL(\))p FM(=)p FL(2)2102 759 y FP(1\()p FO(x)2223 771 y FM(s)p FL(+1)2343 759 y FO(;)g(:)g(:)g(:)g FP(\))p FO(;)2363 869 y FP(\(2.87\))118 1033 y(where)38 b FQ(j)p FO(\013)p FQ(j)h FP(is)f(the)g(length)g(of)h(the)f(m)n(ulti-index.)69 b(Also,)40 b(the)f(v)n(ector)e(\012)h(is)118 1132 y(cyclic.)243 1232 y(The)20 b(corresp)r(onding)d(state)j(on)g FB(O)1294 1244 y FM(n)1358 1232 y FP(is)g(de\014ned)g(b)n(y)g(the)g(follo)n(wing) f(form)n(ula)956 1395 y FO(!)s FP(\()p FO(S)1094 1407 y FM(\013)1142 1395 y FO(S)1198 1361 y FN(\003)1193 1416 y FM(\014)1237 1395 y FP(\))24 b(=)e FO(\016)1417 1407 y FM(\013;\014)1525 1395 y FO(n)1575 1361 y FN(\000j)p FM(\013)p FN(j)1713 1395 y FO(;)118 1559 y FP(where)27 b FO(\016)395 1571 y FM(\013;\014)530 1559 y FP(is)h(one)f(if)h FQ(j)p FO(\013)p FQ(j)c FP(=)e FQ(j)p FO(\014)t FQ(j)p FP(,)29 b(and)e(zero)g(otherwise.)243 1658 y(Indeed,)i(since)g FO(S)793 1670 y FM(\013)841 1658 y FO(S)897 1628 y FN(\003)892 1682 y FM(\014)936 1658 y FP(\012)d FQ(2)g FO(H)1172 1673 y FN(j)p FM(\013)p FN(j\000j)p FM(\014)s FN(j)1391 1658 y FP(,)j(w)n(e)g(ha)n(v)n(e)f(\()p FO(S)1843 1670 y FM(\013)1890 1658 y FO(S)1946 1628 y FN(\003)1941 1682 y FM(\014)1986 1658 y FP(\012)p FO(;)14 b FP(\012\))26 b(=)f(0,)k FQ(j)p FO(\013)p FQ(j)d(6)p FP(=)118 1758 y FQ(j)p FO(\014)t FQ(j)p FP(,)i(and)g(\(2.87\))f(implies)201 1964 y(\()p FO(S)284 1976 y FM(\013)331 1964 y FO(S)387 1929 y FN(\003)382 1984 y FM(\013)429 1964 y FP(\012)p FO(;)14 b FP(\012\))24 b(=)729 1851 y Fz(Z)775 2039 y Fv(Z)820 2023 y Fw(s)820 2056 y(n)873 1964 y FO(\016)910 1976 y FM(\013)953 1984 y Fy(1)989 1964 y FP(\()p FO(x)1068 1976 y FL(1)1106 1964 y FP(\))14 b FO(:)g(:)g(:)g(\016)1300 1976 y FM(\013)1343 1984 y Fw(s)1379 1964 y FP(\()p FO(x)1458 1976 y FM(s)1495 1964 y FP(\))g FO(d\026)1634 1976 y FL(1)1671 1964 y FP(\()p FO(x)1750 1976 y FL(1)1788 1964 y FP(\))g FO(:)g(:)g(:)g(d\026)2038 1976 y FM(s)2074 1964 y FP(\()p FO(x)2153 1976 y FM(s)2189 1964 y FP(\))23 b(=)g FO(n)2382 1929 y FN(\000)p FM(s)2469 1964 y FO(;)118 2190 y FP(where)k FO(s)c FP(=)g FQ(j)p FO(\013)p FQ(j)p FP(,)28 b(and)g(the)f(form)n(ula)g(follo)n(ws.)243 2290 y(P)n(assing)18 b(to)h(a)h(unitarily)g(equiv)-5 b(alen)n(t)19 b(realization,)i(one)e(has)h(the)g(follo)n(wing)118 2389 y(form)n(ulae)27 b(for)g(the)h(op)r(erators)d(in)j FO(H)i FP(=)23 b FO(L)1438 2401 y FL(2)1474 2389 y FP(\()p FJ(T)18 b FQ(\002)g FJ(Z)1725 2359 y FN(1)1725 2410 y FM(n)1789 2389 y FO(;)c(dz)22 b FQ(\012)c FO(d\026)p FP(\),)448 2553 y(\()p FO(S)531 2565 y FM(j)567 2553 y FO(f)9 b FP(\)\()p FO(z)t(;)14 b(x)808 2565 y FL(1)845 2553 y FO(;)g(x)929 2565 y FL(2)966 2553 y FO(;)g(:)g(:)g(:)g FP(\))23 b(=)g FO(z)t(n)1350 2518 y FL(1)p FM(=)p FL(2)1453 2553 y FO(\016)1490 2565 y FM(j)1525 2553 y FP(\()p FO(x)1604 2565 y FL(1)1642 2553 y FP(\))p FO(f)9 b FP(\()p FO(z)t(;)14 b(x)1883 2565 y FL(2)1920 2553 y FO(;)g(x)2004 2565 y FL(3)2042 2553 y FO(;)g(:)g(:)g(:)f FP(\))p FO(:)118 2716 y FP(The)33 b(latter)f(form)h(giv)n(es)f(an)g(explicit)h(decomp)r (osition)f(of)h(the)g(constructed)118 2816 y(represen)n(tation)c(in)n (to)i(a)f(direct)h(in)n(tegral)f(of)h(irreducible)f(inequiv)-5 b(alen)n(t)31 b(rep-)118 2915 y(resen)n(tations)26 b(\()p FO(S)666 2885 y FM(\013)661 2937 y(j)713 2915 y FP(\))i(constructed)g (ab)r(o)n(v)n(e.)118 3040 y FC(Example)34 b FP(22)p FC(.)k FP(\(The)25 b(KMS)f(represen)n(tation\))g(W)-7 b(e)24 b(giv)n(e)g(an)g(explicit)h(form)n(ula)118 3140 y(for)e(the)i(represen) n(tation)d(of)i(the)g(Cun)n(tz)g(algebra)e FB(O)1739 3152 y FM(n)1784 3140 y FP(,)j(corresp)r(onding)d(to)h(the)118 3240 y(unique)28 b(KMS)f(state)h(related)f(to)g(the)h(action)f(of)h (the)g(gauge)e(group.)243 3339 y(In)n(tro)r(duce)36 b(some)g (notations.)63 b(Let)36 b FJ(Z)1488 3309 y FN(1)1488 3360 y FM(n;)o FL(0)1616 3339 y FP(b)r(e)h(the)g(set)g(of)f(all)g (\014nite)h(se-)118 3450 y(quences)32 b FO(\013)g FP(=)f(\()p FO(\013)696 3462 y FL(1)733 3450 y FO(;)14 b(:)g(:)g(:)g(;)g(\013)971 3462 y FM(n)1016 3450 y FO(;)g FP(0)p FO(;)g FP(0)p FO(;)g(:)g(:)g(:)f FP(\);)35 b FJ(Z)1472 3420 y FN(1)1472 3470 y FM(n;)p FL(0)1596 3450 y FP(=)1692 3388 y Fz(S)1761 3475 y FM(k)1815 3450 y FJ(Z)1877 3420 y FM(k)1877 3470 y(n)1916 3450 y FP(,)f(with)f FJ(Z)2228 3420 y FM(k)2228 3470 y(n)2299 3450 y FQ(\032)e FJ(Z)2456 3420 y FM(k)q FL(+1)2456 3470 y FM(n)118 3550 y FP(giv)n(en)i(b)n(y)g(\()p FO(\013)547 3562 y FL(1)585 3550 y FO(;)14 b(:)g(:)g(:)g(;)g(\013)823 3562 y FM(n)868 3550 y FP(\))33 b FQ(7!)g FP(\()p FO(\013)1134 3562 y FL(1)1172 3550 y FO(;)14 b(:)g(:)g(:)f(;)h(\013)1409 3562 y FM(n)1455 3550 y FO(;)g FP(0\).)54 b(Let)34 b FQ(j)p FO(\013)p FQ(j)g FP(b)r(e)h(the)f(n)n(um)n(b)r(er)f(of)118 3649 y(the)27 b(tailing)g(non-zero)e FO(\013)907 3661 y FM(j)970 3649 y FP(in)i FO(\013)p FP(.)37 b(W)-7 b(rite)27 b FO(j)1443 3661 y FM(k)1507 3649 y FP(=)c(\(0)p FO(;)14 b(:)g(:)g(:)f(;)h FP(0)p FO(;)g(j;)g FP(0)p FO(;)g(:)g(:)g(:)f FP(\))27 b(with)h FO(j)k FP(in)118 3749 y(the)f FO(k)s FP(-th)f(place,)g FO(j)j FQ(2)28 b FJ(Z)895 3761 y FM(n)934 3749 y FP(,)j FO(k)g FP(=)c(1,)j(2,)g FO(:)14 b(:)g(:)28 b 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FP(=)22 b(0,)28 b FO(:)14 b(:)g(:)27 b FP(,)h FO(n)18 b FQ(\000)g FP(1.)p eop %%Page: 197 201 197 200 bop 118 100 a FK(2.5.)36 b(Represen)n(tations)26 b(of)i(some)f(n)n(uclear)f(algebras)673 b FP(197)243 333 y(The)23 b(represen)n(tation)f(space)h(in)g(our)g(example)g(will)h (ha)n(v)n(e)e(the)i(form)f FO(H)2463 345 y FL(0)2510 333 y FQ(\012)118 432 y FO(L)175 444 y FL(2)212 432 y FP(\()p FJ(Z)305 402 y FN(1)305 453 y FM(n)370 432 y FO(;)14 b(d\026)p FP(\()p FO(x)p FP(\)\),)32 b(where)e FO(\026)g FP(is)g(an)g(in\014nite)h(pro)r(duct)f(of)g(Haar)f(measures)g (on)118 532 y FJ(Z)179 544 y FM(n)219 532 y FP(.)243 632 y(The)j(space)f FO(H)713 644 y FL(0)782 632 y FP(is)h(generated)e (b)n(y)i(its)g(orthogonal)e(basis)h(consisting)g(of)118 731 y(v)n(ectors)26 b FO(e)p FP(\()p FO(i;)14 b(k)s(;)g(\013)p FP(\),)27 b(where)g FO(\013)d FQ(2)f FP(\000)k(is)g(an)n(y)g(m)n (ulti-index,)h FO(i)22 b FQ(2)i FJ(Z)2126 743 y FM(n)2165 731 y FP(,)k(and)f FO(k)f FP(=)c(0)118 831 y(if)29 b FO(\013)c FQ(6)p FP(=)f FJ(?)k FP(and)h FO(i)24 b FP(=)g(0,)k(and)g FO(k)g FP(=)c(0,)k(1,)g FO(:)14 b(:)g(:)28 b FP(,)h(elsewhere,)e(i.e.,) i(for)f FO(\013)d FP(=)f FJ(?)p FP(,)29 b(or)118 930 y FO(i)23 b FQ(6)p FP(=)f(0.)243 1030 y(The)27 b(op)r(erators)f FO(S)832 1042 y FL(0)869 1030 y FP(,)i FO(:)14 b(:)g(:)27 b FP(,)h FO(S)1146 1042 y FM(n)p FN(\000)p FL(1)1304 1030 y FP(act)f(as)g(follo)n(ws)188 1261 y FO(S)239 1273 y FM(j)274 1261 y FO(e)p FP(\()p FO(i;)14 b(k)s(;)g(\013)p FP(\))19 b FQ(\012)f FO(f)9 b FP(\()p FO(x)p FP(\))23 b(=)g FO(n)1003 1227 y FL(1)p FM(=)p FL(2)1121 1144 y Fz(\032)1225 1210 y FO(e)p FP(\()p FO(j)g FQ(\000)18 b FO(\013)1489 1222 y FL(1)1526 1210 y FO(;)c FP(0)p FO(;)g(\033)s FP(\()p FO(\013)p FP(\)\))p FO(;)85 b(i)22 b FP(=)h(0)p FO(;)14 b(\013)23 b FQ(6)p FP(=)f FJ(?)1225 1310 y FO(e)p FP(\()p FO(i;)14 b(k)20 b FP(+)e(1)p FO(;)c(\013)p FP(\))p FO(;)254 b FP(otherwise)2436 1144 y Fz(\033)1027 1443 y FQ(\012)18 b FO(\016)1147 1455 y FM(j)1182 1443 y FP(\()p FO(x)1261 1455 y FL(1)1299 1443 y FP(\))p FO(f)9 b FP(\()p FO(\033)s FP(\()p FO(x)p FP(\)\))p FO(;)180 1626 y(S)236 1592 y FN(\003)231 1647 y FM(j)274 1626 y FO(e)p FP(\()p FO(i;)14 b(k)s(;)g(\013)p FP(\))19 b FQ(\012)f FO(f)9 b FP(\()p FO(x)p FP(\))23 b(=)g FO(n)1003 1592 y FN(\000)p FL(1)p FM(=)p FL(2)1173 1509 y Fz(\032)1277 1576 y FO(e)p FP(\(0)p FO(;)14 b FP(0)p FO(;)g(\033)1553 1588 y FM(j)s FN(\000)p FM(i)1662 1576 y FP(\()p FO(\013)p FP(\)\))p FO(;)85 b(k)25 b FP(=)e(0)1277 1675 y FO(e)p FP(\()p FO(i;)14 b(k)20 b FQ(\000)e FP(1)p FO(;)c(\013)p FP(\))p FO(;)172 b(k)25 b FQ(6)p FP(=)e(0)2158 1509 y Fz(\033)1027 1808 y FQ(\012)18 b FO(f)9 b FP(\()p FO(\033)1239 1820 y FM(j)1274 1808 y FP(\()p FO(x)p FP(\)\))p FO(:)243 1991 y FP(The)26 b(v)n(ector)g(\012)d(=)f FO(e)p FP(\(0)p FO(;)14 b FP(0)p FO(;)g FJ(?)p FP(\))h FQ(\012)h FP(1\()p FO(x)p FP(\))27 b(is)g(cyclic,)f(and)h(the)g(corresp)r(onding)118 2090 y(state)h(is)693 2273 y FO(!)s FP(\()p FO(S)831 2285 y FM(\013)879 2273 y FO(S)935 2238 y FN(\003)930 2293 y FM(\014)974 2273 y FP(\))c(=)e(\()p FO(S)1200 2285 y FM(\013)1248 2273 y FO(S)1304 2238 y FN(\003)1299 2293 y FM(\014)1344 2273 y FP(\012)p FO(;)14 b FP(\012\))23 b(=)f FO(\016)1680 2285 y FM(\013;\014)1788 2273 y FO(n)1838 2238 y FN(\000j)p FM(\013)p FN(j)1976 2273 y FO(:)243 2455 y FP(Notice)32 b(that)g(the)h(v)n(ector)e(\012)h(is)g(not)h (cyclic)f(with)g(resp)r(ect)g(to)g(the)h(UHF)118 2555 y(subalgebra)24 b(in)j FB(O)687 2567 y FM(n)732 2555 y FP(,)f(and)g(the)h(corresp)r(onding)d(cyclic)i(subspace)f(is)h(a)g (prop)r(er)118 2654 y(subspace)33 b(of)h FO(H)7 b FP(.)55 b(This)34 b(means,)h(in)f(particular,)g(that)g(the)h(represen)n(tation) 118 2754 y(corresp)r(onding)27 b(to)i(the)g(tracial)f(state)h(on)f(UHF) i(cannot)e(b)r(e)h(extended)h(to)e(a)118 2854 y(represen)n(tation)e(of) i FB(O)815 2866 y FM(n)887 2854 y FP(in)g(the)g(same)f(space.)243 2953 y(Also,)35 b(c)n(ho)r(osing)d(another)h(pro)r(duct)h(measure)f(on) h FJ(Z)1946 2923 y FN(1)1946 2974 y FM(n)2010 2953 y FP(,)i(corresp)r(onding)118 3053 y(to)41 b(w)n(eigh)n(ts)e FO(p)583 3065 y FL(0)620 3053 y FP(,)i FO(:)14 b(:)g(:)28 b FP(,)44 b FO(p)918 3065 y FM(n)p FN(\000)p FL(1)1048 3053 y FP(,)1115 2991 y Fz(P)1216 3053 y FO(p)1258 3065 y FM(i)1330 3053 y FP(=)h(1,)e(the)e(form)n(ula)f(similar)g(to)h(that) 118 3169 y(giv)n(en)34 b(ab)r(o)n(v)n(e,)h(but)g(with)g FO(p)1006 3125 y FL(1)p FM(=)p FL(2)1006 3192 y FM(j)1145 3169 y FP(replacing)e FO(n)1557 3138 y FN(\000)p FL(1)p FM(=)p FL(2)1714 3169 y FP(,)j(giv)n(es)d(a)i(represen)n(tation)118 3268 y(corresp)r(onding)19 b(to)i(states)f(constructed)g(as)g(an)h (extension)f(of)h(pro)r(duct)f(states)118 3368 y(on)27 b(UHF)i(whic)n(h)e(ha)n(v)n(e)g(the)h(form)641 3550 y FO(!)s FP(\()p FO(S)779 3562 y FM(\013)826 3550 y FO(S)882 3516 y FN(\003)877 3571 y FM(\014)922 3550 y FP(\))23 b(=)g FO(\016)1102 3562 y FM(\013;\014)1210 3550 y FO(p)1252 3562 y FM(\013)1295 3570 y Fy(1)1345 3550 y FO(:)14 b(:)g(:)g(p)1498 3562 y FM(\013)1541 3570 y Fw(s)1577 3550 y FO(;)180 b(s)23 b FP(=)f FQ(j)p FO(\013)p FQ(j)p FO(:)118 3733 y FC(Example)49 b FP(23)p FC(.)g FP(W)-7 b(e)41 b(giv)n(e)f(another)g (example)g(of)h(a)f(represen)n(tation.)75 b(The)118 3832 y(corresp)r(onding)26 b(state)h(is)524 4085 y FO(!)s FP(\()p FO(S)662 4097 y FM(\013)709 4085 y FO(S)765 4051 y FN(\003)760 4106 y FM(\014)804 4085 y FP(\))d(=)947 3943 y Fz(\()1014 4028 y FO(n)1064 3998 y FN(\000j)p FM(\013)p FN(j)1203 4028 y FO(;)83 b FQ(j)p FO(\013)p FQ(j)23 b FP(=)g FQ(j)p FO(\014)t FQ(j)p FO(;)1667 3966 y Fz(P)1769 4028 y FO(\013)1822 4040 y FM(i)1873 4028 y FP(=)1960 3966 y Fz(P)2062 4028 y FO(\014)2109 4040 y FM(i)2136 4028 y FO(;)1014 4148 y FP(0)p FO(;)230 b FP(otherwise.)p eop %%Page: 198 202 198 201 bop 118 100 a FP(198)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(The)h(represen)n(tation)d(space)i(is)g FJ(Z)1198 345 y FM(n)1255 333 y FQ(\012)18 b FO(l)1363 345 y FL(2)1400 333 y FP(\()p FJ(Z)p FP(\))12 b FQ(\012)18 b FO(L)1678 345 y FL(2)1715 333 y FP(\()p FJ(Z)1808 303 y FN(1)1808 353 y FM(n)1873 333 y FO(;)c(d\026)p FP(\))28 b(with)g(the)f(stan-)118 432 y(dard)g(pro)r(duct)h(measure)e FO(\026)p FP(,)i(and)f(the)h(form)n(ulas)f(are)393 613 y FO(S)444 625 y FM(j)479 613 y FO(e)518 625 y FM(l)561 613 y FQ(\012)18 b FO(e)683 625 y FM(k)742 613 y FQ(\012)g FO(f)9 b FP(\()p FO(x)p FP(\))24 b(=)f FO(n)1148 579 y FL(1)p FM(=)p FL(2)1252 613 y FO(e)1291 625 y FM(l)p FN(\000)p FM(j)1417 613 y FQ(\012)18 b FO(e)1539 625 y FM(k)q FL(+1)1682 613 y FQ(\012)g FO(\016)1802 625 y FM(j)1837 613 y FP(\()p FO(x)1916 625 y FL(1)1954 613 y FP(\))p FO(f)9 b FP(\()p FO(\033)s FP(\()p FO(x)p FP(\)\))p FO(;)402 754 y(s)441 720 y FN(\003)441 775 y FM(j)479 754 y FO(e)518 766 y FM(l)561 754 y FQ(\012)18 b FO(e)683 766 y FM(k)742 754 y FQ(\012)g FO(f)9 b FP(\()p FO(x)p FP(\))24 b(=)f FO(n)1148 720 y FN(\000)p FL(1)p FM(=)p FL(2)1304 754 y FO(e)1343 766 y FM(x)1381 774 y Fy(1)1413 766 y FL(+)p FM(l)1507 754 y FQ(\012)18 b FO(e)1629 766 y FM(k)q FN(\000)p FL(1)1773 754 y FQ(\012)g FO(f)9 b FP(\()p FO(\033)1985 766 y FM(j)2021 754 y FP(\()p FO(x)p FP(\)\))p FO(:)118 935 y FP(The)33 b(state)g(whic)n(h)g(is)g(written)g (ab)r(o)n(v)n(e)f(is)h(obtained)f(from)h(the)h(v)n(ector)d FO(e)2451 947 y FL(0)2510 935 y FQ(\012)118 1034 y FO(e)157 1046 y FL(0)212 1034 y FQ(\012)18 b FP(1\()p FO(x)p FP(\),)29 b(whic)n(h)e(is)h(cyclic.)118 1276 y FH(Commen)m(ts)36 b(to)h(Chapter2)118 1457 y FR(Section)32 b(2.1.)243 1557 y FP(2.1.1.)j(The)27 b(study)g(of)f(a)g(class)g(of)h(the)g(relations)e FO(X)7 b(X)1950 1527 y FN(\003)2010 1557 y FP(=)23 b FO(F)12 b FP(\()p FO(X)2271 1527 y FN(\003)2309 1557 y FO(X)7 b FP(\))26 b(w)n(as)118 1657 y(motiv)-5 b(ated)35 b(b)n(y)f(n)n(umerous)g(examples)f(arising)h(in)g(mathematical)h(ph)n (ysics,)118 1756 y(including)i(the)g(Hermitian)g FO(q)s FP(-plane,)h FO(q)s FP(-CCR,)f(quan)n(tum)g(disk,)i(etc.)64 b(\(see)118 1856 y([164)o(,)22 b(36)o(,)h(63)o(,)f(157)o(,)g(138)o(,)g (139)o(])g(etc.\).)36 b(In)22 b([64],)h(suc)n(h)f(relations)f(are)h (treated)g(as)118 1955 y(a)k(general)g(deformation)f(of)i(CCR.)g(Note)f (that)h(these)g(relations)e(are)h(\\singly)118 2055 y(generated")g (\(connect)i(a)f(single)g(op)r(erator)f FO(X)34 b FP(and)27 b(its)h(adjoin)n(t\).)243 2155 y(In)18 b([288)o(],)i(represen)n (tations)d(of)h(suc)n(h)g(relations)f(w)n(ere)h(studied.)34 b(The)19 b(study)118 2254 y(of)30 b(the)g(relation)f(is)g(based)h(on)f (the)h(study)g(of)g(relations)e(of)i(the)g(form)g FO(AU)35 b FP(=)118 2354 y FO(U)9 b(F)j FP(\()p FO(A)p FP(\))32 b(with)h(a)e(self-adjoin)n(t)h(\(or)f(normal\))g(op)r(erator)f FO(A)p FP(.)50 b(Prop)r(erties)30 b(and)118 2454 y(represen)n(tations) 36 b(of)i(suc)n(h)f(relations)f(w)n(ere)h(discussed)g(in)h([195)o(,)g (288)n(,)g(290)o(,)118 2553 y(291)o(],)27 b(etc.)243 2653 y(F)-7 b(or)18 b(a)h(one-to-one)f(con)n(tin)n(uous)g(mapping)h FO(F)12 b FP(\()p FQ(\001)p FP(\))20 b(of)f(a)g(compact)g(set)g(\001,)i (the)118 2752 y(mapping)f FO(F)12 b FP(\()p FQ(\001)p FP(\))21 b(de\014nes)g(an)f(automorphism)f(of)i FO(C)6 b FP(\(\001\),)23 b(and)d(represen)n(tations)118 2852 y(of)32 b(the)g(relation)f(are)g(represen)n(tations)f(of)i(the)h (corresp)r(onding)d FO(C)2246 2822 y FN(\003)2284 2852 y FP(-algebra)118 2952 y(whic)n(h)c(is)g(the)h(crossed)e(pro)r(duct)h FO(C)6 b FP(\(\003\))16 b FJ(o)f(Z)p FP(.)30 b(In)d(this)f(case,)g (metho)r(ds)g(of)g(the)118 3051 y(study)k(of)f(represen)n(tations)e(go) i(bac)n(k)f(to)h([165)o(],)h(see)f(also)f([79)o(,)h(278)o(])g(and)g (the)118 3151 y(bibliograph)n(y)d(therein.)243 3251 y(In)35 b(our)g(exp)r(osition,)i(w)n(e)e(do)g(not)h(assume)e(that)i(the)g (mapping)f FO(F)12 b FP(\()p FQ(\001)p FP(\))36 b(is)118 3350 y(con)n(tin)n(uous)25 b(and)g(one-to-one;)g(this)h(fact)f(mak)n (es)g(the)h(problem)f(of)g(in)n(tro)r(duc-)118 3450 y(ing)c(and)g (studying)h(the)f(corresp)r(onding)f FQ(\003)p FP(-algebra)e(more)j (complicated)g(\(see,)118 3550 y(e.g.,)27 b([236)o(,)h(12)o(],)g (etc.\).)243 3649 y(W)-7 b(e)25 b(sho)n(w)e(in)i(Prop)r(osition)e(29)h (that)h(the)g(considered)f(class)f(of)i(op)r(erators)118 3749 y(is)37 b(a)f(subset)h(of)g(the)h(class)d(of)i(cen)n(tered)g(op)r (erators)e(studied)i(in)g([177)o(].)65 b(In)118 3848 y(Section)28 b(2.4.2)e(w)n(e)h(discuss)g(cen)n(tered)g(op)r(erators)f (again.)243 3948 y(In)i(the)g(case)f(of)h(the)h(crossed)d(pro)r(duct,)i (the)h(relationships)e(b)r(et)n(w)n(een)h(er-)118 4048 y(go)r(dic)40 b(measures)g(and)h(irreducible)f(represen)n(tations)f(w)n (ere)h(discussed)g(b)n(y)118 4147 y(man)n(y)29 b(authors)e(\(see,)j (e.g.,)f([278)n(])g(and)g(the)h(bibliograph)n(y)d(therein\).)41 b(In)29 b(the)p eop %%Page: 199 203 199 202 bop 118 100 a FK(Commen)n(ts)27 b(to)h(Chapter)f(2)1452 b FP(199)118 333 y(non-bijectiv)n(e)25 b(case,)f(the)i(dynamical)e (system)h(admits)g(a)f(standard)g(in\014nite-)118 432 y(dimensional)30 b(one-to-one)e(realization)h(\(see,)h(e.g.,)h([290)n (]\);)h(di\013eren)n(t)e(condi-)118 532 y(tions)22 b(for)g(a)g (non-bijectiv)n(e)f(dynamical)h(system)g(to)g(b)r(e)h(simple)f(are)f (discussed)118 632 y(in)28 b([291)o(],)f([289)o(])h(and)f(the)h (bibliograph)n(y)e(cited)i(there.)243 732 y(2.1.2.)42 b(F)-7 b(or)29 b(the)i(relationship)d(b)r(et)n(w)n(een)i(cycles)g(of)f (the)i(dynamical)e(sys-)118 831 y(tem)j(and)g(\014nite-dimensional)f (represen)n(tations)e(of)j(the)g(relation,)f(see)h([79)o(].)118 931 y(The)g(connection)g(with)h(the)f(Shark)n(o)n(vsky)e(theorem)i(is)g (discussed)g(in)g([291)o(].)118 1030 y(Basic)j(facts)g(ab)r(out)g (cycles)g(of)g(second)g(order)f(and)h(more)f(general)g(con)n(tin-)118 1130 y(uous)k(mappings)f(of)h(an)g(in)n(terv)-5 b(al)37 b(and)h(their)g(detailed)g(in)n(v)n(estigation)e(can)118 1230 y(b)r(e)c(found,)h(e.g.,)g(in)f([260)n(],)h([261)o(];)h(represen)n (tations)c(of)h(algebraic)f(relations)118 1329 y(arising)c(from)i(con)n (tin)n(uous)e(fraction)h(mapping)g(w)n(ere)g(discussed)g(in)h([267)o (].)243 1429 y(2.1.3.)62 b(The)37 b(partition)f(of)h(all)f(irreducible) h(represen)n(tations)d(in)n(to)j(t)n(w)n(o)118 1529 y(classes,)27 b(degenerate)f(\(with)j FO(U)36 b FP(or)27 b FO(U)1304 1499 y FN(\003)1370 1529 y FP(ha)n(ving)g(non-zero)f(k)n(ernel\))h(and) g(non-)118 1629 y(degenerate)k(\(with)i(unitary)e FO(U)9 b FP(\),)34 b(is)e(similar)f(to)h(the)g(W)-7 b(old)32 b(decomp)r(osition)118 1728 y(for)e(isometries.)44 b(In)31 b(fact,)g(one)f(can)g(sho)n(w)g(that)g(an)n(y)g(represen)n(tation)f (space)118 1828 y(can)24 b(b)r(e)g(decomp)r(osed)f(in)n(to)h(the)g (orthogonal)e(direct)i(sum)g(of)g(t)n(w)n(o)f(subspaces,)118 1928 y FO(H)30 b FP(=)23 b FO(H)374 1940 y FL(0)425 1928 y FQ(\010)14 b FO(H)573 1940 y FL(1)609 1928 y FP(,)26 b(suc)n(h)f(that)h(the)f(op)r(erator)f FO(U)34 b FP(is)25 b(a)g(completely)g(non-unitary)118 2027 y(cen)n(tered)e(partial)h (isometry)f(on)g FO(H)1225 2039 y FL(0)1287 2027 y FP(and)g(unitary)h (on)f FO(H)1914 2039 y FL(1)1952 2027 y FP(.)35 b(The)24 b(description)118 2127 y(of)37 b(cen)n(tered)f(partial)f(isometries)h (is)g(essen)n(tially)g(con)n(tained)g(in)h([177)n(];)k(the)118 2226 y(application)25 b(of)h(this)g(description)g(to)f(the)i(study)f (of)g(op)r(erator)e(relations)h(is)g(a)118 2326 y(mo)r(di\014ed)j(v)n (ersion)e(of)i([191)n(].)243 2426 y(The)39 b(description)g(of)g(the)h (an)n(ti-F)-7 b(o)r(c)n(k)38 b(represen)n(tations)f(for)i(a)g(second-) 118 2526 y(order)c(mapping)i(relies)e(on)i(the)g(formalism)e(of)i(sym)n (b)r(olic)f(dynamics,)i(see,)118 2625 y(e.g.,)27 b([261)o(].)243 2725 y(The)g(exp)r(osition)g(for)g(a)h(unitary)f FO(U)36 b FP(follo)n(ws)27 b([290)n(].)243 2825 y(Some)k(result)g(on)g (description)f(of)i(the)f FO(C)1560 2795 y FN(\003)1599 2825 y FP(-algebra)e(related)i(to)g(second-)118 2925 y(order)26 b(mapping)i(can)f(b)r(e)h(found)g(in)g([167)n(,)g(214)o(,)f (215)o(].)118 3051 y FR(Section)50 b(2.2.)83 b FP(In)44 b(this)g(section)f(w)n(e)g(giv)n(e)f(a)h(n)n(um)n(b)r(er)h(of)f (examples)g(of)118 3150 y FQ(\003)p FP(-algebras)34 b(kno)n(wn)j(from)f (pap)r(ers)h(on)g(mathematical)f(ph)n(ysics)h(\(see,)i(e.g.,)118 3250 y([309)o(])d(and)g(the)h(bibliograph)n(y)d(therein\).)64 b(Our)35 b(purp)r(ose)h(here)g(is)g(to)g(illus-)118 3349 y(trate)30 b(ho)n(w)g(the)h(metho)r(ds)g(dev)n(elop)r(ed)f(in)h (Section)f(2.1)g(can)g(b)r(e)h(generalized)118 3449 y(to)d(a)f(wider)g (class)g(of)g(relations)g(connecting)g(sev)n(eral)f(op)r(erators.)243 3549 y(2.2.1.)58 b(T)-7 b(riples)35 b(of)h(op)r(erators)d FO(A)1319 3561 y FL(1)1356 3549 y FP(,)38 b FO(A)1479 3561 y FL(2)1516 3549 y FP(,)g FO(A)1639 3561 y FL(3)1676 3549 y FP(,)g(satisfying)c(the)i(relations)118 3649 y FQ(f)p FO(A)222 3661 y FL(1)259 3649 y FO(;)14 b(A)358 3661 y FL(2)396 3649 y FQ(g)27 b FP(=)g FO(A)619 3661 y FL(3)657 3649 y FP(,)k FQ(f)p FO(A)815 3661 y FL(2)852 3649 y FO(;)14 b(A)951 3661 y FL(3)988 3649 y FQ(g)28 b FP(=)f FO(A)1212 3661 y FL(1)1250 3649 y FP(,)k FQ(f)p FO(A)1408 3661 y FL(3)1445 3649 y FO(;)14 b(A)1544 3661 y FL(1)1581 3649 y FQ(g)28 b FP(=)f FO(A)1805 3661 y FL(2)1873 3649 y FP(arise)i(as)h(represen)n(ta-)118 3748 y(tions)h(of)f(a)g(natural)g(graded)g(analogue)f(of)h(the)h(Lie)g (algebra)e FO(so)p FP(\(3\))i(\(on)f(the)118 3848 y(de\014nition)f(of)g (graded)f(Lie)h FQ(\003)p FP(-algebras,)e(see,)i(e.g.,)f([243)o(]\).)41 b(The)30 b(irreducible)118 3948 y(represen)n(tations)24 b(of)h(this)h(algebra)e(w)n(ere)h(studied)h(in)g([103)n(];)h(represen)n (tations)118 4047 y(of)h(these)f(relations)g(with)h(other)f(in)n(v)n (olutions)f(w)n(ere)h(studied)h(in)g([201)n(].)243 4147 y(Algebras)40 b(generated)g(b)n(y)h(families)g(of)h(pro)5 b(jections)40 b(satisfying)h(linear)p eop %%Page: 200 204 200 203 bop 118 100 a FP(200)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(relations)h(w)n(ere)g(considered)h(in)g([32)o (].)42 b(In)30 b([90)o(],)f(follo)n(wing)g(the)g(lines)g(of)h([32)o(],) 118 432 y(the)35 b(results)f(of)h([103)o(])g(are)e(applied)i(to)g(the)g (description)f(of)h(four-tuples)f(of)118 541 y(pro)5 b(jections)39 b(suc)n(h)h(that)950 478 y Fz(P)1038 499 y FL(4)1038 565 y FM(i)p FL(=1)1163 541 y FO(P)1216 553 y FM(i)1288 541 y FP(=)j FO(\013I)48 b FP(\(see)40 b(also)f([91)o(]\).) 75 b(Notice)40 b(that)118 640 y(solutions)f(of)g(the)g(latter)g (relation)f(exist)h(only)g(for)g(a)g(coun)n(table)f(n)n(um)n(b)r(er)118 740 y(of)f(the)g(parameter)e FO(\013)p FP(;)41 b(see)c(also)e([231)o (],)k(where)d(the)h(authors)e(studied)i(the)118 839 y(sets)c FO(\013)g FQ(2)g FJ(R)40 b FP(for)33 b(whic)n(h)g(there)g(are)g (orthogonal)e(pro)5 b(jections)32 b FO(P)2200 851 y FL(1)2237 839 y FP(,)i FO(:)14 b(:)g(:)27 b FP(,)35 b FO(P)2529 851 y FM(n)118 939 y FP(satisfying)488 877 y Fz(P)576 897 y FM(n)576 964 y(i)p FL(=1)701 939 y FO(P)754 951 y FM(i)805 939 y FP(=)23 b FO(\013I)7 b FP(.)243 1039 y(Recen)n(t)19 b(results)h(on)f(families)h(of)g(linearly)f(connected)g (pro)5 b(jections)19 b(can)g(b)r(e)118 1138 y(found)30 b(in)f([78)o(,)h(148)n(,)g(149)n(,)g(232)n(,)g(233)n(].)43 b(This)29 b(problem)g(is)g(closely)f(related)h(to)118 1238 y(the)23 b(study)g(of)g(families)g(of)g(self-adjoin)n(t)f(op)r (erators)f(whose)i(sum)g(is)f(zero)g([173)o(,)118 1338 y(174)o(].)35 b(The)24 b(latter)f(problem)g(has)g(in)n(teresting)g (applications)g([147)n(,)h(216)n(,)g(295)o(].)243 1437 y(2.2.2.)35 b(The)28 b(exp)r(osition)f(follo)n(ws)g([190)n(].)243 1537 y(2.2.3.)35 b(The)26 b(class)f(of)h(represen)n(tations)f(describ)r (ed)h(in)g(Section)g(2.2.2)f(can)118 1636 y(easily)35 b(b)r(e)i(extended)f(to)g(more)f(general)f(relations)h(whic)n(h)h (arise)f(from)g(dy-)118 1736 y(namical)28 b(systems)g(on)g(the)h (plane;)f(the)h(metho)r(ds)g(and)f(ideas)g(used)g(here)g(are)118 1836 y(the)36 b(same)f(as)g(in)h(Section)g(2.1.)61 b(Notice)35 b(that)h(some)f(facts)h(from)f(the)i(one-)118 1935 y(dimensional)27 b(dynamics)h(fail)f(to)h(hold)f(in)h(the)h(t)n(w)n(o-dimensional)d (case.)36 b(The)118 2035 y(exp)r(osition)27 b(essen)n(tially)g(follo)n (ws)f([290)o(].)243 2135 y(2.2.4.)35 b(The)28 b(exp)r(osition)f(follo)n (ws)g([212)n(].)243 2234 y(2.2.5.)37 b(The)28 b(Skly)n(anin)g(algebra)e (w)n(as)h(in)n(tro)r(duced)h(and)g(studied)h(in)f([269)o(,)118 2334 y(270)o(];)33 b(these)f(pap)r(ers)f(also)f(con)n(tain)h(classes)f (of)i(represen)n(tations)e(of)h(its)h(real)118 2433 y(forms.)243 2533 y(F)-7 b(or)32 b(represen)n(tations)f(of)h(the)h(quan)n(tum)g (algebra)e FO(U)1934 2545 y FM(q)1970 2533 y FP(\()p FO(sl)r FP(\(2\)\),)j(see,)g(e.g.,)118 2633 y([140)o(,)28 b(143)o(],)h(and)f(the)h(bibliograph)n(y)e(therein.)39 b(In)29 b(man)n(y)f(pap)r(ers,)g(the)h(nota-)118 2732 y(tion)i FO(U)349 2744 y FM(q)386 2732 y FP(\()p FO(sl)r FP(\(2\)\))g(is)g(used)g(for)g(another)f(deformation)g(of)i FO(sl)r FP(\(2\);)g(the)g(algebra)118 2832 y(considered)f(in)h(the)g(b) r(o)r(ok)g(is)f(also)g(denoted)h(b)n(y)1697 2811 y(\024)1683 2832 y FO(U)1740 2844 y FM(q)1776 2832 y FP(\()p FO(sl)1872 2844 y FL(2)1909 2832 y FP(\))g(\(see,)h(e.g.,)g([140)n(]\).)118 2932 y(Our)27 b(exp)r(osition)g(follo)n(ws)g([287)n(].)243 3031 y(A)h(more)e(general)h(class)f(of)i(algebras)d(w)n(as)i(in)n(tro)r (duced)g(in)h([187)o(].)118 3151 y FR(Section)41 b(2.3.)58 b FP(The)36 b(non-standard)e FO(q)s FP(-deformed)g(algebras)g FO(U)2192 3163 y FM(q)2228 3151 y FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)h FP(\))q(\))118 3251 y(de\014ned)34 b(in)g(terms)g(of)f (trilinear)g(relations)g(for)g(generating)f(elemen)n(ts)i(w)n(ere)118 3350 y(in)n(tro)r(duced)e(b)n(y)f(D.)i(F)-7 b(airlie)31 b([82)o(].)50 b(An)32 b(algebra)e(whic)n(h)i(can)f(b)r(e)i(reduced)e (to)118 3450 y FO(U)175 3462 y FM(q)211 3450 y FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)i FP(\)\))27 b(w)n(as)20 b(de\014ned)i(in)f([186)o(].)35 b(It)21 b(is)g(kno)n(wn)g(that)g(Lie)g (algebras)e(of)i(the)118 3550 y(Lie)29 b(groups)e FO(S)5 b(L)p FP(\(2)p FO(;)14 b FJ(C)g FP(\))35 b(and)28 b FO(S)5 b(O)r FP(\(3\))30 b(are)d(isomorphic,)h(but)h(the)g FO(q)s FP(-deformed)118 3649 y(algebras)e FO(U)500 3661 y FM(q)537 3649 y FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)h FP(\)\))36 b(di\013er)29 b(from)g(the)g(quan)n(tum)g(algebras)f FO(U)2206 3661 y FM(q)2242 3649 y FP(\()p FO(sl)r FP(\(2)p FO(;)14 b FJ(C)g FP(\))q(\))118 3749 y(in)n(tro)r(duced)26 b(b)n(y)h(V.)g(Drinfeld)g(and)f(M.)h(Jim)n(b)r(o.)36 b(Moreo)n(v)n(er,)24 b(they)j(ha)n(v)n(e)e(non-)118 3848 y(coinciding)36 b(sets)h(of)f(irreducible)g(\014nite)h(and)g (in\014nite)g(dimensional)f(repre-)118 3948 y(sen)n(tations.)f (Finite-dimensional)24 b(irreducible)g(represen)n(tations)e(whic)n(h,)j (for)118 4048 y FO(q)48 b FQ(!)e FP(1,)e(yield)c(the)i(w)n(ell-kno)n (wn)d(\014nite-dimensional)i(irreducible)f(repre-)118 4147 y(sen)n(tations)35 b(of)h(the)g(Lie)g(algebra)e FO(so)p FP(\(3)p FO(;)14 b FJ(C)h FP(\))43 b(w)n(ere)35 b(describ)r(ed)g(in)i([82)o(].)62 b(The)p eop %%Page: 201 205 201 204 bop 118 100 a FK(Commen)n(ts)27 b(to)h(Chapter)f(2)1452 b FP(201)118 333 y(complete)32 b(classi\014cation)e(of)i (\014nite-dimensional)f(represen)n(tations)f(when)i FO(q)118 432 y FP(is)f(not)g(a)f(ro)r(ot)g(of)h(unit)n(y)g(ha)n(v)n(e)f(b)r(een) h(obtained)g(in)g([110)o(].)47 b(This)31 b(pap)r(er)f(also)118 532 y(con)n(tains)f(a)g(description)g(of)h(some)f(classes)f(of)i (irreducible)f(represen)n(tations)118 632 y(when)f FO(q)j FP(is)c(a)g(ro)r(ot)g(of)h(unit)n(y)f(\(see)h(also)e([109)o(]\).)243 732 y(Irreducible)38 b(represen)n(tations)e(of)j(the)g FQ(\003)p FP(-algebras)d FO(U)1979 744 y FM(q)2015 732 y FP(\()p FO(so)p FP(\(3\)\))k(\()p FO(q)45 b FQ(2)d FJ(R)118 831 y FP(and)d FQ(j)p FO(q)s FQ(j)k FP(=)g(1\))c(determined)h (b)n(y)f(the)h(in)n(v)n(olution)e FO(I)1813 801 y FN(\003)1806 852 y FL(1)1894 831 y FP(=)k FQ(\000)p FO(I)2102 843 y FL(1)2140 831 y FP(,)g FO(I)2248 801 y FN(\003)2241 852 y FL(2)2329 831 y FP(=)g FQ(\000)p FO(I)2537 843 y FL(2)118 931 y FP(w)n(ere)24 b(describ)r(ed)h(in)g([246)o(].)36 b(It)26 b(w)n(as)e(sho)n(wn)g(there)h(that)g(all)g(suc)n(h)g(represen)n (ta-)118 1031 y(tions)k(are)f(analogues)f(of)h(irreducible)h FQ(\003)p FP(-represen)n(tations)d(of)i(the)i(real)e(com-)118 1130 y(pact)33 b(form)f(of)g FO(so)p FP(\(3)p FO(;)14 b FJ(C)h FP(\))39 b(for)32 b FO(q)i(>)d FP(0)h(and)g(its)h(graded)e (analogue)g(for)h FO(q)i(<)d FP(0.)118 1230 y(The)36 b(later)g(w)n(ere)f(classi\014ed)g(in)h([103)o(].)62 b(The)36 b(classi\014cation)f(of)h(irreducible)118 1329 y FQ(\003)p FP(-represen)n(tations)20 b(of)i FO(U)904 1341 y FM(q)940 1329 y FP(\()p FO(so)p FP(\(3\)\),)j(when)d FO(q)k FP(is)c(a)g(ro)r(ot)g(of)g(unit)n(y)-7 b(,)24 b(w)n(as)d(also)g(ob-)118 1429 y(tained)i(indep)r(enden)n(tly)h(in)f ([17)o(].)35 b(It)24 b(w)n(as)e(sho)n(wn)g(in)h([17)o(,)g(109)n(,)g (110)o(,)g(246)o(])g(that)118 1529 y(the)34 b(algebras)e FO(U)654 1541 y FM(q)691 1529 y FP(\()p FO(so)p FP(\(3)p FO(;)14 b FJ(C)h FP(\)\))40 b(ha)n(v)n(e)33 b(irreducible)h (\014nite-dimensional)f(repre-)118 1628 y(sen)n(tations)25 b(whic)n(h)g(ha)n(v)n(e)f(no)h(analogue)f(for)h(the)h(Lie)f(algebra)f FO(so)p FP(\(3)p FO(;)14 b FJ(C)g FP(\))32 b(or)25 b(its)118 1728 y(graded)h(analogue,)g(that)i(is,)g(whic)n(h)g(do)f(not)g(admit)h (the)g(limits)g(as)f FO(q)f FQ(!)e FP(1)j(or)118 1828 y FO(q)f FQ(!)d(\000)p FP(1.)36 b(Some)25 b(classes)f(of)i(irreducible) f(represen)n(tations)e(of)j(the)f FQ(\003)p FP(-algebra)118 1927 y FO(U)175 1939 y FM(q)211 1927 y FP(\()p FO(so)p FP(\(2)p FO(;)14 b FP(1\)\))32 b(corresp)r(onding)e(to)h(the)h(in)n(v)n (olution)f FO(I)1798 1897 y FN(\003)1791 1948 y FL(1)1866 1927 y FP(=)e FQ(\000)p FO(I)2061 1939 y FL(1)2098 1927 y FP(,)k FO(I)2197 1897 y FN(\003)2190 1948 y FL(2)2264 1927 y FP(=)d FO(I)2395 1939 y FL(2)2464 1927 y FP(are)118 2027 y(giv)n(en)f(in)h([92].)44 b(Note)30 b(that)g(all)g(suc)n(h)f (non-trivial)g(represen)n(tations)f(are)h(un-)118 2126 y(b)r(ounded.)35 b(Un)n(b)r(ounded)22 b(represen)n(tations)c(of)j(the)g FQ(\003)p FP(-algebra)e(de\014ned)i(b)n(y)f(the)118 2226 y(in)n(v)n(olution)27 b FO(I)551 2196 y FN(\003)544 2247 y FL(1)612 2226 y FP(=)c FQ(\000)p FO(I)801 2238 y FL(1)838 2226 y FP(,)28 b FO(I)932 2196 y FN(\003)925 2247 y FL(2)993 2226 y FP(=)23 b FQ(\000)p FO(I)1182 2238 y FL(2)1247 2226 y FP(w)n(ere)j(studied)i(in)g([246)o(].)118 2352 y FR(Section)k(2.4)243 2452 y FP(2.4.1.)37 b(F)-7 b(ollo)n(wing)27 b([127)n(])h(w)n(e)g(consider)f(the)i(Wic)n(k)f(analogue)e(of)i(the)g (\\di-)118 2551 y(rect)23 b(pro)r(duct")g(of)g FO(q)s FP(-CCR)h(whic)n(h)f(is)g(an)g(example)g(of)h FO(q)1864 2563 y FM(ij)1922 2551 y FP(-CCR)f(constructed)118 2651 y(in)29 b([43)o(].)42 b(All)29 b(represen)n(tations)e(of)i(the)g (\\direct)f(pro)r(duct")h(are)f(describ)r(ed)h(in)118 2751 y(terms)e(of)h(the)g(one-dimensional)e FO(q)s FP(-CCR.)243 2851 y(2.4.2.)34 b(W)-7 b(e)25 b(consider)f(Wic)n(k)h(analogues)d(of)j (the)g(t)n(wisted)g(canonical)f(com-)118 2950 y(m)n(utation)h (relations)f(in)n(tro)r(duced)h(and)g(studied)h(b)n(y)f(Pusz)g(and)g(W) -7 b(orono)n(wicz)118 3050 y([225)o(],)38 b(and)d(t)n(wisted)h (canonical)f(an)n(ticomm)n(utation)f(relations)h(studied)h(b)n(y)118 3150 y(Pusz)g([224)o(].)63 b(W)-7 b(e)37 b(denote)g(these)f(algebras)f (b)n(y)h FO(\026)p FP(-CCR)h(and)f FO(\026)p FP(-CAR,)h(re-)118 3249 y(sp)r(ectiv)n(ely)-7 b(.)46 b(The)31 b(pro)r(of)g(of)f(Prop)r (osition)g(51)g(is)g(giv)n(en)g(according)f(to)i([127)o(].)118 3349 y(The)20 b(same)g(prop)r(osition)f(for)h FO(\026)t FQ(\000)t FO(C)6 b(AR)21 b FP(\(Theorem)f(36\))f(w)n(as)h(pro)n(v)n(ed) f(in)h([220)o(].)243 3449 y(2.4.3.)35 b(The)28 b(exp)r(osition)f(follo) n(ws)g([202)n(].)243 3549 y(2.4.4.)72 b(Algebra)39 b(of)g(functions)h (on)g(the)g(non-standard)e(three-dimen-)118 3649 y(sional)g(real)g (quan)n(tum)h(sphere)g(w)n(as)e(in)n(tro)r(duced)i(b)n(y)g(M.)g(Noumi)g (and)g(K.)118 3748 y(Mimac)n(hi)25 b([185)n(].)36 b(The)25 b(description)f(of)h(irreducible)f(represen)n(tations)f(of)i(this)118 3848 y(algebra)h(follo)n(ws)h([202)n(].)243 3948 y(2.4.5.)51 b(The)33 b(Heisen)n(b)r(erg)e(relations)h(for)g(the)h(quan)n(tum)g FO(E)2132 3960 y FL(2)2202 3948 y FP(group)f(w)n(as)118 4048 y(in)n(tro)r(duced)38 b(b)n(y)h(W)-7 b(orono)n(wicz)36 b([305)o(].)69 b(The)39 b(classi\014cation)e(of)i(irreducible)118 4147 y(represen)n(tations)34 b(of)i(this)h(algebra)d(is)i(due)h(to)f ([198)n(].)63 b(Notice)36 b(that)g(w)n(e)g(do)p eop %%Page: 202 206 202 205 bop 118 100 a FP(202)443 b FK(Chapter)28 b(2.)36 b(Represen)n(tations)26 b(of)i(dynamical)f FQ(\003)p FK(-algebras)118 333 y FP(not)h(assume)f(the)h(natural)g(sp)r(ectral)f (condition)g(of)h([305)o(],)g(th)n(us)g(obtaining)f(a)118 432 y(wider)g(class)g(of)h(represen)n(tations.)243 532 y(2.4.6.)36 b(The)28 b(notions)g(of)f(a)h(Wic)n(k)g(algebra)e(and)i (Wic)n(k)f(ideal)h(w)n(ere)f(in)n(tro-)118 632 y(duced)e(in)g([127)n (].)36 b(Prop)r(osition)23 b(55)h(w)n(as)f(pro)n(v)n(ed)g(for)i FO(n)d FP(=)h(2)h(in)h([127)o(],)g(and)f(for)118 732 y(general)i FO(n)i FP(in)g([220)n(].)243 832 y(Recen)n(t)19 b(results)h(on)f(the)i(related)e FO(C)1344 802 y FN(\003)1382 832 y FP(-algebras)f(obtained)h(b)n(y)h(the)g(authors)118 932 y(see)27 b([125)o(,)h(223)n(].)118 1057 y FR(Section)k(2.5.)243 1157 y FP(2.5.1.)h(The)22 b(idea)f(of)h(decomp)r(osition)f(of)h(the)g (represen)n(tation)d(space)i(with)118 1257 y(resp)r(ect)36 b(to)g(some)f(comm)n(utativ)n(e)g(family)i(w)n(as)e(used)g(b)n(y)h(G.)h (Mac)n(k)n(ey)d(in)i(a)118 1357 y(general)j(framew)n(ork)f(of)h (imprimitivit)n(y)i(systems)e(whic)n(h)h(resulted)g(in)g(his)118 1456 y(metho)r(d)32 b(of)g(description)g(of)f(represen)n(tations)f(of)i (semi-direct)f(pro)r(ducts)h(of)118 1556 y(lo)r(cally)c(compact)g (groups)e([165)o(,)j(166)n(].)39 b(The)29 b(non-group)d(examples)i(of)g (suc)n(h)118 1655 y(decomp)r(osition)37 b(are)f(the)h(G)-10 b(\027)-52 b(arding{Wigh)n(tman)36 b(decomp)r(osition)h(of)g(CCR)118 1755 y(and)c(CAR)g(with)h(in\014nite)g(n)n(um)n(b)r(er)e(of)h(degrees)f (of)h(freedom)g([86)o(,)g(87)o(];)j(also)118 1855 y(w)n(e)c(men)n(tion) g([6,)g(95)o(,)h(100)n(,)g(112)n(])g(on)f(comm)n(utativ)n(e)f(mo)r (dels)i(for)e(CAR)i(and)118 1954 y(CCR,)22 b([273)o(])g(for)g (AF-algebras,)f([293)n(,)i(175)n(,)f(99)o(,)g(117)o(],)h(and)f(the)h (bibliograph)n(y)118 2054 y(therin,)31 b(for)e(curren)n(t)g(and)h(lo)r (cal)f(curren)n(t)g(algebras,)f(etc.)44 b(The)30 b(term)g(\\com-)118 2154 y(m)n(utativ)n(e)38 b(mo)r(del")h(w)n(as)e(in)n(tro)r(duced)i(in)g ([294)n(].)70 b(As)39 b(a)f(rule,)j(the)e(general)118 2253 y(comm)n(utativ)n(e)27 b(mo)r(del)g(do)r(es)h(not)f(giv)n(e)g(a)g (unitary)g(description)g(of)g(all)g(repre-)118 2353 y(sen)n(tations)20 b(of)h(the)h(corresp)r(onding)d(op)r(erator)g(relations;)j(ho)n(w)n(ev) n(er,)e(the)i(exis-)118 2452 y(tence)j(of)f(a)h(comm)n(utativ)n(e)e(mo) r(del)i(is)g(a)f(prop)r(ert)n(y)f(of)i(the)g(relation)e(re\015ecting) 118 2552 y(the)34 b(structure)f(of)g(its)h(represen)n(tations.)52 b(General)33 b(op)r(erator)e(relations)h(for)118 2652 y(whic)n(h)g(there)f(exists)h(a)f(comm)n(utativ)n(e)g(mo)r(del)h(in)g (terms)g(of)g(m)n(ultiplication)118 2751 y(and)22 b(w)n(eigh)n(ted)f (op)r(erator-v)-5 b(alued)19 b(shift)k(w)n(ere)d(studied)i(in)g([28)o (,)g(192)o(,)g(195)n(])g(\(see)118 2851 y(also)27 b([26)o(]\).)243 2951 y(2.5.2.)51 b(The)33 b(comm)n(utativ)n(e)f(mo)r(del)h(for)f(cen)n (tered)g(op)r(erators)f(w)n(as)h(con-)118 3051 y(structed)d(in)g([177)o (].)41 b(W)-7 b(e)30 b(rewrite)e(the)h(relations)f(whic)n(h)h(de\014ne) g(the)h(class)e(of)118 3150 y(cen)n(tered)g(op)r(erators)f(in)i(the)h (form)e(that)h(enables)g(us)g(to)f(apply)h(the)g(general)118 3250 y(theorem)e(from)g(the)h(previous)f(section.)243 3350 y(2.5.3.)44 b(Represen)n(tations)29 b(of)i(Cun)n(tz)f(algebras)f (o)r(ccup)n(y)h(a)g(sp)r(ecial)g(place)118 3450 y(due)24 b(to)f(their)h(relation)f(to)g(endomorphisms)g(of)g FO(L)p FP(\()p FO(H)7 b FP(\))24 b(preserving)e(the)i(iden-)118 3550 y(tit)n(y)g([11)o(,)f(159)o(],)h(and)g(other)f(applications)f (\(see,)i(e.g.,)g([45)o(]\).)36 b(Their)23 b(represen-)118 3649 y(tations)j(w)n(ere)f(studied)h(in)h(n)n(umerous)e(pap)r(ers)g (\(see,)i(e.g.,)f([48)o(,)g(44)o(,)g(46)o(])g(etc.\).)118 3749 y(W)-7 b(e)24 b(c)n(ho)r(ose)e(appropriate)f(elemen)n(ts)i(in)h (the)f(Cun)n(tz)h(algebra,)e(whic)n(h)h(enables)118 3848 y(us)f(to)h(apply)f(the)g(results)g(of)g(Section)h(2.5.1,)f(and)g (construct)g(a)g(comm)n(utativ)n(e)118 3948 y(mo)r(del)31 b(for)f(its)h(represen)n(tations;)f(w)n(e)g(apply)h(this)g(mo)r(del)f (to)h(the)g(construc-)118 4048 y(tion)25 b(and)h(study)f(of)g(represen) n(tations)f(of)h(the)h(Cun)n(tz)f(algebras.)34 b(The)25 b(results)118 4147 y(of)j(this)g(section)f(are)f(due)i(to)g([47)o(].)p eop %%Page: 203 207 203 206 bop 118 904 a FS(Chapter)46 b(3)118 1226 y(On)f(the)g (complexit)l(y)h(of)f(the)h(description)f(of)118 1375 y(represen)l(tations)i(of)e Fm(\003)p FS(-algebras)118 1830 y FH(3.1)112 b FG(\003)p FH(-Wild)36 b(algebras)i(and)h(relations) 118 2024 y FP(Before)31 b(passing)f(to)i(the)g(complexit)n(y)f(problem) g(of)h(unitary)f(description)g(of)118 2124 y(represen)n(tations)24 b(for)h(concrete)g(classes)f(of)i FQ(\003)p FP(-algebras,)d(w)n(e)i (giv)n(e)g(de\014nitions)118 2224 y(and)33 b(some)g(results)g (concerning)f(the)i(ideology)e(and)h(metho)r(dology)f(of)h(ma-)118 2323 y(jorization)h(of)h FQ(\003)p FP(-algebras)d(and)i FQ(\003)p FP(-wildness)g(in)h(Sections)g(3.1.1)e(and)i(3.1.2.)118 2423 y(Then,)e(in)f(Sections)f(3.1.3{3.1.6)e(w)n(e)i(will)h(giv)n(e)f (a)g(n)n(um)n(b)r(er)g(of)h(examples)f(of)118 2522 y FQ(\003)p FP(-wild)21 b(algebras)e(generated)h(b)n(y)h(pro)5 b(jections)20 b(and)h(idemp)r(oten)n(ts,)i(generated)118 2622 y(b)n(y)29 b(quadratic,)f(cubic)h(and)g(semilinear)f(relations,)h FQ(\003)p FP(-wild)f(group)g(algebras,)118 2722 y(etc.)243 2828 y(W)-7 b(e)20 b(emphasize)f(once)g(again)f(that)i(the)g(fact)f (that)h(some)f(algebra)f(is)h FQ(\003)p FP(-wild)118 2927 y(implies)k(that)g(the)g(problem)g(of)g(unitary)f(description)g (of)h FC(al)t(l)h FP(represen)n(tations)118 3027 y(is)30 b(v)n(ery)g(complicated.)44 b(Ho)n(w)n(ev)n(er,)29 b(the)i(problem)f (of)g(construction)g(and)g(de-)118 3127 y(scription)k(of)h(di\013eren)n (t)g(classes)f(of)h(represen)n(tations)e(b)r(ecomes)h(ev)n(en)h(more) 118 3226 y(attractiv)n(e.)243 3333 y(Belo)n(w)40 b(in)i(this)f(c)n (hapter,)j(w)n(e)d(will)h(write)f(Rep)14 b FA(A)41 b FP(for)g(a)f FQ(\003)p FP(-algebra)f FA(A)118 3432 y FP(instead)28 b(of)f FQ(\003)p FP(-Rep)13 b FA(A)p FP(,)28 b(to)f(simplify)h(the)g(notations.)118 3683 y FR(3.1.1)94 b(Ma)5 b(jorization)33 b(of)g FQ(\003)p FR(-algebras)f(with)g(resp)s (ect)h(to)f(the)h(com-)410 3783 y(plexit)m(y)f(of)f(their)h(represen)m (tations)118 3948 y(1.)52 b FP(Let)33 b FA(A)g FP(b)r(e)g(a)g FQ(\003)p FP(-algebra.)50 b(W)-7 b(e)33 b(recall)f(\(see)h(Section)g (1.1.3\))f(that)h(a)g(pair)118 4048 y(\()159 4026 y(~)150 4048 y FA(A)p FO(;)14 b(\036)9 b FP(:)28 b FA(A)23 b FQ(\000)-48 b(!)571 4026 y FP(~)562 4048 y FA(A)o FP(\),)27 b(where)951 4026 y(~)942 4048 y FA(A)f FP(is)h(a)e FQ(\003)p FP(-algebra)f(and)j FO(\036)f FP(is)h(a)f FQ(\003)p FP(-homomorphism,) 118 4147 y(is)g(called)h(an)f(en)n(v)n(eloping)f FQ(\003)p FP(-algebra)f(if)j(for)f(an)n(y)f FQ(\003)p FP(-represen)n(tation)f FO(\031)12 b FP(:)29 b FA(A)22 b FQ(\000)-48 b(!)1284 4357 y FP(203)p eop %%Page: 204 208 204 207 bop 118 100 a FP(204)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FO(L)p FP(\()p FO(H)7 b FP(\))21 b(of)f(the)i(algebra)d FA(A)h FP(there)g(exists)h(a)f(unique)h FQ(\003)p FP(-represen)n (tation)i(~)-47 b FO(\031)13 b FP(:)2402 311 y(~)2393 333 y FA(A)22 b FQ(\000)-48 b(!)118 432 y FO(L)p FP(\()p FO(H)7 b FP(\))28 b(suc)n(h)f(that)h(the)g(diagram)1110 959 y(~)1101 980 y FA(A)287 b FO(L)p FP(\()p FO(H)7 b FP(\))p 1185 954 239 4 v 1340 952 a Fu(-)1283 1024 y FP(~)-46 b FO(\031)1390 762 y(\031)1189 678 y Fu(@)1272 761 y(@)1355 844 y(@)1405 894 y(@)-83 b(R)1101 566 y FA(A)p 1129 894 4 299 v 1131 894 a Fu(?)1048 765 y FO(\036)118 1135 y FP(is)28 b(comm)n(utativ)n(e.)118 1275 y FR(2.)58 b FP(T)-7 b(o)34 b(v)n(erify)h(whether)f(or)g(not)h(a)g(pair)f(\()p FA(A)p FO(;)14 b(\036)9 b FP(:)30 b FA(A)35 b FQ(\000)-48 b(!)1942 1253 y FP(~)1933 1275 y FA(A)o FP(\))36 b(determines)e(an)118 1374 y(en)n(v)n(eloping)23 b(algebra,)h(it)g(is)h(only)f(su\016cien)n (t)g(to)h(consider)e(unital)i(represen)n(ta-)118 1474 y(tions)j(of)h(the)g(algebra)e FA(A)p FP(.)39 b(Indeed,)29 b(for)f(an)n(y)g(unital)h(represen)n(tation)e FO(\031)32 b FP(of)c FA(A)118 1574 y FP(let)e(there)f(exist)g(a)g(represen)n (tation)j(~)-47 b FO(\031)29 b FP(of)1424 1552 y(~)1415 1574 y FA(A)c FP(making)g(the)g(follo)n(wing)f(diagram)118 1673 y(comm)n(utativ)n(e)1110 2191 y(~)1101 2212 y FA(A)287 b FO(L)p FP(\()p FO(H)7 b FP(\))p 1185 2186 239 4 v 1340 2184 a Fu(-)1283 2256 y FP(~)-46 b FO(\031)1390 1994 y(\031)1189 1910 y Fu(@)1272 1993 y(@)1355 2076 y(@)1405 2126 y(@)-83 b(R)1101 1798 y FA(A)p 1129 2126 4 299 v 1131 2126 a Fu(?)1048 1997 y FO(\036)243 2385 y FP(Supp)r(ose)25 b(that)g FO(\031)12 b FP(:)29 b FA(A)22 b FQ(\000)-48 b(!)23 b FO(L)p FP(\()p FO(H)7 b FP(\))25 b(is)g(not)g(a)g(unital)g (represen)n(tation.)35 b(Then)118 2485 y FO(\031)s FP(\()p FO(e)p FP(\))46 b(=)g FO(E)g FP(is)41 b(a)g(pro)5 b(jection)40 b(in)h FO(L)p FP(\()p FO(H)7 b FP(\),)45 b(and)c(for)f(an)n(y)h FO(a)k FQ(2)h FA(A)41 b FP(w)n(e)g(ha)n(v)n(e)118 2585 y FO(\031)s FP(\()p FO(a)p FP(\))26 b(=)g FO(\031)s FP(\()p FO(eae)p FP(\))f(=)h FO(E)5 b(\031)s FP(\()p FO(a)p FP(\))p FO(E)g FP(.)42 b(The)29 b(space)g FO(H)35 b FP(decomp)r(oses)29 b(in)n(to)g(the)g(direct)118 2684 y(sum)d FO(H)k FP(=)23 b(Im)14 b FO(E)20 b FQ(\010)15 b FP(k)n(er)d FO(E)5 b FP(,)27 b(and)f(the)g(subspaces)f(Im)14 b FO(E)5 b FP(,)26 b(k)n(er)13 b FO(E)31 b FP(are)25 b(in)n(v)-5 b(arian)n(t)118 2784 y(with)31 b(resp)r(ect)f(to)h(the)f(op)r(erators)f FO(\031)s FP(\()p FO(a)p FP(\);)k(moreo)n(v)n(er)27 b(k)n(er)13 b FO(E)33 b FQ(\032)27 b FP(k)n(er)13 b FO(\031)s FP(\()p FO(a)p FP(\),)32 b(and)118 2884 y(according)26 b(to)h(this)h (inclusion,)g(w)n(e)f(ha)n(v)n(e)946 3090 y FO(\031)s FP(\()p FO(a)p FP(\))d(=)1215 2973 y Fz(\022)1277 3041 y FO(\031)1327 3011 y FL(\(1\))1416 3041 y FP(\()p FO(a)p FP(\))83 b(0)1380 3141 y(0)185 b(0)1649 2973 y Fz(\023)1724 3090 y FO(;)118 3312 y FP(where)25 b FO(\031)406 3282 y FL(\(1\))521 3312 y FP(is)h(a)f(unital)h(represen)n(tation)e(of)h FA(A)p FP(.)36 b(Since)26 b(the)g(homomorphism)118 3412 y FO(\036)i FP(is)g(unital,)g(w)n(e)f(ha)n(v)n(e)f FO(\031)905 3382 y FL(\(1\))995 3412 y FP(\()p FO(e)p FP(\))d(=)k(~)-46 b FO(\031)1259 3382 y FL(\(1\))1348 3412 y FP(\()p FO(\036)p FP(\()p FO(e)p FP(\)\),)29 b(and)969 3623 y(~)-47 b FO(\031)t FP(\()p FO(a)p FP(\))23 b(=)1234 3506 y Fz(\022)1299 3574 y FP(~)-46 b FO(\031)1345 3544 y FL(\(1\))1434 3574 y FP(\()p FO(a)p FP(\))84 b(0)1398 3674 y(0)186 b(0)1667 3506 y Fz(\023)118 3848 y FP(is)29 b(the)h(needed)f(homomorphism)f (from)h FA(A)g FP(to)1613 3827 y(~)1604 3848 y FA(A)p FP(.)42 b(Since)34 b(~)-47 b FO(\031)s FP(\()p FO(e)p FP(\))26 b(=)k(~)-46 b FO(\031)s FP(\()p FO(\036)p FP(\()p FO(e)p FP(\)\))27 b(=)118 3948 y FO(E)5 b FP(,)23 b(an)n(y)e(represen)n (tation)g(of)1017 3926 y(~)1008 3948 y FA(A)g FP(that)h(mak)n(es)f(the) h(previous)f(diagram)g(comm)n(u-)118 4048 y(tativ)n(e,)i(coincides)e (with)h(the)g(one)f(presen)n(ted)g(ab)r(o)n(v)n(e.)34 b(Th)n(us)26 b(~)-47 b FO(\031)25 b FP(is)d(determined)118 4147 y(uniquely)-7 b(.)p eop %%Page: 205 209 205 208 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(205)118 333 y FR(3.)38 b FP(If)29 b(\()356 311 y(~)347 333 y FA(A)p FO(;)14 b(\036)p FP(\))29 b(is)f(an)g(en)n(v)n(eloping)f(algebra)g(of)h (the)h(algebra)d FA(A)i FP(and)g(\()2276 311 y(~)2267 333 y FA(A)2327 345 y FL(1)2364 333 y FO(;)14 b( )s FP(\))29 b(is)118 432 y(an)c(en)n(v)n(eloping)f(algebra)f(of)1027 411 y(~)1018 432 y FA(A)o FP(,)j(then)g(\()1354 411 y(~)1345 432 y FA(A)1405 444 y FL(1)1442 432 y FO(;)14 b( )i FQ(\016)e FO(\036)p FP(\))26 b(is)f(an)g(en)n(v)n(eloping)f(algebra)118 532 y(of)k(the)g(algebra)d FA(A)p FP(.)37 b(The)28 b(pro)r(of)f(is)g (eviden)n(t.)118 684 y FR(4.)69 b FP(No)n(w)38 b(w)n(e)g(in)n(tro)r (duce)h(the)g(relation)e(of)i(ma)5 b(jorization)37 b(for)h FQ(\003)p FP(-algebras.)118 784 y(Denote)31 b(b)n(y)f FA(K)h FP(the)g(algebra)e(of)h(compact)g(op)r(erators)e(in)j(a)f (separable)f(\(p)r(os-)118 883 y(sibly)j(\014nite-dimensional\))g(Hilb) r(ert)g(space)g FO(H)1611 895 y FL(0)1648 883 y FP(.)50 b(The)32 b(algebra)e FA(K)22 b FQ(\012)f FA(A)32 b FP(will)118 983 y(o)r(ccasionally)26 b(b)r(e)i(denoted)f(b)n(y)h FO(M)1203 995 y FM(n)1247 983 y FP(\()p FA(A)p FP(\),)g FO(n)23 b FQ(2)h FJ(N)k FQ([)19 b(f1g)p FP(.)243 1084 y(Let)30 b FO(\031)12 b FP(:)29 b FA(A)d FQ(\000)-48 b(!)27 b FO(L)p FP(\()p FO(H)7 b FP(\))30 b(b)r(e)g(a)g(represen)n (tation)e(of)i FA(A)p FP(.)43 b(It)31 b(induces)f(a)f(repre-)118 1183 y(sen)n(tation,)735 1368 y(^)-46 b FO(\031)26 b FP(=)d(id)18 b FQ(\012)g FO(\031)12 b FP(:)28 b FA(K)19 b FQ(\012)f FA(A)23 b FQ(\000)-48 b(!)23 b FO(L)p FP(\()p FO(H)1693 1380 y FL(0)1748 1368 y FQ(\012)18 b FO(H)7 b FP(\))p FO(;)118 1576 y FP(of)22 b(the)g(algebra)e FA(K)7 b FQ(\012)g FA(A)p FP(.)35 b(If)23 b(\()1002 1557 y FJ(^)991 1576 y FA(K)c FQ(\012)f FA(A)o FO(;)c(\036)p FP(\))23 b(is)f(an)f(en)n(v)n(eloping)g FQ(\003)p FP(-algebra)e(of)j FA(K)7 b FQ(\012)g FA(A)118 1675 y FP(then,)22 b(b)n(y)e(de\014nition,) 27 b(^)-47 b FO(\031)24 b FP(uniquely)c(determines)g(the)g(represen)n (tation)j(~)-46 b FO(\031)23 b FP(of)d(the)118 1793 y(en)n(v)n(eloping) h(algebra)817 1774 y FJ(^)806 1793 y FA(K)e FQ(\012)f FA(A)k FP(in)g(the)g(same)g(Hilb)r(ert)h(space)e FO(L)p FP(\()p FO(H)2131 1805 y FL(0)2175 1793 y FQ(\012)7 b FO(H)g FP(\).)35 b(No)n(w)118 1916 y(let)29 b FO( )j FP(b)r(e)d(a)f(homomorphism)g(of)g(the)h(algebra)e FA(B)i FP(in)n(to)f(the)h(algebra)2347 1897 y FJ(^)2335 1916 y FA(K)19 b FQ(\012)f FA(A)p FP(;)118 2016 y(then)924 2200 y(~)-46 b FO(\031)22 b FQ(\016)c FO( )12 b FP(:)28 b FA(B)23 b FQ(7!)g FO(L)p FP(\()p FO(H)1526 2212 y FL(0)1582 2200 y FQ(\012)18 b FO(H)7 b FP(\))118 2385 y(is)28 b(a)f(represen)n (tation)f(of)h FA(B)p FP(.)118 2572 y FR(Lemma)f(13.)38 b FC(L)l(et)26 b FP(\()797 2553 y FJ(^)786 2572 y FA(K)19 b FQ(\012)g FA(A)o FO(;)14 b(\036)p FP(\))28 b FC(b)l(e)f(an)g (enveloping)h FQ(\003)p FC(-algebr)l(a)g(of)f FA(K)12 b FQ(\012)g FA(A)27 b FC(and)118 2672 y FO(\031)165 2684 y FM(j)233 2672 y FQ(2)33 b FP(Rep\()p FA(A)p FP(\))p FC(,)k FO(j)h FP(=)32 b(1)p FC(,)37 b FP(2)p FC(.)54 b(L)l(et)39 b FP(~)-46 b FO(\031)1241 2684 y FM(j)1311 2672 y FC(denote)36 b(its)f(lifting)h(to)f(the)g(enveloping)118 2772 y FQ(\003)p FC(-algebr)l(a)i(as)f(describ)l(e)l(d)h(ab)l(ove.)58 b(If)36 b FO(V)53 b FQ(2)35 b FO(L)p FP(\()p FO(H)7 b FP(\))35 b FC(intertwines)h FO(\031)2224 2784 y FL(1)2297 2772 y FC(and)h FO(\031)2512 2784 y FL(2)2549 2772 y FC(,)118 2871 y(then)23 b FO(I)11 b FQ(\012)t FO(V)42 b FC(intertwines)28 b FP(~)-47 b FO(\031)972 2883 y FL(1)1033 2871 y FC(and)28 b FP(~)-46 b FO(\031)1235 2883 y FL(2)1295 2871 y FP(\()p FO(I)31 b FC(is)24 b(the)f(identity)h(op)l(er)l(ator)g (in)f FO(L)p FP(\()p FO(H)7 b FP(\)\))p FC(.)118 3040 y(Pr)l(o)l(of.)43 b FP(W)-7 b(e)31 b(will)f(pro)n(v)n(e)e(the)j(lemma)f (for)g FO(V)49 b FP(in)n(tert)n(wining)29 b(one)h(represen)n(ta-)118 3140 y(tion)e FO(\031)s FP(,)g(since,)f(b)n(y)h(setting)1060 3370 y FO(\031)f FP(=)1221 3253 y Fz(\022)1282 3319 y FO(\031)1329 3331 y FL(1)1472 3319 y FP(0)1304 3419 y(0)104 b FO(\031)1497 3431 y FL(2)1535 3253 y Fz(\023)1610 3370 y FO(;)118 3599 y FP(and)1071 3731 y FO(V)42 b FP(=)1248 3614 y Fz(\022)1322 3680 y FP(0)94 b FO(U)1310 3780 y(U)103 b FP(0)1524 3614 y Fz(\023)1599 3731 y FO(;)118 3929 y FP(w)n(e)27 b(obtain)h(the)g(result)f(for)g FO(U)36 b FP(in)n(tert)n(wining)27 b FO(\031)1606 3941 y FL(1)1672 3929 y FP(and)g FO(\031)1880 3941 y FL(2)1918 3929 y FP(.)243 4048 y(Let)19 b FB(A)g FP(denote)g(the)g FO(C)924 4018 y FN(\003)963 4048 y FP(-algebra)e(generated)h(b)n(y)23 b(~)-47 b FO(\031)t FP(\()1843 4029 y FJ(^)1832 4048 y FA(K)19 b FQ(\012)f FA(A)o FP(\))i(and)f FB(B)g FP(denote)118 4147 y(its)25 b FO(C)296 4117 y FN(\003)335 4147 y FP(-subalgebra)d (generated)i(b)n(y)29 b(^)-46 b FO(\031)s FP(\()p FA(K)13 b FQ(\012)g FA(A)p FP(\);)27 b(then)e FB(B)g FP(is)g(a)g FO(C)2121 4117 y FN(\003)2159 4147 y FP(-subalgebra)p eop %%Page: 206 210 206 209 bop 118 100 a FP(206)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FP(in)34 b FB(A)p FP(.)54 b(Let)33 b(us)g(sho)n(w)g(that)g(if)h FO(\031)1154 345 y FL(1)1191 333 y FP(,)h FO(\031)1296 345 y FL(2)1366 333 y FQ(2)e FP(Rep\()p FB(A)p FP(\))h(and)f FO(\031)1974 345 y FL(1)2044 333 y Fs(\026)g FB(B)f FP(=)g FO(\031)2351 345 y FL(2)2421 333 y Fs(\026)h FB(B)p FP(,)118 432 y(then)41 b FO(\031)367 444 y FL(1)449 432 y FP(=)i FO(\031)604 444 y FL(2)642 432 y FP(.)75 b(Indeed,)44 b(the)c(represen)n(tations)f FO(\031)1840 444 y FL(1)1904 432 y FQ(\016)31 b FP(~)-46 b FO(\031)43 b FP(and)d FO(\031)2284 444 y FL(2)2349 432 y FQ(\016)31 b FP(~)-47 b FO(\031)44 b FP(of)130 531 y FJ(^)118 550 y FA(K)19 b FQ(\012)f FA(A)27 b FP(are)g(liftings)h(of)f(the)h(represen)n(tation)e FO(\031)1603 562 y FL(1)1659 550 y FQ(\016)c FP(~)-46 b FO(\031)21 b FQ(\016)d FO(\036)24 b FP(=)f FO(\031)2055 562 y FL(2)2110 550 y FQ(\016)g FP(~)-47 b FO(\031)22 b FQ(\016)c FO(\036)p FP(,)28 b(since)118 650 y FB(B)34 b FP(is)g(the)h(closure)e(of)38 b(^)-46 b FO(\031)s FP(\()p FA(K)24 b FQ(\012)e FA(A)p FP(\))34 b(=)k(~)-47 b FO(\031)t FP(\()p FO(\036)p FP(\()p FA(K)24 b FQ(\012)e FA(A)p FP(\)\).)57 b(By)34 b(the)g(de\014nition)h(of)118 750 y(en)n(v)n(eloping)f FQ(\003)p FP(-algebra,)h(suc)n(h)g(a)g(lifting)i (is)e(unique,)j(hence)d FO(\031)2129 762 y FL(1)2203 750 y FP(=)h FO(\031)2351 762 y FL(2)2388 750 y FP(.)61 b(By)118 849 y(Lemma)32 b(11.)49 b(in)32 b(Section)g(1.1.3,)g FB(B)e FP(=)g FB(A)p FP(.)50 b(By)31 b(the)i(assumption,)f FO(I)c FQ(\012)21 b FO(V)49 b FQ(2)123 949 y FP(^)-47 b FO(\031)s FP(\()p FA(K)26 b FQ(\012)f FA(A)p FP(\))463 919 y FN(0)486 949 y FP(,)41 b(so)c FO(I)32 b FQ(\012)25 b FO(V)59 b FQ(2)40 b FB(B)1085 919 y FN(0)1108 949 y FP(,)h(hence)c FO(I)c FQ(\012)24 b FO(V)59 b FQ(2)40 b FB(A)1836 919 y FN(0)1860 949 y FP(,)g(and)e(consequen)n(tly)-7 b(,)118 1072 y FO(I)26 b FQ(\012)18 b FO(V)42 b FQ(2)27 b FP(~)-46 b FO(\031)s FP(\()524 1053 y FJ(^)513 1072 y FA(K)19 b FQ(\012)f FA(A)p FP(\))762 1042 y FN(0)786 1072 y FP(.)p 2514 1072 4 57 v 2518 1019 50 4 v 2518 1072 V 2567 1072 4 57 v 118 1238 a FR(De\014nition)42 b(13.)k FC(We)38 b(say)h(that)f(a)g FQ(\003)p FC(-algebr)l(a)h FA(B)g FC(majorizes)h(a)e FQ(\003)p FC(-algebr)l(a)118 1337 y FA(A)j FP(\()p FC(and)i(denote)f(it)g(by)g FA(B)j FQ(\037)g FA(A)p FP(\),)f FC(if)f(ther)l(e)f(exists)f(an)h(enveloping)h (alge-)118 1460 y(br)l(a)d FP(\()312 1441 y FJ(^)301 1460 y FA(K)19 b FQ(\012)f FA(A)p FO(;)c(\036)p FP(\))40 b FC(of)h(the)f(algebr)l(a)h FA(K)26 b FQ(\012)f FA(A)p FC(,)42 b(and)e(a)h(unital)e(homomorphism)118 1583 y FO( )12 b FP(:)30 b FA(B)j FQ(\000)-47 b(!)488 1564 y FJ(^)477 1583 y FA(K)18 b FQ(\012)h FA(A)34 b FC(such)h(that)f(the)h (functor)g FO(F)21 b FP(:)44 b(Rep)14 b FA(A)31 b FQ(\000)-46 b(!)32 b FP(Rep)14 b FA(B)35 b FC(de\014ne)l(d)118 1682 y(by)30 b(the)g(fol)t(lowing)i(rule)6 b FP(:)220 1865 y FO(F)12 b FP(\()p FO(\031)s FP(\))24 b(=)j(~)-47 b FO(\031)22 b FQ(\016)c FO( )s(;)184 b FC(for)30 b(any)g FO(\031)d FQ(2)c FP(Rep)14 b FA(A)p FO(;)208 1989 y(F)e FP(\()p FO(A)p FP(\))24 b(=)e FO(I)k FQ(\012)18 b FO(A;)184 b FC(for)30 b(any)g(op)l(er)l(ator)h FO(A)f FC(intertwining)g FO(\031)2149 2001 y FL(1)2216 1989 y FC(and)g FO(\031)2424 2001 y FL(2)2462 1989 y FO(;)118 2172 y FC(is)g(ful)t(l.)118 2338 y(R)l(emark)36 b FP(49)p FC(.)i FP(It)25 b(follo)n(ws)e(from)h (the)h(pro)r(of)f(of)h(the)f(previous)g(lemma)g(that,)i(in)118 2437 y(order)h(to)i(v)n(erify)e(that)i FO(F)41 b FP(is)28 b(full,)i(it)f(is)f(su\016cien)n(t)h(to)f(v)n(erify)g(that,)h(for)f(ev) n(ery)118 2537 y FO(\031)43 b FQ(2)e FP(Rep\()p FA(A)p FP(\))d(in)g FO(L)p FP(\()p FO(H)7 b FP(\))38 b(and)f(an)n(y)g(op)r (erator)f FO(A)j FP(in)f FO(L)p FP(\()p FO(H)7 b FP(\),)40 b(the)e(inclusion)118 2637 y FO(I)26 b FQ(\012)18 b FO(A)23 b FQ(2)g FO(F)12 b FP(\()p FO(\031)s FP(\)\()p FA(B)p FP(\))742 2607 y FN(0)795 2637 y FP(implies)28 b FO(I)d FQ(\012)18 b FO(A)24 b FQ(2)f FO(\031)s FP(\()p FA(A)p FP(\))1559 2607 y FN(0)1583 2637 y FP(.)243 2769 y(F)-7 b(or)19 b(some)h(commen)n(ts)g(on)g(De\014nition)h(13,)g(see)f(also)g (Section)g(3.2.3)f(b)r(elo)n(w.)118 2919 y FR(5.)38 b FP(F)-7 b(or)28 b(a)g(class)f(of)35 b FO(C)813 2889 y FN(\003)851 2919 y FP(-algebras,)26 b(it)j(is)f(p)r(ossible)g(to)g (only)g(consider)f(the)h(ho-)118 3018 y(momorphisms)21 b FO( )12 b FP(:)28 b FA(B)c FQ(\000)-49 b(!)23 b FA(K)1055 3005 y FP(^)1042 3018 y FQ(\012)8 b FA(A)21 b FP(in)i(De\014nition)f (13,)h(since)e(for)h(a)g FO(C)2246 2988 y FN(\003)2284 3018 y FP(-algebra)118 3118 y FA(A)p FP(,)27 b FA(K)313 3105 y FP(^)302 3118 y FQ(\012)18 b FA(A)27 b FP(is)g(also)f(a)i FO(C)856 3088 y FN(\003)894 3118 y FP(-algebra,)d(and)j(the)g(unique)f (en)n(v)n(eloping)f(algebra)g(of)118 3218 y(a)36 b FO(C)261 3188 y FN(\003)300 3218 y FP(-algebra)e(is)i(the)h(algebra)e(itself)i (\()p FA(K)1505 3205 y FP(^)1493 3218 y FQ(\012)24 b FA(A)36 b FP(is)h(the)g(completion)f(of)g(the)118 3317 y(algebraic)31 b(tensor)g(pro)r(duct)h(in)h(the)g FO(C)1359 3287 y FN(\003)1397 3317 y FP(-prenorm.)50 b(It)32 b(is)g(kno)n(wn)g (to)g(b)r(e)h(in-)118 3417 y(dep)r(enden)n(t)38 b(of)f(the)h(c)n(hoice) f(of)g(the)h(prenorm\).)65 b(Moreo)n(v)n(er,)37 b(w)n(e)g(ha)n(v)n(e)f (the)118 3517 y(follo)n(wing)27 b(fact.)118 3682 y FR(Lemma)44 b(14.)k FC(L)l(et)40 b FO( )12 b FP(:)32 b FA(B)44 b FQ(\000)-46 b(!)43 b FA(A)e FC(b)l(e)g(a)g(morphism)i(of)59 b FO(C)2066 3652 y FN(\003)2105 3682 y FC(-algebr)l(as.)73 b(If)118 3782 y FO(\031)s FP(\()p FO( )s FP(\()p FA(B)p FP(\)\))426 3752 y FN(0)474 3782 y FP(=)23 b FO(\031)s FP(\()p FA(A)p FP(\))736 3752 y FN(0)789 3782 y FC(for)31 b(any)f FO(\031)d FQ(2)c FP(Rep)14 b FA(A)p FC(,)30 b(then)f FO( )k FC(is)d(a)g(surje)l(ction.)118 3948 y(Pr)l(o)l(of.)43 b FP(Denote)i FA(B)753 3960 y FL(1)843 3948 y FP(=)51 b FO( )s FP(\()p FA(B)p FP(\).)89 b(Assuming)45 b(that)g FA(B)1934 3960 y FL(1)2024 3948 y FQ(6)p FP(=)51 b FA(A)p FP(,)e(w)n(e)44 b(ha)n(v)n(e)118 4048 y(that)c(there)g(is)g(a)g (non-zero)e(con)n(tin)n(uous)h(functional)h FO(f)53 b FQ(2)44 b FA(A)2145 4018 y FN(\003)2223 4048 y FP(suc)n(h)c(that)118 4147 y FO(f)9 b FP(\()p FO(b)p FP(\))48 b(=)g(0)42 b(for)g(all)h FO(b)k FQ(2)i FA(B)1046 4159 y FL(1)1084 4147 y FP(.)82 b(Then)43 b(one)f(can)g(\014nd)h(\(see)g(the)g(pro)r(of)f(of)p eop %%Page: 207 211 207 210 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(207)118 333 y(Lemma)36 b(3.9)f(in)h([156)n(]\))h(a)e(represen)n(tation)f FO(\031)12 b FP(:)31 b FA(A)36 b FQ(\000)-48 b(!)37 b FO(L)p FP(\()p FO(H)7 b FP(\))36 b(and)f(a)h(\014nite-)118 432 y(dimensional)c(op)r (erator)f FO(\032)g FQ(2)g FO(L)p FP(\()p FO(H)7 b FP(\))32 b(suc)n(h)h(that)f FO(f)9 b FP(\()p FO(a)p FP(\))31 b(=)g(T)-7 b(r\()p FO(\032\031)s FP(\()p FO(a)p FP(\)\))33 b(for)f(all)118 532 y FO(a)h FQ(2)h FA(A)p FP(.)55 b(Since)34 b FO(\031)s FP(\()p FO( )s FP(\()p FA(B)p FP(\)\))953 502 y FN(0)1011 532 y FP(=)f FO(\031)s FP(\()p FA(A)p FP(\))1283 502 y FN(0)1307 532 y FP(,)i(w)n(e)f(ha)n(v)n(e)e FO(\031)s FP(\()p FA(B)1846 544 y FL(1)1885 532 y FP(\))1917 502 y FN(00)1993 532 y FP(=)g FO(\031)s FP(\()p FA(A)p FP(\))2264 502 y FN(0)q(0)2307 532 y FP(.)56 b(Then)118 659 y FO(\031)s FP(\()p FA(A)p FP(\))33 b FQ(\022)p 421 587 226 4 v 31 w FO(\031)s FP(\()p FA(B)576 671 y FL(1)615 659 y FP(\))647 601 y FM(W)9 b(O)r(T)856 659 y FP(b)n(y)33 b(the)g(v)n(on)g(Neumann)g (densit)n(y)g(theorem.)53 b(Hence,)118 759 y(for)32 b(ev)n(ery)f FO(a)f FQ(2)i FA(A)f FP(there)h(is)h(a)e(generalized)g(sequence)h FO(b)1924 771 y FM(\013)2002 759 y FQ(2)f FA(B)2161 771 y FL(1)2231 759 y FP(suc)n(h)h(that)118 858 y FO(\031)s FP(\()p FO(a)p FP(\))24 b(=)f(w-lim)13 b FO(\031)s FP(\()p FO(b)722 870 y FM(\013)770 858 y FP(\).)37 b(So)646 1020 y FO(f)9 b FP(\()p FO(a)p FP(\))23 b(=)g(T)-7 b(r)o(\()p FO(\032)14 b FP(w-lim)g FO(\031)s FP(\()p FO(b)1424 1032 y FM(\013)1472 1020 y FP(\)\))827 1145 y(=)23 b(lim)14 b(T)-7 b(r)o(\()p FO(\032\031)s FP(\()p FO(b)1322 1157 y FM(\013)1370 1145 y FP(\)\))24 b(=)e(lim)15 b FO(f)9 b FP(\()p FO(b)1793 1157 y FM(\013)1839 1145 y FP(\))24 b(=)e(0)p FO(:)118 1307 y FP(This)28 b(con)n(tradiction)e(pro)n(v)n(es) g(that)i FA(B)1323 1319 y FL(1)1383 1307 y FP(=)23 b FA(A)p FP(.)p 2514 1307 4 57 v 2518 1254 50 4 v 2518 1307 V 2567 1307 4 57 v 118 1469 a FR(Theorem)g(50.)35 b FC(A)24 b FO(C)825 1439 y FN(\003)864 1469 y FC(-algebr)l(a)h FA(B)f FC(majorizes)i(a)e FO(C)1766 1439 y FN(\003)1805 1469 y FC(-algebr)l(a)h FA(A)f FC(if)h(and)f(only)118 1569 y(if)48 b FA(B)30 b FC(c)l(ontains)g(an)g(ide)l(al)40 b FA(I)30 b FC(such)f(that)38 b FA(B)p FO(=)p FA(I)23 b FQ(')g FA(K)c FQ(\012)f FA(A)p FC(.)118 1718 y(Pr)l(o)l(of.)43 b FP(Let)28 b(us)g(note)f(that)h(an)n(y)f(represen)n(tation)f FO(\031)31 b FP(of)d FA(K)1956 1705 y FP(^)1945 1718 y FQ(\012)18 b FA(A)27 b FP(has)g(the)h(form)118 1818 y FO(\031)e FP(=)d FO(U)9 b FP(id)14 b FQ(\012)g FO(\031)554 1830 y FL(0)592 1818 y FO(U)658 1788 y FN(\003)721 1818 y FP(for)25 b(some)g FO(\031)1099 1830 y FL(0)1160 1818 y FQ(2)e FP(Rep)q(\()p FA(A)p FP(\))i(and)h(unitary)f FO(U)9 b FP(.)36 b(Indeed,)26 b(since)118 1918 y FO(C)183 1887 y FN(\003)222 1918 y FP(-algebra)g FA(K)j FP(is)f(of)h(t)n(yp)r(e) f FO(I)36 b FP(and)28 b(an)n(y)g(irreducible)g(represen)n(tation)f(of)h FA(K)h FP(is)118 2017 y(unitary)g(equiv)-5 b(alen)n(t)28 b(to)h(id)q(,)g(the)g(ab)r(o)n(v)n(e)f(statemen)n(t)h(is)g(true)g(for)g (irreducible)118 2117 y FO(\031)k FP(\(see,)c(e.g.,)g([274)o(]\).)41 b(An)30 b(arbitrary)d(represen)n(tation)g(can)i(b)r(e)g(decomp)r(osed) 118 2216 y(as)f FO(\031)f FP(=)384 2154 y Fz(L)477 2241 y FM(\013)538 2216 y FO(\031)585 2228 y FM(\013)632 2216 y FP(,)i(where)f FO(\031)972 2228 y FM(\013)1048 2216 y FP(is)g(irreducible,)g(and)h(so)e FO(\031)1879 2228 y FM(\013)1951 2216 y FP(=)d FO(U)2097 2228 y FM(\013)2144 2216 y FP(id)19 b FQ(\012)g FO(\031)2363 2228 y FM(\013)2410 2233 y FL(0)2448 2216 y FO(U)2514 2186 y FN(\003)2505 2237 y FM(\013)2552 2216 y FP(.)118 2316 y(Setting)38 b FO(\031)461 2328 y FL(0)538 2316 y FP(=)642 2254 y Fz(L)734 2341 y FM(\013)795 2316 y FO(\031)842 2328 y FM(\013)890 2333 y FL(0)965 2316 y FP(and)f FO(U)48 b FP(=)1345 2254 y Fz(L)1437 2341 y FM(\013)1498 2316 y FO(U)1555 2328 y FM(\013)1640 2316 y FP(w)n(e)37 b(get)g FO(\031)43 b FP(=)c FO(U)9 b FP(id)25 b FQ(\012)f FO(\031)2410 2328 y FL(0)2448 2316 y FO(U)2514 2286 y FN(\003)2552 2316 y FP(.)118 2416 y(Let)30 b FO(F)322 2428 y FM( )381 2416 y FP(:)43 b(Rep\()p FA(A)p FP(\))26 b FQ(\000)-48 b(!)26 b FP(Rep\()p FA(B)p FP(\))k(b)r(e)g(full.)43 b(W)-7 b(e)30 b(will)g(sho)n(w)e(that)i(the)g(functor)118 2515 y FO(F)171 2527 y FM( )231 2515 y FP(:)43 b(Rep\()p FA(K)22 b FQ(\012)f FA(A)p FP(\))29 b FQ(\000)-48 b(!)29 b FP(Rep)q(\()p FA(B)p FP(\))j(is)g(also)e(full.)50 b(Indeed,)33 b(tak)n(e)e(an)g (arbitrary)118 2615 y FO(\031)44 b FQ(2)d FP(Rep\()p FA(K)27 b FQ(\012)e FA(A)p FP(\).)68 b(Then)39 b FO(\031)44 b FP(=)c FO(U)9 b FP(\(id)26 b FQ(\012)f FO(\031)1590 2627 y FL(0)1628 2615 y FP(\))p FO(U)1726 2585 y FN(\003)1764 2615 y FP(.)68 b(Let)39 b FO(V)59 b FQ(2)42 b FO(\031)s FP(\()p FO( )s FP(\()p FA(B)p FP(\)\))2527 2585 y FN(0)2552 2615 y FP(,)118 2715 y(then)e FO(U)385 2684 y FN(\003)423 2715 y FO(V)19 b(U)51 b FQ(2)43 b FP(\(id)26 b FQ(\012)g FO(\031)961 2727 y FL(0)999 2715 y FP(\()p FO( )s FP(\()p FA(B)p FP(\)\)\))1289 2684 y FN(0)1314 2715 y FP(,)42 b(and)d(so)g FO(U)1732 2684 y FN(\003)1769 2715 y FO(V)19 b(U)52 b FP(=)42 b FO(I)33 b FQ(\012)25 b FO(V)2259 2727 y FL(0)2297 2715 y FP(,)42 b(where)118 2814 y FO(V)166 2826 y FL(0)235 2814 y FQ(2)31 b FO(\031)368 2826 y FL(0)406 2814 y FP(\()p FO( )s FP(\()p FA(B)p FP(\)\))664 2784 y FN(0)689 2814 y FP(.)51 b(Then)32 b FO(V)1032 2826 y FL(0)1101 2814 y FQ(2)f FO(\031)1234 2826 y FL(0)1272 2814 y FP(\()p FA(A)p FP(\))1396 2784 y FN(0)1419 2814 y FP(,)j(since)e FO(F)1737 2826 y FM( )1797 2814 y FP(:)43 b(Rep\()p FA(A)p FP(\))31 b FQ(\000)-48 b(!)31 b FP(Rep\()p FA(B)p FP(\))118 2914 y(is)g(full.)48 b(This)30 b(pro)n(v)n(es)f(that)j FO(U)1095 2884 y FN(\003)1133 2914 y FO(V)18 b(U)38 b FQ(2)29 b FP(\(id)21 b FQ(\012)f FO(\031)1632 2926 y FL(0)1670 2914 y FP(\()p FA(A)p FP(\)\))1826 2884 y FN(0)1850 2914 y FP(,)31 b(and)g(so)g FO(V)47 b FQ(2)29 b FO(\031)s FP(\()p FA(A)p FP(\))2528 2884 y FN(0)2552 2914 y FP(.)118 3013 y(No)n(w)e(w)n(e)g(can)h(apply)f(the)h(previous)e(lemma)i(to)f (\014nish)h(the)g(pro)r(of.)p 2514 3013 V 2518 2961 50 4 v 2518 3013 V 2567 3013 4 57 v 243 3175 a(In)k(what)g(follo)n(ws,)g (the)g FO(C)1076 3145 y FN(\003)1114 3175 y FP(-algebra)e FA(K)22 b FQ(\012)f FA(A)31 b FP(will)i(also)d(b)r(e)j(o)r(ccasionally) 118 3275 y(denoted)28 b(b)n(y)f FO(M)628 3287 y FM(n)673 3275 y FP(\()p FA(A)p FP(\),)h FO(n)23 b FQ(2)g FJ(N)28 b FQ([)19 b(f1g)p FP(.)118 3416 y FR(6.)51 b FP(No)n(w)31 b(consider)h(the)g(general)f(case)h(of)g(arbitrary)e FQ(\003)p FP(-algebras.)49 b(The)32 b(fol-)118 3516 y(lo)n(wing)27 b(lemma)g(is)h(a)f(simple)h(mo)r(di\014cation)f(of)h(Theorem)e(6.3.5)h (in)h([178)n(].)118 3665 y FR(Lemma)19 b(15.)33 b FC(L)l(et)c FA(A)21 b FC(b)l(e)h(a)g(unital)30 b FQ(\003)p FC(-algebr)l(a)22 b(and)g(let)29 b FB(B)22 b FC(b)l(e)g(any)g FO(C)2230 3635 y FN(\003)2268 3665 y FC(-algebr)l(a.)118 3765 y(L)l(et)42 b FO(\031)35 b FQ(2)e FP(Rep\()p FB(B)22 b FQ(\012)g FA(A)p FP(\))35 b FC(b)l(e)f(a)h(r)l(epr)l(esentation)g(in)g FO(L)p FP(\()p FO(H)7 b FP(\))p FC(.)53 b(Then)36 b(ther)l(e)e(ar)l(e) 118 3865 y(r)l(epr)l(esentations)f FO(\036)9 b FP(:)29 b FA(A)f FQ(\000)-46 b(!)28 b FO(L)p FP(\()p FO(H)7 b FP(\))32 b FC(and)h FO( )12 b FP(:)29 b FB(B)g FQ(\000)-46 b(!)28 b FO(L)p FP(\()p FO(H)7 b FP(\))32 b FC(such)h(that)f(for)i(al)t (l)118 3964 y FO(a)23 b FQ(2)g FA(A)30 b FC(and)g FO(b)23 b FQ(2)g FB(B)p FC(,)761 4127 y FO(\031)s FP(\()p FO(b)c FQ(\012)f FO(a)p FP(\))23 b(=)g FO( )s FP(\()p FO(b)p FP(\))p FO(\036)p FP(\()p FO(a)p FP(\))h(=)f FO(\036)p FP(\()p FO(a)p FP(\))p FO( )s FP(\()p FO(b)p FP(\))p FO(:)p eop %%Page: 208 212 208 211 bop 118 100 a FP(208)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FC(Pr)l(o)l(of.)43 b FP(Let)27 b FO(a)570 303 y FN(0)616 333 y FQ(2)c FA(A)j FP(and)g(de\014ne)h FO( )1233 345 y FM(a)1269 328 y Fx(0)1305 333 y FP(:)g FB(B)d FQ(\000)-49 b(!)23 b FO(L)p FP(\()p FO(H)7 b FP(\),)27 b FO(b)22 b FQ(7!)h FO(\031)s FP(\()p FO(b)16 b FQ(\012)g FO(a)2234 303 y FN(0)2257 333 y FP(\).)37 b(Let)26 b(us)118 432 y(sho)n(w)31 b(that)h FO( )566 444 y FM(a)602 428 y Fx(0)660 432 y FP(is)g(con)n(tin)n(uous)e(for)h(all)h FO(a)1462 402 y FN(0)1515 432 y FQ(2)e FA(A)p FP(.)48 b(Since)32 b FO( )2006 444 y FM(a)2042 428 y Fx(0)2100 432 y FP(is)g(a)f(mapping) 118 532 y(b)r(et)n(w)n(een)24 b(Banac)n(h)e(spaces,)i(w)n(e)f(can)h (apply)f(the)i(closed)e(graph)f(theorem.)35 b(W)-7 b(e)118 632 y(need)34 b(to)f(sho)n(w)g(that)g(if)h FO(b)940 644 y FM(n)1018 632 y FQ(!)f FP(0)g(in)h FB(B)f FP(and)h FO( )1630 644 y FM(a)1666 627 y Fx(0)1692 632 y FP(\()p FO(b)1760 644 y FM(n)1805 632 y FP(\))f FQ(!)g FO(c)h FP(in)f FO(L)p FP(\()p FO(H)7 b FP(\),)35 b(then)118 731 y FO(c)30 b FP(=)f(0.)49 b(Replacing)31 b FO(b)816 743 y FM(n)893 731 y FP(with)h FO(b)1122 701 y FN(\003)1122 752 y FM(n)1166 731 y FO(b)1202 743 y FM(n)1247 731 y FP(,)h FO(a)1347 701 y FN(0)1402 731 y FP(with)f(\()p FO(a)1671 701 y FN(0)1695 731 y FP(\))1727 701 y FN(\003)1765 731 y FO(a)1809 701 y FN(0)1832 731 y FP(,)h(and)f FO(c)f FP(with)h FO(c)2350 701 y FN(\003)2389 731 y FO(c)p FP(,)g(w)n(e)118 831 y(can)27 b(assume)g(that)h FO(c)23 b FQ(\025)g FP(0.)37 b(Let)28 b FO(\034)37 b FP(b)r(e)28 b(an)g(arbitrary)d(p)r(ositiv)n(e)i (functional)h(on)118 930 y FO(L)p FP(\()p FO(H)7 b FP(\).)37 b(Then)28 b(the)g(functional)762 1112 y FO(\032)9 b FP(:)28 b FB(B)23 b FQ(\000)-48 b(!)23 b FJ(C)15 b FO(;)186 b(b)22 b FQ(7!)h FO(\034)9 b FP(\()p FO(\031)s FP(\()p FO(b)20 b FQ(\012)e FO(a)1843 1078 y FN(0)1866 1112 y FP(\)\))118 1294 y(is)26 b(p)r(ositiv)n(e)f(and,)h(therefore,)f(con)n(tin)n(uous.) 35 b(Hence)26 b FO(\034)9 b FP(\()p FO(c)p FP(\))24 b(=)e(lim)15 b FO(\034)9 b FP(\()p FO(b)2249 1306 y FM(n)2309 1294 y FQ(\012)14 b FO(a)2432 1264 y FN(0)2455 1294 y FP(\))23 b(=)118 1394 y(lim)14 b FO(\032)p FP(\()p FO(b)358 1406 y FM(n)403 1394 y FP(\))37 b(=)f(0,)i(since)d(lim)14 b FO(b)1052 1406 y FM(n)1134 1394 y FP(=)36 b(0.)60 b(So)36 b FO(c)g FP(=)g(0,)i(whic)n(h)d(pro)n(v)n(es)f(that)i FO( )2512 1406 y FM(a)2548 1389 y Fx(0)118 1493 y FP(is)d(con)n(tin)n (uous.)53 b(W)-7 b(e)34 b(can)f(assume)f(that)i FO(\031)j FP(is)c(non-degenerate.)52 b(Consider)118 1593 y(the)38 b(linear)g(subset)f FB(L)k FP(=)f FO(\031)s FP(\()p FB(B)26 b FQ(\012)f FA(A)p FP(\))p FO(H)7 b FP(.)67 b(Ev)n(ery)36 b FO(z)44 b FQ(2)d FB(L)d FP(can)f(b)r(e)i(written)118 1692 y(as)d FO(z)41 b FP(=)412 1630 y Fz(P)500 1651 y FM(n)500 1717 y(i)p FL(=1)625 1692 y FO(\031)s FP(\()p FO(b)743 1704 y FM(i)796 1692 y FQ(\012)24 b FO(a)929 1704 y FM(i)956 1692 y FP(\)\()p FO(x)1067 1704 y FM(i)1096 1692 y FP(\),)39 b(where)d FO(b)1475 1704 y FM(i)1541 1692 y FQ(2)i FB(B)p FP(,)h FO(a)1803 1704 y FM(i)1869 1692 y FQ(2)g FA(A)p FP(,)f FO(x)2131 1704 y FM(i)2197 1692 y FQ(2)h FO(H)7 b FP(.)64 b(Let)118 1792 y FO(z)48 b FP(=)315 1730 y Fz(P)402 1750 y FM(m)402 1817 y(j)s FL(=1)535 1792 y FO(\031)s FP(\()p FO(b)653 1762 y FN(0)653 1814 y FM(i)708 1792 y FQ(\012)27 b FO(a)844 1762 y FN(0)844 1814 y FM(i)872 1792 y FP(\)\()p FO(x)983 1762 y FN(0)983 1814 y FM(i)1011 1792 y FP(\))41 b(b)r(e)h(another)d(presen)n(tation.) 76 b(Let)40 b FO(v)2275 1804 y FM(\026)2361 1792 y FP(b)r(e)h(an)118 1892 y(appro)n(ximate)26 b(iden)n(tit)n(y)i(for)f FB(B)h FP(and)f FO(a)c FQ(2)g FA(A)p FP(.)37 b(Then)245 2135 y FO(\031)s FP(\()p FO(v)367 2147 y FM(\026)431 2135 y FQ(\012)18 b FO(a)p FP(\)\()p FO(z)t FP(\))23 b(=)847 2032 y FM(n)808 2056 y Fz(X)814 2233 y FM(i)p FL(=1)941 2135 y FO(\031)s FP(\()p FO(v)1063 2147 y FM(\026)1109 2135 y FO(b)1145 2147 y FM(i)1190 2135 y FQ(\012)18 b FO(aa)1361 2147 y FM(i)1389 2135 y FP(\)\()p FO(x)1500 2147 y FM(i)1529 2135 y FP(\))23 b(=)1702 2032 y FM(m)1672 2056 y Fz(X)1674 2233 y FM(j)s FL(=1)1805 2135 y FO(\031)s FP(\()p FO(v)1927 2147 y FM(\026)1973 2135 y FO(b)2009 2101 y FN(0)2009 2156 y FM(i)2054 2135 y FQ(\012)18 b FO(aa)2225 2101 y FN(0)2225 2156 y FM(i)2253 2135 y FP(\)\()p FO(x)2364 2101 y FN(0)2364 2156 y FM(i)2392 2135 y FP(\))p FO(:)118 2401 y FP(Since)28 b FO( )389 2413 y FM(a)457 2401 y FP(is)f(con)n(tin)n(uous,)g(w)n(e)g(can)g(pass)g(to)h(the)g (limit)265 2645 y(lim)303 2694 y FM(\026)394 2645 y FO(\031)s FP(\()p FO(v)516 2657 y FM(\026)580 2645 y FQ(\012)18 b FO(a)p FP(\)\()p FO(z)t FP(\))23 b(=)996 2541 y FM(n)957 2566 y Fz(X)963 2743 y FM(i)p FL(=1)1091 2645 y FO(\031)s FP(\()p FO(b)1209 2657 y FM(i)1255 2645 y FQ(\012)18 b FO(aa)1426 2657 y FM(i)1453 2645 y FP(\)\()p FO(x)1564 2657 y FM(i)1593 2645 y FP(\))23 b(=)1767 2541 y FM(m)1736 2566 y Fz(X)1739 2743 y FM(j)s FL(=1)1870 2645 y FO(\031)s FP(\()p FO(b)1988 2610 y FN(0)1988 2665 y FM(i)2034 2645 y FQ(\012)18 b FO(aa)2205 2610 y FN(0)2205 2665 y FM(i)2233 2645 y FP(\)\()p FO(x)2344 2610 y FN(0)2344 2665 y FM(i)2372 2645 y FP(\))p FO(:)118 2919 y FP(So)25 b(the)h(mapping)f FO(\036)p FP(\()p FO(a)p FP(\))9 b(:)29 b FB(L)23 b FQ(\000)-48 b(!)23 b FB(L)p FP(,)j FO(z)g FQ(7!)1413 2857 y Fz(P)1501 2878 y FM(n)1501 2944 y(i)p FL(=1)1627 2919 y FO(\031)s FP(\()p FO(b)1745 2931 y FM(i)1786 2919 y FQ(\012)14 b FO(aa)1953 2931 y FM(i)1980 2919 y FP(\)\()p FO(x)2091 2931 y FM(i)2120 2919 y FP(\))25 b(is)g(correctly)118 3019 y(de\014ned.)55 b(Since)34 b FO(\036)p FP(\()p FO(a)p FP(\))g(=)e(lim)1081 3031 y FM(\026)1139 3019 y FO(\031)s FP(\()p FO(v)1261 3031 y FM(\026)1329 3019 y FQ(\012)22 b FO(a)p FP(\)\()p FO(z)t FP(\),)35 b(it)f(is)f(ob)n(vious)g(that)g FO(\036)p FP(\()p FO(a)p FP(\))i(is)118 3119 y(linear.)45 b(Since)31 b FO( )664 3131 y FM(a)735 3119 y FP(is)g(con)n(tin)n(uous,) f(there)h(is)f FO(M)37 b FP(=)28 b FO(M)9 b FP(\()p FO(a)p FP(\))28 b FQ(2)g FJ(R)2140 3131 y FL(+)2232 3119 y FP(suc)n(h)j(that) 118 3218 y FQ(k)p FO(\031)s FP(\()p FO(b)22 b FQ(\012)f FO(a)p FP(\))p FQ(k)31 b(\024)g FO(M)9 b FQ(k)p FO(b)p FQ(k)p FP(,)32 b FO(b)e FQ(2)i FB(B)p FP(.)52 b(So)32 b FO(\036)p FP(\()p FO(a)p FP(\))i(is)e(a)g(b)r(ounded)h(op)r(erator.) 50 b(Since)118 3318 y FO(\031)32 b FP(is)c(non-degenerate,)e FB(L)i FP(is)g(dense)g(in)h FO(H)35 b FP(and)27 b FO(\036)p FP(\()p FO(a)p FP(\))j(is)d(uniquely)i(extended)118 3418 y(to)f(a)g(b)r(ounded)g(op)r(erator)e(on)i FO(H)35 b FP(\(whic)n(h)28 b(is)g(also)f(denoted)h(b)n(y)f FO(\036)p FP(\()p FO(a)p FP(\)\).)40 b(Then)118 3517 y FO(\036)9 b FP(:)29 b FA(A)22 b FQ(\000)-48 b(!)23 b FO(L)p FP(\()p FO(H)7 b FP(\))24 b(and)h FO( )h FP(=)d FO( )1035 3529 y FL(1)1097 3517 y FP(\()p FO(a)1173 3487 y FN(0)1219 3517 y FP(=)g(1\))h(are)g(the)h(required)f(represen)n(tations.)118 3617 y(Belo)n(w)j(w)n(e)g(use)g(the)h(notations)f FO(\036)d FP(=)e FO(\031)1344 3629 y Ft(A)1425 3617 y FP(and)27 b FO( )g FP(=)22 b FO(\031)1801 3629 y Fj(B)p 2514 3617 4 57 v 2518 3564 50 4 v 2518 3617 V 2567 3617 4 57 v 118 3783 a FR(Prop)s(osition)30 b(65.)41 b FC(L)l(et)29 b FA(A)g FC(b)l(e)h(a)g(unital)g FQ(\003)p FC(-algebr)l(a.)39 b(Then)220 3948 y FP(1.)i FC(for)27 b(every)g FO(\031)f FQ(2)e FP(Rep\()p FA(K)11 b FQ(\012)g FA(A)p FP(\))27 b FC(in)f FO(L)p FP(\()p FO(H)7 b FP(\))26 b FC(ther)l(e)h(ar)l(e)f (unitary)h FO(U)32 b FQ(2)23 b FO(L)p FP(\()p FO(H)7 b FP(\))326 4048 y FC(and)38 b(a)g(r)l(epr)l(esentation)h FO(\031)1172 4060 y FL(0)1247 4048 y FQ(2)f FP(Rep\()p FA(A)p FP(\))g FC(in)g FO(L)p FP(\()p FO(H)1914 4060 y FL(1)1952 4048 y FP(\))g FC(such)g(that)g FO(H)44 b FP(=)326 4147 y FO(H)395 4159 y FL(0)450 4147 y FQ(\012)18 b FO(H)602 4159 y FL(1)669 4147 y FC(and)31 b FO(U)897 4117 y FN(\003)935 4147 y FO(\031)s(U)h FP(=)22 b(id)d FQ(\012)f FO(\031)1379 4159 y FL(0)1416 4147 y FC(.)p eop %%Page: 209 213 209 212 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(209)220 333 y(2.)41 b(pro-)n FO(C)538 303 y FN(\003)577 333 y FP(\()p FA(K)25 b FQ(\012)e FA(A)p FP(\))36 b FQ(')g FA(K)1097 320 y FP(^)1086 333 y FQ(\012)23 b FP(pro-)o FO(C)1387 303 y FN(\003)1425 333 y FP(\()p FA(A)p FP(\))p FC(,)40 b(wher)l(e)e FA(K)1946 320 y FP(^)1935 333 y FQ(\012)23 b FP(pro-)o FO(C)2236 303 y FN(\003)2275 333 y FP(\()p FA(A)p FP(\))37 b FC(is)g(a)326 432 y(unique,sinc)l(e)28 b FA(K)h FC(is)f(nucle)l(ar,)h FP(pro-)o FO(C)1495 402 y FN(\003)1534 432 y FC(-algebr)l(a)g(which)h (is)f(the)f(c)l(omple-)326 532 y(tion)i(of)g(the)g(algebr)l(aic)i (tensor)d(pr)l(o)l(duct)38 b FP(\()p FC(se)l(e)e FP([208)o(]\))p FC(.)118 707 y(Pr)l(o)l(of.)43 b FP(1.)58 b(By)34 b(the)h(previous)e (lemma,)k FO(\031)h FP(=)c FO(\031)1656 719 y Ft(K)1729 707 y FQ(\012)23 b FO(\031)1864 719 y Ft(A)1917 707 y FP(.)57 b(Denote)35 b(b)n(y)g FB(A)f FP(=)p 118 734 225 4 v 118 807 a FO(\031)165 819 y Ft(A)218 807 y FP(\()p FA(A)p FP(\))24 b(the)f FO(C)569 776 y FN(\003)608 807 y FP(-subalgebra)e(in)i FO(L)p FP(\()p FO(H)7 b FP(\))23 b(generated)g(b)n(y)g(the)g(range)f(of)h FO(\031)2342 819 y Ft(A)2395 807 y FP(.)36 b(Let)118 906 y FO(j)14 b FP(:)28 b FB(A)23 b FQ(\000)-48 b(!)23 b FO(L)p FP(\()p FO(H)7 b FP(\))23 b(denote)h(the)g(natural)f(em)n(b)r(edding.)36 b(If)24 b FO(\031)j FQ(6)p FP(=)22 b(0,)i(then)h FO(\031)2350 918 y Ft(K)2423 906 y FQ(6)p FP(=)d(0,)118 1006 y(and)d(b)n(y)f(the)h (simplicit)n(y)g(of)f FA(K)h FP(w)n(e)g(ha)n(v)n(e)e FO(\031)1383 1018 y Ft(K)1433 1006 y FP(\()p FA(K)p FP(\))24 b FQ(')f FA(K)p FP(.)34 b(The)19 b(follo)n(wing)e(diagram)118 1105 y(is)28 b(comm)n(utativ)n(e.)822 1300 y FA(K)19 b FQ(\012)f FA(A)611 b(K)19 b FQ(\012)f FB(A)p 1063 1276 562 4 v 1542 1274 a Fu(-)1208 1245 y FP(id)h FQ(\012)f FO(\031)1426 1257 y Ft(A)1247 1710 y FO(L)p FP(\()p FO(H)7 b FP(\))1036 1499 y FO(\031)988 1415 y Fu(@)1071 1498 y(@)1154 1581 y(@)1204 1631 y(@)-83 b(R)1604 1501 y FO(\031)1651 1513 y Ft(K)1719 1501 y FQ(\012)18 b FO(j)1619 1415 y Fu(\000)1536 1498 y(\000)1453 1581 y(\000)1403 1631 y(\000)-83 b(\011)118 1859 y FP(Since)29 b FA(K)f FP(is)h(of)f(t)n(yp)r(e)g FO(I)7 b FP(,)29 b(the)g(represen)n(tation)d FO(\031)1615 1871 y Ft(K)1684 1859 y FQ(\012)18 b FO(j)30 b FP(=)23 b FO(U)1985 1829 y FN(\003)2023 1859 y FP(id)c FQ(\012)k FP(^)-46 b FO(\031)s(U)9 b FP(,)28 b(where)123 1959 y(^)-47 b FO(\031)39 b FQ(2)d FP(Rep\()p FB(A)p FP(\))f(\(see)g(the)h(pro)r(of) e(of)h(Theorem)f(50\).)59 b(So)34 b FO(\031)39 b FP(=)c FO(U)2183 1929 y FN(\003)2221 1959 y FP(id)23 b FQ(\012)g FO(\031)2448 1971 y FL(0)2486 1959 y FO(U)9 b FP(,)118 2059 y(where)27 b FO(\031)405 2071 y FL(0)466 2059 y FP(=)g(^)-47 b FO(\031)27 b Fs(\026)22 b FA(A)p FP(.)243 2161 y(2.)34 b(T)-7 b(ak)n(e)19 b FO(\031)26 b FQ(2)e FP(Rep\()p FA(K)t FQ(\012)t FA(A)p FP(\).)34 b(Then)21 b FO(H)29 b FP(=)23 b FO(H)1605 2173 y FL(0)1646 2161 y FQ(\012)t FO(H)1784 2173 y FL(1)1841 2161 y FP(and)d FO(\031)27 b FP(=)22 b FO(U)2222 2131 y FN(\003)2260 2161 y FP(id)t FQ(\012)t FO(\031)2449 2173 y FL(0)2486 2161 y FO(U)9 b FP(.)118 2261 y(Denote)37 b(the)h(canonical)e (homomorphism)f(b)n(y)i FO(\036)9 b FP(:)32 b FA(A)38 b FQ(\000)-48 b(!)39 b FP(pro-)n FO(C)2221 2231 y FN(\003)2260 2261 y FP(\()p FA(A)p FP(\).)65 b(By)118 2361 y(the)25 b(de\014nition)f(of)g(an)g(en)n(v)n(eloping)f(algebra,)g(the)i (represen)n(tation)d FO(\031)2268 2373 y FL(0)2330 2361 y FP(admits)118 2460 y(a)36 b(unique)h(lifting)42 b(~)-46 b FO(\031)774 2472 y FL(0)849 2460 y FQ(2)39 b FP(Rep\(pro-)o FO(C)1332 2430 y FN(\003)1371 2460 y FP(\()p FA(A)p FP(\)\))e(in)g FO(L)p FP(\()p FO(H)1828 2472 y FL(1)1865 2460 y FP(\).)65 b(Then)37 b(de\014ne)f(the)118 2560 y(represen)n(tation)43 b(~)-47 b FO(\031)47 b FQ(2)d FP(Rep\()p FA(K)1136 2547 y FP(^)1124 2560 y FQ(\012)26 b FP(pro-)o FO(C)1428 2530 y FN(\003)1466 2560 y FP(\()p FA(A)p FP(\)\))41 b(via)e(the)h(rule)k(~) -46 b FO(\031)s FP(\()p FO(k)30 b FQ(\012)c FO(a)p FP(\))43 b(=)118 2659 y FO(U)184 2629 y FN(\003)222 2659 y FP(id)24 b FQ(\012)29 b FP(~)-47 b FO(\031)451 2671 y FL(0)489 2659 y FO(U)9 b FP(.)62 b(Since)37 b FO(\036)p FP(\()p FA(A)p FP(\))g(is)f(quasi-dense)f(in)h(pro-)p FO(C)1937 2629 y FN(\003)1974 2659 y FP(\()p FA(A)p FP(\),)j(the)e(algebra)118 2759 y FA(K)27 b FQ(\012)e FO(\036)p FP(\()p FA(A)p FP(\))40 b(is)f(quasi-dense)e(in)j FA(K)1252 2746 y FP(^)1240 2759 y FQ(\012)26 b FP(pro-)o FO(C)1544 2729 y FN(\003)1582 2759 y FP(\()p FA(A)p FP(\).)71 b(Th)n(us)39 b(the)g(lifting)44 b(~)-46 b FO(\031)42 b FP(is)118 2859 y(unique.)p 2514 2859 4 57 v 2518 2806 50 4 v 2518 2859 V 2567 2859 4 57 v 243 3049 a(No)n(w)29 b(w)n(e)g(are)g(in)h(a)f(p)r(osition)g(to)h (pro)n(v)n(e)e(the)i(main)g(theorem)f(ab)r(out)g(ma-)118 3148 y(jorization.)118 3323 y FR(Theorem)39 b(51.)44 b FP(1)p FC(.)58 b(If)36 b FP(\()958 3302 y(~)949 3323 y FA(A)p FO(;)14 b(\036)p FP(\))37 b FC(is)f(an)h(enveloping)g FQ(\003)p FC(-algebr)l(a)g(of)73 b FA(A)p FC(,)38 b(then)118 3423 y FP(pro-)o FO(C)331 3393 y FN(\003)369 3423 y FP(\()p FA(A)p FP(\))24 b FQ(')e FP(pro-)o FO(C)817 3393 y FN(\003)855 3423 y FP(\()896 3401 y(~)887 3423 y FA(A)p FP(\))p FC(.)243 3526 y FP(2)p FC(.)37 b FQ(\003)p FC(-A)n(lgebr)l(a)25 b FA(B)i FC(majorizes)h(a)e FQ(\003)p FC(-algebr)l(a)h FA(A)f FC(if)h(and)f(only)h(if)g(ther)l(e)f(exists)118 3625 y(a)h(c)l(ontinuous)e(morphism)j FO( )12 b FP(:)28 b(pro-)o FO(C)1315 3595 y FN(\003)1354 3625 y FP(\()p FA(B)p FP(\))c FQ(\000)-47 b(!)24 b FP(pro-)n FO(C)1852 3595 y FN(\003)1891 3625 y FP(\()p FA(K)11 b FQ(\012)g FA(A)p FP(\))27 b FC(with)g(quasi-)118 3725 y(dense)k(image)37 b FP(\()p FC(such)31 b(that)f(for)h(any)g(r)l(epr)l(esentation)f FO(\031)e FQ(2)c FP(Rep\(pro-)o FO(C)2365 3695 y FN(\003)2404 3725 y FP(\()p FA(K)19 b FQ(\012)118 3825 y FA(A)p FP(\)\))30 b FC(the)g(set)f FO(\031)s FP(\()p FO( )s FP(\(pro-)p FO(C)924 3794 y FN(\003)962 3825 y FP(\()p FA(B)p FP(\)\)\))j FC(is)e(dense)g(in)g FP(Im)14 b FO(\031)s FP(\))p FC(.)243 3927 y FP(3)p FC(.)49 b(If)35 b FP(pro-)n FO(C)663 3897 y FN(\003)702 3927 y FP(\()p FA(A)p FP(\))f FC(is)g(a)g FO(\033)s FC(-)p FO(C)1174 3897 y FN(\003)1212 3927 y FC(-algebr)l(a)41 b FP(\()p FC(for)35 b(example)f(if)69 b FA(A)33 b FC(is)h(\014nitely)118 4027 y(gener)l(ate)l(d)9 b FP(\))p FC(,)46 b(then)41 b FA(B)k FQ(\037)g FA(A)c FC(if)i(and)f(only)g(if)h(ther)l(e)f(exists)f(a)i(c)l(ontinuous)118 4127 y(morphism)31 b FO( )12 b FP(:)28 b(pro-)o FO(C)835 4096 y FN(\003)874 4127 y FP(\()p FA(B)p FP(\))c FQ(\000)-47 b(!)24 b FP(pro-)n FO(C)1372 4096 y FN(\003)1411 4127 y FP(\()p FA(K)19 b FQ(\012)f FA(A)p FP(\))30 b FC(with)g(dense)g (image.)p eop %%Page: 210 214 210 213 bop 118 100 a FP(210)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FC(Pr)l(o)l(of.)43 b FP(1.)34 b(Let)18 b FO( )k FP(denote)c(the)h(canonical)f(morphism)g(form)2015 311 y(~)2006 333 y FA(A)g FP(to)g(pro-)p FO(C)2390 303 y FN(\003)2428 333 y FP(\()2469 311 y(~)2460 333 y FA(A)o FP(\).)118 432 y(F)-7 b(or)30 b(ev)n(ery)f FO(\031)h FQ(2)e FP(Rep\()p FA(A)p FP(\))j(there)f(is)g(a)f(unique)i(lifting)k(~) -47 b FO(\031)31 b FQ(2)d FP(Rep\()2186 411 y(~)2177 432 y FA(A)p FP(\))i(and,)h(b)n(y)118 532 y(the)f(de\014nition)h(of)f (the)g(en)n(v)n(eloping)e(pro-)p FO(C)1501 502 y FN(\003)1539 532 y FP(-algebra,)k(~)-46 b FO(\031)33 b FP(is)d(uniquely)g(lifted)118 632 y(to)h(a)f(represen)n(tation)k(^)-46 b FO(\031)31 b FQ(2)e FP(Rep)q(\(pro-)o FO(C)1394 601 y FN(\003)1432 632 y FP(\()1473 610 y(~)1464 632 y FA(A)p FP(\)\).)47 b(Hence)32 b(\(pro-)p FO(C)2155 601 y FN(\003)2192 632 y FP(\()2233 610 y(~)2224 632 y FA(A)p FP(\))p FO(;)14 b( )24 b FQ(\016)c FO(\036)p FP(\))118 731 y(is)26 b(an)f(en)n(v)n (eloping)f(pro-)p FO(C)933 701 y FN(\003)970 731 y FP(-algebra)g(of)i FA(A)p FP(.)35 b(Then)26 b(pro-)p FO(C)1927 701 y FN(\003)1965 731 y FP(\()p FA(A)p FP(\))d FQ(')f FP(pro-)o FO(C)2412 701 y FN(\003)2451 731 y FP(\()2492 709 y(~)2483 731 y FA(A)p FP(\))118 831 y(b)n(y)27 b(the)h(uniqueness)g(of)f(the)h(pro-) p FO(C)1246 801 y FN(\003)1284 831 y FP(-en)n(v)n(eloping)d(algebra.) 243 949 y(2.)68 b(a\))38 b(Let)g(\()690 930 y FJ(^)679 949 y FA(K)20 b FQ(\012)e FA(A)o FO(;)c(\036)p FP(\))39 b(b)r(e)g(an)f(en)n(v)n(eloping)e FQ(\003)p FP(-algebra)g(of)i FA(K)26 b FQ(\012)f FA(A)p FP(.)68 b(If)118 1072 y FO(\031)26 b FQ(2)e FP(Rep\()457 1053 y FJ(^)446 1072 y FA(K)19 b FQ(\012)f FA(A)p FP(\),)k(then)g FO(\031)k FP(=)d FO(U)13 b FP(~)-46 b FO(\031)1197 1084 y FL(0)1234 1072 y FO(U)1300 1042 y FN(\003)1359 1072 y FP(for)20 b(some)g FO(\031)1727 1084 y FL(0)1787 1072 y FQ(2)k FP(Rep\()p FA(A)p FP(\))d(and)f(unitary) 118 1171 y FO(U)50 b FQ(2)42 b FO(L)p FP(\()p FO(H)7 b FP(\).)70 b(Indeed,)41 b(since)e(the)g(mapping)f(Rep\()p FA(K)26 b FQ(\012)g FA(A)p FP(\))41 b FQ(3)h FO(\024)f FQ(7!)j FP(~)-45 b FO(\024)41 b FQ(2)118 1294 y FP(Rep\()305 1275 y FJ(^)294 1294 y FA(K)19 b FQ(\012)f FA(A)p FP(\))34 b(is)g(bijectiv)n(e,)i FO(\031)h FP(=)f(~)-45 b FO(\024)34 b FP(for)f(some)h FO(\024)f FQ(2)h FP(Rep\()p FA(K)24 b FQ(\012)e FA(A)p FP(\).)56 b(By)33 b(the)118 1394 y(previous)40 b(prop)r(osition,)j FO(\024)i FP(=)g FO(U)9 b FP(id)28 b FQ(\012)e FO(\031)1442 1406 y FL(0)1480 1394 y FO(U)1546 1364 y FN(\003)1584 1394 y FP(,)44 b(where)d(id)g(is)g(the)g(iden)n (tical)118 1493 y(represen)n(tation)f(of)h FA(K)h FP(in)f FO(L)p FP(\()p FO(H)1148 1505 y FL(0)1185 1493 y FP(\))h(and)f FO(\031)1481 1505 y FL(0)1560 1493 y FP(is)g(a)g(represen)n(tation)e (of)j FA(A)f FP(in)118 1593 y FO(L)p FP(\()p FO(H)276 1605 y FL(1)313 1593 y FP(\),)e FO(H)45 b FP(=)38 b FO(H)693 1605 y FL(0)754 1593 y FQ(\012)24 b FO(H)912 1605 y FL(1)950 1593 y FP(.)63 b(Notice)37 b(that)g FO(U)30 b FP(~)-63 b FO(\031)1608 1605 y FL(0)1645 1593 y FO(U)1711 1563 y FN(\003)1786 1593 y FP(is)36 b(a)g(represen)n(tation)f(of)130 1697 y FJ(^)118 1716 y FA(K)19 b FQ(\012)f FA(A)24 b FP(whic)n(h)h(is)g(a)f(lifting)h(of)g FO(\024)p FP(,)g(and)g(b)n(y)f (the)h(uniqueness)g(of)g(suc)n(h)f(a)g(lifting,)1048 1887 y FO(U)13 b FP(~)-46 b FO(\031)1161 1899 y FL(0)1198 1887 y FO(U)1264 1853 y FN(\003)1325 1887 y FP(=)26 b(~)-45 b FO(\024)23 b FP(=)f FO(\031)s(:)243 2059 y FP(b\).)62 b(Assume)36 b(that)h FA(B)g FQ(\037)g FA(A)p FP(,)g(i.e.,)i(there)d (exists)f(a)h FQ(\003)p FP(-homomorphism)118 2182 y FO( )12 b FP(:)28 b FA(B)c FQ(\000)-49 b(!)466 2163 y FJ(^)454 2182 y FA(K)19 b FQ(\012)f FA(A)j FP(suc)n(h)f(that)h(the)g(functor)g FO(F)1516 2194 y FM( )1576 2182 y FP(:)41 b(Rep)q(\()p FA(A)p FP(\))23 b FQ(\000)-49 b(!)23 b FP(Rep)q(\()p FA(B)p FP(\))e(is)g(full.)118 2308 y(Denote)31 b FB(A)d FP(=)f(pro-)o FO(C)803 2277 y FN(\003)841 2308 y FP(\()884 2289 y FJ(^)873 2308 y FA(K)20 b FQ(\012)e FA(A)o FP(\))28 b FQ(')g FP(pro-)o FO(C)1456 2277 y FN(\003)1494 2308 y FP(\()p FA(K)21 b FQ(\012)f FA(A)p FP(\),)31 b(and)g(let)f FO(\036)9 b FP(:)2242 2289 y FJ(^)2231 2308 y FA(K)19 b FQ(\012)f FA(A)27 b FQ(\000)-48 b(!)118 2407 y FB(A)40 b FP(b)r(e)f(the)h(canonical)e(morphism.)72 b(Then)40 b(the)g(functor)f FO(F)21 b FP(:)46 b(Rep\()p FB(A)p FP(\))d FQ(\000)-48 b(!)118 2507 y FP(Rep\()p FA(B)p FP(\),)37 b(Rep\()p FB(A)p FP(\))d FQ(3)h FO(\031)i FQ(7!)c FO(\031)26 b FQ(\016)d FO(\036)g FQ(\016)f FO( )37 b FQ(2)d FP(Rep\()p FA(B)p FP(\),)j(is)d(also)f(full.)57 b(Indeed,)118 2630 y FO(\031)27 b FQ(\016)c FO(\036)36 b FQ(2)g FP(Rep\()620 2611 y FJ(^)609 2630 y FA(K)19 b FQ(\012)f FA(A)p FP(\),)37 b(so)e FO(\031)26 b FQ(\016)d FO(\036)36 b FP(=)g FO(U)13 b FP(~)-46 b FO(\031)1465 2642 y FL(0)1502 2630 y FO(U)1568 2599 y FN(\003)1641 2630 y FP(for)35 b(some)f FO(\031)2038 2642 y FL(0)2111 2630 y FQ(2)i FP(Rep\()p FA(A)p FP(\))g(in)118 2729 y FO(L)p FP(\()p FO(H)7 b FP(\).)37 b(Th)n(us,)508 2901 y FO(F)12 b FP(\()p FO(\031)s FP(\)\()p FA(B)p FP(\))824 2867 y FN(0)872 2901 y FP(=)23 b FO(F)1013 2913 y FM( )1063 2901 y FP(\()p FO(\031)f FQ(\016)c FO(\036)p FP(\)\()p FA(B)p FP(\))1442 2867 y FN(0)1490 2901 y FP(=)23 b FO(F)1631 2913 y FM( )1682 2901 y FP(\()p FO(U)13 b FP(~)-46 b FO(\031)1827 2913 y FL(0)1864 2901 y FO(U)1930 2867 y FN(\003)1968 2901 y FP(\)\()p FA(B)p FP(\))2137 2867 y FN(0)2162 2901 y FO(;)118 3099 y(F)171 3111 y FM( )222 3099 y FP(\()p FO(U)13 b FP(~)-46 b FO(\031)367 3111 y FL(0)404 3099 y FO(U)470 3069 y FN(\003)508 3099 y FP(\)\()p FA(B)p FP(\))677 3069 y FN(0)725 3099 y FP(=)22 b(\()p FO(U)13 b FP(~)-46 b FO(\031)957 3111 y FL(0)995 3099 y FP(\()p FA(B)p FP(\))p FO(U)1198 3069 y FN(\003)1237 3099 y FP(\))1269 3069 y FN(0)1315 3099 y FP(=)23 b(\()p FO(U)13 b FP(~)-46 b FO(\031)1548 3111 y FL(0)1586 3099 y FP(\()1629 3080 y FJ(^)1618 3099 y FA(K)19 b FQ(\012)f FA(A)p FP(\))p FO(U)1933 3069 y FN(\003)1971 3099 y FP(\))2003 3069 y FN(0)2051 3099 y FP(since)25 b(the)h(func-)118 3225 y(tor)41 b FO(F)319 3237 y FM( )410 3225 y FP(is)g(full.)79 b(\()p FO(U)13 b FP(~)-46 b FO(\031)871 3237 y FL(0)908 3225 y FP(\()951 3206 y FJ(^)940 3225 y FA(K)19 b FQ(\012)f FA(A)p FP(\))p FO(U)1255 3194 y FN(\003)1293 3225 y FP(\))1325 3194 y FN(0)1394 3225 y FP(=)46 b FO(\031)s FP(\()p FO(\036)p FP(\()1679 3206 y FJ(^)1668 3225 y FA(K)20 b FQ(\012)e FA(A)p FP(\)\))1950 3194 y FN(0)2019 3225 y FP(=)45 b FO(\031)s FP(\()p FB(A)p FP(\))2307 3194 y FN(0)2331 3225 y FP(,)g(since)118 3350 y FO(\036)p FP(\()210 3331 y FJ(^)199 3350 y FA(K)20 b FQ(\012)e FA(A)o FP(\))28 b(is)g(quasi-dense)e(in)i FB(A)p FP(.)243 3450 y(The)22 b(morphism)f FO(\036)7 b FQ(\016)g FO( )26 b FP(can)c(b)r(e)h(extended) f(to)g(a)f(con)n(tin)n(uous)h(morphism)f FO(\034)118 3550 y FP(of)f(the)f(pro-)p FO(C)553 3519 y FN(\003)591 3550 y FP(\()p FA(B)p FP(\),)j(since)d(if)h FO(\036)1085 3562 y FL(1)1132 3550 y FP(:)28 b FA(B)23 b FQ(\000)-48 b(!)23 b FP(pro-)o FO(C)1615 3519 y FN(\003)1653 3550 y FP(\()p FA(B)p FP(\))e(denotes)e(the)h(canonical)118 3649 y(morphism,)g(then)g FO(\036)752 3661 y FL(1)789 3649 y FP(\()p FA(B)p FP(\))g(is)f(quasi-dense)e(in)i(pro-)p FO(C)1755 3619 y FN(\003)1793 3649 y FP(\()p FA(B)p FP(\),)i(and)e(the) g(top)r(ology)118 3749 y(on)25 b FO(\036)280 3761 y FL(1)318 3749 y FP(\()p FA(B)p FP(\))i(induced)f(from)f(pro-)p FO(C)1197 3719 y FN(\003)1234 3749 y FP(\()p FA(B)p FP(\))i(\(the)f (pro)5 b(jectiv)n(e)24 b(top)r(ology)g(induced)118 3848 y(b)n(y)e(all)g(represen)n(tations)e(of)i FA(B)p FP(\))h(is)f(stronger) f(than)h(the)h(top)r(ology)e(induced)h(b)n(y)118 3948 y(the)j(map)g FO(\036)13 b FQ(\016)g FO( )29 b FP(\(pro)5 b(jectiv)n(e)24 b(top)r(ology)f(induced)j(b)n(y)e(those)h(represen)n (tations)118 4048 y(of)40 b FA(B)h FP(whic)n(h)f(come)g(through)g FB(A)p FP(\).)75 b(Let)41 b(us)f(sho)n(w)g(that)g(the)h(range)e(of)h FO(\034)118 4147 y FP(is)35 b(dense)f(in)h(ev)n(ery)f(represen)n (tation.)56 b(T)-7 b(ak)n(e)34 b(an)g(arbitrary)f(represen)n(tation)p eop %%Page: 211 215 211 214 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(211)118 333 y FO(\031)26 b FQ(2)e FP(Rep\()p FB(A)p FP(\).)35 b(Then)19 b(the)h(natural)e(injection)h FO(j)c FP(:)p 1654 260 299 4 v 27 w FO(\031)s FP(\()p FO(\034)9 b FP(\()p FA(B)p FP(\)\))26 b FQ(\000)-48 b(!)23 b FO(\031)s FP(\()p FB(A)p FP(\))d(satis\014es)118 432 y(the)29 b(conditions)f(of)g(the)h (previous)e(lemma.)40 b(So)28 b FO(\031)s FP(\()p FO(\034)9 b FP(\(pro-)p FO(C)2022 402 y FN(\003)2061 432 y FP(\()p FA(B)p FP(\)\)\))30 b(is)e(dense)118 532 y(in)g(Im)14 b FO(\031)s FP(.)37 b(The)28 b(con)n(v)n(erse)d(statemen)n(t)j(is)f(ob) n(vious.)243 632 y(3.)59 b(If)36 b(pro-)p FO(C)672 601 y FN(\003)709 632 y FP(\()p FA(A)p FP(\))g(is)f(a)g FO(\033)s FP(-)p FO(C)1180 601 y FN(\003)1219 632 y FP(-algebra,)g(then)h(pro-)p FO(C)1981 601 y FN(\003)2018 632 y FP(\()p FA(K)24 b FQ(\012)f FA(A)p FP(\))36 b(is)f(also)118 731 y(a)42 b FO(\033)s FP(-)p FO(C)345 701 y FN(\003)384 731 y FP(-algebra.)80 b(Therefore)41 b(its)i(top)r(ology)f(can)g(b)r(e)h(determined)g(b)n(y)f (a)118 831 y(coun)n(table)26 b(increasing)f(family)h FO(p)1174 843 y FM(n)1219 831 y FP(\()p FQ(\001)p FP(\))h(of)f FO(C)1491 801 y FN(\003)1529 831 y FP(-seminorms.)36 b(W)-7 b(e)26 b(ha)n(v)n(e)f(pro)n(v)n(ed)118 930 y(that)33 b FO(\034)9 b FP(\(pro-)p FO(C)594 900 y FN(\003)632 930 y FP(\()p FA(B)p FP(\)\))34 b(is)f(dense)f(in)h(pro-)p FO(C)1471 900 y FN(\003)1508 930 y FP(\()p FA(K)23 b FQ(\012)e FA(A)p FP(\))33 b(in)g(an)n(y)e FO(C)2157 900 y FN(\003)2196 930 y FP(-seminorm)118 1030 y(\(eac)n(h)k(seminorm)h FO(p)p FP(\()p FQ(\001)p FP(\))g(de\014nes)g(a)f(represen)n(tation)f FO(\031)1854 1042 y FM(p)1929 1030 y FP(of)h(pro-)p FO(C)2245 1000 y FN(\003)2283 1030 y FP(\()p FA(K)24 b FQ(\012)g FA(A)p FP(\))118 1130 y(and)k FO(\031)327 1142 y FM(p)365 1130 y FP(\()p FO(\034)9 b FP(\(pro-)p FO(C)688 1100 y FN(\003)727 1130 y FP(\()p FA(B)p FP(\)\)\))29 b(is)e(dense)h(in)g (Im\()p FO(\031)s FP(\)\).)243 1229 y(Using)40 b(argumen)n(ts)f (similar)h(to)h(Can)n(tor's)e(diagonal)g(metho)r(d)i(w)n(e)f(can)118 1329 y(pro)n(v)n(e)26 b(that)h(it)g(is)g(also)f(dense)h(in)h(the)f(top) r(ology)f(determined)h(b)n(y)g(the)h(family)118 1429 y FO(p)160 1441 y FM(n)205 1429 y FP(\()p FQ(\001)p FP(\).)38 b(This)27 b(completes)h(the)g(pro)r(of.)p 2514 1429 4 57 v 2518 1376 50 4 v 2518 1429 V 2567 1429 4 57 v 118 1591 a FR(Corollary)35 b(8.)41 b FC(The)32 b(r)l(elation)g FQ(\037)f FC(is)g(a)g(quasi-or)l(der)h(r)l(elation,)h(i.e.,)g FA(C)25 b FQ(\037)g FA(B)118 1691 y FC(and)39 b FA(B)23 b FQ(\037)g FA(A)29 b FC(imply)i FA(C)23 b FQ(\037)g FA(A)p FC(.)243 1842 y FP(Indeed,)31 b(if)g FO( )672 1854 y FL(1)719 1842 y FP(:)e(pro-)n FO(C)983 1812 y FN(\003)1022 1842 y FP(\()p FA(C)p FP(\))f FQ(\000)-48 b(!)28 b FP(pro-)o FO(C)1506 1812 y FN(\003)1544 1842 y FP(\()p FA(K)1631 1854 y FL(1)1690 1842 y FQ(\012)20 b FA(B)p FP(\))29 b(=)e FA(K)2056 1854 y FL(1)2126 1829 y FP(^)2114 1842 y FQ(\012)20 b FP(pro-)o FO(C)2412 1812 y FN(\003)2451 1842 y FP(\()p FA(A)p FP(\))118 1942 y(and)i FO( )328 1954 y FL(2)375 1942 y FP(:)27 b(pro-)o FO(C)638 1912 y FN(\003)677 1942 y FP(\()p FA(B)p FP(\))d FQ(\000)-49 b(!)23 b FP(pro-)o FO(C)1173 1912 y FN(\003)1212 1942 y FP(\()p FA(K)1299 1954 y FL(2)1345 1942 y FQ(\012)8 b FA(A)p FP(\))22 b(=)g FA(K)1674 1954 y FL(2)1731 1929 y FP(^)1720 1942 y FQ(\012)7 b FP(pro-)o FO(C)2005 1912 y FN(\003)2043 1942 y FP(\()p FA(A)p FP(\))23 b(ha)n(v)n(e)e(dense)118 2042 y(ranges)26 b(in)i(ev)n(ery)e(represen)n(tation,)g(then)i(the)g (comp)r(osed)f(homomorphism)389 2206 y(\(id)491 2218 y Ft(K)536 2226 y Fy(1)591 2206 y FQ(\012)18 b FO( )728 2218 y FL(2)765 2206 y FP(\))h FQ(\016)f FO( )930 2218 y FL(1)977 2206 y FP(:)27 b(pro-)o FO(C)1240 2172 y FN(\003)1279 2206 y FP(\()p FA(C)p FP(\))c FQ(\000)-48 b(!)23 b FA(K)1595 2218 y FL(1)1663 2193 y FP(^)1651 2206 y FQ(\012)18 b FA(K)1789 2218 y FL(2)1857 2193 y FP(^)1845 2206 y FQ(\012)g FP(pro-)o FO(C)2141 2172 y FN(\003)2180 2206 y FP(\()p FA(A)p FP(\))118 2371 y(also)32 b(has)g(a)g(dense)h(range)e(in)i(ev)n (ery)e(represen)n(tation)g(and)i FA(K)2076 2383 y FL(1)2147 2358 y FP(^)2135 2371 y FQ(\012)22 b FA(K)2277 2383 y FL(2)2347 2371 y FP(is)33 b(also)118 2471 y(the)28 b(algebra)e(of)h (compact)h(op)r(erators.)118 2596 y FC(R)l(emark)34 b FP(50)p FC(.)i FP(If)23 b FA(B)g FQ(\037)g FA(A)e FP(then)i(if)g FO(\031)1220 2608 y FL(1)1280 2596 y FP(and)f FO(\031)1483 2608 y FL(2)1542 2596 y FP(are)g(distinct)g(represen)n(tation)f(of)118 2696 y FA(A)28 b FP(in)g(the)g(same)g(space)f FO(H)7 b FP(,)28 b(then)g(the)g(represen)n(tation)f FO(F)1933 2708 y FM( )1983 2696 y FP(\()p FO(\031)2062 2708 y FL(1)2100 2696 y FP(\))h(and)g FO(F)2375 2708 y FM( )2426 2696 y FP(\()p FO(\031)2505 2708 y FL(2)2543 2696 y FP(\))118 2796 y(are)f(also)f(distinct.)243 2921 y(Indeed,)g(let)f FO( )12 b FP(:)28 b(pro-)o FO(C)981 2891 y FN(\003)1019 2921 y FP(\()p FA(B)p FP(\))c FQ(\000)-48 b(!)23 b FP(pro-)o FO(C)1516 2891 y FN(\003)1554 2921 y FP(\()p FA(K)13 b FQ(\012)g FA(A)p FP(\))27 b(b)r(e)e(a)g(morphism)f(with)118 3021 y(quasi-dense)37 b(range,)j(and)e(let)g FO(\031)1182 3033 y FL(1)1220 3021 y FP(,)j FO(\031)1331 3033 y FL(2)1409 3021 y FQ(2)g FP(Rep\()p FA(A)p FP(\))e(b)r(e)g(distinct)f(represen-) 118 3120 y(tations)e(on)h(the)g(same)f(space)g FO(H)7 b FP(.)64 b(Let)41 b(^)-46 b FO(\031)1497 3132 y FM(i)1563 3120 y FP(=)38 b(id)25 b FQ(\012)f FO(\031)1896 3132 y FM(i)1924 3120 y FP(,)39 b FO(i)f FP(=)f(1,)i(2)d(b)r(e)h(the)118 3220 y(corresp)r(onding)d(represen)n(tations)g(of)i FA(K)24 b FQ(\012)f FA(A)36 b FP(on)f FO(H)1802 3232 y FL(0)1864 3220 y FQ(\012)23 b FO(H)7 b FP(.)61 b(It)37 b(is)e(ob)n(vious)118 3320 y(that)h(^)-47 b FO(\031)348 3332 y FL(1)414 3320 y FQ(6)p FP(=)33 b(^)-46 b FO(\031)555 3332 y FL(2)592 3320 y FP(.)47 b(By)31 b(the)g(de\014nition)g(of)g(an)g(en)n(v)n (eloping)e(algebra,)h(they)h(can)118 3419 y(b)r(e)25 b(lifted)g(to)f(distinct)h(represen)n(tations)i(~)-46 b FO(\031)1454 3431 y FL(1)1491 3419 y FP(,)30 b(~)-47 b FO(\031)1586 3431 y FL(2)1648 3419 y FP(of)25 b(pro-)p FO(C)1954 3389 y FN(\003)1991 3419 y FP(\()p FA(K)12 b FQ(\012)g FA(A)p FP(\).)36 b(Denote)118 3519 y FB(A)23 b FP(=)g(pro-)o FO(C)506 3489 y FN(\003)544 3519 y FP(\()p FA(K)c FQ(\012)f FA(A)p FP(\).)37 b(F)-7 b(or)27 b(ev)n(ery)f FO(x)e FQ(2)f FB(A)28 b FP(de\014ne)795 3684 y FO(p)p FP(\()p FO(x)p FP(\))c(=)f(max)o(\()p FQ(k)p FO(\031)1335 3696 y FL(1)1372 3684 y FP(\()p FO(x)p FP(\))p FQ(k)p FO(;)14 b FQ(k)p FO(\031)1651 3696 y FL(2)1689 3684 y FP(\()p FO(x)p FP(\))p FQ(k)p FP(\))p FO(:)118 3848 y FP(Then)26 b FO(p)p FP(\()p FQ(\001)p FP(\))h(is)f(a)f FO(C)703 3818 y FN(\003)742 3848 y FP(-norm)g(on)h FB(A)p FP(,)g(and)g(w)n(e)g(denote)f(the)i(completion)f(of)g FB(A)g FP(in)118 3948 y(this)j(norm)f(b)n(y)g FB(A)679 3960 y FM(p)717 3948 y FP(.)40 b(If)29 b FO(j)34 b FP(denotes)28 b(the)h(natural)e(morphism)h(from)g FB(A)h FP(to)f FB(A)2513 3960 y FM(p)2552 3948 y FP(,)118 4048 y(then)e(there)e(are)g(represen)n (tations)f FO(\031)1270 4060 y FM(i;p)1375 4048 y FQ(2)h FP(Rep\()p FB(A)1694 4060 y FM(p)1732 4048 y FP(\),)i FO(i)d FP(=)g(1,)i(2,)g(and)g(w)n(e)f(ha)n(v)n(e)118 4147 y(the)k(comm)n(utativ)n(e)f(diagram)p eop %%Page: 212 216 212 215 bop 118 100 a FP(212)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)1022 353 y FB(A)332 b(A)1482 365 y FM(p)p 1110 338 283 4 v 1310 336 a Fu(-)1232 304 y FO(j)1139 565 y FP(~)-46 b FO(\031)1182 577 y FM(i)1112 477 y Fu(@)1195 560 y(@)1278 643 y(@)1328 693 y(@)-83 b(R)1267 772 y FO(L)p FP(\()p FO(H)1425 784 y FL(0)1480 772 y FQ(\012)18 b FO(H)7 b FP(\))p 1467 693 4 299 v 1469 693 a Fu(?)1502 549 y FO(\031)1549 561 y FM(i;p)118 935 y FP(Since)39 b FO( )j FP(has)d(a)f(quasi-dense)g (range)f(and)i FO(\036)p FP(\()p FA(B)p FP(\))h(is)f(quasi-dense)f(in)h (pro-)118 1035 y FO(C)183 1005 y FN(\003)222 1035 y FP(\()p FA(B)p FP(\))30 b(\(where)g FO(\036)9 b FP(:)29 b FA(B)e FQ(\000)-48 b(!)27 b FP(pro-)n FO(C)1213 1005 y FN(\003)1252 1035 y FP(\(\()p FO(B)t FP(\)\)\))k(is)f(the)g(natural)f(morphism\),)h (w)n(e)118 1134 y(ha)n(v)n(e)f(that)h(the)g FQ(\003)p FP(-subalgebra)d FB(B)1189 1146 y FM(p)1254 1134 y FP(=)f FO(j)f FQ(\016)20 b FO( )j FQ(\016)c FO(\036)p FP(\()p FA(B)p FP(\))31 b(is)f(quasi-dense)e(in)i FB(A)2513 1146 y FM(p)2552 1134 y FP(,)118 1234 y(and)20 b(since)f(the)i(latter)e(is)h (a)f FO(C)1023 1204 y FN(\003)1062 1234 y FP(-algebra,)g(the)h (quasi-densit)n(y)f(implies)h(densit)n(y)-7 b(.)118 1334 y(If)30 b FO(F)256 1346 y FM( )307 1334 y FP(\()p FO(\031)386 1346 y FL(1)424 1334 y FP(\))c(=)h FO(F)627 1346 y FM( )677 1334 y FP(\()p FO(\031)756 1346 y FL(2)794 1334 y FP(\),)k(then)f FO(\031)1118 1346 y FL(1)p FM(;p)1236 1334 y FP(=)c FO(\031)1374 1346 y FL(2)p FM(;p)1495 1334 y FP(on)k(the)g(dense)f FQ(\003)p FP(-subalgebra)e FB(B)2536 1346 y FM(p)118 1433 y FP(and)h(as)g(suc)n(h)g(m)n(ust)g(coincide.)39 b(Then)28 b(it)h(follo)n(ws)e(from)h(the)g(comm)n(utativit)n(y)118 1533 y(of)g(the)g(diagram)e(that)32 b(~)-46 b FO(\031)906 1545 y FL(1)966 1533 y FP(=)27 b(~)-46 b FO(\031)1101 1545 y FL(2)1138 1533 y FP(,)28 b(whic)n(h)g(con)n(tradicts)e(the)i (assertion.)118 1670 y FC(R)l(emark)47 b FP(51)p FC(.)f FP(Belo)n(w)36 b(in)h(this)g(section,)h(w)n(e)e(giv)n(e)g(a)g(n)n(um)n (b)r(er)g(of)h(examples)118 1769 y(of)32 b FQ(\003)p FP(-algebras)d(and)i(mappings)h FO( )12 b FP(:)29 b FA(A)h FQ(\000)-49 b(!)31 b FO(M)1577 1781 y FM(n)1621 1769 y FP(\()p FA(B)p FP(\))i(suc)n(h)f(that)g(the)g(functor)118 1869 y FO(F)171 1881 y FM( )231 1869 y FP(:)45 b(Rep)14 b FA(B)43 b FQ(\000)-49 b(!)42 b FP(Rep)14 b FA(A)39 b FP(is)f(full.)72 b(But)39 b(w)n(e)g(do)f(not)h(discuss)g(metho)r(ds)g (of)118 1969 y(construction)31 b(of)g(suc)n(h)g(mappings)f(here.)48 b(This)31 b(is)g(a)g(separate)f(topic)h(to)g(b)r(e)118 2068 y(discussed)c(elsewhere.)243 2170 y(W)-7 b(e)25 b(also)f(do)h(not)g(discuss)g(the)g(question)g(of)g(what)g(is)g(the)h (minimal)f(n)n(um-)118 2270 y(b)r(er)g FO(n)p FP(,)g(for)f(whic)n(h)h (there)g(exists)f(a)h(homomorphism)e FO( )12 b FP(:)28 b FA(A)23 b FQ(\000)-48 b(!)23 b FO(M)2208 2282 y FM(n)2252 2270 y FP(\()p FA(B)p FP(\))j(suc)n(h)118 2369 y(that)36 b(the)f(corresp)r(onding)f(functor)h FO(F)1347 2381 y FM( )1433 2369 y FP(is)g(full.)61 b(W)-7 b(e)36 b(only)f(notice)g(that) h(in)118 2469 y([188)o(],)h(it)f(w)n(as)f(sho)n(wn)f(that)i(for)f(the)h FO(C)1406 2439 y FN(\003)1444 2469 y FP(-algebra)e FB(A)h FP(with)h FO(m)f FP(self-adjoin)n(t)118 2568 y(generators)g FO(a)574 2580 y FL(1)611 2568 y FP(,)j FO(:)14 b(:)g(:)27 b FP(,)40 b FO(a)903 2580 y FM(m)966 2568 y FP(,)g(for)d FO(n)i FQ(\025)f(\000)p FP(3)24 b(+)1579 2503 y FQ(p)p 1648 2503 258 4 v 65 x FP(9)18 b(+)g(8)p FO(m)o FP(,)40 b FO(M)2049 2580 y FM(n)2094 2568 y FP(\()p FB(A)p FP(\))e(is)f(gener-) 118 2668 y(ated)28 b(b)n(y)h(a)f(pair)f(of)i(self-adjoin)n(t)f (generators.)37 b(It)28 b(is)h(sho)n(wn)e(in)i(pap)r(er)f([228)o(],)118 2768 y(that)34 b(this)h(statemen)n(t)f(holds)f(for)h FO(n)f FQ(\025)1404 2702 y(p)p 1473 2702 216 4 v 66 x FO(m)18 b FQ(\000)g FP(1,)36 b(and)d(that)i(this)f(estimate)118 2867 y(is)e(exact,)h(i.e.,)h(there)e(exists)g(a)g(comm)n(utativ)n(e)g FO(C)1708 2837 y FN(\003)1746 2867 y FP(-algebra)e FO(C)6 b FP(\()p FO(K)g FP(\))33 b(with)g FO(m)118 2967 y FP(self-adjoin)n(t)c (generators)e(suc)n(h)j(that)g FO(M)1404 2979 y FM(n)1448 2967 y FP(\()p FO(C)6 b FP(\()p FO(K)g FP(\)\))31 b(is)e(not)h(singly)f (generated)118 3067 y(for)h FO(n)f(<)419 3001 y FQ(p)p 489 3001 V 489 3067 a FO(m)18 b FQ(\000)g FP(1)o(,)32 b(i.e.,)g FO(M)1001 3079 y FM(n)1046 3067 y FP(\()p FO(C)6 b FP(\()p FO(K)g FP(\)\))31 b(is)g(not)g(generated)e(b)n(y)i(a)f(pair)g (of)h(self-)118 3166 y(adjoin)n(t)c(elemen)n(ts.)118 3393 y FR(3.1.2)94 b FQ(\003)p FR(-Wildness)29 b(of)j FQ(\003)p FR(-algebras)118 3550 y(1.)k FP(In)25 b(the)h(theory)f(of)h (represen)n(tations)e(of)h(algebras,)f(it)i(w)n(as)f(suggested)g([71)o (])118 3649 y(to)30 b(consider)f(the)i(represen)n(tation)d(problem)i (to)g(b)r(e)h(wild)f(if)h(it)f(con)n(tains)f(the)118 3749 y(classical)21 b(unsolv)n(ed)h(problem)h(of)f(represen)n(tation)f (theory)-7 b(,)23 b(i.e.,)h(the)f(problem)118 3848 y(to)37 b(describ)r(e,)h(up)f(to)g(similarit)n(y)-7 b(,)38 b(a)e(pair)g(of)g (matrices)g(without)h(relations.)118 3948 y(T)-7 b(o)26 b(de\014ne)h(an)g(analogue)e(of)h(wildness)h(for)f FQ(\003)p FP(-algebras)d(\()p FQ(\003)p FP(-wildness\),)k(it)g(w)n(as)118 4048 y(suggested)18 b([150)o(])h(to)g(c)n(ho)r(ose,)g(for)g(a)g (standard)f FQ(\003)p FP(-wild)g(problem)h(in)g(the)h(theory)118 4147 y(of)33 b FQ(\003)p FP(-represen)n(tations,)e(the)j(problem)e(of)h (describing)f(pairs)g(of)g(self-adjoin)n(t)p eop %%Page: 213 217 213 216 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(213)118 333 y(\(or)31 b(unitary\))f(op)r(erators)f(up)j(to)f(a)f(unitary)h(equiv)-5 b(alence)31 b(\(represen)n(tations)118 432 y(of)h(the)h FQ(\003)p FP(-algebra)c FA(S)798 444 y FL(2)867 432 y FP(\(or)j FA(U)1060 444 y FL(2)1096 432 y FP(\))h(generated)e(b)n(y)h (a)f(pair)h(of)g(free)f(self-adjoin)n(t)118 532 y(\(or)g(unitary\))h (generators\))e(and)i(to)g(regard)e(the)i(problems)g(whic)n(h)g(con)n (tain)118 632 y(the)c(standard)f FQ(\003)p FP(-wild)g(problem)g(as)g FQ(\003)p FP(-wild.)118 772 y FR(2.)33 b FP(One)19 b(can)g(pro)n(v)n(e) e(that)i(the)h(standard)e FQ(\003)p FP(-wild)h(problem)f(con)n(tains)g (as)h(a)f(sub-)118 871 y(problem)31 b(the)h(problem)g(of)f(describing)g FQ(\003)p FP(-represen)n(tations)e(of)j(an)n(y)e(\014nitely)118 971 y(generated)d FQ(\003)p FP(-algebra.)34 b(The)27 b(follo)n(wing)g(theorem)g(holds.)118 1118 y FR(Theorem)k(52.)40 b FA(S)756 1130 y FL(2)817 1118 y FP(=)22 b FJ(C)15 b FQ(h)q FO(a;)f(b)28 b FQ(j)c FO(a)f FP(=)g FO(a)1382 1088 y FN(\003)1420 1118 y FO(;)14 b(b)23 b FP(=)f FO(b)1639 1088 y FN(\003)1677 1118 y FQ(i)i(\037)f FA(S)1890 1130 y FM(m)1976 1118 y FP(=)g FJ(C)15 b FQ(h)p FO(a)2194 1130 y FL(1)2237 1118 y FO(;)f(:)g(:)g(:)g(;)g(a)2466 1130 y FM(m)2552 1118 y FQ(j)118 1218 y FO(a)162 1230 y FM(i)213 1218 y FP(=)22 b FO(a)344 1187 y FN(\003)344 1239 y FM(i)382 1218 y FO(;)28 b(i)23 b FP(=)f(1)p FO(;)14 b(:)g(:)g(:)g(;)g(m)p FQ(i)29 b FC(for)i(any)f FO(m)23 b FP(=)g(1)p FC(,)29 b FP(2)p FC(,)h FO(:)14 b(:)g(:)27 b FC(.)118 1365 y(Pr)l(o)l(of.)43 b FP(F)-7 b(or)30 b(the)g(algebra)982 1343 y(~)968 1365 y FA(S)1037 1377 y FM(m)1100 1365 y FP(,)h(tak)n(e)e(the)i(algebra)d FA(S)1844 1377 y FM(m)1937 1365 y FP(itself,)j FO(n)c FP(=)g FO(m)20 b FP(+)f(2.)118 1464 y(De\014ne)28 b(a)f(homomorphism)g FO( )12 b FP(:)28 b FA(S)1217 1476 y FL(2)1277 1464 y FQ(\000)-48 b(!)23 b FO(M)1481 1476 y FM(m)p FL(+2)1627 1464 y FP(\()p FA(S)1728 1476 y FM(m)1792 1464 y FP(\))k(as)g(follo)n(ws:)457 1957 y FO( )s FP(\()p FO(a)p FP(\))d(=)733 1566 y Fz(2)733 1712 y(6)733 1762 y(6)733 1811 y(6)733 1861 y(6)733 1911 y(6)733 1961 y(6)733 2011 y(6)733 2060 y(6)733 2110 y(6)733 2163 y(4)789 1623 y FO(e)920 1690 y FL(1)p 920 1704 34 4 v 920 1751 a(2)963 1723 y FO(e)739 b Fp(0)1095 1790 y FL(1)p 1095 1804 V 1095 1851 a(3)1138 1823 y FO(e)1264 1919 y FP(.)1297 1944 y(.)1329 1970 y(.)1462 2045 y FL(1)p 1450 2059 59 4 v 1450 2106 a FM(m)1518 2078 y FO(e)1072 2178 y Fp(0)1705 2145 y FL(1)p 1650 2159 143 4 v 1650 2206 a FM(m)p FL(+1)1803 2178 y FO(e)1990 2251 y FL(1)p 1935 2265 V 1935 2312 a FM(m)p FL(+2)2088 2283 y FO(e)2126 1566 y Fz(3)2126 1712 y(7)2126 1762 y(7)2126 1811 y(7)2126 1861 y(7)2126 1911 y(7)2126 1961 y(7)2126 2011 y(7)2126 2060 y(7)2126 2110 y(7)2126 2163 y(5)2195 1957 y FO(;)465 2804 y( )s FP(\()p 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y(7)2144 3061 y(5)2213 2804 y FO(:)243 3341 y FP(One)27 b(can)g(directly)g(c)n(hec)n(k)g(that)h(the)g(functor) g FO(F)1761 3353 y FM( )1839 3341 y FP(is)f(full.)p 2514 3341 4 57 v 2518 3288 50 4 v 2518 3341 V 2567 3341 4 57 v 243 3502 a(Theorem)d(52)h(p)r(ermits)g(one)g(to)g(sa)n(y)g(that)g (the)h(problem)f(of)g(unitary)g(clas-)118 3602 y(si\014cation)36 b(of)g(pairs)f(of)h(self-adjoin)n(t)g(op)r(erators)e(con)n(tains,)k(as) d(a)h(subprob-)118 3701 y(lem,)i(the)f(problem)e(of)h(unitary)f (classi\014cation)g(of)h(represen)n(tation)e(of)i(an)n(y)118 3801 y FQ(\003)p FP(-algebra)c(with)i(a)g(coun)n(table)f(n)n(um)n(b)r (er)h(of)g(generators)e(\(b)r(ecause)i(it)h(is)f(al-)118 3900 y(w)n(a)n(ys)26 b(p)r(ossible)h(to)h(c)n(ho)r(ose)e(these)i (generators)d(to)j(b)r(e)g(self-adjoin)n(t\).)118 4048 y FR(Corollary)40 b(9.)45 b FC(F)-6 b(or)36 b(any)f FO(m)f FP(=)f(1)p FC(,)k FP(2)p FC(,)e FO(:)14 b(:)g(:)28 b FC(,)37 b FA(S)1663 4060 y FL(2)1733 4048 y FQ(\037)c FA(U)1885 4060 y FM(m)1982 4048 y FP(=)g FJ(C)15 b FQ(h)p FO(u)2214 4060 y FL(1)2257 4048 y FO(;)f(:)g(:)g(:)f(;)h(u)2489 4060 y FM(m)2552 4048 y FO(;)118 4147 y(u)166 4117 y FN(\003)166 4168 y FL(1)204 4147 y FO(;)g(:)g(:)g(:)f(;)h(u)436 4117 y FN(\003)436 4168 y FM(m)522 4147 y FQ(j)23 b FO(u)616 4159 y FM(i)643 4147 y FO(u)691 4117 y FN(\003)691 4169 y FM(i)752 4147 y FP(=)g FO(u)888 4117 y FN(\003)888 4169 y FM(i)925 4147 y FO(u)973 4159 y FM(i)1024 4147 y FP(=)f FO(e;)14 b(i)22 b FP(=)h(1)p FO(;)14 b(:)g(:)g(:)f(;)h(m)p FQ(i)p FC(.)p eop %%Page: 214 218 214 217 bop 118 100 a FP(214)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FR(Theorem)j(53.)40 b FA(U)741 345 y FL(2)801 333 y FQ(\037)23 b FA(S)958 345 y FL(2)995 333 y FC(.)118 502 y(Pr)l(o)l(of.)43 b FP(Cho)r(ose)30 b(the)g(en)n(v)n(eloping)f (algebra)1533 480 y(~)1519 502 y FA(S)1588 514 y FL(2)1655 502 y FP(to)h(b)r(e)h(the)f(algebra)f(of)h(frac-)118 602 y(tions)g(of)h(the)f(algebra)f FA(S)930 614 y FL(2)997 602 y FP(\(see,)j(e.g.,)e([89)o(]\))h(with)g(resp)r(ect)f(to)g(the)h (set)g(\006)c(=)118 701 y FQ(f)p FO(a)10 b FQ(\000)g FO(ie;)k(a)c FP(+)g FO(ie;)k(b)c FQ(\000)g FO(ie;)k(b)c FP(+)g FO(ie)p FQ(g)p FP(.)33 b(De\014ne)24 b(the)g(homomorphism)e FO( )12 b FP(:)28 b FA(U)2286 713 y FL(2)2346 701 y FQ(\000)-48 b(!)2482 680 y FP(~)2469 701 y FA(S)2538 713 y FL(2)118 801 y FP(as)27 b(follo)n(ws:)334 986 y FO( )s FP(\()p FO(u)471 998 y FL(1)508 986 y FP(\))d(=)e(\()p FO(a)d FQ(\000)f FO(ie)p FP(\)\()p FO(a)g FP(+)g FO(ie)p FP(\))1206 952 y FN(\000)p FL(1)1295 986 y FO(;)97 b( )s FP(\()p FO(u)1552 998 y FL(2)1589 986 y FP(\))23 b(=)g(\()p FO(b)18 b FQ(\000)g FO(ie)p FP(\)\()p FO(b)g FP(+)g FO(ie)p FP(\))2270 952 y FN(\000)p FL(1)118 1171 y FP(\(the)28 b(Ca)n(yley)f (transformation\).)35 b(The)28 b(rest)f(is)h(ob)n(vious.)p 2514 1171 4 57 v 2518 1118 50 4 v 2518 1171 V 2567 1171 4 57 v 118 1346 a FR(3.)36 b FP(Theorems)26 b(52,)h(53)f(allo)n(w,)g 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FN(\003)522 2874 y FP(-algebras)e(whic)n(h)j(are)g(not)g(wild.)71 b(F)-7 b(or)39 b(example,)j(the)d(group)118 2973 y FO(C)183 2943 y FN(\003)222 2973 y FP(-algebra)27 b FO(C)607 2943 y FN(\003)645 2973 y FP(\()p FO(B)t FP(\()p FO(m;)14 b FP(2\)\))30 b(of)f(the)h(Burnside)e(group)g(with)i(t)n(w)n(o)e (generators)118 3073 y(and)34 b(su\016cien)n(tly)g(large)e(o)r(dd)j FO(m)e FP(is)h(not)g(n)n(uclear,)h(but)f(it)h(is)f(also)f(not)h(wild) 118 3172 y(\(see)28 b(Section)f(3.1.6\).)118 3394 y FR(3.1.3)94 b FQ(\003)p FR(-Wild)24 b(algebras)h(generated)h(b)m(y)g(orthogonal)f (pro)5 b(jections)410 3494 y(and)32 b(idemp)s(oten)m(ts)118 3649 y(1.)53 b FP(F)-7 b(or)32 b(represen)n(tations)f(of)i(the)g FQ(\003)p FP(-algebra)e FB(P)1650 3661 y FL(2)1719 3649 y FP(=)h FJ(C)15 b FQ(h)p FO(p)1944 3661 y FL(1)1987 3649 y FO(;)f(p)2066 3661 y FL(2)2135 3649 y FQ(j)32 b FO(p)2232 3619 y FN(\003)2232 3670 y FL(1)2302 3649 y FP(=)g FO(p)2441 3661 y FL(1)2510 3649 y FP(=)118 3749 y FO(p)160 3719 y FL(2)160 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FO(w)731 4026 y FM(i)782 4014 y FP(=)g FO(w)930 3979 y FN(\003)928 4034 y FM(i)969 4014 y FO(;)14 b(i)23 b FP(=)f(1)p FO(;)14 b FP(2)p FO(;)g FP(3)22 b FQ(j)h FO(w)1474 3979 y FL(2)1472 4034 y FM(i)1535 4014 y FP(=)g FO(e;)14 b(i)22 b FP(=)g(1)p FO(;)14 b FP(2)p FO(;)g FP(3;)1344 4147 y FQ(f)p FO(w)1445 4159 y FL(1)1482 4147 y FO(;)g(w)1578 4159 y FL(2)1615 4147 y FQ(g)23 b FP(=)g FO(w)1827 4159 y FL(1)1864 4147 y FO(w)1923 4159 y FL(2)1980 4147 y FP(+)18 b FO(w)2122 4159 y FL(2)2159 4147 y FO(w)2218 4159 y FL(1)2279 4147 y FP(=)23 b(0)p FQ(g)2451 4080 y Fz(\013)p eop %%Page: 216 220 216 219 bop 118 100 a FP(216)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FC(is)i FQ(\003)p FC(-wild.)118 473 y(Pr)l(o)l(of.)43 b FP(De\014ne)e(the)g(homomorphism)e FO( )k FP(of)e(the)f FQ(\003)p FP(-algebra)e FB(P)2191 485 y FL(3)p FM(;)p FL(2an)n(ti)2434 473 y FP(in)n(to)118 572 y FO(M)199 584 y FL(4)236 572 y FP(\()p FA(U)322 584 y FL(2)359 572 y 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FM(k)24 b FL(times)811 3843 y Fz(3)811 3990 y(7)811 4043 y(5)880 4035 y FO(;)180 b(e)28 b FP(is)f(the)h(iden)n(tit)n(y)g(in)g(the)g(algebra)e FO(C)2279 4001 y FN(\003)2317 4035 y FP(\()p FB(F)2405 4047 y FL(2)2440 4035 y FP(\),)p eop %%Page: 217 221 217 220 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(217)517 420 y FO(J)563 432 y FL(1)623 420 y FP(=)711 253 y Fz(2)711 402 y(4)799 319 y FO(E)860 331 y FL(4)766 419 y FP(0)808 431 y FL(3)p FN(\002)p FL(4)766 519 y FP(0)808 531 y FL(5)p FN(\002)p FL(4)930 253 y Fz(3)930 402 y(5)999 420 y FO(;)97 b(J)1165 432 y FL(2)1225 420 y FP(=)1313 253 y Fz(2)1313 402 y(4)1368 319 y FP(0)1410 331 y FL(4)p FN(\002)p FL(3)1401 419 y FO(E)1462 431 y FL(3)1368 519 y FP(0)1410 531 y FL(5)p FN(\002)p FL(3)1532 253 y Fz(3)1532 402 y(5)1601 420 y FO(;)g(J)1767 432 y FL(3)1828 420 y FP(=)1915 253 y Fz(2)1915 402 y(4)1970 319 y FO(A)2032 331 y FL(1)1970 419 y FO(A)2032 431 y FL(2)1970 519 y FO(A)2032 531 y FL(3)2070 253 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1379 y FL(2)1805 1367 y FP(\),)838 1558 y FO(A)900 1570 y FL(3)960 1558 y FP(=)1048 1486 y Fz(p)p 1131 1486 702 4 v 72 x FO(E)1192 1570 y FL(5)1248 1558 y FQ(\000)18 b FO(A)1393 1529 y FN(\003)1393 1580 y FL(1)1431 1558 y FO(A)1493 1570 y FL(1)1549 1558 y FQ(\000)g FO(A)1694 1529 y FN(\003)1694 1580 y FL(2)1733 1558 y FO(A)1795 1570 y FL(2)1832 1558 y FO(;)118 1749 y(N)48 b FP(is)39 b(c)n(hosen)g(so)f(that)i FQ(k)p FO(A)1015 1719 y FN(\003)1015 1770 y FL(1)1053 1749 y FO(A)1115 1761 y FL(1)1178 1749 y FP(+)26 b FO(A)1331 1719 y FN(\003)1331 1770 y FL(2)1370 1749 y FO(A)1432 1761 y FL(2)1469 1749 y FQ(k)42 b FO(<)g FP(1)d(in)g FO(M)1930 1761 y FL(5)1967 1749 y FP(\()p FO(C)2064 1719 y FN(\003)2103 1749 y FP(\()p FB(F)2191 1761 y FL(2)2226 1749 y FP(\)\).)73 b(Then)118 1849 y FO(J)172 1819 y FN(\003)164 1869 y FL(1)210 1849 y FO(J)256 1861 y FL(1)317 1849 y FP(=)22 b FO(E)465 1861 y FL(4)503 1849 y FP(,)i FO(J)604 1819 y FN(\003)596 1869 y FL(2)642 1849 y FO(J)688 1861 y FL(2)748 1849 y FP(=)f FO(E)897 1861 y FL(3)935 1849 y FP(,)h(and)f FO(J)1193 1819 y FN(\003)1185 1869 y FL(3)1231 1849 y FO(J)1277 1861 y FL(3)1337 1849 y FP(=)g FO(E)1486 1861 y FL(4)1523 1849 y FP(.)36 b(This)23 b(implies)g(that)g(\()p FO(J)2297 1861 y FM(i)2325 1849 y FO(J)2379 1819 y FN(\003)2371 1870 y FM(i)2418 1849 y FP(\))2450 1819 y FL(2)2510 1849 y FP(=)118 1948 y FO(J)164 1960 y FM(i)192 1948 y FO(J)246 1918 y FN(\003)238 1970 y FM(i)284 1948 y FP(,)49 b FO(i)i FP(=)g(1,)d(2,)h(3.)87 b(Moreo)n(v)n(er,)46 b(since)e FO(J)1614 1918 y FN(\003)1606 1969 y FL(1)1653 1948 y FO(J)1699 1960 y FL(2)1787 1948 y FP(=)51 b(0)44 b(and)h FO(J)2214 1960 y FL(2)2251 1948 y FO(J)2305 1918 y FN(\003)2297 1969 y FL(1)2394 1948 y FP(=)51 b(0,)118 2048 y(w)n(e)35 b(see)g(that)g(\()p FO(J)655 2060 y FL(1)693 2048 y FO(J)747 2018 y FN(\003)739 2069 y FL(1)785 2048 y FP(\)\()p FO(J)895 2060 y FL(2)933 2048 y FO(J)987 2018 y FN(\003)979 2069 y FL(2)1025 2048 y FP(\))h(=)f(\()p FO(J)1271 2060 y FL(2)1308 2048 y 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FO(;)1289 3071 y(p)1331 3083 y FM(i)1358 3071 y FO(p)1400 3083 y FM(j)1458 3071 y FP(=)23 b(0)p FO(;)14 b(i)22 b FQ(6)p FP(=)g FO(j)1802 3004 y Fz(\013)1865 3071 y FQ(\037)g FO(C)2017 3037 y FN(\003)2056 3071 y FP(\()p FB(F)2144 3083 y FL(2)2179 3071 y FP(\))p FO(:)118 3267 y FR(5.)35 b FP(The)26 b(problem)e(of)h(unitary)g(classi\014cation)f (of)h(quadruples)f(of)h(orthogonal)118 3366 y(pro)5 b(jections)26 b FO(p)587 3378 y FL(1)624 3366 y FP(,)i FO(p)717 3378 y FL(2)754 3366 y FP(,)g FO(p)847 3378 y FL(3)884 3366 y FP(,)g FO(p)977 3378 y FL(4)1041 3366 y FP(suc)n(h)g(that)608 3558 y FO(\013)p FP(\()p FO(p)735 3570 y FL(1)792 3558 y FP(+)18 b FO(p)917 3570 y FL(2)972 3558 y FP(+)g FO(p)1097 3570 y FL(3)1153 3558 y FP(+)g FO(p)1278 3570 y FL(4)1315 3558 y FP(\))23 b(=)g FO(I)7 b(;)180 b FP(0)22 b FO(<)h(\013)g(<)g FP(1)p FO(;)118 3749 y FP(for)30 b(a)g(\014xed)h FO(\013)e FQ(6)p FP(=)708 3716 y FL(1)p 708 3730 34 4 v 708 3777 a(2)751 3749 y FP(,)j(has)e(only)g(a)g(\014nite)h(n)n(um)n(b)r(er)g(of) f(irreducible)g(represen-)118 3848 y(tations,)39 b(the)e(dimension)f (of)h(whic)n(h)f(dep)r(ends)h(on)g(the)g(parameter)e FO(\013)i FP(\(see)118 3948 y(Section)h(2.2.1\).)68 b(If)38 b FO(\013)j FP(=)1016 3915 y FL(1)p 1016 3929 V 1016 3977 a(2)1059 3948 y FP(,)g(then)e(there)e(is)h(an)g(uncoun)n(table)g (family)g(of)118 4048 y(irreducible)c(represen)n(tations,)g(and)g (their)g(dimension)g(is)g(one)g(or)g(t)n(w)n(o)f(\(see)118 4147 y(Section)28 b(2.2.1\).)p eop %%Page: 218 222 218 221 bop 118 100 a FP(218)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)243 333 y FP(It)i(follo)n(ws)f(directly)g(from)h(Theorem)e(57)h(that)h FC(the)i(pr)l(oblem)h(of)g(unitary)118 432 y(classi\014c)l(ation)25 b(of)g(\014ve)e(ortho)l(gonal)j(pr)l(oje)l(ctions)f FO(r)1691 444 y FL(1)1728 432 y FC(,)h FO(r)1816 444 y FL(2)1853 432 y FC(,)g FO(r)1941 444 y FL(3)1978 432 y FC(,)g FO(r)2066 444 y FL(4)2103 432 y FC(,)g FO(r)2191 444 y FL(5)2252 432 y FC(such)e(that)118 532 y FO(r)155 544 y FL(1)211 532 y FP(+)18 b FO(r)331 544 y FL(2)387 532 y FP(+)g FO(r)507 544 y FL(3)564 532 y FP(+)g FO(r)684 544 y FL(4)740 532 y FP(+)g FO(r)860 544 y FL(5)921 532 y FP(=)k(2)p FO(e)29 b FC(is)h FQ(\003)p FC(-wild)p FP(:)38 b FB(R)1543 544 y FL(5)p FM(;)p FL(2)1656 532 y FQ(\037)22 b FB(P)1798 544 y FL(3)p FM(;)p FN(?)p FL(2)1940 532 y FP(,)28 b(where)266 798 y FB(R)326 810 y FL(5)p FM(;)p FL(2)440 798 y FP(=)22 b FJ(C)581 705 y Fz(D)638 798 y FO(r)675 810 y FL(1)713 798 y FO(;)14 b(:)g(:)g(:)f(;)h(r)934 810 y FL(5)995 798 y FQ(j)23 b FO(r)1080 763 y FL(2)1078 818 y FM(i)1141 798 y FP(=)g FO(r)1266 810 y FM(i)1317 798 y FP(=)f FO(r)1443 763 y FN(\003)1441 818 y FM(i)1482 798 y FO(;)14 b(i)23 b FP(=)g(1)p FO(;)14 b(:)g(:)g(:)f(;)h FP(5;)2006 694 y FL(5)1963 719 y Fz(X)1969 895 y FM(i)p FL(=1)2097 798 y FO(r)2134 810 y FM(i)2185 798 y FP(=)23 b(2)p FO(e)2354 705 y Fz(E)2403 798 y FO(:)243 1055 y FP(Indeed,)28 b(let)f FO( )12 b FP(:)28 b FB(R)832 1067 y FL(5)p FM(;)p FL(2)946 1055 y FQ(\000)-49 b(!)23 b FB(P)1123 1067 y FL(3)p FM(;)p FN(?)p FL(2)1293 1055 y FP(b)r(e)28 b(de\014ned)g(b)n(y)f(the)h(form)n (ulas)849 1239 y FO( )s FP(\()p FO(r)975 1251 y FL(1)1013 1239 y FP(\))c(=)e FO(p;)97 b( )s FP(\()p FO(r)1444 1251 y FL(2)1482 1239 y FP(\))23 b(=)g FO(e)18 b FQ(\000)g FO(p;)476 1363 y( )s FP(\()p FO(r)602 1375 y FL(3)640 1363 y FP(\))23 b(=)g FO(p)825 1375 y FL(1)862 1363 y FO(;)97 b( )s FP(\()p FO(r)1108 1375 y FL(4)1146 1363 y FP(\))23 b(=)g FO(p)1331 1375 y FL(2)1368 1363 y FO(;)97 b( )s FP(\()p FO(r)1614 1375 y FL(5)1652 1363 y FP(\))23 b(=)g FO(e)18 b FQ(\000)g FO(p)1977 1375 y FL(1)2032 1363 y FQ(\000)g FO(p)2157 1375 y FL(2)2194 1363 y FO(:)118 1547 y FP(One)30 b(directly)g(c)n(hec)n(ks)f(that)h(the)g(functor)g FO(F)1531 1559 y FM( )1591 1547 y FP(:)43 b(Rep\()p FB(P)1888 1559 y FL(3)p FM(;)p FN(?)p FL(2)2030 1547 y FP(\))27 b FQ(\000)-48 b(!)27 b FP(Rep\()p FB(R)2452 1559 y FL(5)p FM(;)p FL(2)2543 1547 y FP(\))118 1646 y(is)h(full.)243 1746 y(It)i(follo)n(ws)e(directly)h(from)g(Theorem)g(55)g(that)h FC(the)h(fol)t(lowing)j FQ(\003)p FC(-algebr)l(a)118 1846 y(is)c FQ(\003)p FC(-wild)p FP(,)237 2104 y FB(R)297 2125 y FL(5)p FM(;)360 2102 y Fy(5)p 360 2111 29 3 v 360 2145 a(2)426 2104 y FP(=)22 b FJ(C)567 2012 y Fz(D)624 2104 y FO(r)661 2116 y FL(1)699 2104 y FO(;)14 b(:)g(:)g(:)f(;)h(r)920 2116 y FL(5)981 2104 y FQ(j)23 b FO(r)1066 2070 y FL(2)1064 2125 y FM(i)1127 2104 y FP(=)g FO(r)1252 2116 y FM(i)1303 2104 y FP(=)g FO(r)1430 2070 y FN(\003)1428 2125 y FM(i)1468 2104 y FO(;)28 b(i)23 b FP(=)g(1)p FO(;)14 b(:)g(:)g(:)f(;)h FP(5;)2006 2000 y FL(5)1963 2025 y Fz(X)1969 2202 y FM(i)p FL(=1)2097 2104 y FO(r)2134 2116 y FM(i)2185 2104 y FP(=)2282 2048 y(5)p 2282 2085 42 4 v 2282 2161 a(2)2348 2104 y FO(e)2387 2012 y Fz(E)426 2339 y FP(=)22 b FJ(C)567 2247 y Fz(D)624 2339 y FO(w)683 2351 y FM(i)734 2339 y FP(=)h(2)p FO(r)901 2351 y FM(i)947 2339 y FQ(\000)18 b FO(e;)c(i)22 b FP(=)h(1)p FO(;)14 b(:)g(:)g(:)f(;)h FP(5)22 b FQ(j)h FO(w)1640 2351 y FM(i)1692 2339 y FP(=)f FO(w)1840 2305 y FN(\003)1838 2360 y FM(i)1879 2339 y FO(;)14 b(w)1977 2305 y FL(2)1975 2360 y FM(i)2038 2339 y FP(=)23 b FO(e;)1565 2576 y(i)g FP(=)f(1)p FO(;)14 b(:)g(:)g(:)f(;)h FP(5;)2052 2472 y FL(5)2009 2497 y Fz(X)2015 2674 y FM(i)p FL(=1)2142 2576 y FO(w)2201 2588 y FM(i)2253 2576 y FP(=)22 b(0)2382 2484 y Fz(E)2432 2576 y FO(:)243 2845 y FP(Indeed,)29 b(its)g(quotien)n(t)g FQ(\003)p FP(-algebra)d FB(R)1404 2866 y FL(5)p FM(;)1467 2844 y Fy(3)p 1467 2853 29 3 v 1467 2886 a(2)1505 2866 y FL(+1)1619 2845 y FP(=)e FJ(C)1762 2778 y Fz(\012)1808 2845 y FO(r)1845 2857 y FL(1)1882 2845 y FO(;)14 b(:)g(:)g(:)g(;)g(r)2104 2857 y FL(5)2167 2845 y FQ(j)25 b FO(r)2254 2815 y FL(2)2252 2867 y FM(i)2317 2845 y FP(=)g FO(r)2446 2815 y FN(\003)2444 2867 y FM(i)2510 2845 y FP(=)118 2973 y FO(r)155 2985 y FM(i)183 2973 y FO(;)j(i)k FP(=)g(1)p FO(;)14 b(:)g(:)g(:)f(;)h FP(5;)697 2911 y Fz(P)784 2932 y FL(3)784 2998 y FM(i)p FL(=1)910 2973 y FO(r)947 2985 y FM(i)1007 2973 y FP(=)1114 2941 y FL(3)p 1114 2955 34 4 v 1114 3002 a(2)1157 2973 y FO(e;)28 b(r)1284 2985 y FL(4)1344 2973 y FP(+)21 b FO(r)1467 2985 y FL(5)1538 2973 y FP(=)32 b FO(e)1674 2906 y Fz(\013)1745 2973 y FP(=)g FJ(C)1896 2906 y Fz(\012)1941 2973 y FO(w)2000 2985 y FM(i)2061 2973 y FP(=)g(2)p FO(r)2237 2985 y FM(i)2287 2973 y FQ(\000)22 b FO(e;)14 b(i)31 b FP(=)118 3090 y(1)p FO(;)14 b(:)g(:)g(:)f(;)h FP(5)23 b FQ(j)g FO(w)514 3102 y FM(i)565 3090 y FP(=)g FO(w)714 3060 y FN(\003)712 3112 y FM(i)752 3090 y FO(;)14 b(w)850 3060 y FL(2)848 3112 y FM(i)911 3090 y FP(=)23 b FO(e;)14 b(i)22 b FP(=)h(1)p FO(;)14 b(:)g(:)g(:)f(;)h FP(5;)1519 3028 y Fz(P)1606 3048 y FL(3)1606 3115 y FM(i)p FL(=1)1731 3090 y FO(w)1790 3102 y FL(1)1851 3090 y FP(=)23 b(0)p FO(;)14 b(w)2077 3102 y FL(4)2124 3090 y FP(+)c FO(w)2258 3102 y FL(5)2319 3090 y FP(=)22 b(0)2448 3023 y Fz(\013)2510 3090 y FP(=)118 3198 y FJ(C)172 3131 y Fz(\012)217 3198 y FO(z)256 3210 y FL(1)320 3198 y FP(=)27 b FO(w)471 3210 y FL(1)529 3198 y FP(+)20 b FO(w)673 3210 y FL(2)711 3198 y FO(;)14 b(z)787 3210 y FL(2)851 3198 y FP(=)980 3165 y FL(1)p 952 3179 88 4 v 952 3188 a FN(p)p 1007 3188 34 3 v 48 x FL(3)1050 3198 y FP(\()p FO(w)1141 3210 y FL(1)1199 3198 y FQ(\000)20 b FO(w)1343 3210 y FL(2)1381 3198 y FP(\))p FO(;)14 b(z)1489 3210 y FL(3)1553 3198 y FP(=)27 b FO(w)1704 3210 y FL(4)1769 3198 y FQ(j)g FO(z)1862 3168 y FN(\003)1858 3221 y FM(k)1927 3198 y FP(=)g FO(z)2058 3210 y FM(k)2098 3198 y FO(;)h(z)2192 3168 y FL(2)2188 3221 y FM(k)2256 3198 y FP(=)f FO(e;)g(k)j FP(=)118 3322 y(1)p FO(;)14 b(:)g(:)g(:)f(;)h FP(3;)g FQ(f)p FO(z)504 3334 y FL(1)540 3322 y FO(;)g(z)616 3334 y FL(2)653 3322 y FQ(g)22 b FP(=)h(0)847 3255 y Fz(\013)909 3322 y FP(=)f FB(P)1051 3334 y FL(3)p FM(;)p FL(2an)n(ti)1282 3322 y FP(is)27 b FQ(\003)p FP(-wild.)118 3473 y FR(6.)33 b FP(F)-7 b(or)18 b(a)g(single)g(idemp)r(oten)n(t,)j(the)e(situation)g (is)f(similar)g(to)g(the)h(situation)f(for)118 3572 y(t)n(w)n(o)25 b(orthogonal)f(pro)5 b(jections.)35 b(Consider)26 b(the)g FQ(\003)p FP(-algebra)e FB(Q)2047 3584 y FL(1)2110 3572 y FP(generated)h(b)n(y)118 3672 y(an)c(idemp)r(oten)n(t)g(and)g(its)g (adjoin)n(t,)h FO(q)1258 3684 y FL(1)1295 3672 y FP(,)g FO(q)1380 3642 y FN(\003)1377 3693 y FL(1)1419 3672 y FP(.)34 b(Let)21 b FO(q)1655 3684 y FL(1)1716 3672 y FP(=)h FO(a)1847 3684 y FL(1)1889 3672 y FP(+)5 b FO(ib)2024 3684 y FL(1)2060 3672 y FP(,)23 b FO(q)2146 3642 y FN(\003)2143 3693 y FL(1)2207 3672 y FP(=)g FO(a)2339 3684 y FL(1)2381 3672 y FQ(\000)5 b FO(ib)2516 3684 y FL(1)2552 3672 y FP(,)118 3772 y(where)24 b FO(a)399 3742 y FN(\003)399 3792 y FL(1)461 3772 y FP(=)e FO(a)592 3784 y FL(1)629 3772 y FP(,)k FO(b)714 3742 y FN(\003)714 3792 y FL(1)775 3772 y FP(=)c FO(b)898 3784 y FL(1)935 3772 y FP(.)36 b(The)25 b FQ(\003)p FP(-algebra)e FB(Q)1575 3784 y FL(1)1637 3772 y FP(coincides)h(with)h(the)h(algebra)318 3955 y FJ(C)372 3888 y Fz(\012)418 3955 y FO(a;)14 b(b)22 b FQ(j)h FO(a)g FP(=)g FO(a)802 3921 y FN(\003)840 3955 y FO(;)14 b(b)22 b FP(=)h FO(b)1059 3921 y FN(\003)1097 3955 y FP(;)14 b FQ(f)p FO(a;)g(b)p FQ(g)21 b FP(=)i FO(ab)18 b FP(+)g FO(ba)k FP(=)h(0)p FO(;)k(a)1951 3921 y FL(2)2007 3955 y FQ(\000)18 b FO(b)2126 3921 y FL(2)2186 3955 y FP(=)23 b FO(e)2313 3888 y Fz(\013)2351 3955 y FO(;)118 4147 y FP(where)k FO(a)c FP(=)g(2)555 4080 y Fz(\000)592 4147 y FO(a)636 4159 y FL(1)692 4147 y FQ(\000)785 4115 y FL(1)p 785 4129 34 4 v 785 4176 a(2)828 4147 y FO(e)867 4080 y Fz(\001)905 4147 y FP(,)k FO(b)c FP(=)g(2)p FO(b)1180 4159 y FL(1)1216 4147 y FP(.)p eop %%Page: 219 223 219 222 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(219)243 333 y(Irreducible)27 b(represen)n(tations)g(of)h(the)h(algebra)e FB(Q)1824 345 y FL(1)1890 333 y FP(\(see)h(Section)h(1.2.2\),)118 432 y(up)f(to)g(a)f(unitary)g(equiv)-5 b(alence,)27 b(coincide)g(with)h (one)g(of)f(the)h(follo)n(wing:)243 532 y(1\))h(t)n(w)n(o)g (one-dimensional)f(represen)n(tations)g(giv)n(en)g(b)n(y)h FO(\031)2076 544 y FL(0)2114 532 y FP(\()p FO(q)2183 544 y FL(1)2221 532 y FP(\))d(=)g(0)j(and)118 632 y FO(\031)165 644 y FL(1)203 632 y FP(\()p FO(q)272 644 y FL(1)309 632 y FP(\))24 b(=)e(1;)243 731 y(2\))i(a)g(family)-7 b(,)26 b(dep)r(ending)f(on)f(a)g(parameter)f FO(\013)h(>)e FP(0,)j(of)g(t)n(w)n(o-dimensional)118 831 y(represen)n(tations:)1006 1046 y FO(\031)1053 1058 y FM(\013)1100 1046 y FP(\()p FO(q)1169 1058 y FL(1)1207 1046 y FP(\))e(=)1350 929 y Fz(\022)1411 995 y FP(1)83 b FO(\013)1411 1095 y FP(0)89 b(0)1589 929 y Fz(\023)1664 1046 y FO(:)243 1261 y FP(By)38 b(decomp)r(osing)h(a)f(represen)n(tation)g(of)h(the)g(algebra)f FB(Q)2141 1273 y FL(1)2217 1261 y FP(in)n(to)h(a)g(di-)118 1361 y(rect)26 b(sum)g(of)g(irreducible)f(represen)n(tations)f(on)i(a)g (\014nite)g(dimensional)g(space)118 1461 y FO(H)7 b FP(,)34 b(w)n(e)f(obtain)g(the)g(structure)g(theorem)f(\(see)h([70)o(,)g(115)o (]\))g(for)g(the)g(unitary)118 1560 y(description)27 b(of)h(idemp)r(oten)n(ts)g(in)f(the)h(\014nite)h(dimensional)e(case.) 243 1660 y(There)43 b(is)g(a)g(structure)g(theorem)g(that)h(giv)n(es)e (a)h(description)g(of)h(an)n(y)118 1760 y(b)r(ounded)31 b(idemp)r(oten)n(t)h(on)e(an)n(y)g(separable)f(Hilb)r(ert)i(space)f(in) h(the)h(form)e(of)118 1859 y(an)d(in)n(tegral)g(of)g(irreducible)g (represen)n(tations.)118 2003 y FR(7.)56 b FP(F)-7 b(urther,)35 b(consider)e(the)i(problem)e(of)h(a)g(unitary)f(description)h(of)g (pairs)118 2103 y(of)j(idemp)r(oten)n(ts)g FO(Q)768 2115 y FL(1)805 2103 y FP(,)i FO(Q)933 2115 y FL(2)1007 2103 y FP(\()p FO(Q)1105 2073 y FL(2)1105 2124 y(1)1181 2103 y FP(=)f FO(Q)1350 2115 y FL(1)1387 2103 y FP(,)i FO(Q)1516 2073 y FL(2)1516 2124 y(2)1591 2103 y FP(=)e FO(Q)1760 2115 y FL(2)1797 2103 y FP(\).)65 b(The)37 b(fact)g(that)g(the)118 2203 y(problem)22 b(of)h(a)f(unitary)g(description)g(of)g(pairs)g(of)g (idemp)r(oten)n(ts)h(is)g(di\016cult)g(is)118 2302 y(just)g(a)e (mathematical)h(folklore.)33 b(W)-7 b(e)23 b(will)f(pro)n(v)n(e)e(a)h (corresp)r(onding)f(theorem)118 2402 y(and)27 b(sho)n(w)e(that,)i(ev)n (en)g(if)g(an)f(additional)g(restriction)g(of)g(self-adjoin)n(tness)g (is)118 2502 y(imp)r(osed)36 b(on)g(one)f(of)h(the)g(idemp)r(oten)n(ts) g(\(one)g(of)g(the)g(idemp)r(oten)n(ts)g(is)g(an)118 2601 y(orthogonal)25 b(pro)5 b(jection\),)27 b(the)h(problem)f(do)r(es) h(not)f(b)r(ecome)h(easer.)118 2757 y FR(Theorem)35 b(58.)42 b FC(L)l(et)33 b FB(Q)895 2769 y FL(2)961 2757 y FP(=)28 b FJ(C)15 b FQ(h)q FO(q)1178 2769 y FL(1)1221 2757 y FO(;)f(q)1295 2769 y FL(2)1332 2757 y FO(;)g(q)1409 2727 y FN(\003)1406 2778 y FL(1)1447 2757 y FO(;)g(q)1524 2727 y FN(\003)1521 2778 y FL(2)1591 2757 y FQ(j)29 b FO(q)1683 2727 y FL(2)1680 2778 y(1)1749 2757 y FP(=)g FO(q)1880 2769 y FL(1)1917 2757 y FO(;)14 b(q)1994 2727 y FL(2)1991 2778 y(2)2060 2757 y FP(=)29 b FO(q)2194 2727 y FL(2)2231 2757 y FQ(i)p FC(,)34 b FB(D)2391 2769 y FL(1)p FM(;)p FL(1)2510 2757 y FP(=)118 2857 y FJ(C)15 b FQ(h)p FO(q)s(;)f(q)321 2826 y FN(\003)366 2857 y FO(;)g(p)26 b FQ(j)h FO(q)561 2826 y FL(2)626 2857 y FP(=)f FO(q)s(;)14 b(p)836 2826 y FL(2)900 2857 y FP(=)27 b FO(p)g FP(=)g FO(p)1195 2826 y FN(\003)1233 2857 y FQ(i)p FC(,)33 b FA(S)1392 2869 y FL(2)1456 2857 y FP(=)27 b FJ(C)15 b FQ(h)p FO(a)1678 2869 y FL(1)1721 2857 y FO(;)f(a)1802 2869 y FL(2)1866 2857 y FQ(j)27 b FO(a)1960 2869 y FL(1)2025 2857 y FP(=)f FO(a)2160 2826 y FN(\003)2160 2877 y FL(1)2198 2857 y FO(;)14 b(a)2279 2869 y FL(2)2343 2857 y FP(=)27 b FO(a)2479 2826 y FN(\003)2479 2877 y FL(2)2517 2857 y FQ(i)p FC(.)118 2956 y(Then)f FB(Q)385 2968 y FL(2)445 2956 y FQ(\037)d FB(D)602 2968 y FL(1)p FM(;)p FL(1)715 2956 y FQ(\037)f FA(S)871 2968 y FL(2)908 2956 y FC(,)27 b(so)e(that)g(the)g FQ(\003)p FC(-algebr)l(as)g FB(Q)1795 2968 y FL(2)1857 2956 y FC(and)h FB(D)2083 2968 y FL(1)p FM(;)p FL(1)2198 2956 y FC(ar)l(e)f FQ(\003)p FC(-wild.)118 3112 y(Pr)l(o)l(of.)43 b FP(Because)23 b FB(D)761 3124 y FL(1)p FM(;)p FL(1)876 3112 y FP(is)h(a)g(quotien)n(t)g(algebra)e(of) j(the)f(algebra)f FB(Q)2207 3124 y FL(2)2244 3112 y FP(,)i(w)n(e)f(ha)n (v)n(e)118 3212 y(that)29 b FB(Q)354 3224 y FL(2)415 3212 y FQ(\037)24 b FB(D)573 3224 y FL(1)p FM(;)p FL(1)692 3212 y FP(\(w)n(e)k(c)n(ho)r(ose)f(an)h(en)n(v)n(eloping)f(algebra)g (for)h FB(D)2125 3224 y FL(1)p FM(;)p FL(1)2243 3212 y FP(to)h(b)r(e)f(the)118 3311 y(algebra)23 b FB(D)475 3323 y FL(1)p FM(;)p FL(1)589 3311 y FP(itself,)j FO(n)d FP(=)g(1,)i FO( )12 b FP(:)28 b FB(Q)1234 3323 y FL(2)1294 3311 y FQ(\000)-49 b(!)24 b FB(D)1486 3323 y FL(1)p FM(;)p FL(1)1600 3311 y FP(is)h(the)g(natural)e(epimorphism)118 3411 y(of)28 b(the)g(algebra)d(on)n(to)i(the)h(quotien)n(t)g (algebra\).)243 3510 y(Let)41 b(us)g(sho)n(w)f(that)h FB(D)1006 3522 y FL(1)p FM(;)p FL(1)1142 3510 y FQ(\037)k FA(S)1321 3522 y FL(2)1358 3510 y FP(.)77 b(Construct)41 b(the)g(homomorphism)118 3610 y FO( )12 b FP(:)28 b FB(D)304 3622 y FL(1)p FM(;)p FL(1)417 3610 y FQ(\000)-48 b(!)23 b FO(M)621 3622 y FL(2)658 3610 y FP(\()p FA(S)759 3622 y FL(2)796 3610 y FP(\):)539 3833 y FO( )s FP(\()p FO(q)s FP(\))h(=)812 3716 y Fz(\022)874 3782 y FO(e)84 b(a)1041 3794 y FL(1)1097 3782 y FP(+)18 b FO(ia)1253 3794 y FL(2)873 3882 y FP(0)208 b(0)1290 3716 y Fz(\023)1365 3833 y FO(;)97 b( )s FP(\()p FO(p)p FP(\))23 b(=)1769 3776 y(1)p 1769 3814 42 4 v 1769 3890 a(2)1834 3716 y Fz(\022)1895 3782 y FO(e)83 b(e)1895 3882 y(e)g(e)2055 3716 y Fz(\023)2130 3833 y FO(:)118 4048 y FP(It)34 b(is)g(easy)e(to)i(c)n(hec)n(k)f(that)g (the)h(corresp)r(onding)e(functor)i FO(F)2052 4060 y FM( )2111 4048 y FP(:)44 b(Rep)14 b FA(S)2405 4060 y FL(2)2475 4048 y FQ(\000)-48 b(!)118 4147 y FP(Rep)14 b FB(D)345 4159 y FL(1)p FM(;)p FL(1)463 4147 y FP(is)27 b(full.)p 2514 4147 4 57 v 2518 4095 50 4 v 2518 4147 V 2567 4147 4 57 v eop %%Page: 220 224 220 223 bop 118 100 a FP(220)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FR(Corollary)38 b(11.)43 b FC(The)35 b(algebr)l(a)g FB(Q)1229 345 y FM(n)1275 333 y FC(,)g(for)f FO(n)d FQ(\025)f FP(2)j(\()p FC(the)h(pr)l(oblem)h(of)g(unitary)118 432 y(description)d(of)e FO(n)g FC(idemp)l(otents)g(if)h FO(n)23 b FQ(\025)f FP(2\))p FC(,)30 b(is)g FQ(\003)p FC(-wild.)118 626 y FR(8.)36 b FP(Finally)-7 b(,)25 b(w)n(e)h(sho)n(w)e (that)i(the)g FQ(\003)p FP(-algebra)d FB(Q)1581 638 y FM(n;)p FN(?)1723 626 y FP(\(the)j(problem)f(of)g(unitary)118 726 y(classi\014cation)e(of)h(a)g(family)g(of)g(pairwise)f(orthogonal)f (idemp)r(oten)n(ts)j FO(Q)2364 738 y FL(1)2401 726 y FP(,)g FO(Q)2515 738 y FL(2)2552 726 y FP(,)118 826 y FO(:)14 b(:)g(:)28 b FP(,)f FO(Q)359 838 y FM(n)404 826 y FP(,)h FO(Q)521 838 y FM(i)548 826 y FO(Q)614 838 y FM(j)672 826 y FP(=)23 b(0)k(for)g FO(i)22 b FQ(6)p FP(=)h FO(j)5 b FP(\))28 b(is)f FQ(\003)p FP(-wild)h(for)f FO(n)22 b FQ(\025)h FP(2.)118 1020 y FR(Theorem)31 b(59.)40 b FC(L)l(et)293 1221 y FB(Q)348 1233 y FL(2)p FM(;)p FN(?)480 1221 y FP(=)22 b FJ(C)621 1154 y Fz(\012)667 1221 y FO(q)704 1233 y FL(1)741 1221 y FO(;)14 b(q)815 1233 y FL(2)852 1221 y FO(;)g(q)929 1187 y FN(\003)926 1242 y FL(1)967 1221 y FO(;)g(q)1044 1187 y FN(\003)1041 1242 y FL(2)1105 1221 y FQ(j)24 b FO(q)1192 1187 y FL(2)1189 1242 y(1)1252 1221 y FP(=)f FO(q)1377 1233 y FL(1)1414 1221 y FO(;)28 b(q)1505 1187 y FL(2)1502 1242 y(2)1565 1221 y FP(=)23 b FO(q)1690 1233 y FL(2)1727 1221 y FO(;)28 b(q)1815 1233 y FL(1)1852 1221 y FO(q)1889 1233 y FL(2)1949 1221 y FP(=)23 b FO(q)2074 1233 y FL(2)2111 1221 y FO(q)2148 1233 y FL(1)2209 1221 y FP(=)f(0)2338 1154 y Fz(\013)2377 1221 y FO(:)118 1423 y FC(Then)37 b FB(Q)396 1435 y FL(2)p FM(;)p FN(?)528 1423 y FQ(\037)22 b FA(S)684 1435 y FL(2)721 1423 y FC(,)31 b(i.e.,)g FB(Q)1001 1435 y FL(2)p FM(;)p FN(?)1140 1423 y FC(is)f(a)g(wild)h FQ(\003)p FC(-algebr)l(a.)118 1617 y(Pr)l(o)l(of.)43 b FP(Let)36 b(us)f(de\014ne)h(a)f(homomorphism)f FO( )12 b FP(:)31 b FB(Q)1743 1629 y FL(2)p FM(;)p FN(?)1888 1617 y FQ(\000)-49 b(!)37 b FO(M)2105 1629 y FL(3)2141 1617 y FP(\()p FA(S)2242 1629 y FL(2)2280 1617 y FP(\))e(as)g(fol-)118 1717 y(lo)n(ws:)366 1997 y FO( )s FP(\()p FO(q)492 2009 y FL(1)530 1997 y FP(\))23 b(=)673 1830 y Fz(2)673 1980 y(4)730 1897 y FO(e)85 b(e)f(a)1021 1909 y FL(1)1077 1897 y FP(+)18 b FO(ia)1233 1909 y FL(2)728 1996 y FP(0)83 b(0)208 b(0)728 2096 y(0)83 b(0)208 b(0)1269 1830 y Fz(3)1269 1980 y(5)1339 1997 y FO(;)97 b( )s FP(\()p FO(q)1585 2009 y FL(2)1622 1997 y FP(\))24 b(=)1765 1830 y Fz(2)1765 1980 y(4)1821 1897 y FP(0)82 b FQ(\000)p FO(e)g FQ(\000)p FO(e)1821 1996 y FP(0)114 b FO(e)148 b(e)1821 2096 y FP(0)113 b(0)144 b(0)2235 1830 y Fz(3)2235 1980 y(5)2304 1997 y FO(:)118 2303 y FP(One)20 b(can)g(directly)g(c)n(hec)n(k)f(that) h([)p FO( )s FP(\()p FO(q)1263 2315 y FM(k)1305 2303 y FP(\)])1360 2273 y FL(2)1420 2303 y FP(=)j FO( )s FP(\()p FO(q)1634 2315 y FM(k)1675 2303 y FP(\),)f FO(k)k FP(=)d(1,)e(2,)g FO( )s FP(\()p FO(q)2207 2315 y FL(1)2245 2303 y FP(\))14 b FO( )s FP(\()p FO(q)2417 2315 y FL(2)2455 2303 y FP(\))23 b(=)118 2403 y FO( )s FP(\()p FO(q)244 2415 y FL(2)282 2403 y FP(\))14 b FO( )s FP(\()p FO(q)454 2415 y FL(1)492 2403 y FP(\))33 b(=)g(0,)i(and)f(that)g(the)g(functor)g FO(F)1606 2415 y FM( )1666 2403 y FP(:)44 b(Rep)14 b FA(S)1960 2415 y FL(2)2030 2403 y FQ(\000)-48 b(!)33 b FP(Rep)14 b FB(Q)2376 2415 y FL(2)p FM(;)p FN(?)2519 2403 y FP(is)118 2503 y(full.)p 2514 2503 4 57 v 2518 2450 50 4 v 2518 2503 V 2567 2503 4 57 v 118 2744 a FR(Corollary)25 b(12.)34 b FC(The)24 b(pr)l(oblem)h(of)f(unitary)f(classi\014c)l(ation) h(of)h(p)l(airs)f(of)g(c)l(om-)118 2843 y(muting)29 b(idemp)l(otents)h (is)h FQ(\003)p FC(-wild.)118 3038 y FR(Corollary)45 b(13.)h FC(The)40 b FQ(\003)p FC(-algebr)l(a)g FB(Q)1321 3050 y FM(n;)p FN(?)1478 3038 y FP(=)f FJ(C)15 b FQ(h)p FO(q)1706 3050 y FL(1)1749 3038 y FO(;)f(:)g(:)g(:)f(;)h(q)1970 3050 y FM(n)2055 3038 y FQ(j)40 b FO(q)2158 3008 y FL(2)2155 3059 y FM(i)2236 3038 y FP(=)f FO(q)2377 3050 y FM(i)2405 3038 y FO(;)14 b(i)39 b FP(=)118 3137 y(1)p FO(;)14 b(:)g(:)g(:)f(;)h (n)p FP(;)28 b FO(q)482 3149 y FM(i)509 3137 y FO(q)546 3149 y FM(j)606 3137 y FP(=)c(0)29 b FC(for)i FO(i)24 b FQ(6)p FP(=)g FO(j)5 b FQ(i)31 b FP(\()p FC(the)g(pr)l(oblem)g(of)h (unitary)e(classi\014c)l(ation)i(of)118 3237 y FO(n)e FC(p)l(airwise)h(ortho)l(gonal)g(idemp)l(otents)7 b FP(\))31 b FC(is)f FQ(\003)p FC(-wild)g(for)h FO(n)23 b FQ(\025)f FP(2)p FC(.)118 3431 y FR(Corollary)42 b(14.)j FC(The)37 b FQ(\003)p FC(-algebr)l(a)g FJ(C)15 b FQ(h)q FO(q)1380 3443 y FL(1)1423 3431 y FO(;)f(:)g(:)g(:)g(;)g(q)1645 3443 y FM(n)1725 3431 y FQ(j)36 b FO(q)1824 3401 y FL(2)1821 3453 y FM(i)1896 3431 y FP(=)f FO(q)2033 3443 y FM(i)2061 3431 y FO(;)28 b(i)35 b FP(=)g(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n)p FP(;)118 3531 y FO(q)155 3543 y FL(1)212 3531 y FP(+)20 b FQ(\001)14 b(\001)g(\001)20 b FP(+)f FO(q)535 3543 y FM(n)607 3531 y FP(=)27 b FO(e)p FQ(i)k FP(\()p FC(the)i(pr)l(oblem)g (of)f(unitary)g(classi\014c)l(ation)h(of)51 b FO(n)31 b FC(idem-)118 3630 y(p)l(otents)38 b FO(Q)477 3642 y FL(1)514 3630 y FC(,)g FO(:)14 b(:)g(:)28 b FC(,)78 b FO(Q)871 3642 y FM(n)954 3630 y FC(such)38 b(that)g FO(Q)1395 3642 y FL(1)1457 3630 y FP(+)24 b FQ(\001)14 b(\001)g(\001)24 b FP(+)g FO(Q)1822 3642 y FM(n)1905 3630 y FP(=)38 b FO(I)7 b FP(\))38 b FC(is)g FQ(\003)p FC(-wild)h(for)118 3730 y FO(n)23 b FQ(\025)g FP(3)p FC(.)118 3924 y(Pr)l(o)l(of.)43 b FP(If)37 b FO(m)h FP(=)g(3,)h(the)e(condition)f FO(q)1350 3936 y FL(1)1412 3924 y FP(+)24 b FO(q)1538 3936 y FL(2)1600 3924 y FP(+)g FO(q)1726 3936 y FL(3)1801 3924 y FP(=)38 b FO(e)f FP(implies)g(that)f(the)118 4024 y(idemp)r(oten)n(ts)h FO(q)635 4036 y FL(1)672 4024 y FP(,)i FO(q)771 4036 y FL(2)808 4024 y FP(,)g FO(q)907 4036 y FL(3)981 4024 y FP(are)c(pairwise)g(orthogonal.)61 b(Then)37 b(the)g(algebra)118 4124 y(under)28 b(consideration)e(coincides)h(with)h(the)g(algebra)e FB(Q)1891 4136 y FL(2)p FM(;)p FN(?)1999 4124 y FP(.)p 2514 4124 V 2518 4071 50 4 v 2518 4124 V 2567 4124 4 57 v eop %%Page: 221 225 221 224 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(221)118 333 y FR(3.1.4)94 b FQ(\003)p FR(-Wild)30 b(semilinear)f(relations)118 488 y(1.)58 b FP(In)35 b(Sections)g(1.3.2-1.3.5)d(w)n(e)j(studied)g (represen)n(tations)e(of)i(semilinear)118 588 y(relations.)40 b(In)29 b(particular,)f(the)i(structure)e(of)h(pairs)f(of)h(op)r (erators)f FO(A)d FP(=)g FO(A)2513 557 y FN(\003)2552 588 y FP(,)118 687 y FO(B)i FP(=)c FO(B)363 657 y FN(\003)429 687 y FP(whic)n(h)k(satisfy)h(the)g(semilinear)e(relation)988 830 y FM(n)948 855 y Fz(X)948 1034 y FM(k)q FL(=1)1082 934 y FO(f)1123 946 y FM(k)1164 934 y FP(\()p FO(A)p FP(\))14 b FO(B)19 b(g)1426 946 y FM(k)1466 934 y FP(\()p FO(A)p FP(\))24 b(=)f(0)658 b(\(3.1\))118 1193 y(w)n(as)27 b(studied.)243 1294 y(This)g(relation)g(corresp)r(onds)f(to)h(the)h(c)n (haracteristic)e(function)846 1541 y(\010\()p FO(t;)14 b(s)p FP(\))23 b(=)1227 1437 y FM(n)1187 1462 y Fz(X)1187 1641 y FM(k)q FL(=1)1321 1541 y FO(f)1362 1553 y FM(k)1403 1541 y FP(\()p FO(t)p FP(\))14 b FO(g)1551 1553 y FM(k)1592 1541 y FP(\()p FO(s)p FP(\))23 b(=)g(0)556 b(\(3.2\))118 1821 y(\(w)n(e)30 b(supp)r(ose)g(that)g(\010\()p FO(t;)14 b(s)p FP(\))27 b(=)p 1121 1749 231 4 v 27 w(\010\()p FO(s;)14 b(t)p FP(\),)31 b FO(t)p FP(,)g FO(s)c FQ(2)g FJ(R)p FP(\).)50 b(If)31 b(the)f(graph)f(\()p FJ(R)2351 1791 y FL(1)2394 1821 y FO(;)14 b FP(\000)27 b(=)118 1921 y FQ(f)p FP(\()p FO(t;)14 b(s)p FP(\))34 b FQ(2)h FJ(R)508 1891 y FL(2)586 1921 y FP(:)f(\010\()p FO(t;)14 b(s)p FP(\))35 b(=)f(0)p FQ(g)p FP(\))f(has)h(only)g(connected)g(comp)r (onen)n(ts)g(of)g(the)118 2059 y(form)360 2041 y Fo(r)418 2059 y FP(,)515 2041 y Fo(r)p 515 2043 4 4 v 510 2038 V 505 2033 V 501 2029 V 498 2024 V 494 2020 V 491 2015 V 489 2011 V 487 2007 V 485 2003 V 483 2000 V 482 1996 V 482 1992 V 481 1989 V 481 1986 V 482 1982 V 483 1979 V 484 1976 V 485 1974 V 487 1971 V 490 1968 V 490 1968 V 492 1966 V 495 1964 V 497 1962 V 500 1960 V 502 1959 V 505 1958 V 507 1957 V 510 1956 V 512 1956 V 515 1956 V 517 1956 V 520 1956 V 522 1957 V 524 1958 V 527 1959 V 529 1960 V 532 1962 V 534 1964 V 537 1966 V 539 1968 V 515 2043 V 519 2038 V 524 2033 V 528 2029 V 531 2024 V 535 2020 V 538 2015 V 540 2011 V 542 2007 V 544 2003 V 546 2000 V 547 1996 V 547 1992 V 548 1989 V 548 1986 V 547 1982 V 546 1979 V 545 1976 V 544 1974 V 542 1971 V 539 1968 V 573 2059 a FP(,)e(or)775 2041 y Fo(r)p 775 2043 125 4 v 99 w(r)958 2059 y FP(,)g(then)g(irreducible)e(represen)n (tations)g(of)h(relation)118 2159 y(\(3.1\))c(are)g(one-)g(and)g(t)n(w) n(o-dimensional)f(and)i(w)n(ere)e(describ)r(ed)i(in)g(1.3.5.)118 2312 y FR(2.)36 b FP(W)-7 b(e)28 b(sho)n(w)f(that)h(all)f(other)g (relations)g(are)f FQ(\003)p FP(-wild.)118 2481 y FR(Prop)s(osition)33 b(67.)42 b FC(If)32 b(the)g(gr)l(aph)h(of)f(semiline)l(ar)h(r)l (elation)39 b FP(\(3.1\))32 b FC(c)l(ontains)118 2619 y(a)e(sub)l(gr)l(aph)570 2602 y Fo(r)p 570 2603 4 4 v 565 2598 V 561 2593 V 557 2589 V 553 2584 V 550 2580 V 547 2576 V 544 2571 V 542 2567 V 540 2563 V 539 2560 V 538 2556 V 537 2552 V 537 2549 V 537 2546 V 537 2542 V 538 2539 V 539 2536 V 541 2534 V 543 2531 V 545 2528 V 545 2528 V 548 2526 V 550 2524 V 553 2522 V 555 2520 V 558 2519 V 560 2518 V 563 2517 V 565 2517 V 568 2516 V 570 2516 V 573 2516 V 575 2517 V 578 2517 V 580 2518 V 583 2519 V 585 2520 V 588 2522 V 590 2524 V 593 2526 V 595 2528 V 570 2603 V 575 2598 V 579 2593 V 583 2589 V 587 2584 V 590 2580 V 593 2576 V 596 2571 V 598 2567 V 600 2563 V 601 2560 V 602 2556 V 603 2552 V 603 2549 V 603 2546 V 603 2542 V 602 2539 V 601 2536 V 599 2534 V 597 2531 V 595 2528 V 570 2603 125 4 v 100 w(r)753 2619 y FC(or)901 2602 y Fo(r)p 901 2603 V 100 w(r)p 1026 2603 V 100 w(r)1192 2619 y FC(,)g(then)g(the)g(r)l(elation)g(is)g FQ(\003)p FC(-wild.)118 2800 y(Pr)l(o)l(of.)43 b FP(W)-7 b(e)24 b(assume)e(that)h(the)h(functions)f FO(f)1507 2812 y FM(k)1548 2800 y FP(\()p FQ(\001)p FP(\))h(and)e FO(g)1855 2812 y FM(k)1896 2800 y FP(\()p FQ(\001)p FP(\))i(are)e(p)r (olynomials)118 2900 y(and)28 b(pro)n(v)n(e)d(that)j(the)g FQ(\003)p FP(-algebra)512 3146 y FA(A)572 3158 y FL(\000)640 3146 y FP(=)23 b FJ(C)782 3054 y Fz(D)838 3146 y FO(a)g FP(=)g FO(a)1037 3112 y FN(\003)1075 3146 y FO(;)14 b(b)22 b FP(=)h FO(b)1294 3112 y FN(\003)1355 3146 y FQ(j)1441 3043 y FM(n)1402 3068 y Fz(X)1401 3246 y FM(k)q FL(=1)1536 3146 y FO(f)1577 3158 y FM(k)1617 3146 y FP(\()p FO(a)p FP(\))14 b FO(b)g(g)1829 3158 y FM(k)1869 3146 y FP(\()p FO(a)p FP(\))24 b(=)f(0)2131 3054 y Fz(E)118 3457 y FP(is)28 b FQ(\003)p FP(-wild)f(if)h(\000)23 b FQ(\033)735 3439 y Fo(r)p 735 3441 4 4 v 731 3436 V 726 3431 V 722 3427 V 719 3422 V 715 3418 V 712 3413 V 710 3409 V 708 3405 V 706 3401 V 704 3397 V 703 3394 V 703 3390 V 702 3387 V 702 3384 V 703 3380 V 704 3377 V 705 3374 V 706 3372 V 708 3369 V 711 3366 V 711 3366 V 713 3364 V 716 3362 V 718 3360 V 721 3358 V 723 3357 V 726 3356 V 728 3355 V 730 3354 V 733 3354 V 735 3354 V 738 3354 V 740 3354 V 743 3355 V 745 3356 V 748 3357 V 750 3358 V 753 3360 V 755 3362 V 758 3364 V 760 3366 V 735 3441 V 740 3436 V 745 3431 V 749 3427 V 752 3422 V 756 3418 V 759 3413 V 761 3409 V 763 3405 V 765 3401 V 767 3397 V 768 3394 V 768 3390 V 769 3387 V 769 3384 V 768 3380 V 767 3377 V 766 3374 V 765 3372 V 763 3369 V 760 3366 V 735 3441 125 4 v 100 w(r)918 3457 y FP(\()p FO(\025)998 3469 y FL(1)1036 3457 y FO(;)14 b(\025)1121 3469 y FL(2)1182 3457 y FQ(2)23 b FJ(R)p FP(,)34 b FO(\025)1419 3469 y FL(1)1480 3457 y FQ(6)p FP(=)22 b FO(\025)1615 3469 y FL(2)1653 3457 y FP(\).)243 3558 y(De\014ne)28 b(a)f FQ(\003)p FP(-homomorphism)f FO( )12 b FP(:)28 b FA(A)1402 3570 y FL(\000)1469 3558 y FQ(\000)-48 b(!)23 b FO(M)1673 3570 y FL(3)1710 3558 y FP(\()p FA(S)1811 3570 y FL(2)1848 3558 y FP(\))28 b(as)f(follo)n(ws:)389 3843 y FO( )s FP(\()p FO(a)p FP(\))d(=)666 3676 y Fz(0)666 3825 y(@)738 3742 y FO(\025)786 3754 y FL(1)824 3742 y FO(e)124 b FP(0)165 b(0)780 3842 y(0)124 b FO(\025)994 3854 y FL(1)1031 3842 y FO(e)g FP(0)780 3941 y(0)165 b(0)124 b FO(\025)1201 3953 y FL(2)1239 3941 y FO(e)1277 3676 y Fz(1)1277 3825 y(A)1364 3843 y FO(;)97 b( )s FP(\()p FO(b)p FP(\))23 b(=)1752 3676 y Fz(0)1752 3825 y(@)1824 3742 y FO(a)1868 3754 y FL(1)2010 3742 y FO(e)105 b(e)1845 3842 y(e)f(a)2032 3854 y FL(2)2153 3842 y FP(0)1845 3941 y FO(e)124 b FP(0)103 b(0)2194 3676 y Fz(1)2194 3825 y(A)2281 3843 y FO(:)243 4124 y FP(It)28 b(is)f(easy)g(to)g(c)n(hec)n(k)g(that)h(the)g(functor)f FO(F)1588 4136 y FM( )1666 4124 y FP(is)h(full.)p 2514 4124 4 57 v 2518 4071 50 4 v 2518 4124 V 2567 4124 4 57 v eop %%Page: 222 226 222 225 bop 118 100 a FP(222)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FR(Prop)s(osition)23 b(68.)34 b FC(If)24 b(the)g(gr)l(aph)g FP(\000)g FC(of)g(semiline)l(ar)h(r)l(elation)30 b FP(\(3.1\))24 b FC(c)l(ontains)118 432 y(a)31 b(sub)l(gr)l(aph)572 414 y Fo(r)p 572 416 125 4 v 100 w(r)p 697 416 V 99 w(r)536 489 y FM(\025)575 497 y Fy(1)661 489 y FM(\025)700 497 y Fy(2)785 489 y FM(\025)824 497 y Fy(3)893 432 y FC(,)h FO(\025)998 444 y FL(1)1060 432 y FQ(6)p FP(=)25 b FO(\025)1198 444 y FL(2)1260 432 y FQ(6)p FP(=)g FO(\025)1398 444 y FL(3)1460 432 y FQ(6)p FP(=)g FO(\025)1598 444 y FL(1)1635 432 y FP(;)32 b FO(\025)1738 444 y FL(1)1775 432 y FC(,)g FO(\025)1880 444 y FL(2)1917 432 y FC(,)g FO(\025)2022 444 y FL(3)2084 432 y FQ(2)26 b FJ(R)p FC(,)37 b(then)31 b(the)118 565 y(r)l(elation)g(is)f FQ(\003)p FC(-wild.)118 747 y(Pr)l(o)l(of.)43 b FP(W)-7 b(e)29 b(pro)n(v)n(e)e(that)i FA(A)988 759 y FL(\000)1057 747 y FP(=)24 b FJ(C)1200 679 y Fz(\012)1245 747 y FO(a)h FP(=)f FO(a)1447 717 y FN(\003)1485 747 y FO(;)14 b(b)24 b FP(=)g FO(b)1707 717 y FN(\003)1770 747 y FQ(j)1817 684 y Fz(P)1905 705 y FM(n)1905 772 y(k)q FL(=1)2044 747 y FO(f)2085 759 y FM(k)2125 747 y FP(\()p FO(a)p FP(\))14 b FO(b)g(g)2337 759 y FM(k)2377 747 y FP(\()p FO(a)p FP(\))25 b(=)118 855 y(0)160 787 y Fz(\013)227 855 y FP(is)i FQ(\003)p FP(-wild)g(if)h(\000)23 b FQ(\033)816 837 y Fo(r)p 816 838 V 100 w(r)p 941 838 V 99 w(r)781 911 y FM(\025)820 919 y Fy(1)905 911 y FM(\025)944 919 y Fy(2)1029 911 y FM(\025)1068 919 y Fy(3)1134 855 y FP(\()p FO(\025)1214 867 y FL(1)1252 855 y FO(;)14 b(\025)1337 867 y FL(2)1375 855 y FO(;)g(\025)1460 867 y FL(3)1521 855 y FQ(2)23 b FJ(R)p FP(,)34 b FO(\025)1758 867 y FL(1)1818 855 y FQ(6)p FP(=)23 b FO(\025)1954 867 y FL(2)2015 855 y FQ(6)p FP(=)f FO(\025)2150 867 y FL(3)2211 855 y FQ(6)p FP(=)h FO(\025)2347 867 y FL(1)2384 855 y FP(\).)243 997 y(W)-7 b(e)23 b(construct)f(the)i FQ(\003)p FP(-homomorphism)d FO( )12 b FP(:)28 b FA(A)1708 1009 y FL(\000)1775 997 y FQ(\000)-48 b(!)23 b FO(M)1979 1009 y FL(7)2016 997 y FP(\()p FA(S)2117 1009 y FL(2)2154 997 y FP(\))h(as)e(follo)n(ws:)426 1489 y FO( )s FP(\()p FO(a)p FP(\))i(=)702 1123 y Fz(0)702 1269 y(B)702 1319 y(B)702 1369 y(B)702 1418 y(B)702 1468 y(B)702 1518 y(B)702 1568 y(B)702 1618 y(B)702 1671 y(@)775 1189 y FO(\025)823 1201 y FL(1)861 1189 y FO(e)124 b FP(0)165 b(0)g(0)h(0)f(0)h(0)816 1289 y(0)124 b FO(\025)1030 1301 y FL(1)1068 1289 y FO(e)g FP(0)165 b(0)h(0)f(0)h(0)816 1389 y(0)g(0)124 b FO(\025)1238 1401 y FL(2)1275 1389 y FO(e)g FP(0)166 b(0)f(0)h(0)816 1488 y(0)g(0)f(0)124 b FO(\025)1445 1500 y FL(2)1483 1488 y FO(e)g FP(0)165 b(0)h(0)816 1588 y(0)g(0)f(0)g(0)124 b FO(\025)1652 1600 y FL(2)1690 1588 y FO(e)g FP(0)166 b(0)816 1687 y(0)g(0)f(0)g(0)h(0)124 b FO(\025)1860 1699 y FL(3)1897 1687 y FO(e)h FP(0)816 1787 y(0)166 b(0)f(0)g(0)h(0)f(0)124 b FO(\025)2067 1799 y FL(3)2105 1787 y FO(e)2143 1123 y Fz(1)2143 1269 y(C)2143 1319 y(C)2143 1369 y(C)2143 1418 y(C)2143 1468 y(C)2143 1518 y(C)2143 1568 y(C)2143 1618 y(C)2143 1671 y(A)2230 1489 y FO(;)438 2220 y( )s FP(\()p FO(b)p FP(\))23 b(=)706 1854 y Fz(0)706 2000 y(B)706 2049 y(B)706 2099 y(B)706 2149 y(B)706 2199 y(B)706 2249 y(B)706 2299 y(B)706 2348 y(B)706 2401 y(@)904 1920 y FP(0)208 b(0)83 b FO(e)105 b(e)f(a)1610 1932 y FL(1)1665 1920 y FP(+)18 b FO(ia)1821 1932 y FL(2)1941 1920 y FP(0)102 b(0)904 2020 y(0)208 b(0)82 b(0)103 b FO(e)231 b(e)209 b FP(0)102 b(0)905 2119 y FO(e)210 b FP(0)82 b(0)102 b(0)227 b(0)210 b FO(e)103 b FP(0)905 2219 y FO(e)211 b(e)84 b FP(0)102 b(0)227 b(0)208 b(0)83 b(2)p FO(e)778 2318 y(a)822 2330 y FL(1)878 2318 y FQ(\000)18 b FO(ia)1034 2330 y FL(2)1155 2318 y FO(e)84 b FP(0)102 b(0)227 b(0)208 b(0)102 b(0)904 2418 y(0)208 b(0)83 b FO(e)104 b FP(0)227 b(0)208 b(0)102 b(0)904 2518 y(0)208 b(0)82 b(0)h(2)p FO(e)207 b FP(0)h(0)102 b(0)2146 1854 y Fz(1)2146 2000 y(C)2146 2049 y(C)2146 2099 y(C)2146 2149 y(C)2146 2199 y(C)2146 2249 y(C)2146 2299 y(C)2146 2348 y(C)2146 2401 y(A)2232 2220 y FO(:)243 2713 y FP(It)28 b(is)f(easy)g(to)g(c)n(hec)n(k)g(that)h(the)g(functor)f FO(F)1588 2725 y FM( )1666 2713 y FP(is)h(full.)p 2514 2713 4 57 v 2518 2660 50 4 v 2518 2713 V 2567 2713 4 57 v 118 2957 a FR(3.1.5)94 b FQ(\003)p FR(-Wild)30 b(quadratic)j(and)g (cubic)f(relations)118 3120 y(1.)80 b FP(The)42 b(relation)f(\()p FO(I)845 3132 y FL(0)883 3120 y FP(\))i(0)k(=)g(0)41 b(de\014nes)h(the)h(standard)e(wild)h FQ(\003)p FP(-algebra)118 3219 y FA(S)187 3231 y FL(2)259 3219 y FP(=)35 b FJ(C)15 b FQ(h)p FO(a;)f(b)40 b FQ(j)35 b FO(a)704 3189 y FN(\003)777 3219 y FP(=)g FO(a;)27 b(b)1007 3189 y FN(\003)1080 3219 y FP(=)35 b FO(b)p FQ(i)p FP(.)58 b(By)34 b(Theorem)g(52,)i(the)f (theory)f(of)h(its)118 3319 y FQ(\003)p FP(-represen)n(tations)29 b(con)n(tains)j FQ(\003)p FP(-represen)n(tations)d(of)j(ev)n(ery)f (\014nitely)h(gener-)118 3419 y(ated)c FQ(\003)p FP(-algebra.)118 3584 y FR(2.)42 b FP(The)30 b(relation)f(\()p FO(I)783 3596 y FL(1)821 3584 y FP(\))h FO(a)927 3554 y FL(2)990 3584 y FP(=)c FO(e)j FP(de\014nes)h(the)g FQ(\003)p FP(-algebra)d FA(D)f FP(=)g FJ(C)15 b FQ(h)q FO(a;)f(b)31 b FQ(j)c FO(a)2446 3554 y FN(\003)2510 3584 y FP(=)118 3684 y FO(a;)h(b)249 3653 y FN(\003)310 3684 y FP(=)22 b FO(b;)28 b(a)528 3653 y FL(2)588 3684 y FP(=)22 b FO(e)p FQ(i)p FP(.)118 3866 y FR(Prop)s(osition)30 b(69.)41 b FC(The)31 b FQ(\003)p FC(-algebr)l(a)f FA(D)g FC(is)g FQ(\003)p FC(-wild.)118 4048 y(Pr)l(o)l(of.)43 b FP(W)-7 b(e)32 b(will)g(sho)n(w)e(that)i FA(D)e FQ(\037)f FB(P)1327 4060 y FL(3)p FM(;)p FN(?)p FL(2)1498 4048 y FP(=)h FJ(C)14 b FQ(h)q FO(p)1721 4060 y FL(1)1764 4048 y FO(;)g(p)1843 4060 y FL(2)1880 4048 y FO(;)g(p)1959 4060 y FL(3)2025 4048 y FQ(j)30 b FO(p)2120 4018 y FN(\003)2120 4069 y FM(i)2187 4048 y FP(=)f FO(p)2323 4060 y FM(i)2351 4048 y FO(;)f(p)2444 4018 y FL(2)2444 4069 y FM(i)2510 4048 y FP(=)118 4147 y FO(p)160 4159 y FM(i)188 4147 y FO(;)f(p)280 4159 y FL(1)317 4147 y FO(p)359 4159 y FL(2)429 4147 y FP(=)33 b FO(p)569 4159 y FL(2)606 4147 y FO(p)648 4159 y FL(1)717 4147 y FP(=)g(0)p FQ(i)p FP(.)54 b(T)-7 b(o)33 b(sho)n(w)g(this,)i(w)n(e)e(de\014ne)h(a)f FQ(\003)p FP(-homomorphism)p eop %%Page: 223 227 223 226 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(223)118 333 y FO( )26 b FP(:)d FA(D)g FQ(\000)-48 b(!)23 b FB(P)514 345 y FL(3)p FM(;)p FN(?)p FL(2)684 333 y FP(as)k(follo)n(ws:)723 492 y FO( )s FP(\()p FO(a)p FP(\))c(=)g FO(p)18 b FQ(\000)g FP(\()p FO(e)h FQ(\000)f FO(p)p FP(\))23 b(=)f(2)p FO(p)c FQ(\000)g FO(e;)731 665 y( )s FP(\()p FO(b)p FP(\))23 b(=)g FO(p)1041 677 y FL(1)1096 665 y FP(+)1189 609 y(1)p 1189 646 42 4 v 1189 722 a(2)1241 665 y FO(p)1283 677 y FL(2)1338 665 y FP(+)1431 609 y(1)p 1431 646 V 1431 722 a(3)1483 665 y(\()p FO(e)18 b FQ(\000)g FO(p)1697 677 y FL(1)1753 665 y FQ(\000)g FO(p)1878 677 y FL(2)1915 665 y FP(\))p FO(:)118 848 y FP(It)26 b(is)f(easy)g(to)g(c)n(hec)n(k)f (that)i(the)g(corresp)r(onding)d(functor)j FO(F)1978 860 y FM( )2051 848 y FP(:)d(Rep)14 b FB(P)2310 860 y FL(3)p FM(;)p FN(?)p FL(2)2475 848 y FQ(\000)-48 b(!)118 948 y FP(Rep)14 b FA(D)28 b FP(is)f(full.)p 2514 948 4 57 v 2518 895 50 4 v 2518 948 V 2567 948 4 57 v 118 1109 a FR(3.)75 b FP(No)n(w)40 b(w)n(e)g(giv)n(e)g(a)g(criterion,)j(in) d(terms)h(of)f(the)h(co)r(e\016cien)n(ts,)i(for)d(the)118 1209 y(quadratic)33 b FQ(\003)p FP(-algebra)e FA(A)i FP(=)g FJ(C)15 b FQ(h)p FO(a;)f(b)39 b FQ(j)33 b FO(a)g FP(=)g FO(a)1570 1179 y FN(\003)1608 1209 y FO(;)28 b(b)33 b FP(=)g FO(b)1862 1179 y FN(\003)1900 1209 y FO(;)27 b(P)2003 1221 y FL(2)2041 1209 y FP(\()p FO(a;)14 b(b)p FP(\))33 b(=)g FO(\013a)2450 1179 y FL(2)2510 1209 y FP(+)118 1309 y FO(\014)t(b)205 1278 y FL(2)262 1309 y FP(+)19 b FO(q)s(=i)p FP([)p FO(a;)14 b(b)p FP(])k(+)i FO(\015)5 b FQ(f)p FO(a;)14 b(b)p FQ(g)j FP(+)i FO(\016)s(a)h FP(+)f FO(\017b)g FP(+)g FO(\037I)33 b FP(=)26 b(0)p FQ(i)p FP(,)j FO(\013)p FP(,)i FO(\014)t FP(,)f FO(q)s FP(,)g FO(\015)5 b FP(,)29 b FO(\016)s FP(,)h FO(\017)p FP(,)g FO(\037)c FQ(2)g FJ(R)118 1408 y FP(to)i(b)r(e)g FQ(\003)p FP(-wild.)118 1556 y FR(Theorem)46 b(60.)i FC(A)41 b FQ(\003)p FC(-algebr)l(a)i FA(A)e FC(is)h FQ(\003)p FC(-wild)g(if)h(and)f(only)g(if)h(one)f(of)h(the)118 1655 y(fol)t(lowing)32 b(c)l(onditions)f(holds)7 b FP(:)217 1803 y(1)p FC(.)42 b FO(\013)23 b FP(=)g FO(\014)k FP(=)c FO(\015)k FP(=)c FO(q)j FP(=)d FO(\016)j FP(=)d FO(\017)f FP(=)h FO(\037)g FP(=)g(0;)217 1969 y(2)p FC(.)326 1902 y Fz(\000)364 1969 y FO(\037)18 b FQ(\000)533 1936 y FM(\016)565 1911 y Fy(2)p 527 1950 77 4 v 527 1998 a FL(4)p FM(\013)613 1902 y Fz(\001)665 1969 y FO(\013)24 b(<)e FP(0)p FC(,)85 b FO(\014)28 b FP(=)22 b FO(\015)28 b FP(=)22 b FO(q)27 b FP(=)22 b FO(\017)h FP(=)g(0;)217 2148 y(3)p FC(.)326 2081 y Fz(\000)364 2148 y FO(\037)18 b FQ(\000)534 2115 y FM(\017)562 2090 y Fy(2)p 527 2129 74 4 v 527 2177 a FL(4)p FM(\014)611 2081 y Fz(\001)663 2148 y FO(\014)27 b(<)c FP(0)p FC(,)85 b FO(\013)23 b FP(=)g FO(\015)k FP(=)c FO(q)j FP(=)d FO(\016)j FP(=)c(0;)217 2338 y(4)p FC(.)42 b FO(\015)374 2308 y FL(2)434 2338 y FP(=)22 b FO(\014)t(\013)i FQ(6)p FP(=)f(0)p FO(;)98 b(\013)p FP(\()p FO(\037)19 b FQ(\000)1155 2305 y FM(\016)1187 2280 y Fy(2)p 1149 2319 77 4 v 1149 2366 a FL(4)p FM(\013)1235 2338 y FP(\))24 b FO(<)e FP(0)p FC(,)1540 2305 y FM(\016)1572 2280 y Fy(2)p 1540 2319 65 4 v 1551 2366 a FM(\013)1638 2338 y FP(=)1735 2305 y FM(\017)1763 2280 y Fy(2)p 1735 2319 61 4 v 1745 2366 a FM(\014)1805 2338 y FC(,)86 b FO(q)26 b FP(=)c(0)p FC(.)118 2485 y(Pr)l(o)l(of.)43 b FP(The)34 b FQ(\003)p FP(-algebra)d(with)k(t)n(w)n(o)e(self-adjoin)n (t)g(v)-5 b(ariables)32 b(and)i(quadratic)118 2585 y(relations)d(is)g (wild)h(i\013)h(there)e(exists)h(a)f(c)n(hange)g(of)g(v)-5 b(ariables)31 b(suc)n(h)g(that)h(the)118 2685 y(algebra)39 b(can)i(b)r(e)g(transformed)f(to)h(the)h FQ(\003)p FP(-algebra)c FA(S)1906 2697 y FL(2)1988 2685 y FP(=)45 b FJ(C)15 b FQ(h)q FO(a;)f(b)50 b FQ(j)c FO(a)f FP(=)118 2784 y FO(a)162 2754 y FN(\003)200 2784 y FO(;)28 b(b)k FP(=)g FO(b)452 2754 y FN(\003)490 2784 y FQ(i)i FP(or)e(the)i FQ(\003)p FP(-algebra)d FA(D)h FP(=)h FJ(C)14 b FQ(h)q FO(a;)g(b)38 b FQ(j)32 b FO(a)h FP(=)f FO(a)1891 2754 y FN(\003)1929 2784 y FO(;)c(b)k FP(=)g FO(b)2181 2754 y FN(\003)2219 2784 y FO(;)c(a)2314 2754 y FL(2)2384 2784 y FP(=)k FO(e)p FQ(i)p FP(.)118 2884 y(One)f(can)f(de\014ne)i(suc)n(h)f(a)f(quadratic)g FQ(\003)p FP(-algebras)e(b)n(y)j(imp)r(osing)g(one)f(of)h(the)118 2984 y(conditions)c(1{4.)p 2514 2984 4 57 v 2518 2931 50 4 v 2518 2984 V 2567 2984 4 57 v 118 3145 a FR(4.)35 b FP(No)n(w)24 b(w)n(e)h(consider)e(a)h(pair)g(of)h(self-adjoin)n(t)f (op)r(erators)f(whic)n(h)h(satisfy)g(the)118 3245 y(cubic)k(semilinear) f(relation:)169 3404 y FO(\017)p FQ(f)p FO(A)307 3370 y FL(2)343 3404 y FO(;)14 b(B)t FQ(g)k FP(+)g FO(i\016)s FP([)p FO(A)744 3370 y FL(2)782 3404 y FO(;)c(B)t FP(])k(+)g(2)p FO(\026AB)t(A)h FP(+)f FO(i\015)5 b FP([)p FO(A;)14 b(B)t FP(])k(+)g(2)p FO(\014)t FQ(f)p FO(A;)c(B)t FQ(g)j FP(+)i FO(\013B)27 b FP(=)c(0)p FO(:)2404 3504 y FP(\(3.3\))118 3664 y(T)-7 b(o)30 b(relation)g(\(3.3\))g(there)g(corresp)r(onds)f(the) h FQ(\003)p FP(-algebra)e FA(A)1994 3676 y FL(3)2059 3664 y FP(=)f FJ(C)15 b FQ(h)p FO(a;)f(b)34 b FQ(j)28 b FO(a)f FP(=)118 3764 y FO(a)162 3733 y FN(\003)200 3764 y FO(;)h(b)23 b FP(=)f FO(b)433 3733 y FN(\003)471 3764 y FO(;)28 b(\017)p FQ(f)p FO(a)642 3733 y FL(2)678 3764 y FO(;)14 b(b)p FQ(g)f FP(+)g FO(i\016)s FP([)p FO(a)1020 3733 y FL(2)1057 3764 y FO(;)h(b)p FP(])f(+)g(2)p FO(\026)h(aba)f FP(+)g FO(i\015)5 b FP([)p FO(a;)14 b(b)p FP(])f(+)g(2)p FO(\014)t FQ(f)p FO(a;)h(b)p FQ(g)f FP(+)g FO(\013b)20 b FP(=)j(0)p FQ(i)p FP(.)118 3863 y(The)28 b(corresp)r(onding)d(c)n(haracteristic)h(function)i(is)g(the)g(follo)n (wing:)584 4023 y(\010\()p FO(t;)14 b(s)p FP(\))23 b(=)g FO(\017)p FP(\()p FO(t)1021 3988 y FL(2)1076 4023 y FP(+)18 b FO(s)1198 3988 y FL(2)1235 4023 y FP(\))h(+)f FO(i\016)s FP(\()p FO(t)1500 3988 y FL(2)1556 4023 y FQ(\000)g FO(s)1678 3988 y FL(2)1715 4023 y FP(\))832 4147 y(+)g(2)p FO(\026)c(ts)k FP(+)g FO(i\015)5 b FP(\()p FO(t)18 b FQ(\000)g FO(s)p FP(\))h(+)f(2)p FO(\014)t FP(\()p FO(t)g FP(+)g FO(s)p FP(\))h(+)f FO(\013;)p eop %%Page: 224 228 224 227 bop 118 100 a FP(224)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FO(\013)p FP(,)g FO(\014)t FP(,)g FO(\015)5 b FP(,)28 b FO(\016)e FQ(2)d FJ(R)p FP(.)243 432 y(It)39 b(follo)n(ws)g(from)g (the)h(general)e(theory)g(of)i(semilinear)e(relations)h(that)118 532 y(the)j(corresp)r(onding)d FQ(\003)p FP(-algebra)g(is)i FQ(\003)p FP(-wild)g(if)h(and)f(only)g(if)h(the)f(equation)118 632 y(\010\()p FO(t;)14 b(s)p FP(\))23 b(=)g(0)e(has)g(either)g(t)n(w)n (o)g(solutions)f(of)h(the)h(form)f(\()p FO(t)1861 644 y FL(1)1899 632 y FO(;)14 b(t)1966 644 y FL(1)2003 632 y FP(\),)23 b(\()p FO(t)2143 644 y FL(1)2180 632 y FO(;)14 b(t)2247 644 y FL(2)2284 632 y FP(\),)23 b(where)118 731 y FO(t)148 743 y FL(1)209 731 y FQ(6)p FP(=)g FO(t)327 743 y FL(2)364 731 y FP(,)28 b(or)f(t)n(w)n(o)h(solutions)f(of)h(the)g (form)f(\()p FO(t)1518 743 y FL(1)1556 731 y FO(;)14 b(t)1623 743 y FL(2)1660 731 y FP(\),)28 b(\()p FO(t)1805 743 y FL(1)1843 731 y FO(;)14 b(t)1910 743 y FL(3)1947 731 y FP(\),)29 b(where)e FO(t)2301 743 y FL(1)2338 731 y FP(,)h FO(t)2419 743 y FL(2)2456 731 y FP(,)h FO(t)2538 743 y FL(3)118 831 y FP(are)e(distinct.)243 930 y(The)32 b(equation)f(\010\()p FO(t;)14 b(s)p FP(\))30 b(=)g(0)i(decomp)r(oses)f (in)h(a)f(natural)g(w)n(a)n(y)g(in)n(to)g(the)118 1030 y(system)c(of)h(t)n(w)n(o)f(equations:)385 1112 y Fz(\032)489 1178 y FP(\010)549 1190 y FL(1)586 1178 y FP(\()p FO(t;)14 b(s)p FP(\))23 b(=)g FO(\017)14 b(t)945 1148 y FL(2)1000 1178 y FP(+)k(2)p FO(\026)c(ts)k FP(+)g FO(\017s)1432 1148 y FL(2)1487 1178 y FP(+)g(2)p FO(\014)t(t)h FP(+)f(2)p FO(\014)t(s)g FP(+)g FO(\013)23 b FP(=)g(0)p FO(;)489 1278 y FP(\010)549 1290 y FL(2)586 1278 y FP(\()p FO(t;)14 b(s)p FP(\))23 b(=)g FO(\016)s FP(\()p FO(t)969 1248 y FL(2)1025 1278 y FQ(\000)18 b FO(s)1147 1248 y FL(2)1184 1278 y FP(\))h(+)f FO(\015)5 b FP(\()p FO(t)18 b FQ(\000)g FO(s)p FP(\))24 b(=)e(0)p FO(:)243 1433 y FP(First)34 b(w)n(e)h(assume)f(that)h FO(\016)j FP(=)c FO(\015)40 b FP(=)35 b(0)f(\(\010)1588 1445 y FL(2)1625 1433 y FP(\()p FO(t;)14 b(s)p FP(\))36 b FQ(\021)e FP(0\).)59 b(The)34 b(equation)118 1532 y(\010)178 1544 y FL(1)215 1532 y FP(\()p FO(t;)14 b(s)p FP(\))39 b(=)f(0)f(de\014nes)g(a)f(curv)n(e)g (of)h(degree)e(t)n(w)n(o.)64 b(Suc)n(h)37 b(curv)n(es)f(ha)n(v)n(e)f (the)118 1632 y(follo)n(wing)27 b(in)n(v)-5 b(arian)n(ts:)742 1831 y FO(I)778 1843 y FL(1)838 1831 y FP(=)23 b(2)p FO(\017;)96 b(I)1157 1843 y FL(2)1218 1831 y FP(=)1306 1711 y Fz(\014)1306 1760 y(\014)1306 1810 y(\014)1306 1860 y(\014)1341 1780 y FO(\017)91 b(\026)1333 1880 y(\026)g(\017)1516 1711 y Fz(\014)1516 1760 y(\014)1516 1810 y(\014)1516 1860 y(\014)1567 1831 y FP(=)23 b FO(\017)1689 1797 y FL(2)1744 1831 y FQ(\000)18 b FO(\026)1877 1797 y FL(2)1914 1831 y FO(;)555 2113 y(I)591 2125 y FL(3)652 2113 y FP(=)740 1943 y Fz(\014)740 1993 y(\014)740 2043 y(\014)740 2093 y(\014)740 2142 y(\014)740 2192 y(\014)776 2013 y FO(\017)92 b(\026)85 b(\014)768 2112 y(\026)93 b(\017)f(\014)767 2212 y(\014)c(\014)f(\013)1090 1943 y Fz(\014)1090 1993 y(\014)1090 2043 y(\014)1090 2093 y(\014)1090 2142 y(\014)1090 2192 y(\014)1140 2113 y FP(=)23 b(\()p FO(\017)1294 2079 y FL(2)1350 2113 y FQ(\000)18 b FO(\026)1483 2079 y FL(2)1520 2113 y FP(\))c FO(\013)19 b FQ(\000)f FP(2)p FO(\014)1814 2079 y FL(2)1865 2113 y FP(\()p FO(\017)g FQ(\000)g FO(\026)p FP(\))p FO(:)118 2367 y FP(If)28 b FO(I)237 2379 y FL(2)299 2367 y FQ(6)p FP(=)23 b(0,)k(then)i(the)f(equation)f(\010)1211 2379 y FL(1)1249 2367 y FP(\()p FO(t;)14 b(s)p FP(\))24 b(=)f(0)k(de\014nes)h(a)f(cen)n(tral)g(curv)n(e.)37 b(By)118 2466 y(an)d(a\016ne)h(transformation)d(of)j(v)-5 b(ariables,)35 b(the)f(equation)g(can)g(b)r(e)h(reduced)118 2566 y(in)n(to)27 b(the)h(follo)n(wing)f(form:)775 2758 y(\()p FO(\017)19 b FP(+)f FO(\026)p FP(\))1040 2742 y(^)1039 2758 y FO(t)1069 2723 y FL(2)1125 2758 y FP(+)g(\()p FO(\017)g FQ(\000)g FO(\026)p FP(\))f(^)-45 b FO(s)1510 2723 y FL(2)1566 2758 y FP(+)1659 2701 y FO(I)1695 2713 y FL(3)p 1659 2738 74 4 v 1659 2815 a FO(I)1695 2827 y FL(2)1765 2758 y FP(=)23 b(0)p FO(:)486 b FP(\(3.4\))243 2952 y(a\))27 b(Let)h FO(I)529 2964 y FL(2)590 2952 y FO(>)22 b FP(0.)243 3051 y(If)28 b FO(I)362 3063 y FL(3)422 3051 y FP(=)23 b(0,)k(then)h(the)g FQ(\003)p FP(-algebra)d FA(A)1354 3063 y FL(3)1419 3051 y FP(is)i(not)h(wild.)243 3151 y(If)k FO(I)366 3163 y FL(3)433 3151 y FO(>)d FP(0,)j(then)g(the)g(set) g(of)f(solutions)g(of)h(\(3.4\))f(is)g(empt)n(y)-7 b(.)49 b(Therefore)118 3251 y(the)28 b FQ(\003)p FP(-algebra)d FA(A)681 3263 y FL(3)746 3251 y FP(is)i(not)h(wild.)243 3350 y(If)g FO(I)362 3362 y FL(3)424 3350 y FO(<)23 b FP(0,)28 b(then)h(equation)e(\(3.4\))h(b)r(ecomes)g(an)g(equation)f(of) h(an)g(ellipse.)118 3450 y(Therefore)i(there)g(are)g(t)n(w)n(o)g (solutions)g(\()p FO(t)1427 3462 y FL(1)1465 3450 y FO(;)14 b(t)1532 3462 y FL(2)1569 3450 y FP(\),)32 b(\()p FO(t)1718 3462 y FL(1)1755 3450 y FO(;)14 b(t)1822 3462 y FL(3)1859 3450 y FP(\))31 b(suc)n(h)g(that)g FO(t)2326 3462 y FL(2)2392 3450 y FQ(6)p FP(=)d FO(t)2515 3462 y FL(3)2552 3450 y FP(.)118 3550 y(Hence,)g(the)g FQ(\003)p FP(-algebra)d FA(A)951 3562 y FL(3)1015 3550 y FP(is)j FQ(\003)p FP(-wild.)243 3649 y(b\))d(Let)h FO(I)529 3661 y FL(2)589 3649 y FO(<)d FP(0)i(then)h(the)f(equation)g(\010\()p FO(t;)14 b(s)p FP(\))23 b(=)g(0)i(is)g(of)g(h)n(yp)r(erb)r(olic)g(t)n(yp)r(e.)243 3749 y(If)d FO(I)356 3761 y FL(3)417 3749 y FP(=)g(0,)h(then)f(w)n(e)g (ha)n(v)n(e)e(a)h(pair)h(of)f(in)n(tersecting)g(straigh)n(t)g(lines.)35 b(Hence)118 3848 y(it)28 b(is)g(easy)e(to)i(see)f(that)h(the)g FQ(\003)p FP(-algebra)d FA(A)1446 3860 y FL(3)1510 3848 y FP(is)j FQ(\003)p FP(-wild.)243 3948 y(If)21 b FO(I)355 3960 y FL(3)416 3948 y FQ(6)p FP(=)h(0,)g(then)f(w)n(e)g(ha)n(v)n(e)e (an)i(equation)f(of)h(a)f(h)n(yp)r(erb)r(ola.)34 b(This)20 b(equation)118 4048 y(has)27 b(no)g(more)g(than)h(one)f(solution)g (only)g(in)h(the)g(case)f(where)g FO(\017)c FP(=)f(0.)37 b(Hence,)118 4147 y(if)28 b FO(\017)23 b FQ(6)p FP(=)g(0,)k(then)h(the) g FQ(\003)p FP(-algebra)d FA(A)1183 4159 y FL(3)1247 4147 y FP(is)j(wild.)p eop %%Page: 225 229 225 228 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(225)243 333 y(c\))26 b(Let)g FO(I)521 345 y FL(2)581 333 y FP(=)d(0.)36 b(Then)26 b(the)g(equation)f(\010)1523 345 y FL(1)1560 333 y FP(\()p FO(t;)14 b(s)p FP(\))24 b(=)f(0)i(has)g(parab)r(olic)g(t)n(yp)r(e)118 432 y(\(the)j(curv)n(e)f(is)g(not)h(cen)n(tral\).)243 533 y(Let)f FO(I)427 545 y FL(3)488 533 y FP(=)c(0.)36 b(Then)28 b FO(\017)23 b FP(=)f FO(\026)28 b FP(and)625 717 y(\010)685 729 y FL(1)722 717 y FP(\()p FO(t;)14 b(s)p FP(\))23 b(=)g FO(\017)p FP(\()p FO(t)18 b FP(+)g FO(s)p FP(\))1271 682 y FL(2)1327 717 y FP(+)g(2)p FO(\014)t FP(\()p FO(t)h FP(+)f FO(s)p FP(\))h(+)f FO(\013)23 b FP(=)g(0)p FO(:)243 902 y FP(If)32 b FO(\014)381 871 y FL(2)440 902 y FQ(\000)21 b FO(\013\017)30 b(<)g FP(0,)j(the)g (equation)e(\010)1387 914 y FL(1)1424 902 y FP(\()p FO(t;)14 b(s)p FP(\))31 b(=)f(0)h(has)h(no)g(solutions,)g(and)118 1001 y(the)c FQ(\003)p FP(-algebra)d FA(A)681 1013 y FL(3)746 1001 y FP(is)i(not)h(wild.)243 1102 y(If)35 b FO(\014)384 1071 y FL(2)445 1102 y FQ(\000)22 b FO(\013\017)35 b(<)g FP(0,)h(then)f FO(t)23 b FP(+)g FO(s)35 b FP(=)f FQ(\000)p FO(\014)t(=\017)p FP(,)i(and)e(it)i(is)e(ob)n(vious)g(that)g (the)118 1201 y FQ(\003)p FP(-algebra)25 b FA(A)538 1213 y FL(3)603 1201 y FP(is)i(not)h(wild.)243 1301 y(F)-7 b(or)40 b FO(\014)456 1271 y FL(2)521 1301 y FQ(\000)27 b FO(\013\017)46 b(>)f FP(0,)g(the)c(equation)g(\010)1535 1313 y FL(1)1572 1301 y FP(\()p FO(t;)14 b(s)p FP(\))46 b(=)f(0)c(describ)r(es)g(a)g(pair)118 1401 y(of)33 b(parallel)e(lines.) 52 b(Hence)32 b(the)h(set)g(of)f(solutions)g(con)n(tains)g(t)n(w)n(o)g (solutions:)118 1501 y(\()p FO(t)180 1513 y FL(1)218 1501 y FO(;)14 b(t)285 1513 y FL(2)322 1501 y FP(\),)43 b(\()p FO(t)482 1513 y FL(1)519 1501 y FO(;)14 b(t)586 1513 y FL(3)623 1501 y FP(\))40 b(suc)n(h)g(that)f FO(t)1116 1513 y FL(2)1197 1501 y FQ(6)p FP(=)j FO(t)1334 1513 y FL(3)1371 1501 y FP(.)73 b(Therefore)39 b(the)h FQ(\003)p FP(-algebra)d FA(A)2443 1513 y FL(3)2519 1501 y FP(is)118 1600 y FQ(\003)p FP(-wild.)243 1701 y(If)42 b FO(I)376 1713 y FL(2)460 1701 y FP(=)47 b(0,)e FO(I)718 1713 y FL(3)802 1701 y FQ(6)p FP(=)i(0,)e(then)d(the)g(equation)f(\010)1797 1713 y FL(1)1835 1701 y FP(\()p FO(t;)14 b(s)p FP(\))47 b(=)f(0)41 b(de\014nes)h(a)118 1800 y(parab)r(ola.)69 b(In)38 b(this)h(case,)i(if)e FO(\014)1166 1770 y FL(2)1230 1800 y FQ(\000)25 b FO(\017\013)h FQ(\000)f FP(4)p FO(\014)t(t)42 b(>)f FP(0,)g(w)n(e)d(ha)n(v)n(e)g(t)n(w)n(o)g(solu-)118 1900 y(tions)29 b(of)g(the)g(equation)f(\010)963 1912 y FL(1)1000 1900 y FP(\()p FO(t;)14 b(s)p FP(\))26 b(=)e(0)29 b(for)f(one)h(v)-5 b(alue)28 b FO(t)p FP(,)i(and)e(the)h FQ(\003)p FP(-algebra)118 2000 y FA(A)178 2012 y FL(3)243 2000 y FP(is)e FQ(\003)p FP(-wild.)243 2100 y(No)n(w)g(consider)f(the)i (case)f(\010)1134 2112 y FL(1)1171 2100 y FP(\()p FO(t;)14 b(s)p FP(\))24 b FQ(\021)e FP(0,)28 b FO(\016)e FQ(6)p FP(=)c(0.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)720 2284 y(\010)780 2296 y FL(2)817 2284 y FP(\()p FO(t;)14 b(s)p FP(\))24 b(=)e(\()p FO(t)d FQ(\000)f FO(s)p FP(\)\()p FO(\016)s(t)h FP(+)f FO(\016)s(s)h FP(+)f FO(\015)5 b FP(\))23 b(=)f(0)p FO(:)118 2468 y FP(In)37 b(that)h(case)e(the)h(equation)g(\010)1167 2480 y FL(2)1204 2468 y FP(\()p FO(t;)14 b(s)p FP(\))39 b(=)f(0)f(de\014nes)g(a)f(pair)h(of)g(in)n(tersect-)118 2568 y(ing)29 b(and)f(non)h(coinciding)f(straigh)n(t)g(lines.)40 b(Therefore)28 b(the)h FQ(\003)p FP(-algebra)d FA(A)2453 2580 y FL(3)2519 2568 y FP(is)118 2667 y FQ(\003)p FP(-wild.)243 2768 y(Consider)h(the)j(case)e(\010)972 2780 y FL(1)1009 2768 y FP(\()p FO(t;)14 b(s)p FP(\))g(\010)1253 2780 y FL(2)1290 2768 y FP(\()p FO(t;)g(s)p FP(\))26 b FQ(6)p FP(=)e(0.)40 b(It)29 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FO(\017)1359 3595 y FL(2)1415 3625 y FP(+)18 b FO(\026)1548 3595 y FL(2)1585 3625 y FP(\))24 b FQ(6)p FP(=)e(0.)243 3725 y(By)28 b(an)f(a\016ne)h(c)n(hange)g(of)g(v)-5 b(ariables,)27 b(system)h(\(3.5\))g(can)g(b)r(e)g(reduced)g(to)118 3825 y(the)g(follo)n(wing)f(form:)490 3944 y Fz(\032)603 3990 y FP(~)594 4011 y(\010)654 4023 y FL(1)691 4011 y FP(\()724 3996 y(~)723 4011 y FO(t;)18 b FP(~)-46 b FO(s)p FP(\))24 b(=)e FO(\017)1021 3996 y FP(~)1020 4011 y FO(t)1050 3981 y FL(2)1105 4011 y FP(+)c(2)p FO(\026)1295 3996 y FP(~)1294 4011 y FO(t)s FP(~)-45 b FO(s)18 b FP(+)g FO(\017)s FP(~)-45 b FO(s)1537 3981 y FL(2)1592 4011 y FP(+)19 b(2)1730 3989 y(~)1718 4011 y FO(\014)7 b FP(~)-46 b FO(s)19 b FP(+)26 b(~)-50 b FO(\013)23 b FP(=)g(0)p FO(;)603 4096 y FP(~)594 4117 y(\010)654 4129 y FL(2)691 4117 y FP(\()724 4102 y(~)723 4117 y FO(t;)18 b FP(~)-46 b FO(s)p FP(\))24 b(=)973 4102 y(~)972 4117 y FO(t)1002 4087 y FL(2)1058 4117 y FQ(\000)d FP(~)-45 b FO(s)1180 4087 y FL(2)1240 4117 y FP(=)23 b(0)p FO(;)p eop %%Page: 226 230 226 229 bop 118 100 a FP(226)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FP(where)379 311 y(~)367 333 y FO(\014)43 b FP(=)37 b FO(\014)29 b FQ(\000)24 b FP(\()p FO(\017)h FP(+)f FO(\026)p FP(\))p FO(\015)5 b(=)p FP(\(2)p FO(\016)r FP(\),)48 b(~)-51 b FO(\013)39 b FP(=)f FO(\013)25 b FP(+)f(\()p FO(\017)g FP(+)g FO(\026)p FP(\))p FO(\015)1954 303 y FL(2)1991 333 y FO(=)p FP(\(2)p FO(\016)2147 303 y FL(2)2184 333 y FP(\))h FQ(\000)f FP(2)p FO(\014)t(\015)5 b(=\016)s FP(.)118 432 y(Solutions)24 b(of)g(this)g(system)g(are)f (solutions)h(of)g(the)g(equation)g(of)2141 411 y(~)2132 432 y(\010)2192 444 y FL(1)2229 432 y FP(\()2262 417 y(~)2261 432 y FO(t;)18 b FP(~)-46 b FO(s)p FP(\))24 b(=)e(0,)118 532 y(where)30 b(~)-45 b FO(s)23 b FP(=)30 b(~)-49 b FO(p)27 b FP(and)k(~)-45 b FO(s)23 b FP(=)f FQ(\000)954 517 y FP(~)953 532 y FO(t)p FP(.)243 632 y(Therefore,)28 b(for)h FA(A)833 644 y FL(3)900 632 y FP(to)g(b)r(e)h FQ(\003)p FP(-wild)f(it)h(is)f(necessary)f(and)h (su\016cien)n(t)h(that)118 731 y(the)f(equation)610 710 y(~)601 731 y(\010\()694 716 y(~)693 731 y FO(t;)17 b FP(~)-45 b FO(s)p FP(\))24 b(=)f(0)28 b(ha)n(v)n(e)f(t)n(w)n(o)g (solutions)g(of)h(the)h(form:)37 b(\()2210 716 y(~)2209 731 y FO(t)2239 743 y FL(0)2277 731 y FO(;)2315 716 y FP(~)2314 731 y FO(t)2344 743 y FL(0)2381 731 y FP(\))28 b(and)118 831 y(\()151 815 y(~)150 831 y FO(t)180 843 y FL(0)218 831 y FO(;)14 b FQ(\000)321 815 y FP(~)320 831 y FO(t)350 843 y FL(0)386 831 y FP(\).)44 b(It)29 b(is)h(easy)f(to)g(sho)n(w)g(that)h(this)g(condition)f(is)g (ful\014lled)i(in)f(one)f(of)118 930 y(the)f(follo)n(wing)f(cases.)238 1082 y(i.)42 b FO(\017)22 b(>)h FP(0,)83 b FO(\026)23 b(>)f FP(0,)83 b(2)p FO(\016)s(\014)22 b FQ(\000)c FO(\017\015)28 b FP(=)22 b(0,)83 b FO(\013\016)1593 1052 y FL(2)1649 1082 y FQ(\000)18 b FO(\017\015)1814 1052 y FL(2)1874 1082 y FO(<)k FP(0;)215 1285 y(ii.)42 b FO(\026)23 b FQ(6)p FP(=)f(0,)83 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FP(=)23 b FJ(C)15 b FQ(h)p FO(a)29 b FP(=)23 b FO(a)1225 3848 y FN(\003)1263 3883 y FO(;)14 b(b)22 b FP(=)h FO(b)1482 3848 y FN(\003)1543 3883 y FQ(j)g FO(aba)f FP(=)h FO(bab)p FQ(i)118 4048 y FC(is)36 b FQ(\003)p FC(-wild.)55 b(Mor)l(e)l(over,)38 b(its)e(quotient)e(algebr)l(a)j FJ(C)15 b FQ(h)p FO(a)39 b FP(=)32 b FO(a)1950 4018 y FN(\003)1988 4048 y FO(;)14 b(b)33 b FP(=)f FO(b)2227 4018 y FN(\003)2298 4048 y FQ(j)h FO(aba)f FP(=)118 4147 y FO(bab)22 b FP(=)h(0)p FQ(i)29 b FC(is)i FQ(\003)p FC(-wild.)p eop %%Page: 227 231 227 230 bop 118 100 a FK(3.1.)36 b FQ(\003)p FK(-Wild)28 b(algebras)d(and)j(relations)1095 b FP(227)118 333 y FC(Pr)l(o)l(of.)43 b FP(Let)20 b(a)f(homomorphism)g FO( )12 b FP(:)28 b FB(B)1339 345 y FL(2)1399 333 y FQ(\000)-48 b(!)23 b FO(M)1603 345 y FL(4)1640 333 y FP(\()p FA(S)1741 345 y FL(2)1778 333 y FP(\))d(b)r(e)g(de\014ned)g(as)f(follo)n(ws:)443 672 y FO( )s FP(\()p FO(a)p FP(\))k(=)719 506 y Fz(0)719 655 y(@)792 568 y FP(0)84 b FO(e)793 668 y(e)g FP(0)1041 619 y Fp(0)845 775 y(0)137 b(0)1099 506 y Fz(1)1099 655 y(A)1186 672 y FO(;)97 b( )s FP(\()p FO(b)p FP(\))23 b(=)1574 456 y Fz(0)1574 602 y(B)1574 652 y(B)1574 705 y(@)1700 573 y Fp(0)1936 522 y FO(e)84 b FP(0)1935 622 y(0)e(0)1648 721 y FO(e)i FP(0)1646 821 y(0)f(0)1895 721 y FO(a)1939 733 y FL(1)2081 721 y FO(e)1917 821 y(e)103 b(a)2103 833 y FL(2)2141 456 y Fz(1)2141 602 y(C)2141 652 y(C)2141 705 y(A)2227 672 y FO(:)118 1012 y FP(One)30 b(can)g(c)n(hec)n(k)f(that)i FO( )s FP(\()p FO(aba)p FP(\))c(=)h(0,)i(and)g FO( )k FP(de\014nes)c(a)g(homomorphism)f(of)118 1112 y(the)f(quotien)n(t)g(algebra.)35 b(The)27 b(constructed)g (functor)h FO(F)1872 1124 y FM( )1950 1112 y FP(is)g(full.)p 2514 1112 4 57 v 2518 1059 50 4 v 2518 1112 V 2567 1112 4 57 v 118 1309 a FC(R)l(emark)k FP(53)p FC(.)i FP(The)20 b FQ(\003)p FP(-wild)f FQ(\003)p FP(-algebra)e FB(B)1390 1321 y FL(2)1451 1309 y FP(=)22 b FJ(C)15 b FQ(h)p FO(a)29 b FP(=)23 b FO(a)1829 1279 y FN(\003)1867 1309 y FO(;)14 b(b)23 b FP(=)f 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FM(?)1621 1811 y FQ(i)p FP(.)42 b(The)30 b FQ(\003)p FP(-algebra)c FA(C)2303 1823 y FL(2)2370 1811 y FP(is)j(not)118 1911 y FQ(\003)p FP(-wild)e(\(see)h(Section)f(3.2.1\).)118 2147 y FR(3.1.6)94 b FQ(\003)p FR(-Wild)30 b(groups.)41 b(P)m(erio)s(dic)32 b(groups)f(are)h(not)g FQ(\003)p FR(-wild)118 2307 y FP(In)23 b(this)g(section)g(w)n(e)f(study)h(the)g (complexit)n(y)f(of)h(description)f(of)h(unitary)g(rep-)118 2407 y(resen)n(tations)29 b(\()p FQ(\003)p FP(-represen)n(tations)f(of) j(group)e(algebras\))g(for)h(some)g(discrete)118 2507 y(coun)n(table)k(groups)g FO(G)p FP(.)59 b(F)-7 b(or)35 b(groups)e FO(G)1421 2519 y FL(1)1494 2507 y FP(and)i FO(G)1728 2519 y FL(2)1765 2507 y FP(,)i(for)d(whic)n(h)h FO(C)2269 2476 y FN(\003)2308 2507 y FP(\()p FO(G)2405 2519 y FL(1)2443 2507 y FP(\))g FQ(\037)118 2606 y FO(C)183 2576 y FN(\003)222 2606 y FP(\()p FO(G)319 2618 y FL(2)356 2606 y FP(\),)42 b(w)n(e)c(will)h(write)f(b)r(elo)n(w)g FO(G)1288 2618 y FL(1)1367 2606 y FQ(\037)j FO(G)1538 2618 y FL(2)1575 2606 y FP(.)70 b(If)38 b FO(C)1826 2576 y FN(\003)1865 2606 y FP(\()p FO(G)p FP(\))h(is)g FQ(\003)p FP(-wild,)h(then)118 2706 y(the)30 b(description)f(of)h(unitary)f (represen)n(tations)e(of)j(suc)n(h)f(a)g(group)g(con)n(tains,)118 2805 y(as)37 b(a)h(subproblem,)i(the)f(description)e(of)h(represen)n (tations)e(of)i(an)n(y)f(coun)n(t-)118 2905 y(able)32 b(group;)i(further)f(in)g(the)g(b)r(o)r(ok)f(w)n(e)g(will)h(call)f (them)h FQ(\003)p FP(-wild)f(\(from)g(the)118 3005 y(viewp)r(oin)n(t)c (of)f(complexit)n(y)g(of)h(their)f(unitary)g(represen)n(tations\).)243 3108 y(Belo)n(w,)e(w)n(e)g(giv)n(e)g(a)h(n)n(um)n(b)r(er)g(of)f (examples)h(of)f(b)r(oth)i FQ(\003)p FP(-wild)e(groups,)g(and)118 3208 y(groups)h(that)i(are)f(not)g FQ(\003)p FP(-wild.)118 3369 y FR(1.)36 b FP(Let)28 b(us)g(giv)n(e)e(a)h(n)n(um)n(b)r(er)h(of)f (examples)g(of)h FQ(\003)p FP(-wild)f(groups.)118 3509 y FC(Example)48 b FP(24)p FC(.)g FP(It)40 b(follo)n(ws)e(directly)h (from)g(the)h(list)g(of)f FQ(\003)p FP(-wild)g FQ(\003)p FP(-algebras)118 3609 y(giv)n(en)25 b(ab)r(o)n(v)n(e,)g(that)h(the)g (groups)f FB(F)1232 3621 y FM(n)1275 3609 y FP(,)h FO(n)d FQ(\025)g FP(2,)j(and)f FJ(Z)1797 3621 y FM(n)1851 3609 y FQ(\003)14 b FJ(Z)1969 3621 y FM(m)2026 3609 y FP(,)26 b FO(n)d FQ(\025)g FP(2,)j FO(m)d FQ(\025)f FP(3,)118 3709 y(are)27 b FQ(\003)p FP(-wild.)118 3849 y FR(2.)36 b FP(The)28 b(follo)n(wing)e(simple)i(statemen)n(t)g(holds.)118 4027 y FR(Prop)s(osition)j(71.)40 b FC(If)31 b(a)f(gr)l(oup)h FO(G)f FC(c)l(ontains)g(a)g(normal)h(sub)l(gr)l(oup)f FO(N)39 b FC(such)118 4127 y(that)30 b FO(G=)-5 b(N)32 b FP(=)22 b FO(G)641 4139 y FL(1)679 4127 y FC(,)30 b(then)g FO(G)23 b FQ(\037)g FO(G)1160 4139 y FL(1)1197 4127 y FC(.)p eop %%Page: 228 232 228 231 bop 118 100 a FP(228)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FC(Pr)l(o)l(of.)43 b FP(Denote)29 b(b)n(y)f FO(\036)9 b FP(:)29 b FO(G)c FQ(\000)-48 b(!)25 b FO(G)1170 345 y FL(1)1236 333 y FP(the)k(mapping)f(whic)n(h)h(maps)f(an)h(elemen)n(t) 118 432 y FO(g)i FQ(2)d FO(G)j FP(to)g(its)f(conjugacy)g(class)f FO(\036)p FP(\()p FO(g)s FP(\))g FQ(2)f FO(G)1512 444 y FL(1)1578 432 y FP(=)g FO(G=)-5 b(N)9 b FP(.)45 b(Then,)32 b(in)n(tro)r(duce)e(a)118 532 y(unitary)d FQ(\003)p FP(-homomorphism) 186 709 y FO( )12 b FP(:)28 b FO(L)p FP(\()p FO(G)p FP(\))c FQ(3)f FO(f)9 b FP(\()p FQ(\001)p FP(\))24 b FQ(7!)f FO( )s FP(\()p FO(f)9 b FP(\)\()p FO(g)1101 721 y FL(1)1138 709 y FP(\))23 b(=)1384 630 y Fz(X)1281 812 y FM(g)10 b FL(:)22 b FM(\036)p FL(\()p FM(g)r FL(\)=)p FM(g)1573 820 y Fy(1)1620 709 y FO(f)9 b FP(\()p FO(g)s FP(\))23 b FQ(2)h FO(L)p FP(\()p FO(G)2033 721 y FL(1)2070 709 y FP(\))f FQ(\032)g FO(C)2278 674 y FN(\003)2316 709 y FP(\()p FO(G)2413 721 y FL(1)2451 709 y FP(\))p FO(;)118 966 y FP(and)38 b(use)f(the)h(same)f(notation)h FO( )i FP(for)e(its)g(unique)f(extension)h(to)f(a)h(unital)118 1065 y FQ(\003)p FP(-homomorphism)27 b(from)h FO(C)1037 1035 y FN(\003)1075 1065 y FP(\()p FO(G)p FP(\))h(to)f FO(C)1400 1035 y FN(\003)1439 1065 y FP(\()p FO(G)1536 1077 y FL(1)1574 1065 y FP(\).)39 b(It)29 b(is)f(clear)f(that)i(the)g (corre-)118 1165 y(sp)r(onding)e(functor)h FO(F)813 1177 y FM(\036)867 1165 y FP(:)41 b(Rep)14 b FO(C)1154 1135 y FN(\003)1193 1165 y FP(\()p FO(G)1290 1177 y FL(1)1327 1165 y FP(\))24 b FQ(\000)-49 b(!)23 b FP(Rep)14 b FO(C)1728 1135 y FN(\003)1767 1165 y FP(\()p FO(G)p FP(\))28 b(is)g(full.)p 2514 1165 4 57 v 2518 1112 50 4 v 2518 1165 V 2567 1165 4 57 v 243 1327 a(The)f(prop)r(osition)g(ab)r(o)n(v)n(e)f(implies)i (the)g(follo)n(wing)e(statemen)n(t.)118 1479 y FR(Corollary)j(15.)38 b FC(A)26 b(gr)l(oup)h FO(G)f FC(that)h(p)l(ossesses)g(a)g(normal)g (sub)l(gr)l(oup)f FO(N)36 b FC(such)118 1579 y(that)30 b FO(G=)-5 b(N)32 b FP(=)22 b FB(F)632 1591 y FL(2)697 1579 y FC(is)30 b FQ(\003)p FC(-wild.)118 1730 y(Example)46 b FP(25)p FC(.)g FP(An)38 b(extension)e FO(G)h FP(of)g(an)n(y)f(group)g FO(G)1829 1742 y FL(1)1904 1730 y FP(b)n(y)g FB(F)2084 1742 y FL(2)2120 1730 y FP(,)j(is)e(a)f FQ(\003)p FP(-wild)118 1830 y(group.)243 1956 y(Notice)22 b(that)h(the)g(pro)r(of)f(of)h (wildness)f(of)h FJ(Z)1587 1968 y FL(2)1627 1956 y FQ(\003)8 b FJ(Z)1738 1968 y FL(3)1792 1956 y FP(giv)n(en)22 b(in)h(Section)f (3.1.3,)118 2055 y(is)28 b(not)f(reduced)g(to)h(the)g(c)n(hec)n(k)f(of) g(the)h(conditions)f(of)h(the)g(corollary)d(ab)r(o)n(v)n(e.)243 2155 y(Since)33 b(the)h(ma)5 b(jorization)32 b(is)h(a)g(quasi-order)e (relation)i(\(Section)h(3.1.1\),)118 2255 y(the)28 b(follo)n(wing)f (corollary)e(from)i(Prop)r(osition)f(71)h(holds.)118 2406 y FR(Corollary)41 b(16.)j FC(If)36 b(a)h(gr)l(oup)f FO(G)g FC(p)l(ossesses)g(a)g(normal)h(sub)l(gr)l(oup)e FO(N)45 b FC(such)118 2506 y(that)30 b FO(G=)-5 b(N)32 b FP(=)22 b FO(G)641 2518 y FL(1)709 2506 y FC(is)30 b FQ(\003)p FC(-wild,)g(then)g FO(G)g FC(is)g FQ(\003)p FC(-wild.)118 2658 y(Example)i FP(26)p FC(.)k FP(The)23 b(group)e FO(S)5 b(L)p FP(\(2)p FO(;)14 b FJ(Z)n FP(\))i(is)22 b FQ(\003)p FP(-wild,)h(since)f FO(S)5 b(L)p FP(\(2)p FO(;)14 b FJ(Z)o FP(\))p FO(=)o FQ(f)p FO(e;)f FQ(\000)p FO(e)p FQ(g)j FP(=)118 2757 y FO(P)c(S)5 b(L)p FP(\(2)p FO(;)14 b FJ(Z)n FP(\))k(=)k FJ(Z)666 2769 y FL(2)715 2757 y FQ(\003)c FJ(Z)837 2769 y FL(3)868 2757 y FP(.)118 2883 y FC(Example)33 b FP(27)p FC(.)k FP(The)24 b(braid)f(group)f FO(B)1277 2895 y FL(2)1338 2883 y FP(is)h FQ(\003)p FP(-wild,)h(since)f FO(B)1948 2895 y FL(2)1985 2883 y FO(=)p FJ(Z)16 b FP(=)23 b FJ(Z)2254 2895 y FL(2)2295 2883 y FQ(\003)10 b FJ(Z)2408 2895 y FL(3)2439 2883 y FP(,)25 b(its)118 2983 y(group)31 b FQ(\003)p FP(-algebra)e(is)i FJ(C)15 b FP([)p FO(B)949 2995 y FL(2)992 2983 y FP(])30 b(=)f FJ(C)15 b FQ(h)q FO(u;)f(v)38 b FQ(j)30 b FO(u)p FP(,)d FO(v)k FP(are)c(unitary)o FO(;)14 b(uv)s(u)29 b FP(=)g FO(v)s(uv)s FQ(i)h FP(=)118 3082 y FJ(C)15 b FQ(h)p FO(w)45 b FP(=)36 b FO(uv)s(u;)14 b(z)39 b FP(=)d FO(uv)j FQ(j)d FO(w)r FP(,)29 b FO(z)h FP(are)d(unitary)p FO(;)14 b(w)1635 3052 y FL(2)1709 3082 y FP(=)35 b FO(z)1852 3052 y FL(3)1889 3082 y FQ(i)p FP(,)j(and)d(its)h(quotien)n(t)118 3182 y(algebra)26 b(is)h FJ(C)15 b FQ(h)q FO(w)r(;)f(z)33 b FQ(j)23 b FO(w)r FP(,)28 b FO(z)j FP(are)c(unitary)o FO(;)14 b(w)1479 3152 y FL(2)1540 3182 y FP(=)23 b FO(z)1671 3152 y FL(3)1730 3182 y FP(=)g FO(e)p FQ(i)g FP(=)f FJ(C)15 b FP([)p FJ(Z)2138 3194 y FL(2)2193 3182 y FQ(\003)j FJ(Z)2315 3194 y FL(3)2346 3182 y FP(].)118 3350 y FR(3.)35 b FP(It)23 b(lo)r(oks)f(attractiv)n(e) h(to)g(in)n(v)n(estigate)e(whether)i(the)h(follo)n(wing)e(groups)g(are) 118 3450 y FQ(\003)p FP(-wild)27 b(or)g(not:)243 3550 y(a\))19 b(Co)n(xeter)f(groups)g(whic)n(h)i(are)e(not)h(a\016ne)h (\(except)f(for)2055 3532 y Fo(r)p 2055 3533 125 4 v 99 w(r)100 b(r)p 2179 3533 V 2084 3515 a FN(1)59 b(1)2365 3550 y FP(whic)n(h)118 3649 y(is)28 b FQ(\003)p FP(-wild,)f(see)g (Section)h(3.1.3\);)243 3749 y(b\))g(non-elemen)n(tary)e(h)n(yp)r(erb)r (olic)h(groups;)118 3848 y(and)h(man)n(y)f(other)g(kno)n(wn)f(groups,)h (whic)n(h)g(are)g(not)g(amenable.)243 3948 y(The)34 b(structure)f(of)h (quotien)n(t)g(groups)f(for)g(the)h(groups)f(listed)h(ab)r(o)n(v)n(e)f (is)118 4048 y(a)c(sub)5 b(ject)29 b(the)g(authors)f(are)f(not)i (familiar)g(enough)f(with;)i(th)n(us)f(w)n(e)f(cannot)118 4147 y(estimate)g(prop)r(erly)e(ho)n(w)h(complicated)g(the)h(listed)g (tasks)f(are.)p eop %%Page: 229 233 229 232 bop 118 100 a FK(3.2.)36 b(Classes)27 b(of)g(non-self-adjoin)n (t)g(op)r(erators)853 b FP(229)118 333 y FR(4.)36 b FP(Consider)27 b(some)g(examples)g(of)g(groups)g(whic)n(h)g(are)g(not)g FQ(\003)p FP(-wild.)243 432 y(If)40 b FO(G)g FP(is)g(amenable,)i(then)e FO(C)1212 402 y FN(\003)1250 432 y FP(\()p FO(G)p FP(\))h(is)f(n)n (uclear.)72 b(Th)n(us,)43 b(w)n(e)c(ha)n(v)n(e)f(the)118 532 y(follo)n(wing)27 b(statemen)n(t.)118 683 y FR(Prop)s(osition)j (72.)41 b FC(If)30 b FO(G)g FC(is)g(an)g(amenable)h(gr)l(oup,)f(then)g FO(G)g FC(is)g(not)f FQ(\003)p FC(-wild.)243 834 y FP(Also,)e(the)h (follo)n(wing)e(theorem)i(holds.)118 986 y FR(Theorem)g(62.)39 b FC(L)l(et)27 b FO(G)h FC(b)l(e)g(a)g(p)l(erio)l(dic)i(gr)l(oup,)f (i.e.,)h(any)e(element)f FO(g)f FQ(2)d FO(G)28 b FC(is)118 1085 y(p)l(erio)l(dic.)41 b(Then)30 b FO(G)g FC(is)g(not)g FQ(\003)p FC(-wild.)118 1236 y(Pr)l(o)l(of.)43 b FP(Supp)r(ose)30 b(that)f FO(G)h FP(is)f FQ(\003)p FP(-wild,)g(i.e.,)h(there)f(exists)g (a)g(homomorphism)118 1336 y FO(\036)9 b FP(:)30 b FO(G)e FQ(\000)-48 b(!)29 b FO(U)508 1348 y FM(n)553 1336 y FP(\()p FO(C)650 1306 y FN(\003)688 1336 y FP(\()p FB(F)776 1348 y FL(2)812 1336 y FP(\)\))i(suc)n(h)g(that)g(the)g(functor)g FO(F)21 b FP(:)43 b(Rep)14 b FB(F)2073 1348 y FL(2)2137 1336 y FQ(\000)-49 b(!)29 b FP(Rep)14 b FO(G)31 b FP(is)118 1436 y(full.)42 b(Consider)28 b(a)h(family)g(of)g(one-dimensional)e (represen)n(tations)g FO(h)2305 1448 y FM(t)2364 1436 y FP(of)h(the)118 1535 y(group)j FB(F)414 1547 y FL(2)479 1535 y FP(=)f FQ(h)p FO(u;)14 b(v)s FQ(i)32 b FP(in)g(the)g(space)f FJ(C)53 b FP(suc)n(h)31 b(that)h FO(h)1787 1547 y FM(t)1816 1535 y FP(\()p FO(u)p FP(\))f(=)e(1,)k FO(h)2199 1547 y FM(t)2228 1535 y FP(\()p FO(v)s FP(\))e(=)e FO(e)2499 1505 y FM(it)2552 1535 y FP(,)118 1644 y FO(t)23 b FQ(2)h FP(\(0)p FO(;)14 b FP(2)p FO(\031)s FP(],)28 b(and)f(denote)h(b)n(y)f FO(U)1128 1656 y FM(t)1157 1644 y FP(\()p FO(g)s FP(\))d(=)1377 1622 y(^)1376 1644 y FO(h)1424 1656 y FM(t)1471 1644 y FQ(\016)19 b FO(\036)p FP(\()p FO(g)s FP(\),)28 b FO(g)e FQ(2)e FO(G)p FP(,)k(the)g(matrix)f(with)118 1744 y(en)n(tries)22 b(that)i(are)d(con)n(tin)n(uous)h(functions)i(in)f FO(t)p FP(.)35 b(Since)23 b(the)h(functor)e FO(F)35 b FP(is)23 b(full,)118 1843 y(the)31 b(represen)n(tations)842 1821 y(^)842 1843 y FO(h)890 1855 y FM(t)915 1863 y Fy(1)971 1843 y FQ(\016)20 b FO(\036)30 b FP(and)1277 1821 y(^)1276 1843 y FO(h)1324 1855 y FM(t)1349 1863 y Fy(2)1406 1843 y FQ(\016)19 b FO(\036)31 b FP(of)f(the)g(group)f FO(G)h FP(are)f(unitarily)118 1943 y(inequiv)-5 b(alen)n(t)30 b(for)g(all)f FO(t)861 1955 y FL(1)926 1943 y FQ(6)p FP(=)d FO(t)1047 1955 y FL(2)1085 1943 y FP(.)44 b(Since)30 b(the)g(irreducible)g(represen)n(tations)e(of)118 2042 y(the)23 b(group)f FO(G)i FP(in)f(a)f(\014nite-dimensional)h(space)f (are)g(uniquely)h(de\014ned,)i(up)e(to)118 2142 y(a)31 b(unitary)f(equiv)-5 b(alence,)32 b(b)n(y)e(their)h(c)n(haracters)e (\(see,)j(for)e(example,)i([135)n(]\),)118 2242 y(there)h(exists)f FO(g)i FQ(2)e FO(G)i FP(suc)n(h)e(that)h(T)-7 b(r)14 b FO(U)1364 2254 y FM(t)1389 2262 y Fy(1)1425 2242 y FP(\()p FO(g)s FP(\))32 b FQ(6)p FP(=)f(T)-7 b(r)13 b FO(U)1816 2254 y FM(t)1841 2262 y Fy(2)1878 2242 y FP(\()p FO(g)s FP(\).)52 b(Then)33 b(T)-7 b(r)14 b FO(U)2439 2254 y FM(t)2468 2242 y FP(\()p FO(g)s FP(\))118 2341 y(is)36 b(a)g(con)n(tin)n(uous)g(function)h(in)f FO(t)p FP(,)j(whic)n(h)d(is)h(not)f(a)g(constan)n(t.)62 b(Order)35 b(the)118 2441 y(eigen)n(v)-5 b(alues)29 b FO(k)593 2453 y FM(i)621 2441 y FP(\()p FO(t)p FP(\),)i FO(i)26 b FP(=)h(1,)i FO(:)14 b(:)g(:)28 b FP(,)i FO(n)p FP(,)g(of)g(the)g(matrix)f FO(U)1862 2453 y FM(t)1891 2441 y FP(\()p FO(g)s FP(\))h(b)n(y)g(the)g (v)-5 b(alue)30 b(of)118 2541 y(the)35 b(argumen)n(t.)57 b(Then)34 b(there)h(exists)f FO(i)g FP(suc)n(h)g(that)h FO(k)1859 2553 y FM(i)1887 2541 y FP(\()p FO(t)p FP(\))g FQ(6)p FP(=)f(const.)58 b(Since)118 2640 y(for)38 b(a)h(unitary)f (matrix)g(the)h(problem)f(of)h(\014nding)g(eigen)n(v)-5 b(alues)37 b(is)i(stable,)118 2740 y(the)26 b(eigen)n(v)-5 b(alue)24 b FO(k)696 2752 y FM(i)724 2740 y FP(\()p FO(t)p FP(\))i(is)f(a)g(non-trivial)f(con)n(tin)n(uous)g(function)i(of)f FO(t)g FP(on)g(some)118 2839 y(in)n(terv)-5 b(al)21 b(\()p FO(t)474 2851 y FL(1)511 2839 y FO(;)14 b(t)578 2851 y FL(2)615 2839 y FP(\).)36 b(But)21 b(then)h(there)f(exists)g FO(t)1506 2851 y FL(0)1565 2839 y FP(suc)n(h)g(that)g FO(k)1962 2851 y FM(i)1990 2839 y FP(\()p FO(t)2052 2851 y FL(0)2090 2839 y FP(\))g(is)g(not)h(a)f(ro)r(ot)118 2939 y(of)30 b(unit)n(y)-7 b(.)45 b(Since)30 b FO(G)h FP(is)f(a)f(p)r(erio)r(dic)h(group,)g(for)f(eac)n(h)h FO(g)g FQ(2)d FO(G)k FP(there)f(exists)f(a)118 3050 y(p)r(o)n(w)n(er)d FO(N)9 b FP(\()p FO(g)s FP(\))27 b(suc)n(h)g(that)h FO(g)980 3020 y FM(N)6 b FL(\()p FM(g)r FL(\))1152 3050 y FP(=)22 b FO(e)p FP(;)27 b(but)h(on)f(the)h(other)e(hand,)h FO(U)2250 3007 y FM(N)6 b FL(\()p FM(g)r FL(\))2241 3071 y FM(t)2266 3079 y Fy(0)2423 3050 y FQ(6)p FP(=)22 b(1.)118 3150 y(The)28 b(obtained)f(con)n(tradiction)f(completes)i(the)g(pro)r(of.)p 2514 3150 4 57 v 2518 3097 50 4 v 2518 3150 V 2567 3150 4 57 v 118 3312 a FR(Corollary)d(17.)34 b FC(Ther)l(e)24 b(ar)l(e)g(gr)l(oups)f(which)i(ar)l(e)f(not)f FQ(\003)p FC(-wild)g(and)h(not)f(amen-)118 3412 y(able.)37 b(Those)24 b(ar)l(e,)h(for)e(example,)j(Burnside)d(gr)l(oups)f FO(B)t FP(\()p FO(m;)14 b(n)p FP(\))23 b FC(which)h(ar)l(e)f(not)118 3511 y(amenable)31 b(for)g(o)l(dd)f FO(n)23 b FQ(\025)g FP(665)28 b FC(and)i FO(m)23 b FQ(\025)g FP(2)29 b(\()p FC(se)l(e)37 b FP([189)n(,)28 b(2)o(]\))p FC(.)118 3750 y FH(3.2)104 b(On)26 b(the)f(complexit)m(y)e(of)i(the)g(description)f (of)i(classes)373 3866 y(of)38 b(non-self-adjoin)m(t)g(op)s(erators)118 4048 y FP(The)d(b)r(orderline)f(b)r(et)n(w)n(een)h(the)g(theory)f(of)h (op)r(erators)e(and)i(the)g(theory)f(of)118 4147 y(op)r(erator)25 b(algebras)f(and)i(their)h(represen)n(tations)d(can)i(b)r(e)h(view)n (ed)f(as)f(a)h(riv)n(er)p eop %%Page: 230 234 230 233 bop 118 100 a FP(230)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FP(with)f(n)n(umerous)e(t)n(w)n(o-w)n(a)n(y)e(bridges)j(joining)f (the)i(banks)e(\(see,)i(for)e(example,)118 432 y([81)o(],)d(and)f (others\).)34 b(One)20 b(of)g(these)h(bridges)e(is)i(discussed)f(in)h (this)f(section:)33 b(w)n(e)118 532 y(consider)22 b(an)h(application)g (of)g(the)h(theory)e(of)h(represen)n(tations)f(of)h FQ(\003)p FP(-algebras)118 632 y(to)28 b(a)f(study)h(of)f(classes)f(of)i(op)r (erators)e(that)h(are)g(singled)g(out)h(algebraically)-7 b(.)243 731 y(Let)24 b FO(X)30 b FP(b)r(e)24 b(a)f(b)r(ounded)i (non-self-adjoin)n(t)d(op)r(erator)h(acting)g(in)h(a)f(Hilb)r(ert)118 831 y(space)28 b FO(H)7 b FP(.)39 b(W)-7 b(e)28 b(consider)g(classes)f (of)h(op)r(erators)e(whic)n(h)j(satisfy)f(p)r(olynomial)118 930 y(relations)f FO(P)509 942 y FM(j)544 930 y FP(\()p FO(X)r(;)14 b(X)760 900 y FN(\003)797 930 y FP(\))23 b(=)g(0,)k FO(j)h FP(=)23 b(1)p FO(;)14 b(:)g(:)g(:)27 b(;)14 b(m)p FP(,)27 b(and)h(more)f(general)f(relations.)243 1030 y(F)-7 b(or)36 b(ev)n(ery)f(suc)n(h)h(class)g(of)g(relations)f (there)i(corresp)r(onds)d(a)i FQ(\003)p FP(-algebra)118 1130 y FA(A)i FP(=)g FJ(C)15 b FQ(h)p FO(x)q(;)f(x)537 1100 y FN(\003)619 1130 y FQ(j)39 b FO(P)734 1142 y FM(j)769 1130 y FP(\()p FO(x;)14 b(x)932 1100 y FN(\003)971 1130 y FP(\))39 b(=)f(0)p FO(;)27 b(j)43 b FP(=)38 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(m)p FQ(i)p FP(.)65 b(If)37 b(the)g(class)f(of)h(op)r (er-)118 1229 y(ators)32 b(is)h(giv)n(en)f(b)n(y)g(non-p)r(olynomial)g (relations,)h(then)h(the)f(corresp)r(onding)118 1329 y FQ(\003)p FP(-algebra)22 b(is)i(giv)n(en)f(in)i(a)e(more)h (complicated)f(w)n(a)n(y)-7 b(.)35 b(Eac)n(h)23 b(represen)n(tation)f FO(\031)118 1429 y FP(of)27 b(the)g FQ(\003)p FP(-algebra)d FA(A)i FP(determines)h(the)g(b)r(ounded)g(op)r(erators)e FO(X)k FP(=)23 b FO(\031)s FP(\()p FO(x)p FP(\))28 b(and)118 1528 y FO(X)194 1498 y FN(\003)255 1528 y FP(=)22 b FO(\031)s FP(\()p FO(x)471 1498 y FN(\003)510 1528 y FP(\))28 b(suc)n(h)g(that) 746 1707 y FO(P)799 1719 y FM(j)834 1707 y FP(\()p FO(X)r(;)14 b(X)1050 1673 y FN(\003)1088 1707 y FP(\))23 b(=)g(0)p FO(;)179 b(j)28 b FP(=)23 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(m:)457 b FP(\(3.6\))243 1886 y(Con)n(v)n(ersely)-7 b(,)24 b(giv)n(en)i(op)r (erators)e FO(X)32 b FP(and)26 b FO(X)1597 1856 y FN(\003)1661 1886 y FP(suc)n(h)g(that)h FO(P)2079 1898 y FM(j)2114 1886 y FP(\()p FO(X)r(;)14 b(X)2330 1856 y FN(\003)2367 1886 y FP(\))24 b(=)e(0,)118 1986 y FO(j)28 b FP(=)23 b(1,)i FO(:)14 b(:)g(:)27 b FP(,)f FO(m)p FP(,)g(one)f(can)g(uniquely)h (de\014ne)g(a)f(represen)n(tation)f(of)h(the)h(whole)118 2085 y(algebra)g FA(A)p FP(.)243 2185 y(The)k(problem)f(to)g(describ)r (e)h(the)g(class)f(of)h(op)r(erators)e(whic)n(h)h(satisfy)h(re-)118 2284 y(lations)25 b(\(3.6\),)i(up)f(to)g(a)f(unitary)h(equiv)-5 b(alence,)26 b(is)f(equiv)-5 b(alen)n(t)26 b(to)g(the)g(one)g(of)118 2384 y(describing)h(represen)n(tations)e(of)j(the)g(corresp)r(onding)e FQ(\003)p FP(-algebra)f FA(A)p FP(.)243 2484 y(F)-7 b(or)23 b(suc)n(h)h(algebras,)f(w)n(e)g(estimate)h(the)h(complexit)n(y)e(of)h (the)h(corresp)r(ond-)118 2583 y(ing)32 b(problem)h(of)f(the)h FQ(\003)p FP(-represen)n(tations)d(theory)-7 b(,)33 b(i.e.,)h(the)f (complexit)n(y)f(of)118 2683 y(the)c(unitary)f(description)g(of)h(the)g (corresp)r(onding)d(class)i(of)h(op)r(erators.)243 2783 y(W)-7 b(e)25 b(consider)f(classes)f(of)i(op)r(erators)e(connected)i (with)g(quadratic,)f(semi-)118 2882 y(linear)c(cubic,)i(and)f(some)f (other)g(relations)f(\(Section)i(3.2.1\).)34 b(Then)20 b(w)n(e)h(study)118 2982 y(complexit)n(y)34 b(of)h(the)g(unitary)f (description)g(of)h(partial)f(isometries,)h(w)n(eakly)118 3081 y(cen)n(tered)29 b(op)r(erators,)f(and)h(algebraic)f(op)r(erators) f(\(Section)j(3.2.2\).)42 b(In)29 b(Sec-)118 3181 y(tion)24 b(3.2.3,)g(w)n(e)g(sp)r(eak)g(ab)r(out)g(the)g(complexit)n(y)g(of)g (description)g(of)g(classes)f(of)118 3281 y(op)r(erators)i(whic)n(h)i (are)f(de\014ned)h(not)g(b)n(y)g(p)r(olynomial)f(equalities)g(but)i (rather)118 3380 y(b)n(y)34 b(conditions)h(similar)f(to)g (inequalities,)i(or)e(other)g(non-algebraic)e(condi-)118 3480 y(tions;)e(namely)-7 b(,)30 b(w)n(e)f(consider)g(h)n(yp)r(onormal) f(op)r(erators)f(and)j(pairs)e(of)i(com-)118 3580 y(m)n(uting)e (completely)f(non-unitary)g(isometries.)118 3795 y FR(3.2.1)94 b(Classes)36 b(of)h(non-self-adjoin)m(t)f(op)s(erators)h(singled)e(out) i(b)m(y)410 3894 y(a)32 b(quadratic)i(or)d(a)i(cubic)f(relation)118 4048 y FP(A)20 b(normal)f(op)r(erator)f FO(X)27 b FP(is)20 b(an)f(op)r(erator)f(suc)n(h)i(that)g FO(X)7 b(X)1911 4018 y FN(\003)1971 4048 y FP(=)23 b FO(X)2135 4018 y FN(\003)2172 4048 y FO(X)7 b FP(.)34 b(Normal)118 4147 y(op)r(erators)e(mak)n(e)i(the)h(most)f(studied)h(region)e(in)h(the)h (terrain)e(of)h(b)r(ounded)p eop %%Page: 231 235 231 234 bop 118 100 a FK(3.2.)36 b(Classes)27 b(of)g(non-self-adjoin)n (t)g(op)r(erators)853 b FP(231)118 333 y(linear)29 b(op)r(erators.)41 b(Irreducible)28 b(normal)h(op)r(erators)f(are)g(one-dimensional.)118 432 y(The)k(sp)r(ectral)f(theorem)g(giv)n(es)g(a)g(pro)r(cedure)f(for)i (assem)n(bling)e(an)n(y)h(normal)118 532 y(op)r(erator)26 b(from)h(irreducible)g(ones.)118 682 y FR(1.)33 b FP(Let)19 b(us)g(ha)n(v)n(e)e(a)i(pair)f(of)g(op)r(erators)f FO(X)25 b FP(and)18 b FO(X)1635 651 y FN(\003)1692 682 y FP(whic)n(h)g(satisfy) h(a)f(quadratic)118 781 y(relation)27 b(of)g(the)h(form)821 964 y FO(P)874 976 y FL(2)912 964 y FP(\()p FO(X)r(;)14 b(X)1128 930 y FN(\003)1165 964 y FP(\))23 b(=)g FO(P)1373 930 y FN(\003)1361 984 y FL(2)1411 964 y FP(\()p FO(X)r(;)14 b(X)1627 930 y FN(\003)1664 964 y FP(\))24 b(=)e(0)p FO(:)532 b FP(\(3.7\))118 1147 y(A)28 b(common)f(form)g(of)h(suc)n(h)f (a)g(relation)g(is)h(the)g(follo)n(wing:)258 1329 y FO(P)311 1341 y FL(2)348 1329 y FP(\()p FO(X)r(;)14 b(X)564 1295 y FN(\003)601 1329 y FP(\))24 b(=)e FO(a)p FP(\()p FO(X)896 1295 y FL(2)952 1329 y FP(+)c(\()p FO(X)1143 1295 y FN(\003)1180 1329 y FP(\))1212 1295 y FL(2)1250 1329 y FP(\))h(+)f FO(i)1413 1295 y FN(\000)p FL(1)1501 1329 y FO(b)p FP(\()p FO(X)1645 1295 y FL(2)1700 1329 y FQ(\000)g FP(\()p FO(X)1891 1295 y FN(\003)1929 1329 y FP(\))1961 1295 y FL(2)1999 1329 y FP(\))g(+)g FO(c)p FP([)p FO(X)r(;)c(X)2375 1295 y FN(\003)2412 1329 y FP(])308 1464 y(+)k FO(d)p FQ(f)p FO(X)r(;)c(X)660 1430 y FN(\003)697 1464 y FQ(g)k FP(+)g FO(e)p FP(\()p FO(X)24 b FP(+)18 b FO(X)1163 1430 y FN(\003)1201 1464 y FP(\))h(+)f FO(i)1364 1430 y FN(\000)p FL(1)1452 1464 y FO(f)9 b FP(\()p FO(X)25 b FQ(\000)18 b FO(X)1787 1430 y FN(\003)1824 1464 y FP(\))h(+)f FO(g)s(I)30 b FP(=)22 b(0;)185 b(\(3.8\))118 1647 y(here)28 b FO(a)p FP(,)i FO(b)p FP(,)e FO(c)p FP(,)i FO(d)p FP(,)f FO(e)p FP(,)g FO(f)9 b FP(,)29 b FO(g)e FQ(2)f FJ(R)p FP(.)46 b(W)-7 b(e)29 b(no)n(w)f(giv)n(e)g(a)h(criterion)e(for)i(the)g (relation)118 1747 y(\(that)37 b(is)e(the)h(corresp)r(onding)e FQ(\003)p FP(-algebra\))g(to)i(b)r(e)g FQ(\003)p FP(-wild)f(in)h(terms) g(of)f(the)118 1846 y(co)r(e\016cien)n(ts.)118 2013 y FR(Theorem)g(63.)43 b FC(The)34 b(c)l(orr)l(esp)l(onding)g FQ(\003)p FC(-algebr)l(a)g(is)g FQ(\003)p FC(-wild)f(if)h(and)g(only)g (if)118 2112 y(one)c(of)h(the)f(fol)t(lowing)i(c)l(onditions)f(hold)9 b FP(:)220 2278 y(1)p FO(:)41 b(a)23 b FP(=)f FO(b)h FP(=)g FO(c)g FP(=)f FO(d)h FP(=)g FO(e)g FP(=)f FO(f)32 b FP(=)23 b FO(g)i FP(=)e(0;)220 2493 y(2)p FO(:)326 2401 y Fz(\020)375 2493 y FO(g)e FQ(\000)639 2437 y FO(e)678 2407 y FL(2)p 529 2474 295 4 v 529 2550 a FP(2\()p FO(a)d FP(+)g FO(d)p FP(\))834 2401 y Fz(\021)883 2493 y FP(\()p FO(a)h FP(+)f FO(d)p FP(\))24 b FO(<)e FP(0)p FC(,)85 b FO(d)19 b FQ(\000)f FO(a)23 b FP(=)f FO(b)h FP(=)g FO(c)g FP(=)f FO(f)32 b FP(=)23 b(0;)220 2770 y(3)p FO(:)326 2678 y Fz(\020)375 2770 y FO(g)e FQ(\000)633 2714 y FO(f)683 2683 y FL(2)p 529 2751 V 529 2827 a FP(2\()p FO(d)e FQ(\000)f FO(a)p FP(\))834 2678 y Fz(\021)883 2770 y FP(\()p FO(d)h FQ(\000)f FO(a)p FP(\))24 b FO(<)e FP(0)p FC(,)85 b FO(a)18 b FP(+)g FO(d)24 b FP(=)e FO(b)h FP(=)g FO(c)g FP(=)f FO(e)h FP(=)g(0;)220 3046 y(4)p FO(:)41 b(b)362 3012 y FL(2)422 3046 y FP(=)22 b(\()p FO(d)584 3012 y FL(2)640 3046 y FQ(\000)c FO(a)767 3012 y FL(2)805 3046 y FP(\))23 b FQ(6)p FP(=)g(0)p FO(;)98 b FP(\()p FO(a)18 b FP(+)g FO(d)p FP(\))1363 2954 y Fz(\020)1414 3046 y FO(g)j FQ(\000)1677 2990 y FO(e)1716 2960 y FL(2)p 1568 3027 V 1568 3103 a FP(2\()p FO(a)d FP(+)g FO(d)p FP(\))1872 2954 y Fz(\021)1945 3046 y FO(<)k FP(0)p FC(,)424 3233 y FO(e)463 3203 y FL(2)p 336 3270 254 4 v 336 3346 a FP(\()p FO(a)c FP(+)g FO(d)p FP(\))622 3289 y(=)803 3233 y FO(f)853 3203 y FL(2)p 719 3270 V 719 3346 a FP(\()p FO(d)h FQ(\000)f FO(a)p FP(\))982 3289 y FC(,)116 b FO(c)23 b FP(=)f(0)p FC(.)243 3500 y FP(This)d(theorem)g(follo)n(ws)g(from)g(Theorem)g(60,)i (b)n(y)e(the)h(c)n(hange)f(of)g(v)-5 b(ariables)118 3599 y FO(X)29 b FP(=)23 b FO(A)7 b FP(+)g FO(iB)t FP(,)24 b FO(X)664 3569 y FN(\003)725 3599 y FP(=)e FO(A)7 b FQ(\000)g FO(iB)t FP(.)36 b(It)22 b(is)g(easy)g(to)f(see)h(that)h(the)f (relation)f(satis\014ed)118 3699 y(b)n(y)32 b(the)h(co)r(e\016cien)n (ts)f(is)g(the)h(follo)n(wing:)45 b FO(\013)32 b FP(=)e FO(a)22 b FP(+)f FO(d)p FP(,)34 b FO(\014)h FP(=)c FO(d)22 b FQ(\000)f FO(a)p FP(,)33 b FO(\015)j FP(=)31 b(2)p FO(b)p FP(,)118 3798 y FO(q)26 b FP(=)d(2)p FO(c)p FP(,)k FO(\017)c FP(=)f(2)p FO(f)9 b FP(,)27 b FO(\037)c FP(=)g FO(g)s FP(.)118 3948 y FR(2.)52 b FP(No)n(w)33 b(w)n(e)f(will)h (consider)f(some)h(classes)e(of)i(non-self-adjoin)n(t)f(op)r(erators) 118 4048 y(whic)n(h)c(satisfy)f(a)h(cubic)g(relation.)37 b(A)n(t)28 b(\014rst)f(w)n(e)h(will)g(pass)f(to)h(a)f(pair)g(of)h (self-)118 4147 y(adjoin)n(t)c(op)r(erators,)f FO(A)p FP(,)i FO(B)t FP(,)g(b)n(y)e(the)i(c)n(hange)e(of)h(v)-5 b(ariables,)23 b FO(X)29 b FP(=)23 b FO(A)11 b FP(+)g FO(iB)28 b FP(and)p eop %%Page: 232 236 232 235 bop 118 100 a FP(232)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FO(X)194 303 y FN(\003)255 333 y FP(=)22 b FO(A)13 b FQ(\000)g FO(iB)t FP(.)36 b(Let)25 b(the)g(self-adjoin)n(t)f(op)r (erators)f FO(A)i FP(and)g FO(B)k FP(satisfy)24 b(a)h(cubic)118 432 y(semi-linear)i(relation)f(\(linear)h(in)h FO(B)t FP(\).)38 b(The)27 b(usual)h(form)f(of)g(suc)n(h)h(a)f(relation)118 532 y(with)h(the)g(condition)g FO(P)868 544 y FL(3)905 532 y FP(\()p FO(A;)14 b(B)t FP(\))24 b(=)f FO(P)1312 502 y FN(\003)1300 553 y FL(3)1350 532 y FP(\()p FO(A;)14 b(B)t FP(\))28 b(is)g(the)g(follo)n(wing:)445 716 y FO(P)498 728 y FL(3)536 716 y FP(\()p FO(A;)14 b(B)t FP(\))24 b(=)e FO(\013B)h FP(+)18 b(2)p FO(\014)t FQ(f)p FO(A;)c(B)t FQ(g)k FP(+)g FO(\017)p FQ(f)p FO(A)1681 682 y FL(2)1718 716 y FO(;)c(B)t FQ(g)k FP(+)g(2)p FO(\026AB)t(A)865 851 y FP(+)g FO(i\015)5 b FP([)p FO(A;)14 b(B)t FP(])19 b(+)f FO(i\016)s FP([)p FO(A)1493 817 y FL(2)1530 851 y FO(;)c(B)t FP(])23 b(=)g(0)p FO(;)571 b FP(\(3.9\))118 1035 y FO(\013)p FP(,)28 b FO(\014)t FP(,)g FO(\015)5 b FP(,)28 b FO(\016)e FQ(2)d FJ(R)618 1005 y FL(1)661 1035 y FP(.)243 1135 y(It)32 b(is)g(easy)g(to)g(see)g(that,)i(in)e (terms)g(of)g FO(X)39 b FP(and)32 b FO(X)1853 1105 y FN(\003)1891 1135 y FP(,)h(the)g(relation)e(\(3.9\))118 1235 y(tak)n(es)c(the)h(form:)253 1419 y FO(P)306 1431 y FL(3)344 1419 y FP(\()p FO(X)r(;)14 b(X)560 1385 y FN(\003)597 1419 y FP(\))24 b(=)e FO(i)769 1385 y FN(\000)p FL(1)858 1419 y FO(a)p FP(\()p FO(X)j FQ(\000)18 b FO(X)1187 1385 y FN(\003)1224 1419 y FP(\))h(+)f FO(i)1387 1385 y FN(\000)p FL(1)1476 1419 y FO(b)p FP(\()p FO(X)1620 1385 y FL(2)1675 1419 y FQ(\000)g FP(\()p FO(X)1866 1385 y FN(\003)1904 1419 y FP(\))1936 1385 y FL(2)1973 1419 y FP(\))648 1554 y(+)g FO(i)760 1520 y FN(\000)p FL(1)849 1554 y FP(\()p FO(c)g FP(+)g FO(d)p FP(\)\()p FO(X)1201 1520 y FL(3)1257 1554 y FQ(\000)g FP(\()p FO(X)1448 1520 y FN(\003)1486 1554 y FP(\))1518 1520 y FL(3)1556 1554 y FP(\))648 1689 y(+)g FO(i)760 1655 y FN(\000)p FL(1)849 1689 y FP(\()p FO(c)g FQ(\000)g FO(d)p FP(\)\()p FO(X)7 b(X)1277 1655 y FN(\003)1315 1689 y FO(X)25 b FQ(\000)18 b FO(X)1568 1655 y FN(\003)1605 1689 y FO(X)7 b(X)1757 1655 y FN(\003)1794 1689 y FP(\))648 1824 y(+)18 b FO(i)760 1789 y FN(\000)p FL(1)849 1824 y FO(d)892 1756 y Fz(\000)930 1824 y FQ(f)p FO(X)1048 1789 y FL(2)1084 1824 y FO(;)c(X)1197 1789 y FN(\003)1234 1824 y FQ(g)k FP(+)g FQ(f)p FO(X)r(;)c FP(\()p FO(X)1635 1789 y FN(\003)1672 1824 y FP(\))1704 1789 y FL(2)1742 1824 y FQ(g)1784 1756 y Fz(\001)648 1958 y FP(+)k FO(f)9 b FP([)p FO(X)r(;)14 b(X)988 1924 y FN(\003)1025 1958 y FP(])k(+)g FO(g)1192 1891 y Fz(\000)1230 1958 y FP([)p FO(X)1329 1924 y FN(\003)1366 1958 y FO(;)c(X)1479 1924 y FL(2)1516 1958 y FP(])k(+)g([\()p FO(X)1771 1924 y FN(\003)1809 1958 y FP(\))1841 1924 y FL(2)1879 1958 y FO(;)c(X)7 b FP(])2015 1891 y Fz(\001)2075 1958 y FP(=)23 b(0)p FO(;)148 b FP(\(3.10\))118 2143 y(where)27 b FO(a)c FP(=)g FO(\013=)p FP(2,)k FO(b)c FP(=)f FO(\014)t FP(,)28 b FO(c)23 b FP(=)g FO(\017=)p FP(4,)j FO(d)e FP(=)e FO(\026=)p FP(4,)27 b FO(f)32 b FP(=)22 b FO(\015)5 b(=)p FP(2,)27 b FO(g)e FP(=)e FO(\016)s(=)p FP(4.)243 2243 y(W)-7 b(rite)37 b FO(I)519 2255 y FL(1)596 2243 y FP(=)i(8)p FO(c)p FP(,)g FO(I)876 2255 y FL(2)953 2243 y FP(=)g FO(c)1093 2213 y FL(2)1155 2243 y FQ(\000)25 b FP(4)p FO(d)1330 2213 y FL(2)1367 2243 y FP(,)40 b FO(I)1466 2255 y FL(3)1543 2243 y FP(=)f FO(a)p FP(\()p FO(c)1759 2213 y FL(2)1821 2243 y FQ(\000)24 b FP(4)p FO(d)1995 2213 y FL(2)2032 2243 y FP(\))i FQ(\000)e 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3823 y FP(\))23 b FQ(6)p FP(=)g(0)29 b FC(and)h(one)g(of)h(the)f(fol)t(lowing)i(c)l(onditions)f(holds)7 b FP(:)358 3992 y(\()p FO(a)p FP(\))42 b FO(c)23 b(>)g FP(0)p FC(,)30 b FO(d)23 b FP(=)g(0)p FC(,)29 b FO(g)s(b)18 b FQ(\000)g FP(2)p FO(cf)31 b FP(=)23 b(0)p FC(,)29 b FP(2)p FO(ag)1645 3962 y FL(2)1700 3992 y FQ(\000)18 b FO(cf)1869 3962 y FL(2)1929 3992 y FO(<)k FP(0)p FC(,)367 4127 y FP(\()p FO(b)p FP(\))41 b FO(d)24 b FQ(6)p FP(=)e(0)p FO(;)14 b(I)777 4139 y FL(4)837 4127 y FQ(6)p FP(=)23 b(0)p FC(,)30 b FO(ac)1102 4096 y FL(2)1157 4127 y FQ(\000)18 b FO(I)1276 4139 y FL(3)1332 4127 y FQ(\000)g FP(\()p FO(d)-14 b(f)28 b FQ(\000)18 b FO(g)s(b)g FQ(\000)g FP(2)p FO(I)1886 4139 y FL(4)1923 4127 y FP(\)\(2)p FO(f)9 b(c)2115 4096 y FL(2)2152 4127 y FO(=g)2237 4096 y FL(2)2273 4127 y FP(\))24 b(=)e(0)p FC(.)p eop %%Page: 233 237 233 236 bop 118 100 a FK(3.2.)36 b(Classes)27 b(of)g(non-self-adjoin)n (t)g(op)r(erators)853 b FP(233)118 333 y FR(3.)56 b FP(A)35 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b(for)f(irreducible)118 831 y(represen)n(tations)27 b(w)n(e)h(ha)n(v)n(e)g FO(X)1087 801 y FN(\003)1125 831 y FO(X)j FP(=)25 b FO(\025I)7 b FP(,)29 b FO(\025)d FQ(\025)f FP(0.)40 b(Then)29 b(either)g FO(X)i FP(=)25 b FO(X)2448 801 y FN(\003)2510 831 y FP(=)118 930 y(0,)33 b(or)f FO(\025)f(>)g FP(0)h(and)g FO(X)38 b FP(=)30 b FO(e)979 900 y FM(i\036)1047 860 y FQ(p)p 1116 860 49 4 v 70 x FO(\025)p FP(,)k(or)d FO(X=)1438 860 y FQ(p)p 1507 860 V 70 x FO(\025)h FP(is)h(a)f(unilateral)f(shift.) 52 b(There)118 1030 y(exists)27 b(a)h(corresp)r(onding)d(structure)i (theorem.)118 1178 y FR(4.)44 b FP(No)n(w)29 b(w)n(e)h(consider)f (another)g(similar)g(class)g(of)h(non-self-adjoin)n(t)f(op)r(era-)118 1277 y(tors)e FO(X)i FQ(2)24 b FO(L)p FP(\()p FO(H)7 b FP(\))27 b(suc)n(h)g(that)h([)p FO(X)1152 1247 y FL(2)1189 1277 y FO(;)14 b(X)1302 1247 y FN(\003)1339 1277 y FP(])23 b(=)g(0,)k(i.e.,)1054 1456 y FO(X)1130 1421 y FL(2)1166 1456 y FO(X)1242 1421 y FN(\003)1303 1456 y FP(=)22 b FO(X)1466 1421 y FN(\003)1504 1456 y FO(X)1580 1421 y FL(2)1616 1456 y FO(:)118 1634 y FP(T)-7 b(aking)31 b(the)h(adjoin)n (ts,)g(w)n(e)f(get)h(\()p FO(X)1265 1604 y FN(\003)1303 1634 y FP(\))1335 1604 y FL(2)1372 1634 y FO(X)k FP(=)30 b FO(X)7 b FP(\()p FO(X)1756 1604 y FN(\003)1793 1634 y FP(\))1825 1604 y FL(2)1862 1634 y FP(.)49 b(Let)32 b FO(X)k FP(=)30 b FO(A)21 b FP(+)g FO(iB)t FP(,)118 1734 y FO(A)30 b FP(=)f FO(A)366 1704 y FN(\003)404 1734 y FP(,)k FO(B)h FP(=)29 b FO(B)718 1704 y FN(\003)756 1734 y FP(.)48 b(Then)32 b(the)g(op)r(erators)d(in)j(this)g(class)e (are)h(selected)g(b)n(y)118 1833 y(the)d(follo)n(wing)f(relation)954 2012 y([)p FO(A)1039 1977 y FL(2)1076 2012 y FO(;)14 b(B)t FP(])24 b(=)e([)p FO(B)t(;)14 b(A)1503 1977 y FL(2)1541 2012 y FP(])23 b(=)g(0)p FO(:)623 b FP(\(3.11\))118 2190 y(Irreducible)38 b(represen)n(tations)e(of)j(a)f(pair)g FO(A)p FP(,)j FO(B)i FP(whic)n(h)38 b(satis\014es)g(relation)118 2290 y(\(3.11\))o(,)32 b(are)e(one-)g(and)h(t)n(w)n(o-dimensional.)45 b(These)31 b(represen)n(tations,)f(up)h(to)118 2389 y(a)38 b(unitary)f(equiv)-5 b(alence,)40 b(are)d(the)h(follo)n(wing:)57 b(one-dimensional)36 b FO(A)41 b FP(=)f FO(a)p FP(,)118 2489 y FO(B)31 b FP(=)25 b FO(b)p FP(,)30 b FO(a)p FP(,)g FO(b)c FQ(2)h FJ(R)p FP(;)37 b(t)n(w)n(o-dimensional)27 b FO(A)g FP(=)f FO(a)1585 2422 y Fz(\000)1637 2459 y FL(1)49 b(0)1637 2508 y(0)23 b FN(\000)p FL(1)1792 2422 y Fz(\001)1830 2489 y FP(,)30 b FO(B)g FP(=)c FO(b)2103 2422 y Fz(\000)2155 2462 y FL(0)d(1)2155 2512 y(1)g(0)2258 2422 y Fz(\001)2296 2489 y FP(,)30 b FO(a)c(>)g FP(0,)118 2589 y FO(b)d(>)f FP(0.)118 2736 y FR(5.)45 b FP(In)31 b(Section)f(3.1.5)f(w)n(e)i(considered)e(the)i(algebra)e FA(B)1887 2748 y FL(2)1952 2736 y FP(=)f FJ(C)15 b FQ(h)p FO(x;)f(y)37 b FQ(j)28 b FO(xy)s(x)g FP(=)118 2836 y FO(y)s(xy)s FQ(i)p FP(.)55 b(In)n(tro)r(duce)33 b(an)g(in)n(v)n (olution)f(b)n(y)i(setting)f FO(x)1708 2806 y FM(?)1779 2836 y FP(=)g FO(y)s FP(;)j(then)e(represen)n(ta-)118 2936 y(tions)27 b(of)g(the)h(arising)e FQ(\003)p FP(-algebra)e(are)j (related)f(to)h(the)h(class)e(of)h(op)r(erators)f FO(X)118 3035 y FP(suc)n(h)h(that)996 3214 y FO(X)7 b(X)1148 3179 y FN(\003)1185 3214 y FO(X)30 b FP(=)22 b FO(X)1447 3179 y FN(\003)1485 3214 y FO(X)7 b(X)1637 3179 y FN(\003)1673 3214 y FO(:)118 3392 y FP(Let)29 b FO(X)h FP(=)24 b FO(U)9 b(C)34 b FP(\()p FO(U)j FP(is)29 b(a)e(partial)h(isometry)-7 b(,)28 b FO(C)i FQ(\025)24 b FP(0,)k(k)n(er)13 b FO(U)32 b FP(=)24 b(k)n(er)13 b FO(C)6 b FP(\))29 b(b)r(e)f(the)118 3492 y(p)r(olar)f(decomp)r(osition)g(of)g(the)h(op)r(erator)e FO(X)7 b FP(.)36 b(Then)794 3670 y FO(U)9 b(C)925 3636 y FL(3)986 3670 y FP(=)22 b FO(C)1138 3636 y FL(3)1176 3670 y FO(U)1242 3636 y FN(\003)1280 3670 y FO(;)96 b(U)1465 3636 y FN(\003)1503 3670 y FO(C)1568 3636 y FL(3)1629 3670 y FP(=)23 b FO(C)1782 3636 y FL(3)1819 3670 y FO(U;)118 3848 y FP(whic)n(h)h(implies)h(that)f FO(X)31 b FP(is)24 b(a)g(quasi-normal)e(op)r(erator,)i(and)g(therefore,)g(self-)118 3948 y(adjoin)n(t,)37 b FO(X)508 3918 y FN(\003)580 3948 y FP(=)e FO(X)7 b FP(.)58 b(Irreducible)34 b FQ(\003)p FP(-represen)n(tations)e(of)j(the)g(algebra)e FA(B)2537 3960 y FL(2)118 4048 y FP(equipp)r(ed)24 b(with)g(the)g(in)n(v)n (olution)f FO(x)1226 4018 y FM(?)1288 4048 y FP(=)f FO(y)k FP(are)d(all)g(one-dimensional,)g FO(X)30 b FP(=)22 b FO(\025)p FP(,)118 4147 y FO(\025)i FQ(2)f FJ(R)p FP(.)p eop %%Page: 234 238 234 237 bop 118 100 a FP(234)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FR(3.2.2)94 b(P)m(artial)25 b(isometries,)c(w)m(eakly)k(cen)m (tered)f(op)s(erators)g(and)g(al-)410 432 y(gebraic)32 b(op)s(erators)118 586 y(1.)j FP(P)n(assing)22 b(to)i(relations)f(of)h (degree)f(four,)h(w)n(e)g(consider)f(only)h(the)g(follo)n(wing)118 686 y FQ(\003)p FP(-algebras)37 b(\(classes)h(of)i(op)r(erators\):)59 b FA(W)42 b FP(=)h FJ(C)15 b FQ(h)p FO(x;)f(x)1849 656 y FN(\003)1936 686 y FQ(j)43 b FP([)p FO(xx)2119 656 y FN(\003)2158 686 y FO(;)14 b(x)2242 656 y FN(\003)2280 686 y FO(x)p FP(])44 b(=)e(0)p FQ(i)118 786 y FP(\(w)n(eakly)32 b(cen)n(tered)g(op)r(erators\),)h FA(P)f FP(=)f FJ(C)15 b FQ(h)p FO(x)q(;)f(x)1610 756 y FN(\003)1686 786 y FQ(j)32 b FP(\()p FO(x)1820 756 y FN(\003)1859 786 y FO(x)p FP(\))1938 756 y FL(2)2008 786 y FP(=)f FO(x)2151 756 y FN(\003)2190 786 y FO(x)p FQ(i)i FP(\(partial)118 885 y(isometries\),)e FA(WP)c FP(=)h FJ(C)15 b FQ(h)p FO(x;)f(x)1060 855 y FN(\003)1132 885 y FQ(j)28 b FP([)p FO(xx)1300 855 y FN(\003)1339 885 y FO(;)14 b(x)1423 855 y FN(\003)1461 885 y FO(x)p FP(])29 b(=)e(0)p FO(;)h FP(\()p FO(x)1824 855 y FN(\003)1862 885 y FO(x)p FP(\))1941 855 y FL(2)2007 885 y FP(=)g FO(x)2147 855 y FN(\003)2185 885 y FO(x)p FQ(i)k FP(\(w)n(eakly)118 985 y(cen)n(tered)27 b(op)r(erators)f(whic)n (h)h(are)g(partial)g(isometries\).)243 1085 y(The)g(follo)n(wing)g (theorem)g(holds.)118 1252 y FR(Theorem)k(65.)40 b FC(The)31 b FQ(\003)p FC(-algebr)l(a)f FA(W)g FC(is)g FQ(\003)p FC(-wild.)118 1420 y(Pr)l(o)l(of.)43 b FP(De\014ne)28 b(a)g(homomorphism)e FO( )12 b FP(:)28 b FA(W)23 b FQ(\000)-48 b(!)23 b FO(M)1722 1432 y FL(3)1758 1420 y FP(\()p FB(F)1846 1432 y FL(2)1882 1420 y FP(\))28 b(b)n(y)667 1709 y FO( )s FP(\()p FO(x)p FP(\))c(=)947 1542 y Fz(0)947 1691 y(@)1152 1603 y FP(0)366 b(0)234 b(2)p FO(e)1059 1708 y FP(\(1)p FO(=)p FP(2\))p FO(e)142 b FP(\()1462 1639 y FQ(p)p 1531 1639 42 4 v 69 x FP(3)p FO(=)p FP(2\))p FO(v)126 b FP(0)1020 1813 y(\()1052 1744 y FQ(p)p 1121 1744 V 69 x FP(3)p FO(=)p FP(2\))p FO(u)82 b FQ(\000)p FP(\(1)p FO(=)p FP(2\))p FO(uv)103 b FP(0)1916 1542 y Fz(1)1916 1691 y(A)2003 1709 y FO(:)243 1993 y FP(T)-7 b(o)27 b(sho)n(w)g(that)g(this)h(is)g(a) f(homomorphism,)g(w)n(e)g(calculate)g(that)286 2272 y FO( )s FP(\()p FO(x)p FP(\))14 b FO( )s FP(\()p FO(x)604 2238 y FN(\003)644 2272 y FP(\))23 b(=)787 2105 y Fz(0)787 2255 y(@)859 2171 y FP(2)p FO(e)82 b FP(0)h(0)879 2271 y(0)103 b FO(e)84 b FP(0)879 2371 y(0)101 b(0)84 b FO(e)1188 2105 y Fz(1)1188 2255 y(A)1275 2272 y FO(;)97 b( )s FP(\()p FO(x)1531 2238 y FN(\003)1570 2272 y FP(\))14 b FO( )s FP(\()p FO(x)p FP(\))24 b(=)1896 2105 y Fz(0)1896 2255 y(@)1970 2171 y FO(e)84 b FP(0)102 b(0)1968 2271 y(0)84 b FO(e)104 b FP(0)1968 2371 y(0)83 b(0)f(2)p FO(e)2298 2105 y Fz(1)2298 2255 y(A)2384 2272 y FO(:)118 2555 y FP(Therefore)26 b([)p FO( )s FP(\()p FO(x)p FP(\))14 b FO( )s FP(\()p FO(x)835 2525 y FN(\003)875 2555 y FP(\))p FO(;)g( )s FP(\()p FO(x)1080 2525 y FN(\003)1120 2555 y FP(\))g FO( )s FP(\()p FO(x)p FP(\)])24 b(=)f(0.)243 2655 y(The)33 b(homomorphism)f FO( )37 b FP(induces)c(the)h(functor)f FO(F)1899 2667 y FM( )1959 2655 y FP(:)43 b(Rep)14 b FO(C)2248 2625 y FN(\003)2287 2655 y FP(\()p FB(F)2375 2667 y FL(2)2410 2655 y FP(\))33 b FQ(\000)-48 b(!)118 2755 y FP(Rep)14 b FA(W)p FP(.)37 b(W)-7 b(e)28 b(will)g(sho)n(w)e (that)i FO(F)1161 2767 y FM( )1240 2755 y FP(is)f(full.)243 2866 y(It)h(follo)n(ws)e(from)h FO(C)6 b(X)942 2836 y FN(\003)980 2866 y FO(X)30 b FP(=)1190 2845 y(^)1166 2866 y FO(X)1242 2836 y FN(\003)1303 2845 y FP(^)1280 2866 y FO(X)6 b(C)34 b FP(that)890 3129 y FO(C)29 b FP(=)1066 2962 y Fz(0)1066 3111 y(@)1138 3028 y FO(C)1197 3040 y FL(11)1351 3028 y FO(C)1410 3040 y FL(12)1608 3028 y FP(0)1138 3128 y FO(C)1197 3140 y FL(21)1351 3128 y FO(C)1410 3140 y FL(22)1608 3128 y FP(0)1183 3227 y(0)170 b(0)127 b FO(C)1623 3239 y FL(33)1694 2962 y Fz(1)1694 3111 y(A)1780 3129 y FO(:)118 3428 y FP(F)-7 b(rom)32 b(the)h(relations)f FO(C)6 b(X)38 b FP(=)1124 3407 y(^)1100 3428 y FO(X)7 b(C)f FP(,)34 b FO(C)6 b(X)1439 3398 y FN(\003)1508 3428 y FP(=)1628 3407 y(^)1604 3428 y FO(X)1680 3398 y FN(\003)1718 3428 y FO(C)g FP(,)34 b(w)n(e)f(ha)n(v)n(e)e(that)i FO(C)2408 3440 y FL(12)2510 3428 y FP(=)118 3528 y FO(C)177 3540 y FL(21)272 3528 y FP(=)24 b(0,)29 b FO(C)514 3540 y FL(11)609 3528 y FP(=)24 b FO(C)757 3540 y FL(22)852 3528 y FP(=)h FO(C)1001 3540 y FL(33)1096 3528 y FP(=)1204 3507 y(~)1185 3528 y FO(C)6 b FP(,)29 b(and)1483 3507 y(~)1464 3528 y FO(C)7 b(U)33 b FP(=)1724 3507 y(^)1709 3528 y FO(U)1794 3507 y FP(~)1775 3528 y FO(C)6 b FP(,)1911 3507 y(~)1892 3528 y FO(C)g(V)44 b FP(=)2151 3507 y(^)2138 3528 y FO(V)2224 3507 y FP(~)2205 3528 y FO(C)6 b FP(.)40 b(Hence,)118 3627 y(w)n(e)30 b(can)f(conclude)h(that)g(the)h(functor)f FO(F)1413 3639 y FM( )1493 3627 y FP(is)g(full.)45 b(Therefore,)29 b(the)i(algebra)118 3727 y FA(W)d FP(is)f FQ(\003)p FP(-wild.)p 2514 3727 4 57 v 2518 3674 50 4 v 2518 3727 V 2567 3727 4 57 v 243 3897 a(Therefore,)d(the)i(problem)f(of)g(unitary)g (description)g(of)g(w)n(eakly)f(cen)n(tered)118 3996 y(op)r(erators)i(is)h FQ(\003)p FP(-wild.)118 4147 y FR(2.)36 b FP(F)-7 b(or)27 b(partial)g(isometries,)g(the)h(follo)n (wing)e(theorem)h(holds.)p eop %%Page: 235 239 235 238 bop 118 100 a FK(3.2.)36 b(Classes)27 b(of)g(non-self-adjoin)n (t)g(op)r(erators)853 b FP(235)118 333 y FR(Theorem)31 b(66.)40 b FC(The)31 b FQ(\003)p FC(-algebr)l(a)f FA(P)g FC(is)g FQ(\003)p FC(-wild.)118 495 y(Pr)l(o)l(of.)43 b FP(W)-7 b(e)20 b(will)f(sho)n(w)g(that)g FA(P)k FQ(\037)g FO(C)1275 465 y FN(\003)1313 495 y FP(\()p FB(F)1401 507 y FL(2)1437 495 y FP(\).)34 b(The)19 b(homomorphism)f FO( )12 b FP(:)28 b FA(P)23 b FQ(\000)-48 b(!)118 594 y FO(M)199 606 y FL(3)236 594 y FP(\()p FO(C)333 564 y FN(\003)372 594 y FP(\()p FB(F)460 606 y FL(2)495 594 y FP(\)\))28 b(is)g(constructed)f(b)n(y)718 874 y FO( )s FP(\()p FO(x)p FP(\))d(=)998 708 y Fz(0)998 857 y(@)1071 708 y FQ(p)p 1140 708 42 4 v 69 x FP(3)o FO(=)p FP(4)14 b FO(u)1452 708 y FQ(p)p 1521 708 V 69 x FP(3)o FO(=)p FP(2)g FO(e)125 b FP(0)1107 876 y(3)p FO(=)p FP(4)14 b FO(v)122 b FQ(\000)p FP(1)p FO(=)p FP(2)14 b FO(v)s(u)1705 846 y FN(\003)1824 876 y FP(0)1110 976 y(1)p FO(=)p FP(2)g FO(e)265 b FP(0)228 b(0)1865 708 y Fz(1)1865 857 y(A)1952 874 y FO(:)118 1150 y FP(It)28 b(is)g(easy)e(to)i(v)n(erify)f(that)860 1423 y FO( )s FP(\()p FO(x)996 1388 y FN(\003)1035 1423 y FP(\))14 b FO( )s FP(\()p FO(x)p FP(\))24 b(=)1361 1256 y Fz(0)1361 1405 y(@)1435 1322 y FO(e)84 b FP(0)e(0)1433 1422 y(0)i FO(e)g FP(0)1433 1521 y(0)f(0)f(0)1724 1256 y Fz(1)1724 1405 y(A)1810 1423 y FO(:)118 1705 y FP(Therefore)26 b(\()p FO( )s FP(\()p FO(x)662 1675 y FN(\003)702 1705 y FP(\))14 b FO( )s FP(\()p FO(x)p FP(\)\))948 1675 y FL(2)1010 1705 y FP(=)22 b FO( )s FP(\()p FO(x)1233 1675 y FN(\003)1272 1705 y FP(\))14 b FO( )s FP(\()p FO(x)p FP(\).)243 1805 y(The)27 b(induced)h(functor)g FO(F)1064 1817 y FM( )1124 1805 y FP(:)41 b(Rep)14 b FO(C)1411 1775 y FN(\003)1450 1805 y FP(\()p FB(F)1538 1817 y FL(2)1573 1805 y FP(\))23 b FQ(\000)-48 b(!)23 b FP(Rep)14 b FA(P)27 b FP(is)h(full.)p 2514 1805 4 57 v 2518 1752 50 4 v 2518 1805 V 2567 1805 4 57 v 243 1970 a(Therefore,)23 b(the)g(problem)g(of)g (the)g(description)g(of)g(partial)f(isometries)h(up)118 2069 y(to)28 b(a)f(unitary)g(equiv)-5 b(alence)27 b(is)h FQ(\003)p FP(-wild.)118 2217 y FR(3.)36 b FP(The)28 b(follo)n(wing)e (theorem)i(holds.)118 2379 y FR(Theorem)j(67.)40 b FC(The)31 b FQ(\003)p FC(-algebr)l(a)f FA(WP)f FC(is)h FQ(\003)p FC(-wild.)118 2541 y(Pr)l(o)l(of.)43 b FP(W)-7 b(e)23 b(will)g(again)e(sho)n(wn)h(that)h FA(WP)f FQ(\037)h FO(C)1637 2511 y FN(\003)1675 2541 y FP(\()p FB(F)1763 2553 y FL(2)1799 2541 y FP(\).)35 b(De\014ne)23 b(a)g(homomor-)118 2641 y(phism)28 b FO( )12 b FP(:)28 b FA(WP)23 b FQ(\000)-49 b(!)23 b FO(M)862 2653 y FL(4)899 2641 y FP(\()p FB(F)987 2653 y FL(2)1023 2641 y FP(\))28 b(as)e(follo)n(ws:)656 2970 y FO( )s FP(\()p FO(x)p FP(\))e(=)936 2754 y Fz(0)936 2900 y(B)936 2950 y(B)936 3003 y(@)1008 2754 y FQ(p)p 1078 2754 42 4 v 1078 2823 a FP(3)o FO(=)p FP(4)14 b FO(u)1389 2754 y FQ(p)p 1458 2754 V 69 x FP(3)p FO(=)p FP(2)g FO(e)124 b FP(0)83 b(0)1045 2923 y(3)p FO(=)p FP(4)14 b FO(v)121 b FQ(\000)p FP(1)p FO(=)p FP(2)14 b FO(v)s(u)1642 2892 y FN(\003)1761 2923 y FP(0)83 b(0)1047 3022 y(1)p FO(=)p FP(2)14 b FO(e)266 b FP(0)227 b(0)83 b(0)1115 3122 y(0)335 b(0)229 b FO(e)84 b FP(0)1927 2754 y Fz(1)1927 2900 y(C)1927 2950 y(C)1927 3003 y(A)2014 2970 y FO(:)118 3300 y FP(It)20 b(is)g(easy)f(to)h(sho)n(w)f(that)h (the)h(corresp)r(onding)d(functor)i FO(F)1911 3312 y FM( )1970 3300 y FP(:)42 b(Rep)14 b FO(C)2258 3270 y FN(\003)2296 3300 y FP(\()p FB(F)2384 3312 y FL(2)2420 3300 y FP(\))23 b FQ(\000)-48 b(!)118 3400 y FP(Rep)14 b FA(WI)28 b FP(is)f(full.)p 2514 3400 4 57 v 2518 3347 50 4 v 2518 3400 V 2567 3400 4 57 v 243 3565 a(Th)n(us,)f(the)g (problem)g(of)g(the)g(description)f(of)h(partial)g(isometries,)f(whic)n (h)118 3665 y(are)i(w)n(eakly)f(cen)n(tered)h(op)r(erators,)f(is)h FQ(\003)p FP(-wild.)118 3795 y FC(R)l(emark)39 b FP(54)p FC(.)i FP(F)-7 b(or)27 b FO(n)c(<)f FQ(1)p FP(,)28 b(consider)f(the)h FQ(\003)p FP(-algebra)354 3973 y FA(WP)510 3993 y FM(n)578 3973 y FP(=)23 b FJ(C)720 3906 y Fz(\012)765 3973 y FO(x;)14 b(x)896 3939 y FN(\003)958 3973 y FQ(j)23 b FO(x)28 b FP(is)g(a)f(partial)g(isometry)-7 b(,)27 b(and)1018 4117 y([)p FO(x)1088 4083 y FM(j)1124 4117 y FO(x)1171 4083 y FN(\003)1209 4077 y FM(j)1244 4117 y FO(;)14 b(x)1328 4083 y FN(\003)1367 4077 y FM(k)1408 4117 y FO(x)1455 4083 y FM(k)1496 4117 y FP(])23 b(=)g(0)p FO(;)k FQ(8)p FO(k)s(;)14 b(j)27 b FP(=)22 b(1)p FO(;)14 b(:)g(:)g(:)f(;)h(n)2276 4050 y Fz(\013)2315 4117 y FO(:)p eop %%Page: 236 240 236 239 bop 118 100 a FP(236)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)118 333 y FQ(\003)p FP(-Algebra)37 b(of)i(w)n(eakly)f(cen)n(tered)g(op)r (erators)f(is)i FA(WP)i FP(=)h FA(WP)2176 352 y FL(1)2214 333 y FP(.)71 b(In)39 b([33)o(],)118 432 y(Theorem)27 b(67)g(w)n(as)g(extended)i(to)f(these)g(algebras:)35 b(it)29 b(w)n(as)e(pro)n(v)n(ed)f(that)i(the)118 532 y FQ(\003)p FP(-algebra)d FA(WP)634 552 y FM(n)707 532 y FP(is)i FQ(\003)p FP(-wild)g(for)g(an)n(y)g FO(n)c(<)f FQ(1)p FP(.)243 632 y(On)h(the)i(other)e(hand,)i(the)f FQ(\003)p FP(-algebra)d(of)j(cen)n(tered)f(op)r(erators)f(is)i(n)n (uclear)118 731 y(\(see)e(Section)g(2.5.2\),)g(and)g(the)g FQ(\003)p FP(-algebra)e(of)i(cen)n(tered)f(partial)g(isometries)g(is) 118 831 y(of)28 b(t)n(yp)r(e)f FO(I)35 b FP(and)27 b(admits)g(a)g (complete)h(description)f(of)g(its)h FQ(\003)p FP(-represen)n(tations) 118 930 y(\(Section)g(2.1.3\))118 1062 y FR(4.)35 b FP(W)-7 b(e)25 b(also)e(consider)h(the)g(complexit)n(y)g(problem)g(of)h(a)f (unitary)f(description)118 1162 y(for)k(algebraic)f(op)r(erators,)g (i.e.,)h(represen)n(tations)f(of)i(the)g FQ(\003)p FP(-algebra)834 1343 y FA(A)894 1355 y FM(R)944 1363 y Fw(n)1012 1343 y FP(=)22 b FJ(C)15 b FQ(h)q FO(x;)f(x)1317 1309 y FN(\003)1385 1343 y FQ(j)23 b FO(R)1494 1355 y FM(n)1539 1343 y FP(\()p FO(x)p FP(\))h(=)f(0)p FQ(i)p FO(;)118 1524 y FP(where)28 b FO(R)422 1536 y FM(n)496 1524 y FP(is)g(a)g(p)r(olynomial)g(in)h(one) f(v)-5 b(ariable)27 b(with)i(complex)f(co)r(e\016cien)n(ts.)118 1623 y(F)-7 b(or)29 b(brevit)n(y)-7 b(,)30 b(w)n(e)g(assume)f(that)h (the)g(p)r(olynomial)f(do)r(es)h(not)g(ha)n(v)n(e)e(m)n(ultiple)118 1723 y(ro)r(ots.)118 1887 y FR(Prop)s(osition)i(73.)41 b FC(L)l(et)266 2089 y FA(A)326 2101 y FM(R)376 2109 y Fy(3)435 2089 y FP(=)22 b FJ(C)576 2021 y Fz(\012)622 2089 y FO(x;)14 b(x)753 2054 y FN(\003)815 2089 y FQ(j)23 b FO(R)924 2101 y FL(3)961 2089 y FP(\()p FO(x)p FP(\))1096 2042 y FL(def)1110 2089 y FP(=)36 b(\()p FO(x)19 b FQ(\000)f FO(\013)1445 2101 y FL(1)1482 2089 y FO(e)p FP(\)\()p FO(x)h FQ(\000)f FO(\013)1787 2101 y FL(2)1825 2089 y FO(e)p FP(\)\()p FO(x)h FQ(\000)f FO(\013)2130 2101 y FL(3)2167 2089 y FO(e)p FP(\))23 b(=)g(0)p FO(;)876 2222 y(\013)929 2234 y FL(1)967 2222 y FO(;)14 b(\013)1057 2234 y FL(2)1094 2222 y FO(;)g(\013)1184 2234 y FL(3)1244 2222 y FQ(2)24 b FJ(C)15 b FO(;)34 b(\013)1487 2234 y FM(k)1551 2222 y FQ(6)p FP(=)22 b FO(\013)1691 2234 y FM(l)1746 2222 y FC(for)31 b FO(k)26 b FQ(6)p FP(=)d FO(l)2063 2155 y Fz(\013)2101 2222 y FO(:)118 2403 y FC(Then)31 b FA(A)395 2415 y FM(R)445 2423 y Fy(3)504 2403 y FQ(\037)22 b FB(Q)646 2415 y FL(2)p FM(;)p FN(?)755 2403 y FC(,)30 b(and)h(c)l(onse)l(quently,)e(the)h FQ(\003)p FC(-algebr)l(a)h FA(A)2025 2415 y FM(R)2075 2423 y Fy(3)2141 2403 y FC(is)f FQ(\003)p FC(-wild.)118 2568 y(Pr)l(o)l(of.)43 b FP(De\014ne)30 b(a)f(homomorphism)f FO( )12 b FP(:)28 b FA(A)1473 2580 y FM(R)1523 2588 y Fy(3)1585 2568 y FQ(\000)-48 b(!)26 b FB(Q)1766 2580 y FL(2)p FM(;)p FN(?)1904 2568 y FP(as)i(follo)n(ws:)39 b FO( )s FP(\()p FO(x)p FP(\))27 b(=)118 2667 y FO(\013)171 2679 y FL(1)208 2667 y FO(q)245 2679 y FL(1)296 2667 y FP(+)13 b FO(\013)427 2679 y FL(2)464 2667 y FO(q)501 2679 y FL(2)551 2667 y FP(+)g FO(\013)682 2679 y FL(3)719 2667 y FP(\()p FO(e)g FQ(\000)g FO(q)918 2679 y FL(1)968 2667 y FQ(\000)g FO(q)1083 2679 y FL(2)1120 2667 y FP(\))p FO(:)25 b FP(It)h(is)e(easy)g(to)h(c)n (hec)n(k)f(that)h(the)g(functor)g FO(F)2524 2679 y FM( )118 2767 y FP(is)j(full.)p 2514 2767 4 57 v 2518 2714 50 4 v 2518 2767 V 2567 2767 4 57 v 118 2932 a FR(Corollary)38 b(18.)43 b FC(If)34 b FO(R)869 2944 y FM(n)948 2932 y FC(is)g(a)g(p)l(olynomial)i(with)e(thr)l(e)l(e)f(and)i(mor)l(e)e (distinct)118 3032 y(r)l(o)l(ots)d(then)f(the)h FQ(\003)p FC(-algebr)l(a)h FA(A)1056 3044 y FM(R)1106 3052 y Fw(n)1180 3032 y FC(is)f FQ(\003)p FC(-wild.)118 3248 y FR(3.2.3)94 b(Hyp)s(onormal)23 b(op)s(erators)i(and)h(pairs)f(of)g(comm)m(uting)d (com-)410 3347 y(pletely)31 b(non-unitary)h(isometries)118 3500 y FP(No)n(w)i(w)n(e)g(will)g(consider)f(classes)g(of)h (non-self-adjoin)n(t)f(op)r(erators)f(that)j(are)118 3600 y(giv)n(en)23 b(b)n(y)g(a)g(non-p)r(olynomial)f(equalit)n(y)-7 b(,)24 b(e.g.,)g(an)f(inequalit)n(y)-7 b(,)24 b(or)f(other)f(non-)118 3700 y(algebraic)17 b(conditions.)34 b(F)-7 b(or)18 b(suc)n(h)h(op)r (erators)d(w)n(e)j(will)g(study)g(the)g(complexit)n(y)118 3799 y(of)28 b(the)g(problem)f(to)g(describ)r(e)g(the)h(op)r(erators)e (up)i(to)f(unitary)h(equiv)-5 b(alence.)118 3948 y FR(1.)44 b FP(Let)30 b(Z)g(b)r(e)h(a)f(h)n(yp)r(onormal)f(op)r(erator,)g(i.e.,)i FO(Z)6 b(Z)1766 3918 y FN(\003)1823 3948 y FQ(\000)20 b FO(Z)1971 3918 y FN(\003)2009 3948 y FO(Z)33 b FQ(\025)27 b FP(0.)44 b(W)-7 b(e)31 b(will)118 4048 y(sho)n(w)22 b(that)h(the)g(problem)f(to)h(describ)r(e)f(the)h(class)f(of)h(h)n(yp)r (onormal)e(op)r(erators)118 4147 y(con)n(tains)33 b(the)h(description)f (problem)g(for)g(one)g(non-self-adjoin)n(t)g(op)r(erator,)p eop %%Page: 237 241 237 240 bop 118 100 a FK(3.2.)36 b(Classes)27 b(of)g(non-self-adjoin)n (t)g(op)r(erators)853 b FP(237)118 333 y FO(X)45 b FQ(2)40 b FO(L)p FP(\()p FO(H)7 b FP(\),)39 b(whic)n(h)e(do)r(es)g(not)g (satisfy)g(an)n(y)f(relations,)i(or)e(the)i(same)e(for)118 432 y(a)g(pair)f(of)g(self-adjoin)n(t)h(op)r(erators.)59 b(T)-7 b(o)36 b(do)f(that,)j(w)n(e)e(use)f(W)-7 b(ogen's)36 b(con-)118 532 y(struction)c(\(see)h([303)n(]\).)53 b(W)-7 b(e)33 b(consider)e(op)r(erators)g FO(Z)37 b FQ(2)32 b FO(L)p FP(\()2043 470 y Fz(L)2135 490 y FN(1)2135 557 y FL(1)2219 532 y FO(H)7 b FP(\))33 b(of)f(the)118 632 y(follo)n(wing)27 b(form:)395 1051 y FO(Z)i FP(=)569 710 y Fz(0)569 856 y(B)569 906 y(B)569 955 y(B)569 1005 y(B)569 1055 y(B)569 1105 y(B)569 1155 y(B)569 1208 y(@)658 773 y FP(0)658 873 y FO(I)127 b FP(0)471 b Fp(0)641 973 y FO(X)90 b FP(2)p FO(I)111 b FP(0)821 1072 y(0)104 b(3)p FO(I)117 b FP(0)989 1172 y(0)110 b(3)p FO(I)123 b FP(0)813 1285 y Fp(0)1139 1269 y FP(.)1172 1294 y(.)1204 1319 y(.)1319 1269 y(.)1352 1294 y(.)1384 1319 y(.)1499 1269 y(.)1531 1294 y(.)1564 1319 y(.)1591 710 y Fz(1)1591 856 y(C)1591 906 y(C)1591 955 y(C)1591 1005 y(C)1591 1055 y(C)1591 1105 y(C)1591 1155 y(C)1591 1208 y(A)1678 1051 y FO(;)180 b FQ(k)p FO(X)7 b FQ(k)21 b(\024)i FP(1)p FO(=)p FP(2)p FO(:)118 1470 y FP(The)34 b(op)r(erator)e FO(Z)39 b FP(is)34 b(a)f(h)n(yp)r(onormal)f(op)r(erator.)54 b(It)34 b(is)f(easy)g(to)g(pro)n(v)n(e)f(the)118 1570 y(follo)n(wing)27 b(prop)r(osition.)118 1708 y FR(Prop)s(osition)i(74.) 39 b FC(A)n(n)27 b(op)l(er)l(ator)j FB(Y)e FC(is)h(intertwining)f(for)i (the)e(p)l(airs)h FO(Z)6 b FC(,)29 b FO(Z)2537 1677 y FN(\003)118 1807 y FC(and)299 1786 y FP(~)282 1807 y FO(Z)6 b FC(,)420 1786 y FP(~)403 1807 y FO(Z)466 1777 y FN(\003)536 1807 y FC(if)33 b(and)g(only)g(if)65 b FB(Y)28 b FP(=)g FO(Y)39 b FQ(\012)20 b FO(I)1459 1819 y FN(1)1529 1807 y FC(,)34 b(wher)l(e)f FO(Y)51 b FC(is)32 b(an)h(intertwining)118 1907 y(op)l(er)l(ator)26 b(for)f FO(X)7 b FC(,)25 b FO(X)769 1877 y FN(\003)831 1907 y FC(and)1020 1886 y FP(~)996 1907 y FO(X)6 b FC(,)1146 1886 y FP(~)1122 1907 y FO(X)1198 1877 y FN(\003)1235 1907 y FC(,)26 b(c)l(orr)l(esp)l(ondingly)34 b FP(\()p FC(that)25 b(is)g FO(Y)18 b(X)30 b FP(=)2431 1886 y(~)2407 1907 y FO(X)7 b(Y)18 b FC(,)118 2006 y FO(Y)h(X)261 1976 y FN(\003)321 2006 y FP(=)433 1985 y(~)409 2006 y FO(X)485 1976 y FN(\003)522 2006 y FO(Y)g FP(\))p FC(.)118 2144 y FR(2.)56 b FP(Let)35 b FO(S)456 2156 y FL(1)493 2144 y FP(,)h FO(S)603 2156 y FL(2)674 2144 y FP(b)r(e)f(isometries)e (without)i(unitary)e(parts,)j(and)e([)p FO(S)2291 2156 y FL(1)2328 2144 y FO(;)14 b(S)2416 2156 y FL(2)2453 2144 y FP(])34 b(=)118 2244 y([)p FO(S)197 2214 y FN(\003)192 2264 y FL(1)235 2244 y FO(;)14 b(S)328 2214 y FN(\003)323 2264 y FL(2)366 2244 y FP(])33 b(=)g(0.)54 b(W)-7 b(e)34 b(will)g(sho)n(w)f(that)h(this)g(description)f(problem)g(con)n(tains) 118 2343 y(the)22 b(description)f(problem)f(for)h(a)g(pair)g(of)g (unitary)g(op)r(erators)e FO(U)9 b FP(,)23 b FO(V)42 b FQ(2)23 b FO(L)p FP(\()p FO(H)7 b FP(\).)118 2443 y(De\014ne)28 b(the)g(op)r(erators)e FO(S)937 2455 y FL(1)974 2443 y FP(,)i FO(S)1076 2455 y FL(2)1136 2443 y FQ(2)23 b FO(L)p FP(\()1303 2381 y Fz(L)1395 2401 y FN(1)1395 2468 y FL(1)1480 2443 y FO(H)7 b FP(\))27 b(in)h(the)g(follo)n(wing)f(w)n(a) n(y:)680 2921 y FO(S)731 2933 y FL(1)791 2921 y FP(=)879 2530 y Fz(0)879 2676 y(B)879 2726 y(B)879 2776 y(B)879 2825 y(B)879 2875 y(B)879 2925 y(B)879 2975 y(B)879 3025 y(B)879 3075 y(B)879 3128 y(@)952 2594 y FP(0)92 b(0)952 2693 y(0)g(0)1651 2652 y Fp(0)952 2793 y FO(I)98 b FP(0)92 b(0)84 b(0)952 2892 y(0)91 b FO(I)99 b FP(0)84 b(0)1219 2992 y FO(I)91 b FP(0)111 b(0)119 b(0)1220 3092 y(0)83 b FO(I)118 b FP(0)h(0)1078 3205 y Fp(0)1476 3189 y FP(.)1508 3213 y(.)1540 3239 y(.)1797 3189 y(.)1830 3213 y(.)1862 3239 y(.)1889 2530 y Fz(1)1889 2676 y(C)1889 2726 y(C)1889 2776 y(C)1889 2825 y(C)1889 2875 y(C)1889 2925 y(C)1889 2975 y(C)1889 3025 y(C)1889 3075 y(C)1889 3128 y(A)1976 2921 y FO(;)244 3770 y(S)295 3782 y FL(2)355 3770 y FP(=)443 3329 y Fz(0)443 3475 y(B)443 3525 y(B)443 3575 y(B)443 3625 y(B)443 3675 y(B)443 3724 y(B)443 3774 y(B)443 3824 y(B)443 3874 y(B)443 3924 y(B)443 3973 y(B)443 4027 y(@)627 3383 y FP(0)290 b(2)1001 3353 y FN(\000)p FL(1)p FM(=)p FL(2)1157 3383 y FO(I)2262 3425 y Fp(0)627 3487 y FP(0)194 b FQ(\000)p FP(2)970 3457 y FN(\000)p FL(1)p FM(=)p FL(2)1125 3487 y FO(V)19 b(U)1258 3457 y FN(\003)516 3591 y FP(2)558 3561 y FN(\000)p FL(1)p FM(=)p FL(2)714 3591 y FO(U)288 b FP(0)390 b(0)290 b(2)1865 3561 y FN(\000)p FL(1)p FM(=)p FL(2)2021 3591 y FO(I)515 3694 y FP(2)557 3664 y FN(\000)p FL(1)p FM(=)p FL(2)713 3694 y FO(V)298 b FP(0)390 b(0)194 b FQ(\000)p FP(2)1834 3664 y FN(\000)p FL(1)p FM(=)p FL(2)1989 3694 y FO(V)19 b(U)2122 3664 y FN(\003)1380 3849 y FP(2)1422 3819 y FN(\000)p FL(1)p FM(=)p FL(2)1577 3849 y FO(U)288 b FP(0)2247 3791 y(.)2280 3816 y(.)2312 3841 y(.)1379 4004 y(2)1421 3974 y FN(\000)p FL(1)p FM(=)p FL(2)1577 4004 y FO(V)297 b FP(0)2247 3946 y(.)2280 3971 y(.)2312 3996 y(.)1050 4117 y Fp(0)1468 4101 y FP(.)1500 4126 y(.)1532 4151 y(.)1899 4101 y(.)1932 4126 y(.)1964 4151 y(.)2247 4101 y(.)2280 4126 y(.)2312 4151 y(.)2340 3329 y Fz(1)2340 3475 y(C)2340 3525 y(C)2340 3575 y(C)2340 3625 y(C)2340 3675 y(C)2340 3724 y(C)2340 3774 y(C)2340 3824 y(C)2340 3874 y(C)2340 3924 y(C)2340 3973 y(C)2340 4027 y(A)2426 3770 y FO(:)p eop %%Page: 238 242 238 241 bop 118 100 a FP(238)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)243 333 y FP(It)g(is)f(easy)g(to)g(pro)n(v)n(e)f(the)i(follo)n(wing)f(prop) r(osition.)118 510 y FR(Prop)s(osition)f(75.)38 b FC(A)n(n)26 b(op)l(er)l(ator)37 b FB(Y)27 b FC(is)g(intertwining)g(for)g(the)g(p)l (airs)h FO(S)2401 522 y FM(i)2428 510 y FC(,)g FO(S)2537 480 y FN(\003)2532 531 y FM(i)118 609 y FC(and)297 588 y FP(~)283 609 y FO(S)334 621 y FM(i)362 609 y FC(,)435 588 y FP(~)421 609 y FO(S)477 579 y FN(\003)472 631 y FM(i)515 609 y FC(,)35 b FO(i)29 b FP(=)g(1)p FC(,)34 b FP(2)p FC(,)h(c)l(orr)l(esp)l(ondingly,)h(if)e(and)g(only)g(if)68 b FB(Y)30 b FP(=)f FO(Y)40 b FQ(\012)21 b FO(I)2479 621 y FN(1)2549 609 y FC(,)118 720 y(wher)l(e)27 b FO(Y)44 b FC(is)26 b(an)g(intertwining)g(op)l(er)l(ator)h(for)g FO(U)9 b FC(,)26 b FO(V)45 b FC(and)1937 699 y Fz(e)1924 720 y FO(U)9 b FC(,)2053 699 y Fz(e)2042 720 y FO(V)19 b FC(,)27 b(c)l(orr)l(esp)l(ond-)118 820 y(ingly)38 b FP(\()p FC(that)30 b(is)g FO(Y)19 b(U)32 b FP(=)876 799 y(~)861 820 y FO(U)9 b(Y)19 b FC(,)30 b FO(Y)18 b(U)1181 790 y FN(\003)1242 820 y FP(=)1344 799 y(~)1330 820 y FO(U)1396 790 y FN(\003)1434 820 y FO(Y)h FC(,)30 b FO(Y)18 b(V)42 b FP(=)1813 799 y(~)1800 820 y FO(V)19 b(Y)f FC(,)31 b FO(Y)18 b(V)2122 790 y FN(\003)2183 820 y FP(=)2284 799 y(~)2271 820 y FO(V)2338 790 y FN(\003)2376 820 y FO(Y)h FP(\))p FC(.)243 997 y FP(In)44 b(Section)g(3.2.3)e(w)n(e)i(pro) n(v)n(ed)e(the)j(complexit)n(y)e(of)h(the)g(description)118 1096 y(of)f(some)f(op)r(erators)f(classes.)82 b(This)42 b(pro)r(of)h(is)f(similar)h(to)f(the)i(pro)r(of)e(of)118 1196 y FQ(\003)p FP(-wildness)36 b(of)h(the)g FQ(\003)p FP(-algebras)d(ab)r(o)n(v)n(e)h(in)i(this)g(c)n(hapter.)64 b(But)37 b(the)g(op)r(er-)118 1296 y(ators)31 b FO(Z)38 b FP(and)32 b FO(S)643 1308 y FL(1)680 1296 y FP(,)i FO(S)788 1308 y FL(2)857 1296 y FP(do)e(not)h(b)r(elong)e(to)i FA(K)22 b FQ(\012)f FO(L)p FP(\()p FO(H)7 b FP(\).)51 b(It)32 b(seems)g(that)g(the)118 1395 y(de\014nitions)39 b(of)h(ma)5 b(jorization)37 b(and)i FQ(\003)p FP(-wildness)f(\(see)h (Sections)g(3.1.1)f(and)118 1495 y(3.1.2\))31 b(are)h(restrictiv)n(e)f (in)h(the)h(case)e FO(n)g FP(=)g FQ(1)p FP(.)51 b(F)-7 b(or)31 b(the)i(un)n(b)r(ounded)g(op)r(er-)118 1595 y(ator)e(the)h (case)e FO(n)g FP(=)f FQ(1)j FP(is)g(essen)n(tial)e(\(see,)j(e.g.,)f ([245)o(]\).)49 b(Therefore,)32 b(these)118 1694 y(de\014nitions)c (should)f(probably)g(b)r(e)h(extended.)118 1955 y FH(Commen)m(ts)36 b(to)h(Chapter)h(3)118 2144 y FR(Section)32 b(3.1.)243 2247 y FP(3.1.1,)f(3.1.2.)46 b(In)32 b(the)g(theory)e(of)h(represen)n (tations)f(of)h(algebras,)f(it)i(w)n(as)118 2347 y(suggested)c([71)o(]) h(to)g(regard)e(a)h(represen)n(tation)f(problem)h(as)h(wild,)g(if)g(it) h(con-)118 2446 y(tains)41 b(the)h(classical)d(unsolv)n(ed)i(problem)f (of)h(represen)n(tation)f(theory:)63 b(to)118 2546 y(describ)r(e,)43 b(up)d(to)f(a)g(similarit)n(y)-7 b(,)42 b(a)d(pair)g(of)h(matrices)f (without)h(relations.)118 2646 y(This)24 b(unsolv)n(ed)f(problem)g(of)h (represen)n(tation)e(theory)h(con)n(tains)f(in)i(itself)g(the)118 2745 y(problem)33 b(to)g(describ)r(e,)h(up)f(to)g(similarit)n(y)-7 b(,)33 b FO(n)g FP(matrices)g(without)g(relations)118 2845 y(for)26 b(an)n(y)f FO(n)e FQ(2)h FJ(N)t FP(,)32 b(and)27 b(therefore,)e(it)i(con)n(tains)f(the)g(problem)g(of)g (description,)118 2944 y(up)g(to)g(similarit)n(y)-7 b(,)25 b(of)h(represen)n(tations)e(of)i(an)n(y)f(\014nitely)h(generated)f (algebra.)243 3048 y(Numerous)38 b(examples)g(of)h(wild)g(problems)f (of)h(represen)n(tation)e(theory)118 3147 y(can)27 b(b)r(e)h(found,)g (e.g.,)g(in)f([13,)g(88)o(],)h(also)f(see)g(bibliograph)n(y)f(therein.) 243 3251 y(T)-7 b(o)30 b(de\014ne)g(an)g(analogue)f(of)h(wildness)g (for)g FQ(\003)p FP(-algebras)e(\()p FQ(\003)p FP(-wildness\),)i(it)118 3350 y(w)n(as)i(suggested)f(in)i([150)n(])g(to)f(c)n(ho)r(ose,)h(for)f (a)g(standard)g FQ(\003)p FP(-wild)g(problem)g(in)118 3450 y(the)c(theory)e(of)i FQ(\003)p FP(-represen)n(tations,)d(the)i (problem)g(of)g(description)g(of)h(a)f(pair)118 3550 y(of)38 b(self-adjoin)n(t)f(\(or)g(unitary\))h(op)r(erators)d(up)j(to)g (a)f(unitary)g(equiv)-5 b(alence,)118 3649 y(or)35 b(whic)n(h)g(is)g (the)h(same,)h(represen)n(tations)c(of)i(the)h(free)f FQ(\003)p FP(-algebra)e FA(S)2396 3661 y FL(2)2468 3649 y FP(\(or)118 3749 y FA(U)172 3761 y FL(2)209 3749 y FP(\))j(generated)e(b)n(y)h(a)g(pair)g(of)g(self-adjoin)n(t)g(\(or)g (unitary\))g(generators.)59 b(It)118 3848 y(w)n(as)25 b(also)h(suggested)f(to)h(regard)e(as)i(wild)g(problems)g(the)g(ones)g (that)h(con)n(tain)118 3948 y(a)k(standard)g FQ(\003)p FP(-wild)g(problem;)i(it)f(w)n(as)f(pro)n(v)n(en)f(that)i(the)g (standard)e FQ(\003)p FP(-wild)118 4048 y(problem)k(con)n(tains,)h(as)e (a)h(sub-problem,)h(the)g(problem)e(of)h(description)g(of)118 4147 y FQ(\003)p FP(-represen)n(tations)25 b(of)i(an)n(y)g(\014nitely)h (or)f(coun)n(tably)g(generated)f FQ(\003)p FP(-algebra.)p eop %%Page: 239 243 239 242 bop 118 100 a FK(Commen)n(ts)27 b(to)h(Chapter)f(3)1452 b FP(239)243 333 y(A)27 b(n)n(um)n(b)r(er)f(of)h(pap)r(ers)f([145)n(,)h (211)n(,)g(146)o(])f(etc.)37 b(are)26 b(dev)n(oted)g(to)g(elab)r(orat-) 118 432 y(ing)c(the)g(meaning)f(of)h(the)g(statemen)n(t)g (\\description)e(of)i FQ(\003)p FP(-represen)n(tations)d(of)118 532 y(a)25 b FQ(\003)p FP(-algebra)e FA(A)h FP(con)n(tains,)h(as)g(a)g (sub-problem,)g(the)g(description)g(of)g FQ(\003)p FP(-repre-)118 632 y(sentations)g(of)g(a)g FQ(\003)p FP(-algebra)e FA(B)p FP(".)36 b(The)25 b(approac)n(h)e(to)j(the)f(estimation)g(of)g(the)118 731 y(complexit)n(y)31 b(of)g FQ(\003)p FP(-represen)n(tations)e(based) i(on)g(the)h(concepts)f(of)g(ma)5 b(joriza-)118 831 y(tion)33 b(relation)f(for)g FQ(\003)p FP(-algebras)e(\(De\014nition)j(13\),)h (and)e FQ(\003)p FP(-wildness)g(\(De\014ni-)118 930 y(tion)25 b(14\))f(Used)g(in)h(the)g(b)r(o)r(ok)f(is)h(due)g(to)f(S.)h(Krugly)n (ak)d(and)i(is)h(exp)r(ounded)g(in)118 1030 y([151)o(,)k(152)o(].)41 b(Theorem)29 b(50)f(on)h(ma)5 b(jorization)28 b(for)g FO(C)1794 1000 y FN(\003)1833 1030 y FP(-algebras)f(and)i(Corol-)118 1130 y(lary)18 b(8)g(establishing)g(that)h(the)h(ma)5 b(jorization)17 b(of)h FQ(\003)p FP(-algebras)e(is)j(a)f(quasi-order) 118 1229 y(relation)26 b(are)f(also)h(outlined)h(there.)36 b(Pro)r(ofs)25 b(giv)n(en)h(in)h(the)f(b)r(o)r(ok)h(are)e(due)i(to)118 1329 y(S.)h(P)n(op)r(o)n(vyc)n(h.)243 1435 y(In)43 b(the)g(b)r(o)r(ok)f (w)n(e)h(do)f(not)h(discuss)g(relations)e(b)r(et)n(w)n(een)i(the)g (notions)118 1534 y(of)33 b(ma)5 b(jorization)31 b(and)h(Morita)g (equiv)-5 b(alence)33 b(\(on)f(Morita)g(equiv)-5 b(alence)32 b(for)118 1634 y FQ(\003)p FP(-algebras,)25 b(see)i([238)o(,)g(52)o(,)h (160)o(],)g(etc.\))243 1739 y(F)-7 b(or)18 b(represen)n(tations)f(of)i (\014nite-dimensional)g(algebras)e(\(and)i(for)g(a)f(wider)118 1839 y(class)33 b(of)h(matrix)g(problems)f(as)g(w)n(ell\),)j(it)e(w)n (as)f(sho)n(wn)g(in)i([75)o(],)h(that)e(these)118 1939 y(problems)h(can)f(b)r(e)i(sub)r(divided)g(in)n(to)f(\\tame")f(and)h (\\wild")f(\(for)h(accurate)118 2038 y(de\014nitions,)d(see)e([71)o (]\).)47 b(W)-7 b(e)32 b(do)e(not)h(discuss)f(here)h(what)g(it)g(means) f(that)h(a)118 2138 y FQ(\003)p FP(-algebra)24 b(is)i(tame;)g(ho)n(w)n (ev)n(er,)e(if)j(one)f(c)n(ho)r(oses)e(t)n(yp)r(e)j(I)f FQ(\003)p FP(-algebras)d(\(or)i(ev)n(en)118 2237 y(n)n(uclear)31 b FQ(\003)p FP(-algebras\))e(to)i(b)r(e)i(\\)p FQ(\003)p FP(-tame",)e(then)h(there)f(exists)h(a)f(large)f(set)i(of)118 2337 y(in)n(termediate)g FQ(\003)p FP(-algebras,)f(whic)n(h)i(are)e (neither)i FQ(\003)p FP(-tame,)g(nor)f FQ(\003)p FP(-wild)g(\(see,)118 2437 y(e.g.,)27 b(Section)h(3.1.6\).)243 2542 y(In)d(Sections)f (3.1.3{3.1.6,)f(a)h(n)n(um)n(b)r(er)h(of)f(examples)h(of)f FQ(\003)p FP(-wild)h(problems)118 2642 y(are)d(giv)n(en.)35 b(F)-7 b(or)23 b(more)f(examples)h(of)g FQ(\003)p FP(-wild)g(problems,) g(see)g(also)g([258)n(],)i([16)o(],)118 2742 y([33)o(],)j(etc.)243 2878 y(3.1.3.)34 b(The)25 b(exp)r(osition)g(of)f(topics)h(on)g FQ(\003)p FP(-wildness)e(of)i FQ(\003)p FP(-algebras)d(gener-)118 2978 y(ated)33 b(b)n(y)g(orthogonal)e(pro)5 b(jections)32 b(and)h(idemp)r(oten)n(ts)h(essen)n(tially)e(follo)n(ws)118 3077 y([150)o(],)22 b([151)o(,)f(152)o(].)34 b(The)22 b(pro)r(of)e(of)h FQ(\003)p FP(-wildness)f(of)h FQ(\003)p FP(-algebras)d FB(R)2119 3089 y FL(5)p FM(;)p FL(2)2209 3077 y FP(,)23 b(and)e FB(R)2470 3098 y FL(5)p FM(;)2532 3075 y Fy(5)p 2532 3084 29 3 v 2532 3118 a(2)118 3177 y FP(in)k(Subsection)f(5)g(of)g(3.1.3)f(is)h(giv)n(en)f(b)n(y)h(S.)h (Krugly)n(ak,)d(Y)-7 b(u.)25 b(Samo)-9 b(\025)-32 b(\020lenk)n(o)22 b(and)118 3277 y(A.)28 b(Piry)n(atinsk)-5 b(a)n(y)n(a.)243 3413 y(3.1.4.)52 b(F)-7 b(or)32 b(facts)h(on)g FQ(\003)p FP(-wildness)f(of)h(semi-linear)f(relations)g(\(Prop)r(osi-)118 3513 y(tions)20 b(67,)f(68\))g(see)h([35)o(,)g(248)n(].)35 b(The)20 b(pro)r(of)f(giv)n(en)g(here)g(is)h(due)g(to)g(S.)g(Krugly)n (ak.)243 3649 y(3.1.5.)34 b FQ(\003)p FP(-Wildness)24 b(of)g(description)g(of)g(pairs)g(of)g(self-adjoin)n(t)g(op)r(erators,) 118 3749 y FO(A)p FP(,)39 b FO(B)t FP(,)g(suc)n(h)d(that)h FO(B)823 3719 y FL(2)898 3749 y FP(=)g FO(I)7 b FP(,)39 b(up)e(to)f(a)g(unitary)g(equiv)-5 b(alence,)39 b(follo)n(ws)c(di-)118 3848 y(rectly)29 b(from)h([150)n(].)44 b(W)-7 b(e)30 b(giv)n(e)f(a)g(simple)h(criterion)e(of)i FQ(\003)p FP(-wildness)f(for) g(pairs)118 3948 y(of)e(self-adjoin)n(t)f(op)r(erators)e(connected)j(b) n(y)f(a)g(quadratic)f(relation)h(\(A.)h(Piry-)118 4048 y(atinsk)-5 b(a)n(y)n(a\),)26 b(or)g(b)n(y)g(a)h(cubic)g(semi-linear)f (relation)g(in)h(terms)f(of)h(co)r(e\016cien)n(ts)118 4147 y(of)h(the)g(relation.)p eop %%Page: 240 244 240 243 bop 118 100 a FP(240)560 b FK(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)f(of)h(represen)n(tations)243 333 y FP(3.1.6.)35 b(W)-7 b(e)27 b(giv)n(e)f(only)h(the)g(simplest)g (examples)f(of)h FQ(\003)p FP(-wild)g(groups.)35 b(The)118 432 y(pro)r(of)20 b(of)g(Theorem)f(62)h(that)g(p)r(erio)r(dic)g(groups) f(are)g(not)h FQ(\003)p FP(-wild)g(can)g(b)r(e)h(found)118 532 y(in)28 b([131)o(].)243 643 y(In)f(Section)f(3.1,)h(w)n(e)f(listed) h(a)g(n)n(um)n(b)r(er)f(of)h(examples)f(of)h FQ(\003)p FP(-algebras)d(and)118 742 y(mappings)39 b FO( )12 b FP(:)32 b FA(A)43 b FQ(\000)-49 b(!)43 b FO(M)951 754 y FM(n)996 742 y FP(\()p FA(B)p FP(\))d(suc)n(h)g(that)f(the)h(functor) g FO(F)2073 754 y FM( )2132 742 y FP(:)46 b(Rep)14 b FA(B)43 b FQ(\000)-48 b(!)118 842 y FP(Rep)14 b FA(A)32 b FP(is)f(full.)51 b(Ho)n(w)n(ev)n(er,)31 b(w)n(e)g(do)h(not)f(discuss) h(metho)r(ds)g(of)g(construction)118 942 y(of)i(suc)n(h)f(mappings.)54 b(This)34 b(is)f(a)g(separate)f(topic;)37 b(it)d(needs)f(the)h(adv)-5 b(anced)118 1041 y(language)35 b(of)h FQ(\003)p FP(-categories)e([240)n (],)39 b(and)d FQ(\003)p FP(-quiv)n(ers)e([145)o(],)39 b([255)n(],)g(etc.)63 b(W)-7 b(e)118 1141 y(also)33 b(do)g(not)h (discuss)f(the)h(question)f(of)h(what)f(is)h(the)g(minimal)g(n)n(um)n (b)r(er)f FO(n)118 1241 y FP(for)e(whic)n(h)g(there)g(exists)g(a)f (homomorphism)g FO( )12 b FP(:)30 b FA(A)e FQ(\000)-48 b(!)29 b FO(M)2018 1253 y FM(n)2063 1241 y FP(\()p FA(B)p FP(\))j(suc)n(h)f(that)118 1340 y(the)26 b(corresp)r(onding)d(functor)j FO(F)1130 1352 y FM( )1206 1340 y FP(is)f(full.)37 b(W)-7 b(e)26 b(only)f(notice)g(that)h(in)g([188)n(],)g(it)118 1440 y(w)n(as)d(sho)n(wn)h(that)h(for)e(the)i FO(C)1028 1410 y FN(\003)1067 1440 y FP(-algebra)d FB(A)i FP(with)h FO(m)f FP(self-adjoin)n(t)g(generators,)118 1539 y FO(a)162 1551 y FL(1)199 1539 y FP(,)36 b FO(:)14 b(:)g(:)27 b FP(,)37 b FO(a)486 1551 y FM(m)549 1539 y FP(,)h(the)d(algebra)f FO(M)1140 1551 y FM(n)1184 1539 y FP(\()p FB(A)p FP(\))i(for)f FO(n)g FQ(\025)h(\000)p FP(3)22 b(+)1887 1474 y FQ(p)p 1956 1474 258 4 v 65 x FP(9)c(+)g(8)p FO(m)o FP(,)37 b(is)e(singly)118 1639 y(generated,)25 b(i.e.,)i(generated)e(b)n(y)h(a) f(pair)h(of)g(self-adjoin)n(t)f(generators.)34 b(It)27 b(w)n(as)118 1739 y(sho)n(wn)32 b(in)g([228)o(])h(that)f(the)h (estimate)f FO(n)f FQ(\025)1526 1673 y(p)p 1595 1673 216 4 v 66 x FO(m)18 b FQ(\000)g FP(1)32 b(holds,)i(and)e(that)h(this) 118 1838 y(estimate)28 b(is)h(exact,)f(i.e.,)g(there)g(exists)g(a)g (comm)n(utativ)n(e)g FO(C)2011 1808 y FN(\003)2049 1838 y FP(-algebra)f FO(C)6 b FP(\()p FO(K)g FP(\))118 1938 y(with)40 b FO(m)f FP(self-adjoin)n(t)g(generators)e(suc)n(h)i(that)g FO(M)1755 1950 y FM(n)1800 1938 y FP(\()p FO(C)6 b FP(\()p FO(K)g FP(\)\))40 b(is)f(not)g(singly)118 2038 y(generated)27 b(for)g FO(n)c(<)784 1972 y FQ(p)p 853 1972 V 66 x FO(m)c FQ(\000)f FP(1)o(.)118 2221 y FR(Section)26 b(3.2.)34 b FP(In)22 b(Section)h(3.2)e(w)n(e)i(consider)e(an)h(application)g(of)h (the)g(theory)118 2320 y(of)e(represen)n(tations)e(of)i FQ(\003)p FP(-algebras)d(to)j(a)f(study)h(of)g(classes)f(of)h(op)r (erators)e(that)118 2420 y(are)27 b(singled)g(out)h(algebraically)-7 b(.)243 2531 y(The)23 b(problem)g(to)g(describ)r(e)g(the)h(class)e(of)i (op)r(erators)d(whic)n(h)i(satisfy)g(rela-)118 2630 y(tions)c(up)g(to)g (a)g(unitary)f(equiv)-5 b(alence)19 b(is)g(equiv)-5 b(alen)n(t)19 b(to)g(the)g(one)g(of)g(describing)118 2730 y(represen)n(tations)26 b(of)h(the)h(corresp)r(onding)e FQ(\003)p FP(-algebra)f FA(A)p FP(.)243 2841 y(F)-7 b(or)26 b(suc)n(h)g(algebras)f(w)n(e)h (estimate)h(the)g(complexit)n(y)f(of)h(the)g(corresp)r(ond-)118 2941 y(ing)36 b(problem)f(of)h FQ(\003)p FP(-represen)n(tation)e (theory)-7 b(,)37 b(i.e.,)i(the)d(complexit)n(y)g(of)f(the)118 3040 y(unitary)27 b(description)g(of)h(the)g(corresp)r(onding)d(class)i (of)h(op)r(erators.)243 3151 y(The)j(con)n(ten)n(t)h(of)f(Section)h (3.2)f(is)g(directly)h(related)f(to)h(the)g(pap)r(ers)f([50)o(,)118 3251 y(51)o(,)f(106)o(,)f(204)o(,)h(302)n(,)g(303)o(,)g(21)o(,)f(22)o (,)h(23)o(,)g(54)o(])g(and)g(the)g(bibliograph)n(y)e(therein.)118 3350 y(In)h(particular,)f(it)h(w)n(as)f(pro)n(v)n(en)f(that)j(in)f(an)n (y)f(factor)g(of)g(in\014nite)i(t)n(yp)r(e)f(there)118 3450 y(exists)h(a)f(generator,)g(whic)n(h)g(is)h(a)g(partial)f (isometry)-7 b(,)30 b(and)f(whic)n(h)h(is,)h(at)e(the)118 3550 y(same)22 b(time,)i(a)e(w)n(eakly)f(cen)n(tered,)i(or)f(h)n(yp)r (onormal,)g(or)f(subnormal,)i(etc.)f(op-)118 3649 y(erator;)30 b(this)g(indicates)f(that)i(the)f(description)f(of)h(suc)n(h)g(op)r (erators)e(is)i(com-)118 3749 y(plicated.)61 b(Moreo)n(v)n(er,)35 b(the)h(constructions)e(used)i(in)g(the)g(pro)r(ofs,)h(in)e(some)118 3848 y(\(but)d(not)g(all\))f(cases)f(can)h(b)r(e)h(applied)g(to)f(the)h (pro)r(of)e(of)i FQ(\003)p FP(-wildness)e(of)h(the)118 3948 y(corresp)r(onding)39 b(class)h(of)g(op)r(erators)f(\(see)i (Section)g(3.2.3,)h(item)g(1\).)76 b(Es-)118 4048 y(sen)n(tially)-7 b(,)29 b(the)g FQ(\003)p FP(-wildness)f(of)h(the)g(corresp)r(onding)e (class)h(of)h(op)r(erators)e(\(for)118 4147 y(example,)h(h)n(yp)r (onormal\))f(means)h(that)g(one)g(can)g(c)n(ho)r(ose)f(suc)n(h)h(an)g (op)r(erator)p eop %%Page: 241 245 241 244 bop 118 100 a FK(Commen)n(ts)27 b(to)h(Chapter)f(3)1452 b FP(241)118 333 y(\(h)n(yp)r(onormal\))27 b(to)g(b)r(e)h(a)g (generator)d(of)j(the)f FQ(\003)p FP(-wild)h FO(C)1847 303 y FN(\003)1885 333 y FP(-algebra.)243 457 y(3.2.1.)36 b(Simple)29 b(criteria)e(of)h FQ(\003)p FP(-wildness)f(for)g(classes)g (of)h(non)f(self-adjoin)n(t)118 557 y(op)r(erators,)i(for)h(whic)n(h)g FO(X)37 b FP(and)30 b FO(X)1228 527 y FN(\003)1296 557 y FP(are)f(related)h(b)n(y)g(a)g(quadratic)f(or)h(cubic)118 656 y(semilinear)d(relation,)g(follo)n(w)f(the)i(results)f(exp)r (ounded)h(in)g(Section)g(3.1.5.)243 756 y(Quasi-normal)c(op)r(erators)h (ha)n(v)n(e)h(a)h(rather)e(simple)j(structure)e([50)o(,)h(104)o(],)118 856 y(etc.)37 b(The)28 b(study)f(of)h(the)g(classes)e(of)h(op)r (erators)f(considered)g(in)i(items)g(4)f(and)118 955 y(5)g(of)h(3.2.1)e(is)i(also)e(not)i(to)r(o)f(complicated.)243 1080 y(3.2.2.)65 b(On)37 b(the)h(complexit)n(y)e(of)i(the)g(unitary)f (description)f(of)i(partial)118 1179 y(isometries,)e(see)f([106)o(,)g (104)o(].)60 b(The)35 b FQ(\003)p FP(-wildness)f(of)h(the)h(classes)e (of)h(w)n(eakly)118 1279 y(cen)n(tered)22 b(op)r(erators)e(and)i(w)n (eakly)f(cen)n(tered)g(partial)g(isometries)g(are)g(pro)n(v)n(ed)118 1379 y(in)28 b([146)o(,)f(210)o(].)243 1478 y(On)i(algebraic)e(op)r (erators,)h(see)h([34)o(].)42 b(In)29 b([33)o(],)h(the)g(pro)r(of)f(of) g FQ(\003)p FP(-wildness)118 1578 y(of)36 b(the)h(description)e(of)h (non)g(self-adjoin)n(t)f(op)r(erators,)i FO(X)7 b FP(,)37 b FO(X)2128 1548 y FN(\003)2166 1578 y FP(,)h(suc)n(h)e(that)118 1678 y([)p FO(X)217 1647 y FM(j)251 1678 y FO(X)327 1647 y FN(\003)365 1641 y FM(j)400 1678 y FO(;)14 b(X)513 1647 y FN(\003)550 1641 y FM(k)591 1678 y FO(X)667 1647 y FM(k)707 1678 y FP(])32 b(=)f([)p FO(X)957 1647 y FM(j)992 1678 y FO(X)1068 1647 y FN(\003)1105 1641 y FM(j)1140 1678 y FO(;)14 b(X)1253 1647 y FM(k)1293 1678 y FO(X)1369 1647 y FN(\003)1406 1641 y FM(k)1447 1678 y FP(])32 b(=)f([)p FO(X)1697 1647 y FN(\003)1735 1641 y FM(j)1770 1678 y FO(X)1846 1647 y FM(j)1880 1678 y FO(;)14 b(X)1993 1647 y FN(\003)2030 1641 y FM(k)2071 1678 y FO(X)2147 1647 y FM(k)2187 1678 y FP(])32 b(=)f(0,)j(1)d FQ(\024)118 1777 y FO(j;)14 b(k)29 b FQ(\024)c FO(n)p FP(,)30 b(is)f(giv)n(en)f (for)h(a)g(\014xed)g FO(n)d FQ(\025)f FP(1.)41 b(On)29 b(the)h(other)e(hand,)i(the)g(class)e(of)118 1877 y(cen)n(tered)i(op)r (erators)f(\(for)h(whic)n(h)h(these)g(relations)e(hold)i(for)f(all)g FO(j)5 b FP(,)32 b FO(k)f FQ(\025)d FP(1\),)118 1976 y(is)g(not)f FQ(\003)p FP(-wild)g(\(see)h(2.5.2\).)243 2101 y(3.2.3.)78 b(As)43 b(A.)f(Piry)n(atinsk)-5 b(a)n(y)n(a)39 b(noticed,)45 b(the)e(construction)e(in)h([303)o(])118 2201 y(means)27 b(essen)n(tially)g(that)h(the)g(class)e(of)i(h)n(yp)r (onormal)e(op)r(erators)g(is)h FQ(\003)p FP(-wild.)243 2300 y(The)32 b FQ(\003)p FP(-wildness)g(of)h(pairs)f(of)g(comm)n (uting)h(partial)e(isometries)h(can)g(b)r(e)118 2400 y(obtained)g(directly)g(from)g([31)o(].)51 b(The)32 b(pro)r(of)f(giv)n (en)h(in)g(the)h(b)r(o)r(ok)f(is)g(due)g(to)118 2500 y(D.)c(Proskurin.)p eop %%Page: 242 246 242 245 bop 118 100 a FP(242)p eop %%Page: 243 247 243 246 bop 118 900 a FS(Bibliograph)l(y)189 1313 y FE([1])41 b(B.)25 b(Ab)r(desselam,)h(J.)g(Bec)n(k)n(ers,)h(A.)e(Chak)l(abarti,)j (and)e(N.)g(Deb)r(ergh,)h Fi(On)g(a)i(defor-)305 1392 y(mation)g(of)h Fq(sl)q FE(\(2\))f Fi(with)h(p)l(ar)l(agr)l(assmanian)j (variables)p FE(,)28 b(J.)f(Ph)n(ys.)g(A:)g(Math.)g(Gen.)305 1471 y Fh(29)c FE(\(1996\),)i(6729{6736.)189 1555 y([2])41 b(S.)30 b(I.)g(Adian,)i Fi(The)g(Burnside)g(pr)l(oblem)h(and)g (identities)e(in)h(gr)l(oups)p FE(,)h(Ergebnisse)305 1634 y(der)23 b(Math.)h(und)g(ihrer)f(Grenzb)r(eb)r(eite,)i(v)n(ol.)e (95,)h(Springer,)f(Berlin,)f(1979,)j(T)-6 b(ransl.)305 1713 y(from)21 b(Russian)j(edn.:)31 b(Nauk)l(a,)24 b(Mosco)n(w,)g (1975.)189 1797 y([3])41 b(N.)21 b(I.)h(Akhiezer,)h Fi(Classic)l(al)j (moment)e(pr)l(oblem)p FE(,)g(Oliv)n(er)d(and)i(Bo)n(yd,)g(1965,)g(T)-6 b(ransl.)305 1876 y(from)21 b(Russian)j(edn:)32 b(Fizmatgiz,)22 b(Mosco)n(w,)i(1961.)189 1960 y([4])41 b(N.)17 b(I.)g(Akhiezer)i(and)f (I.)g(M.)f(Glazman,)h Fi(The)i(the)l(ory)h(of)f(line)l(ar)h(op)l(er)l (ators)h(in)e(Hilb)l(ert)305 2039 y(sp)l(ac)l(e)p FE(,)27 b(Ungar,)f(New)g(Y)-6 b(ork,)26 b(1961,)h(T)-6 b(ransl.)25 b(from)f(Russian)i(edn.:)36 b(Gostekhizdat,)305 2118 y(Mosco)n(w,)23 b(1950.)189 2202 y([5])41 b(S.)17 b(A.)g(Amitsur)e(and) j(J.)f(Levitski,)i Fi(Minimal)h(identities)f(for)h(algebr)l(as)p FE(,)g(Pro)r(c.)d(Amer.)305 2281 y(Math.)23 b(So)r(c.)h Fh(1)g FE(\(1950\),)h(449{463.)189 2365 y([6])41 b(H.)18 b(Araki,)h Fi(Hamiltonian)j(formalism)h(and)f(c)l(anonic)l(al)g(r)l (elations)g(in)f(quantum)h(\014eld)305 2444 y(the)l(ory)p FE(,)h(J.)h(Math.)f(Ph)n(ys.)h Fh(1)f FE(\(1960\),)i(no.)f(4,)f (492{504.)189 2528 y([7])41 b(R.)27 b(Arens,)i Fi(R)l(epr)l (esentations)h(of)g FD(\003)p Fi(-algebr)l(as)p FE(,)g(Duk)n(e)f(Math.) f(J.)g Fh(14)f FE(\(1947\),)k(269{)305 2607 y(282.)189 2691 y([8])41 b(W.)32 b(B.)f(Arv)n(eson,)k Fi(Op)l(er)l(ator)f(algebr)l (as)h(and)f(invariant)g(subsp)l(ac)l(es)p FE(,)h(Ann)e(Math.)305 2770 y Fh(100)22 b FE(\(1974\),)j(433{532.)189 2854 y([9])p 305 2854 212 4 v 264 w(,)37 b Fi(A)n(n)e(invitation)h(to)f Fq(C)1196 2830 y Fg(\003)1232 2854 y Fi(-algebr)l(as)p FE(,)i(Graduate)f(texts)g(in)e(mathematics,)305 2933 y(v)n(ol.)23 b(39,)h(Springer,)f(Berlin,)f(1976.)153 3017 y([10])p 305 3017 V 265 w(,)38 b Fi(The)e(harmonic)i(analysis)f (of)f(automorphism)j(gr)l(oups)p FE(,)g(Pro)r(c.)c(Symp.)305 3096 y(Pure)23 b(Math.)h Fh(38)f FE(\(1982\),)i(199{269.)153 3180 y([11])p 305 3180 V 265 w(,)33 b Fi(Continuous)i(analo)l(gues)g (of)e(Fo)l(ck)h(sp)l(ac)l(e)p FE(,)g(Mem.)c(Amer.)g(Math.)i(So)r(c.,) 305 3259 y(v)n(ol.)23 b(80,)h(Amer.)d(Math.)j(So)r(c.,)g(Pro)n (vidence,)g(R.I.,)f(1989.)153 3343 y([12])42 b(V.)30 b(Arzumanian)f(and)i(A.)f(V)-6 b(ershik,)32 b Fi(Star-algebr)l(as)h (asso)l(ciate)l(d)g(with)g(endomor-)305 3422 y(phisms)p FE(,)21 b(Op)r(erator)g(algebras)f(and)h(group)f(represen)n(tations,)i (Pro)r(c)e(In)n(t.)h(Conf.,)f(v)n(ol.)305 3501 y(I.)j(\(Boston\),)i (Pitman,)e(1984,)h(pp.)g(17{27.)153 3585 y([13])42 b(M.)22 b(Auslander,)g(I.)h(Reiten,)g(and)g(S.)g(O.)f(Smalo,)g Fi(R)l(epr)l(esentation)j(the)l(ory)g(of)g(Artin)305 3664 y(algebr)l(as)p FE(,)18 b(Cam)n(bridge)d(Studies)i(in)f(Adv)l (anced)i(Mathematics,)f(v)n(ol.)f(36,)i(Cam)n(bridge)305 3742 y(Univ.)23 b(Press,)f(1995.)153 3827 y([14])42 b(T.)26 b(Y)-6 b(a.)27 b(Azizo)n(v)g(and)h(I.)e(S.)h(Ioh)n(vido)n(v,)i Fi(Line)l(ar)f(op)l(er)l(ators)j(in)e(sp)l(ac)l(es)g(with)g(an)g(in-) 305 3905 y(de\014nite)24 b(mertic)p FE(,)e(J.)g(Wiley)h(and)h(Sons,)f (New)f(Y)-6 b(ork,)23 b(1989,)g(T)-6 b(ransl.)22 b(from)f(Russian)305 3984 y(edn.:)31 b(Nauk)l(a,)24 b(Mosco)n(w,)g(1986.)153 4068 y([15])42 b(O.)21 b(V.)f(Bagro,)i Fi(Pairs)i(of)g(self-adjoint)g (op)l(er)l(ators)i(c)l(onne)l(cte)l(d)e(by)f(a)h(cubic)f(r)l(elation)p FE(,)305 4147 y(Ukrain.)g(Mat.)g(Zh.)g Fh(47)g FE(\(1995\),)i(no.)f(5,) g(600{602,)h(\(Russian\).)1284 4357 y FP(243)p eop %%Page: 244 248 244 247 bop 118 100 a FP(244)1866 b FK(Bibliograph)n(y)153 333 y FE([16])42 b(O.)18 b(V.)g(Bagro)g(and)h(S.)g(A.)f(Krugly)n(ak,)h Fi(R)l(epr)l(esentations)j(of)f(involutive)f(quivers)h(and)305 412 y(wild)26 b(pr)l(oblems)p FE(,)f(Preprin)n(t)e(KVIUS,)g(Kiev,)h (1995,)g(\(Russian\).)153 501 y([17])p 305 501 212 4 v 265 w(,)34 b Fi(R)l(epr)l(esentations)g(of)g(D.)f(Fairlie)h(algebr)l (as)p FE(,)h(Preprin)n(t)c(KVIUS,)h(Kiev,)305 580 y(1996,)24 b(\(Russian\).)153 669 y([18])42 b(B.)35 b(A.)h(Barnes,)i Fi(A)f(note)g(on)h(sep)l(ar)l(ating)g(families)g(of)f(r)l(epr)l (esentations)p FE(,)j(Pro)r(c.)305 748 y(Amer.)21 b(Math.)j(So)r(c.)g Fh(87)f FE(\(1983\),)i(95{98.)153 837 y([19])42 b(H.)28 b(Bart,)i(T.)e(Ehrhardt,)j(and)e(B.)g(Silb)r(ermann,)f Fi(Zer)l(o)j(sums)h(of)e(idemp)l(otents)i(in)305 916 y(Banach)27 b(algebr)l(as)p FE(,)d(In)n(tegr.)g(Equat.)g(Op)r(er.)f (Theory)i Fh(19)e FE(\(1994\),)j(125{134.)153 1005 y([20])42 b(A.)21 b(O.)h(Barut)g(and)h(R.)e(Raczk)l(a,)i Fi(The)l(ory)i(of)g(gr)l (oup)g(r)l(epr)l(esentations)g(and)g(applic)l(a-)305 1084 y(tions)p FE(,)e(PWN,)g(W)-6 b(arsza)n(w)n(a,)25 b(1977.)153 1173 y([21])42 b(H.)23 b(Benc)n(k)n(e,)j Fi(Gener)l(ators)h(of)f Fq(W)1192 1150 y Fg(\003)1228 1173 y Fi(-algebr)l(as)p FE(,)e(T)-6 b(ohoku)26 b(Math.)e(J.)g Fh(22)g FE(\(1970\),)i(541{)305 1252 y(546.)153 1341 y([22])p 305 1341 V 265 w(,)f Fi(Gener)l(ators)j(of)f Fq(W)1105 1318 y Fg(\003)1141 1341 y Fi(-algebr)l(as)g(II)p FE(,)f(T)-6 b(ohoku)26 b(Math.)f(J.)g Fh(24)g FE(\(1972\),)i(371{)305 1420 y(381.)153 1509 y([23])p 305 1509 V 265 w(,)22 b Fi(Gener)l(ators)k(of)e Fq(W)1097 1486 y Fg(\003)1133 1509 y Fi(-algebr)l(as)h(III)p FE(,)e(T)-6 b(ohoku)23 b(Math.)g(J.)f Fh(24)f FE(\(1972\),)k(383{)305 1588 y(388.)153 1677 y([24])42 b(Y)-6 b(u.)22 b(M.)f(Berezansky)-6 b(,)23 b Fi(Exp)l(ansion)j(in)e(eigenfunctions)g(of)h(self-adjoint)f(op)l(er)l (ators)p FE(,)305 1756 y(T)-6 b(ransl.)20 b(Math.)i(Monogr.,)f(v)n(ol.) g(17,)h(Amer.)d(Math.)j(So)r(c.,)g(Pro)n(vidence,)g(R.I.,)f(1968,)305 1835 y(T)-6 b(ransl.)22 b(from)g(Russian)h(edn.:)32 b(Nauk)n(o)n(v)l(a) 25 b(Dumk)l(a,)d(Kiev,)i(1965.)153 1924 y([25])p 305 1924 V 265 w(,)33 b Fi(Self-adjoint)f(op)l(er)l(ators)j(in)d(sp)l(ac)l (es)i(of)f(functions)g(of)g(in\014nitely)e(many)305 2003 y(variables)p FE(,)19 b(T)-6 b(rans.)18 b(Math.)h(Monographs,)g(v)n (ol.)f(63,)i(AMS,)d(Pro)n(vidence,)k(R.I.,)d(1986,)305 2082 y(T)-6 b(ransl.)22 b(from)g(Russian)h(edn.:)32 b(Kiev,)23 b(Nauk)n(o)n(v)l(a)i(Dumk)l(a,)e(1978.)153 2171 y([26])42 b(Y)-6 b(u.)29 b(M.)f(Berezansky)j(and)f(Y)-6 b(u.)29 b(G.)g(Kondrat'ev,)j Fi(Sp)l(e)l(ctr)l(al)g(metho)l(ds)h(in)d (in\014nite)305 2250 y(dimensional)e(analysis)p FE(,)e(Klu)n(w)n(er)f (Acad.)g(Publ.,)g(Dordrec)n(h)n(t,)g(1992,)i(T)-6 b(ransl.)24 b(from)305 2329 y(Russian)f(edn.:)31 b(Nauk)n(o)n(v)l(a)26 b(Dumk)l(a,)c(Kiev,)h(1988.)153 2418 y([27])42 b(Y)-6 b(u.)31 b(M.)g(Berezansky)-6 b(,)34 b(G.)e(Lassner,)h(and)f(V.)f(S.)h (Y)-6 b(ak)n(o)n(vlev,)34 b Fi(On)f(de)l(c)l(omp)l(osition)305 2497 y(of)e(p)l(ositive)g(functionals)h(on)f(c)l(ommutative)g(nucle)l (ar)h FD(\003)p Fi(-algebr)l(as)p FE(,)f(Ukrain.)e(Mat.)305 2576 y(Zh)n(urn.)23 b Fh(39)g FE(\(1987\),)i(no.)f(5,)f(638{641,)i (\(Russian\).)153 2665 y([28])42 b(Y)-6 b(u.)24 b(M.)h(Berezansky)-6 b(,)26 b(V.)f(L.)f(Ostro)n(vsky)-8 b(\025)-27 b(\020,)25 b(and)h(Y)-6 b(u.)25 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)24 b Fi(Eigenfunc-)305 2744 y(tion)h(exp)l(ansion)h(of)g(families)f(of)h (c)l(ommuting)g(op)l(er)l(ators)h(and)f(r)l(epr)l(esentations)h(of)305 2823 y(c)l(ommutation)j(r)l(elations)p FE(,)f(Ukr.)d(Math.)h(Zh.)g Fh(40)f FE(\(1988\),)k(no.)d(1,)g(106{109,)j(\(Rus-)305 2902 y(sian\).)153 2991 y([29])42 b(Y)-6 b(u.)22 b(M.)f(Berezansky)-6 b(,)24 b(Z.)e(G.)h(Sheftel,)g(and)g(G.)f(F.)g(Us,)g Fi(F)-5 b(unctional)25 b(analysis)h(I,)e(II)p FE(,)305 3070 y(Op)r(erator)f (Theory)-6 b(,)24 b(Adv.)e(Appl.,)h(v)n(ol.)f(85,)i(86,)f(Birkh\177)-35 b(auser)23 b(V)-6 b(erlag,)22 b(Basel,)h(1996,)305 3149 y(T)-6 b(ransl.)22 b(from)g(Russian)h(edn.:)32 b(Vyshc)n(ha)24 b(Shk)n(ola,)g(Kiev,)f(1990.)153 3238 y([30])42 b(F.)28 b(A.)g(Berezin)h(and)g(G.)f(I.)h(Kats,)h Fi(Lie)f(gr)l(oups)j(with)e(c) l(ommuting)h(and)g(antic)l(om-)305 3317 y(muting)23 b(p)l(ar)l(ameters) p FE(,)g(Matem.)d(Sb)r(ornik)h Fh(82)f FE(\(1970\),)j(no.)e(3,)g (343{359,)i(\(Russian\).)153 3406 y([31])42 b(C.)32 b(A.)h(Berger,)i (L.)d(A.)h(Coburn,)i(and)f(A.)e(Leb)r(o)n(w,)k Fi(R)l(epr)l(esentation) f(and)h(index)305 3485 y(the)l(ory)21 b(for)f Fq(C)679 3462 y Fg(\003)715 3485 y Fi(-algebr)l(as)h(gener)l(ate)l(d)g(by)f(c)l (ommuting)h(isometries)p FE(,)e(J.)f(F)-6 b(unct.)19 b(Anal.)305 3564 y Fh(27)k FE(\(1978\),)i(51{99.)153 3653 y([32])42 b(Y)-6 b(u.)22 b(N.)g(Bespalo)n(v,)h Fi(Col)t(le)l (ctions)i(of)g(orthopr)l(oje)l(ctions)i(satisfying)d(r)l(elations)p FE(,)g(Ukr.)305 3732 y(Mat.)f(Zh)n(urn.)h Fh(44)e FE(\(1992\),)k(no.)d (3,)h(309{317,)h(\(Russian\).)153 3821 y([33])p 305 3821 V 265 w(,)d Fi(A)n(lgebr)l(aic)h(op)l(er)l(ators,)j(p)l(artial)f (isometries,)f(and)g(wild)h(pr)l(oblems)p FE(,)e(Meth-)305 3900 y(o)r(ds)g(F)-6 b(unct.)25 b(Anal.)e(T)-6 b(op)r(ol.)24 b Fh(3)f FE(\(1997\),)i(no.)f(1,)f(28{45.)153 3990 y([34])42 b(Y)-6 b(u.)24 b(N.)g(Bespalo)n(v)i(and)f(Y)-6 b(u.)25 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)24 b Fi(A)n(lgebr)l(aic)j(op) l(er)l(ators)i(and)f(p)l(airs)g(of)305 4068 y(selfadjoint)33 b(op)l(er)l(ators)j(c)l(onne)l(cte)l(d)e(by)e(a)h(p)l(olynomial)j(r)l (elation)p FE(,)e(F)-6 b(unct.)32 b(Anal.)f(i)305 4147 y(Prilozhen.)23 b Fh(25)g FE(\(1991\),)i(no.)f(4,)f(72{74,)i (\(Russian\).)p eop %%Page: 245 249 245 248 bop 118 100 a FK(Bibliograph)n(y)1865 b FP(245)153 333 y FE([35])42 b(Y)-6 b(u.)33 b(N.)g(Bespalo)n(v,)j(Y)-6 b(u.)34 b(S.)f(Samo)-8 b(\025)-27 b(\020lenk)n(o,)36 b(and)e(V.)f(S.)g(Sh)n(ul'man,)i Fi(On)f(families)305 412 y(of)29 b(op)l(er)l(ators)i(c)l(onne)l(cte)l(d)e(by)f(semi-line)l (ar)h(r)l(elations)p FE(,)g(Applications)e(of)g(Metho)r(ds)305 490 y(of)k(F)-6 b(unctional)32 b(Analysis)e(in)h(Mathematical)g(Ph)n (ysics,)i(Inst.)f(Math.)f(Acad.)g(Sci.)305 569 y(Ukraine,)23 b(Kiev,)g(1991,)h(\(Russian\),)g(pp.)g(28{51.)153 653 y([36])42 b(L.)33 b(C.)g(Biedenharn,)j Fi(The)f(quantum)h(gr)l(oup)g Fq(su)1650 661 y Ff(q)1683 653 y FE(\(2\))g Fi(and)g(a)f Fq(q)r Fi(-analo)l(gue)h(of)f(the)305 732 y(b)l(oson)26 b(op)l(er)l(ators)p FE(,)g(J.)d(Ph)n(ys.)h(A)f Fh(22)g FE(\(1989\),)i(L873{L878.)153 816 y([37])42 b(M.)28 b(Sh.)i(Birman)e (and)i(M.)e(A.)h(Solom)n(y)n(ak,)i Fi(The)g(sp)l(e)l(ctr)l(al)i(the)l (ory)e(of)g(self-adjoint)305 895 y(op)l(er)l(ators)c(in)f(Hilb)l(ert)f (sp)l(ac)l(e)p FE(,)f(Izdat.)h(Leningrad.)f(Univ,)f(Leningrad,)h(1980.) 153 979 y([38])42 b(B.)22 b(E.)h(Blac)n(k)l(adar,)g Fi(A)i(simple)h (unital)g(pr)l(oje)l(ctionless)g Fq(C)1833 956 y Fg(\003)1869 979 y Fi(-algebr)l(a)p FE(,)d(J.)g(Op)r(er.)g(The-)305 1058 y(ory)h Fh(5)f FE(\(1981\),)j(63{71.)153 1142 y([39])42 b(V.)16 b(M.)g(Bondarenk)n(o)j(and)e(Y)-6 b(u.)17 b(A.)f(Drozd,)i Fi(R)l(epr)l(esentations)j(typ)l(e)e(of)h(\014nite)f(gr)l(oups)p FE(,)305 1221 y(Zap.)k(Nauc)n(hn.)i(Sem.)d(LOMI)i Fh(71)f FE(\(1977\),)i(24{42,)f(\(Russian\).)153 1305 y([40])42 b(A.)26 b(B\177)-35 b(ottc)n(her,)30 b(I.)d(Goh)n(b)r(erg,)i(Y)-6 b(u.)27 b(Karlo)n(vic)n(h,)h(N.)f(Krupnik,)g(S.)g(Ro)r(c)n(h,)i(B.)e (Silb)r(er-)305 1384 y(man,)21 b(and)h(I.)g(Spitk)n(o)n(vsky)-6 b(,)24 b Fi(Banach)h(algebr)l(as)g(gener)l(ate)l(d)f(by)g Fq(N)31 b Fi(idemp)l(otents)25 b(and)305 1462 y(applic)l(ations)p FE(,)g(Op)r(erator)f(Theory)g(Adv.)f(Appl.)g Fh(90)g FE(\(1996\),)j(19{54.)153 1546 y([41])42 b(N.)23 b(Bourbaki,)g Fi(Gr)l(oup)l(es)28 b(et)d(algebr)l(es)g(de)h(Lie)f(IV{VI)p FE(,)f(Hermann,)f(P)n(aris,)f(1968.)153 1630 y([42])42 b(M.)25 b(Bo)6 b(_)-25 b(zejk)n(o)25 b(and)i(R.)e(Sp)r(eic)n(her,)h Fi(A)n(n)i(example)g(of)g(a)g(gener)l(alize)l(d)g(Br)l(ownian)i(mo-)305 1709 y(tion.)p FE(,)23 b(Comm)n(un.)e(Math.)j(Ph)n(ys.)f Fh(37)g FE(\(1991\),)i(519{531.)153 1793 y([43])p 305 1793 212 4 v 265 w(,)i Fi(Completely)i(p)l(ositive)g(maps)h(on)f (Coxeter)f(gr)l(oups,)j(deforme)l(d)f(c)l(ommu-)305 1872 y(tation)25 b(r)l(elations,)i(and)f(op)l(er)l(ator)i(sp)l(ac)l(es)p FE(,)c(Math.)g(Ann.)g Fh(300)e FE(\(1994\),)j(97{120.)153 1956 y([44])42 b(O.)21 b(Bratteli)i(and)g(P)-6 b(.)21 b(E.)h(T.)g(J\034rgensen,)h Fi(Endomorphisms)28 b(of)c Fe(B)p FE(\()p Fe(H)q FE(\))p Fi(.)g(II.)h(Finitely)305 2035 y(c)l(orr)l(elate)l(d)i(states)e(on)h Fe(O)982 2043 y Ff(n)1025 2035 y FE(,)d(J.)g(F)-6 b(unct.)25 b(Anal.)e Fh(145)g FE(\(1997\),)i(no.)e(2,)h(323{373.)153 2119 y([45])p 305 2119 V 265 w(,)40 b Fi(Isometries,)h(shifts,)f(Cuntz)e (algebr)l(as)h(and)f(multir)l(esolution)i(wavelet)305 2198 y(analysis)26 b(of)g(sc)l(ale)g Fq(N)7 b FE(,)23 b(In)n(t.)i(Equat.)f(Op)r(er.)f(Theory)i Fh(28)e FE(\(1997\),)i (382{443.)153 2282 y([46])p 305 2282 V 265 w(,)d Fi(Iter)l(ate)l(d)j (function)g(systems)f(and)h(p)l(ermutation)h(r)l(epr)l(esentations)f (of)g(the)305 2361 y(Cuntz)g(algebr)l(a)p FE(,)f(Mem.)e(Amer.)g(Math.)i (So)r(c.,)f(AMS,)g(1998.)153 2445 y([47])42 b(O.)29 b(Bratteli,)h(P)-6 b(.)29 b(E.)g(T.)g(J\034rgensen,)j(and)e(V.)f(Ostro)n(vskyi,)h Fi(R)l(epr)l(esentation)i(the-)305 2524 y(ory)21 b(and)i(numeric)l(al)g (AF-invariants:)31 b(The)22 b(r)l(epr)l(esentations)h(and)f(c)l(entr)l (alizers)h(of)305 2603 y(c)l(ertain)k(states)h(on)g Fq(O)907 2615 y Ff(d)943 2603 y FE(,)e Fd(arXiv:math.OA/9907036)p FE(,)32 b(to)26 b(app)r(ear)h(in)e(Mem.)g(Amer.)305 2681 y(Math.)e(So)r(c.)153 2765 y([48])42 b(O.)18 b(Bratteli,)i(P)-6 b(.)19 b(E.)f(T.)h(J\034rgensen,)i(and)e(G.)g(L.)g(Price,)g Fi(Endomorphisms)25 b(of)c Fe(B)p FE(\()p Fe(H)q FE(\),)305 2844 y(Pro)r(c.)i(Symp.)f(Pure)i(Math.)g Fh(59)f FE(\(1996\),)i (93{138.)153 2928 y([49])42 b(O.)16 b(Bratteli)h(and)g(D.)g(W.)f (Robinson,)j Fi(Op)l(er)l(ator)h(algebr)l(as)g(and)g(quantum)g (statistic)l(al)305 3007 y(me)l(chanics)p FE(,)k(Bo)r(oks)g(and)g (Monographs)g(in)g(Ph)n(ysics,)f(Springer-V)-6 b(elag,)23 b(1979.)153 3091 y([50])42 b(A.)21 b(Bro)n(wn,)h Fi(On)h(a)i(class)f (of)g(op)l(er)l(ators)p FE(,)g(Pro)r(c)e(Amer.)e(Math.)i(So)r(c.)g Fh(4)g FE(\(1953\),)i(723{)305 3170 y(728.)153 3254 y([51])p 305 3254 V 265 w(,)30 b Fi(The)g(unitary)h(e)l(quivalenc)l(e)f(of)h (binormal)g(op)l(er)l(ators)p FE(,)h(Amer.)27 b(J.)i(Math.)305 3333 y Fh(76)23 b FE(\(1954\),)i(no.)e(2,)h(414{434.)153 3417 y([52])42 b(L.)31 b(Bro)n(wn,)j(P)-6 b(.)32 b(Green,)i(and)f(M.)e (A.)g(Rie\013el,)j Fi(Stable)f(isomorphism)j(ans)e(str)l(ong)305 3496 y(Morita)26 b(e)l(qivalenc)l(e)f(of)h Fq(C)1013 3472 y Fg(\003)1049 3496 y Fi(-algebr)l(as)p FE(,)e(P)n(aci\014c)g(J.)f (Math.)h Fh(71)f FE(\(1977\),)i(349{363.)153 3580 y([53])42 b(I.)33 b(M.)g(Burban)h(and)g(A.)f(U.)g(Klim)n(yk,)h Fi(On)h(sp)l(e)l(ctr)l(al)h(pr)l(op)l(erties)g(of)f Fq(q)r Fi(-oscil)t(lator)305 3659 y(op)l(er)l(ators)p FE(,)25 b(Lett.)f(Math.)g(Ph)n(ys.)g Fh(29)e FE(\(1993\),)k(13{18.)153 3743 y([54])42 b(S.)24 b(L.)g(Campb)r(ell,)f Fi(Line)l(ar)k(op)l(er)l (ators)h(for)f(which)g Fq(T)1722 3719 y Fg(\003)1758 3743 y Fq(T)36 b Fi(and)28 b Fq(T)10 b(T)2077 3719 y Fg(\003)2139 3743 y Fi(c)l(ommute)27 b(\(II\))p FE(,)305 3822 y(P)n(aci\014c)d(Journ.)g(Math.)f Fh(53)g FE(\(1974\),)i(no.)f(2,) f(355{361.)153 3906 y([55])42 b(V.)21 b(Chari)g(and)h(A.)f(N.)g (Pressly)-6 b(,)21 b Fi(A)i(guide)h(to)g(quantum)g(gr)l(oups)p FE(,)g(Cam)n(bridge)c(Univ.)305 3984 y(Press,)i(Cam)n(bridge,)h(1994.) 153 4068 y([56])42 b(M.)17 b(D.)g(Choi,)h Fi(The)j(ful)t(l)f Fq(C)997 4045 y Fg(\003)1033 4068 y Fi(-algebr)l(a)h(of)f(the)g(fr)l(e) l(e)g(gr)l(oup)i(of)e(two)h(gener)l(ators)p FE(,)e(P)n(aci\014c)305 4147 y(J.)k(Math.)h Fh(87)f FE(\(1980\),)i(no.)e(1,)h(41{48.)p eop %%Page: 246 250 246 249 bop 118 100 a FP(246)1866 b FK(Bibliograph)n(y)153 333 y FE([57])42 b(J.)21 b(M.)g(Cohen,)i Fq(C)781 309 y Fg(\003)817 333 y Fi(-algebr)l(as)h(without)g(idemp)l(otents)p FE(,)f(J.)e(F)-6 b(unct.)23 b(Anal.)e Fh(33)g FE(\(1979\),)305 412 y(211{216.)153 492 y([58])42 b(A.)23 b(Connes,)h Fi(Non-c)l(ommutative)i(ge)l(ometry)p FE(,)d(Acad.)h(Press,)e(New)i(Y) -6 b(ork,)23 b(1994.)153 572 y([59])42 b(J.)31 b(Cun)n(tz,)j Fi(Simple)f Fq(C)926 548 y Fg(\003)962 572 y Fi(-algebr)l(as)g(gener)l (ate)l(d)h(by)e(isometries)p FE(,)g(Comm)n(un.)e(Math.)305 651 y(Ph)n(ys.)23 b Fh(57)g FE(\(1977\),)i(173{185.)153 731 y([60])42 b(Ch.)27 b(W.)h(Curtis)g(and)g(I.)g(Reiner,)h Fi(R)l(epr)l(esentation)h(the)l(ory)g(of)g(\014nite)f(gr)l(oups)i(and) 305 810 y(asso)l(ciative)26 b(algebr)l(as)p FE(,)e(Wiley)-6 b(,)24 b(New)f(Y)-6 b(ork,)23 b(1962.)153 890 y([61])42 b(Y)-6 b(u.)19 b(L.)h(Daletski)-8 b(\025)-27 b(\020,)20 b Fi(F)-5 b(unctional)23 b(inte)l(gr)l(als)f(r)l(elate)l(d)i(with)e(op) l(er)l(ator)i(evolution)f(e)l(qua-)305 969 y(tions)p FE(,)g(Usp)r(ekhi)h(Mat.)f(Nauk)h Fh(17)f FE(\(1962\),)i(no.)f(5,)f (3{115,)i(\(Russian\).)153 1049 y([62])42 b(A.)19 b(Y)-6 b(u.)20 b(Daletsky)h(and)f(Y)-6 b(u.)20 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)20 b Fi(A)i(nonc)l(ommutative)h(moment)g(pr) l(ob-)305 1128 y(lem)p FE(,)g(F)-6 b(un)n(ts.)24 b(Anal.)f(Prilozh.)g Fh(21)g FE(\(1987\),)i(no.)f(2,)f(72{73,)h(\(Russian\).)153 1208 y([63])42 b(E.)25 b(V.)g(Damaskinsky)h(and)g(P)-6 b(.)26 b(P)-6 b(.)26 b(Kulish,)f Fi(Deforme)l(d)j(oscil)t(lators)h(and) g(their)e(ap-)305 1287 y(plic)l(ations)p FE(,)d(Zap.)g(Nauc)n(hn.)g (Sem.)f(LOMI)g Fh(189)g FE(\(1991\),)i(37{74,)f(\(Russian\).)153 1367 y([64])42 b(C.)25 b(Dask)l(alo)n(y)n(anis,)h Fi(Gener)l(alize)l(d) i(deforme)l(d)h(oscil)t(lator)f(and)h(nonline)l(ar)f(algebr)l(as)p FE(,)305 1446 y(J.)23 b(Ph)n(ys.)g(A)h Fh(24)f FE(\(1991\),)i (L789{L794.)153 1526 y([65])42 b(K.)25 b(R.)h(Da)n(vidson,)h Fq(C)890 1503 y Fg(\003)926 1526 y Fi(-algebr)l(as)i(by)e(example)p FE(,)g(Amer.)e(Math.)h(So)r(c.,)h(Pro)n(vidence,)305 1605 y(R.I.,)22 b(1997.)153 1685 y([66])42 b(C.)26 b(Da)n(vis,)g Fi(Sep)l(ar)l(ation)k(of)f(two)f(line)l(ar)h(subsp)l(ac)l(es)p FE(,)f(Acta)g(Sci.)e(Math.)g(Szeged)i Fh(19)305 1764 y FE(\(1958\),)d(172{187.)153 1844 y([67])42 b(C.)25 b(Delb)r(ecq)h(and)h(C.)e(Quesne,)h Fi(A)i(cubic)e(deformation)j(of)f Fq(su)p FE(\(2\),)e(Mo)r(dern)g(Ph)n(ys.)305 1923 y(Lett.)e(A)g Fh(8)f FE(\(1993\),)i(961{966.)153 2003 y([68])p 305 2003 212 4 v 265 w(,)f Fi(R)l(epr)l(esentation)i(the)l(ory)h(and)f Fq(q)r Fi(-b)l(oson)h(r)l(e)l(alizations)g(of)f(Witten)-7 b('s)25 b Fq(su)p FE(\(2\))305 2082 y Fi(and)h Fq(su)p FE(\(1)p Fq(;)12 b FE(1\))26 b Fi(deformations)p FE(,)f(Ph)n(ys.)e (Lett.)i(B)e Fh(300)g FE(\(1993\),)i(227{233.)153 2162 y([69])42 b(J.)e(Dixmier,)j Fi(L)l(es)f Fq(C)914 2139 y Fg(\003)950 2162 y Fi(-algebr)l(as)g(et)f(leur)h(r)l(epr)l (esentations)p FE(,)k(Gauthier-Villars,)305 2241 y(P)n(aris,)22 b(1969.)153 2321 y([70])42 b(D.)415 2305 y(\024)411 2321 y(Z.)33 b(Dok)n(o)n(vi)n(\024)-33 b(c,)37 b Fi(Unitary)d(similarity)g (of)h(pr)l(oje)l(ctors)p FE(,)i(Aequationes)e(Math.)e Fh(42)305 2400 y FE(\(1991\),)25 b(220{224.)153 2480 y([71])42 b(P)-6 b(.)40 b(Dono)n(v)l(an)i(and)g(M.)e(R.)g(F)-6 b(reislic)n(h,)44 b Fi(Some)e(evidenc)l(e)f(for)h(an)g(extension)g(of) 305 2559 y(the)31 b(Br)l(auer{Thr)l(al)t(l)j(c)l(onje)l(ctur)l(e)p FE(,)e(Sonderforsc)n(h)n(ungsb)r(ereic)n(h)f(Theor.)f(Math.)f Fh(40)305 2638 y FE(\(1972\),)c(24{26.)153 2718 y([72])42 b(R.)i(S.)g(Doran)h(and)g(V.)f(A.)g(Bel\014,)50 b Fi(Char)l (acterization)d(of)e Fq(C)2081 2695 y Fg(\003)2117 2718 y Fi(-algebr)l(as.)h(The)305 2797 y(Gelfand{Naimark)22 b(the)l(or)l(ems)p FE(,)f(Pure)d(and)h(applied)f(mathematics,)g(v)n (ol.)g(101,)i(Mar-)305 2876 y(cel)k(Dekk)n(er,)f(Inc.,)h(New)g(Y)-6 b(ork,)23 b(Basel,)g(1986.)153 2956 y([73])42 b(R.)18 b(G.)i(Douglas,)g Fi(Banach)i(algebr)l(a)h(te)l(chniques)e(in)h(op)l (er)l(ator)h(the)l(ory)p FE(,)d(Acad.)g(Press,)305 3035 y(New)j(Y)-6 b(ork,)24 b(London,)g(1972.)153 3115 y([74])42 b(V.)32 b(G.)g(Drinfeld,)h Fi(Hopf)h(algebr)l(as)h(and)g(the)f(quantum) g(Yang{Baxter)g(e)l(quation)p FE(,)305 3194 y(So)n(viet)24 b(Math.)g(Dokl.)f Fh(32)g FE(\(1985\),)i(no.)f(1,)f(254{258.)153 3274 y([75])42 b(Y)-6 b(u.)40 b(A.)g(Drozd,)45 b Fi(T)-5 b(ame)42 b(and)g(wild)h(matrix)e(pr)l(oblems)p FE(,)46 b(Represen)n(tations)d(and)305 3353 y(quadratic)20 b(forms,)e(Inst.)h (Mat.)g(AN)g(UkrSSR,)g(Kiev,)g(1979,)i(pp.)e(39{74,)i(\(Russian\).)153 3433 y([76])42 b(H.)22 b(A.)g(Dy)n(e,)h Fi(On)h(gr)l(oups)j(of)e(me)l (asur)l(e)h(pr)l(eserving)f(tr)l(ansformations.)h(I)p FE(,)d(Amer.)e(J.)305 3512 y(Math.)i Fh(81)g FE(\(1959\),)i(119{159.) 153 3593 y([77])42 b(K.)17 b(Dyk)n(ema)h(and)h(A.)e(Nica,)i Fi(On)h(the)h(Fo)l(ck)g(r)l(epr)l(esentation)h(of)e(the)h Fq(q)r Fi(-c)l(ommutation)305 3671 y(r)l(elations)p FE(,)j(J.)f(Reine)h (Angew.)g(Math.)g Fh(440)e FE(\(1993\),)j(201{212.)153 3752 y([78])42 b(T.)23 b(Ehrhardt,)i(V.I.)e(Rabano)n(vic)n(h,)j(Y)-6 b(u.S.)24 b(Samoilenk)n(o,)g(and)g(B.)g(Silb)r(ermann,)f Fi(On)305 3830 y(the)36 b(de)l(c)l(omp)l(osition)i(of)f(the)f(identity) f(into)h(a)g(sum)h(of)g(idemp)l(otents)p FE(,)h(Metho)r(ds)305 3909 y(F)-6 b(unct.)24 b(Anal.)f(T)-6 b(op)r(ol.)24 b Fh(7)f FE(\(2001\),)i(no.)f(4,)f(1{6.)153 3990 y([79])42 b(E.)19 b(G.)h(E\013ros)f(and)i(F.)e(Hahn,)h Fi(L)l(o)l(c)l(al)t(ly)k (c)l(omp)l(act)f(tr)l(ansformation)h(gr)l(oups)f(and)g Fq(C)2513 3966 y Fg(\003)2549 3990 y Fi(-)305 4068 y(algebr)l(as)p FE(,)i(Mem.)e(Amer.)g(Math.)h(So)r(c,)i(v)n(ol.)e(75,)h(Amer.)e(Math.)h (So)r(c.,)h(Pro)n(vidence,)305 4147 y(R.I.,)d(1967.)p eop %%Page: 247 251 247 250 bop 118 100 a FK(Bibliograph)n(y)1865 b FP(247)153 333 y FE([80])42 b(G.)19 b(G.)g(Emc)n(h,)g Fi(A)n(lgebr)l(aic)j(metho)l (ds)h(in)f(statistic)l(al)g(me)l(chanics)g(and)h(quantum)f(\014eld)305 412 y(the)l(ory)p FE(,)h(Wiley{In)n(terscience,)j(1972.)153 491 y([81])42 b(J.)18 b(Ernest,)h Fi(Charting)h(the)h(op)l(er)l(ator)h (terr)l(ain)p FE(,)d(Mem.)e(Amer.)f(Math.)i(So)r(c.,)i(v)n(ol.)d(171,) 305 570 y(Amer.)k(Math.)j(So)r(c.,)g(Pro)n(vidence,)g(R.I.,)f(1976.)153 650 y([82])42 b(D.)31 b(B.)g(F)-6 b(airlie,)32 b Fi(Quantum)h (deformations)h(of)g Fq(S)t(U)7 b FE(\(2\),)33 b(J.)e(Ph)n(ys.)h(A:)f (Math.)g(and)305 729 y(Gen.)24 b Fh(23)f FE(\(1990\),)i(L183{L186.)153 809 y([83])42 b(T.)34 b(Finc)n(k,)39 b(S.)c(Ro)r(c)n(h,)j(and)e(B.)f (Silb)r(ermann,)h Fi(Two)h(pr)l(oje)l(ctions)h(the)l(or)l(ems)g(and)305 888 y(symb)l(ol)f(c)l(alculus)g(for)g(op)l(er)l(ators)h(with)f(massive) g(lo)l(c)l(al)h(sp)l(e)l(ctr)l(a)p FE(,)h(Math.)c(Nac)n(hr.)305 967 y Fh(162)22 b FE(\(1993\),)j(167{185.)153 1047 y([84])42 b(M.)23 b(Flato,)h(J.)f(Simon,)g(H.)g(Snellman,)g(and)h(D.)g (Sternheimer,)f Fi(Simple)j(facts)g(ab)l(out)305 1134 y(analytic)32 b(ve)l(ctors)g(and)h(inte)l(gr)l(ability)p FE(,)e(Ann.)g(Scien)n(t.)g(de)g(l')1952 1118 y(\023)1946 1134 y(Ecole)g(Norm.)d(Sup.)j Fh(5)305 1213 y FE(\(1972\),)25 b(423{434.)153 1293 y([85])42 b(M.)17 b(F)-6 b(ragoulopulou,)20 b Fi(A)n(n)h(intr)l(o)l(duction)h(to)e(the)h(r)l(epr)l(esentation)g (the)l(ory)g(of)g(top)l(olo)l(g-)305 1372 y(ic)l(al)i FD(\003)p Fi(-algebr)l(as)p FE(,)f(Sc)n(hriftenreihne)f(des)g(Math.)g (Inst.)g(der)f(Univ.)g(M)r(\177)-37 b(unster,)21 b(v)n(ol.)g(48,)305 1451 y(Univ.)i(M)r(\177)-37 b(unster,)23 b(1988.)153 1531 y([86])42 b(L.)31 b(G)-9 b(\027)-44 b(arding)32 b(and)g(A.)f(Wigh)n(tman,)i Fi(R)l(epr)l(esentations)h(of)f(the)g (antic)l(ommutation)305 1610 y(r)l(elations)p FE(,)24 b(Pro)r(c.)f(Nat.)h(Acad.)g(Sci.)f(USA)g Fh(40)g FE(\(1954\),)i(no.)f (9,)f(617{622.)153 1690 y([87])p 305 1690 212 4 v 265 w(,)k Fi(R)l(epr)l(esentations)j(of)e(the)g(c)l(ommutation)i(r)l (elations)p FE(,)e(Pro)r(c.)f(Nat.)f(Acad.)305 1769 y(Sci.)d(USA)g Fh(40)g FE(\(1954\),)i(no.)f(9,)f(623{626.)153 1849 y([88])42 b(P)-6 b(.)28 b(Gabriel)g(and)g(A.)g(V.)g(Roiter,)h Fi(R)l(epr)l (esentations)i(of)f(\014nite-dimensional)g(alge-)305 1927 y(br)l(as)p FE(,)24 b(Springer-V)-6 b(erlag,)22 b(Berlin,)g(1997.)153 2007 y([89])42 b(P)-6 b(.)31 b(Gabriel)h(and)h (M.)e(Zisman,)h Fi(Calculus)j(of)e(fr)l(actions)h(and)h(homotopy)g(the) l(ory)p FE(,)305 2086 y(Springer,)23 b(Berlin{Heidelb)r(erg{New)h(Y)-6 b(ork,)23 b(1967.)153 2166 y([90])42 b(D.)17 b(V.)h(Galinsky)-6 b(,)19 b Fi(R)l(epr)l(esentations)j(of)f FD(\003)p Fi(-algebr)l(as)g (gener)l(ate)l(d)g(by)f(ortho)l(gonal)j(pr)l(o-)305 2245 y(je)l(ctions)d(satisfying)g(a)h(line)l(ar)h(r)l(elation)p FE(,)e(Metho)r(ds)f(F)-6 b(unct.)19 b(Anal.)f(T)-6 b(op)r(ol.)18 b Fh(4)g FE(\(1998\),)305 2324 y(no.)23 b(3,)h(27{32.)153 2404 y([91])42 b(D.)20 b(V.)g(Galinsky)h(and)h(M.)e(A.)g(Murato)n(v,)i Fi(On)g(r)l(epr)l(esentations)j(of)e(algebr)l(as)h(gener-)305 2483 y(ate)l(d)i(by)f(sets)g(of)h(thr)l(e)l(e)g(and)h(four)f(orthopr)l (oje)l(ctions)p FE(,)g(Sp)r(ectral)e(and)g(ev)n(olutionary)305 2561 y(problems.)e(v)n(ol.)h(8,)g(T)-6 b(a)n(vria,)23 b(Simferop)r(ol,)f(1998,)i(pp.)g(15{22.)153 2641 y([92])42 b(O.)25 b(M.)h(Ga)n(vrilik)f(and)i(A.)f(U.)g(Klim)n(yk,)e Fi(R)l(epr)l(esentations)30 b(of)e(the)g Fq(q)r Fi(-deforme)l(d)h(al-) 305 2720 y(gebr)l(as)d Fq(U)569 2728 y Ff(q)603 2720 y FE(\()p Fq(so)697 2729 y Fc(2)p Ff(;)p Fc(1)781 2720 y FE(\))p Fi(,)g Fq(U)904 2728 y Ff(q)938 2720 y FE(\()p Fq(so)1032 2729 y Fc(3)p Ff(;)p Fc(1)1116 2720 y FE(\),)d(J.)h(Math.)f (Ph)n(ys.)h Fh(35)f FE(\(1994\),)i(no.)f(2,)f(3670{3686.)153 2800 y([93])42 b(I.)24 b(M.)f(Gel'fand)i(and)f(V.)g(A.)g(P)n(onomarev,) g Fi(R)l(emarks)j(on)f(classi\014c)l(ation)i(of)e(a)h(p)l(air)305 2879 y(of)22 b(c)l(ommuting)h(line)l(ar)g(tr)l(ansformations)i(in)d(a)g (\014nite-dimensional)h(sp)l(ac)l(e)p FE(,)f(F)-6 b(unct.)305 2958 y(Anal.)23 b(i)g(Prilozh.)g Fh(3)g FE(\(1969\),)i(no.)f(4,)f (81{82,)h(\(Russian\).)153 3038 y([94])p 305 3038 V 265 w(,)41 b Fi(Quadruples)g(of)e(subsp)l(ac)l(es)h(of)f(a)h (\014nite-dimensional)f(ve)l(ctor)g(sp)l(ac)l(e)p FE(,)305 3117 y(Dokl.)23 b(Ak)l(ad.)h(Nauk)g(SSSR)g Fh(197)e FE(\(1971\),)j(no.) f(4,)g(762{765,)h(\(Russian\).)153 3197 y([95])42 b(I.)18 b(M.)f(Gelfand)i(and)g(N.)f(Y)-6 b(a.)18 b(Vilenkin,)h Fi(Gener)l(alize)l(d)j(functions)p FE(,)d(v)n(ol.)f(4,)h(Academic)305 3276 y(Press,)30 b(New)h(Y)-6 b(ork,)31 b(1964,)h(T)-6 b(ransl.)29 b(from)g(Russian)g(edn.:)45 b(Fizmatgiz,)31 b(Mosco)n(w,)305 3354 y(1961.)153 3434 y([96])42 b(J.)24 b(Glimm,)d Fi(A)26 b(Stone{Weierstr)l(ass)h(the)l(or)l(em)h(for)e Fq(C)1757 3411 y Fg(\003)1793 3434 y Fi(-algebr)l(as)p FE(,)f(Ann.)f(Math.)g Fh(72)305 3513 y FE(\(1960\),)h(216{144.)153 3593 y([97])42 b(I.)24 b(Goh)n(b)r(erg,)i(P)-6 b(.)25 b(Lancaster,)h(and)f(L.)g(Ro)r(dman,)f Fi(Matric)l(es)i(and)i (inde\014nite)e(sc)l(alar)305 3672 y(pr)l(o)l(ducts)p FE(,)f(Op)r(er.)e(Theory)h(Adv.)f(Appl.,)g(v)n(ol.)g(8,)h(Birkhauser)f (V)-6 b(erlag,)23 b(1983.)153 3752 y([98])42 b(I.)24 b(Goh)n(b)r(erg)h(and)f(B.)g(Reic)n(hstein,)g Fi(On)i(classi\014c)l (ation)h(of)f(normal)h(matric)l(es)g(in)e(an)305 3831 y(inde\014nite)31 b(sc)l(alar)h(pr)l(o)l(duct)p FE(,)h(In)n(tegral)e (Equat.)f(Op)r(er.)f(Theory)j Fh(13)d FE(\(1990\),)k(365{)305 3910 y(394.)153 3990 y([99])42 b(G.)27 b(A.)f(Goldin,)i(R.)e(Menik)n (o\013,)i(and)g(D.)e(H.)h(Sharp,)h Fi(Particle)g(statistics)h(fr)l(om)g (in-)305 4068 y(duc)l(e)l(d)e(r)l(epr)l(esentations)h(of)f(a)f(lo)l(c)l (al)i(curr)l(ent)f(gr)l(oup)p FE(,)e(J.)g(Math.)f(Ph)n(ys.)g Fh(21)g FE(\(1980\),)305 4147 y(no.)f(4,)h(650{664.)p eop %%Page: 248 252 248 251 bop 118 100 a FP(248)1866 b FK(Bibliograph)n(y)118 333 y FE([100])42 b(V.)21 b(Y)-6 b(a.)22 b(Golo)r(dets,)h Fi(Classi\014c)l(ation)j(of)e(r)l(epr)l(esentations)i(of)e(the)g(antic) l(ommutation)305 412 y(r)l(elations)p FE(,)g(Russ.)f(Math.)g(Surv)n (eys)i Fh(24)e FE(\(1969\),)i(1{63.)118 496 y([101])42 b(K.)23 b(R.)g(Go)r(o)r(dearl)i(and)f(P)-6 b(.)24 b(Menal,)f Fi(F)-5 b(r)l(e)l(e)26 b(and)h(r)l(esidual)t(ly)g(\014nite-dimensional) f Fq(C)2513 472 y Fg(\003)2549 496 y Fi(-)305 575 y(algebr)l(as)p FE(,)e(J.)f(F)-6 b(unct.)25 b(Anal.)e Fh(9)g FE(\(1990\),)i(no.)f(2,)f (391{410.)118 659 y([102])42 b(R.)29 b(W.)g(Go)r(o)r(dman,)i Fi(A)n(nalytic)g(and)h(entir)l(e)e(ve)l(ctors)i(for)f(r)l(epr)l (esentations)h(of)f(Lie)305 738 y(gr)l(oups)p FE(,)24 b(T)-6 b(rans.)23 b(Amer.)f(Math.)i(So)r(c.)g Fh(143)e FE(\(1969\),)j(55{76.)118 822 y([103])42 b(M.)36 b(F.)h(Goro)r(dni)-8 b(\025)-27 b(\020)37 b(and)h(G.)f(B.)f(P)n(o)r(dk)n(olzin,)41 b Fi(Irr)l(e)l(ducible)f(r)l(epr)l(esentations)f(of)f(a)305 901 y(gr)l(ade)l(d)25 b(Lie)d(algebr)l(a)p FE(,)h(Sp)r(ectral)f(Theory) f(of)g(Op)r(erators)h(and)g(In\014nite-dimensional)305 980 y(Analysis,)17 b(Inst.)h(Math.)f(Acad.)h(Sci.)f(UkrSSR,)f(Kiev,)j (1984,)g(pp.)e(66{76,)i(\(Russian\).)118 1064 y([104])42 b(P)-6 b(.)23 b(Halmos,)e Fi(A)k(Hilb)l(ert)g(sp)l(ac)l(e)i(pr)l(oblem) f(b)l(o)l(ok)p FE(,)e(V)-6 b(an)24 b(Nostrand,)f(Princeton,)h(1967.)118 1148 y([105])p 305 1148 212 4 v 265 w(,)f Fi(Two)k(subsp)l(ac)l(es)p FE(,)d(T)-6 b(rans.)23 b(Amer.)f(Math.)i(So)r(c.)g Fh(144)e FE(\(1969\),)j(381{389.)118 1233 y([106])42 b(P)-6 b(.)25 b(R.)h(Halmos)e(and)j(J.)e(E.)h(McLaughlin,)g Fi(Partial)j(isometries)p FE(,)d(P)n(aci\014c)h(J.)f(Math.)305 1311 y Fh(13)d FE(\(1963\),)i (585{596.)118 1396 y([107])42 b(P)-6 b(.)28 b(de)g(la)g(Harp)r(e,)h Fi(Op)l(er)l(ator)j(algebr)l(as,)f(fr)l(e)l(e)f(gr)l(oups)i(and)e (other)h(gr)l(oups)p FE(,)f(Recen)n(t)305 1475 y(adv)l(ances)i(in)f(op) r(erator)h(algebras,)g(Orl)n(\023)-33 b(eans,)32 b(1992,)h(Asterisque,) g(v)n(ol.)d(232,)j(So)r(c.)305 1553 y(Math.)23 b(F)-6 b(rance,)24 b(1995,)h(pp.)e(121{153.)118 1638 y([108])p 305 1638 V 265 w(,)40 b Fi(T)-5 b(opics)38 b(on)h(ge)l(ometric)f(gr)l (oup)h(the)l(ory)p FE(,)h(Preliminary)35 b(v)n(ersion,)40 b(1998,)305 1717 y(h)n(ttp://www.unige.c)n(h/math/biblio/preprin)n (t/1998/geogroup/.)118 1801 y([109])i(M.)25 b(Ha)n(vli)n(\024)-33 b(cek,)26 b(A.)g(U.)f(Klim)n(yk,)g(and)h(E.)g(P)n(elan)n(to)n(v\023)-35 b(a,)28 b Fi(Nonstandar)l(d)i Fq(U)2244 1809 y Ff(q)2278 1801 y FE(\()p Fq(so)2372 1810 y Fc(3)2407 1801 y FE(\))e Fi(and)305 1880 y Fq(U)353 1888 y Ff(q)387 1880 y FE(\()p Fq(so)481 1889 y Fc(4)516 1880 y FE(\))p Fi(:)52 b(tensor)35 b(pr)l(o)l(ducts)i(of)e(r)l(epr)l(esentations,)k(oscil)t(lator)d(r)l(e) l(alizations)h(and)305 1959 y(r)l(o)l(ots)27 b(of)e(unity)p FE(,)e(Czec)n(h)i(J.)e(Ph)n(ys.)h Fh(47)f FE(\(1997\),)i(no.)e(1,)h (13{16.)118 2043 y([110])42 b(M.)19 b(Ha)n(vli)n(\024)-33 b(cek,)21 b(A.)e(U.)g(Klim)n(yk,)g(and)i(S.)e(P)n(o)l(\024)-32 b(sta,)22 b Fi(R)l(epr)l(esentations)i(of)e(the)g(cyclic)l(al)t(ly)305 2122 y(symmetric)32 b Fq(q)r Fi(-deforme)l(d)i(algebr)l(a)g Fq(so)1349 2130 y Ff(q)1383 2122 y FE(\(3\),)g(J.)d(Math.)h(Ph)n(ys.)f Fh(40)g FE(\(1999\),)k(no.)d(4,)305 2201 y(1365{1382.)118 2285 y([111])42 b(A.)23 b(Heb)r(ec)n(k)n(er,)i(S.)e(Sc)n(hrec)n(k)n(en) n(b)r(erg,)j(J.)e(Sc)n(h)n(w)n(enk,)h(W.)f(W)-6 b(eic)n(h,)24 b(and)h(J.)f(W)-6 b(ess,)24 b Fi(R)l(ep-)305 2364 y(r)l(esentations)32 b(of)h(a)f Fq(q)r Fi(-deforme)l(d)h(Heisenb)l(er)l(g)f(algebr)l(a)p FE(,)h(Z.)d(Ph)n(ys.)h(C)f Fh(64)g FE(\(1994\),)305 2443 y(355{359.)118 2527 y([112])42 b(G.)21 b(C.)f(Hegerfeldt)i(and)f(O.)g (Melsheimer,)e Fi(The)24 b(form)g(of)f(r)l(epr)l(esentations)h(of)g (CCR)305 2606 y(for)c(Bose)h(\014elds)h(and)f(c)l(onne)l(ction)g(with)g (\014nitely)f(many)h(de)l(gr)l(e)l(es)g(of)g(fr)l(e)l(e)l(dom)p FE(,)f(Com-)305 2685 y(m)n(un.)i(Math.)i(Ph)n(ys.)f Fh(12)g FE(\(1969\),)i(no.)f(4,)f(304{323.)118 2769 y([113])42 b(I.)35 b(Hernstein,)j Fi(Nonc)l(ommutative)f(rings)p FE(,)g(Math.)e(Asso)r(c.)g(Amer.,)h(Wiley)-6 b(,)38 b(New)305 2848 y(Y)-6 b(ork,)23 b(1968.)118 2932 y([114])42 b(A.)19 b(S.)h(Holev)n(o,)h Fi(Pr)l(ob)l(abilistic)j(and)f(staistic)l(al)g(asp) l(e)l(cts)h(of)e(quantum)i(the)l(ory)p FE(,)d(North)305 3011 y(Holand,)32 b(Amsterdam,)d(1982,)k(T)-6 b(ransl.)30 b(from)e(Russian)i(edn.:)45 b(Nauk)l(a,)33 b(Mosco)n(w,)305 3090 y(1980.)118 3174 y([115])42 b(Kh.)33 b(D.)g(Ikramo)n(v,)j Fi(On)e(a)i(c)l(anonic)l(al)g(form)g(of)f(pr)l(oje)l(ctions)h(with)g(r) l(esp)l(e)l(ct)f(to)g(a)305 3253 y(unitary)25 b(similarity)p FE(,)d(Zh)n(urn.)h(Vyc)n(hisl.)g(Mat.)g(i)f(Matem.)g(Fiziki)h Fh(36)f FE(\(1980\),)j(no.)e(1,)305 3332 y(3{5,)h(\(Russian\).)118 3416 y([116])42 b(A.)24 b(Inoue,)i Fi(L)l(o)l(c)l(al)t(ly)i Fq(C)913 3392 y Fg(\003)949 3416 y Fi(-algebr)l(as)p FE(,)d(Mem.)e(F)-6 b(acult)n(y)27 b(Sci.)d(Kyush)n(u)i(Univ.)e(\(Ser.)h (A.\))305 3495 y Fh(25)e FE(\(1971\),)i(197{235.)118 3579 y([117])42 b(R.)31 b(S.)g(Ismagilo)n(v,)h Fi(R)l(epr)l (esentations)i(of)g(in\014nite-dimensional)f(gr)l(oups)p FE(,)i(T)-6 b(ransl.)305 3658 y(Math.)23 b(Monogr.,)g(v)n(ol.)h(152,)g (Amer.)d(Math.)j(So)r(c.,)f(Pro)n(vidence,)i(R.I.,)d(1996.)118 3742 y([118])42 b(N.)f(Jacobson,)48 b Fi(Structur)l(e)43 b(of)g(rings)p FE(,)k(Amer.)40 b(Math.)i(So)r(c.)h(Coll.)e(Publ.,)46 b(v)n(ol.)305 3821 y(XXXVI)r(I,)23 b(Amer.)f(Math.)h(So)r(c.,)h(Pro)n (vidence,)g(R.I.,)f(1956.)118 3905 y([119])42 b(A.)25 b(Jan)n(tzen,)k Fi(L)l(e)l(ctur)l(es)f(on)g(quantum)h(gr)l(oups)p FE(,)f(Amer.)c(Math.)i(So)r(c.,)h(Pro)n(vidence,)305 3984 y(R.I.,)22 b(1996.)118 4068 y([120])42 b(M.)17 b(Jim)n(b)r(o,)h Fi(A)j Fq(q)r Fi(-di\013er)l(enc)l(e)f(analo)l(gue)i(of)f Fq(U)7 b FE(\()p Fh(g)q FE(\))21 b Fi(and)h(the)e(Yang{Baxter)h(e)l (quation)p FE(,)305 4147 y(Lett.)j(Math.)g(Ph)n(ys.)f Fh(10)g FE(\(1985\),)i(no.)f(1,)f(63{69.)p eop %%Page: 249 253 249 252 bop 118 100 a FK(Bibliograph)n(y)1865 b FP(249)118 333 y FE([121])42 b(V.)20 b(Jones)h(and)g(V.)f(S.)h(Sunder,)g Fi(Intr)l(o)l(duction)k(to)e(subfactors)p FE(,)e(London)h(Math.)e(So)r (c.)305 412 y(Lect.)k(Note.)g(Ser.,)f(v)n(ol.)g(234,)h(Cam)n(bridge)f (Univ.)g(Press,)g(Cam)n(bridge,)f(1994.)118 496 y([122])42 b(C.)18 b(Jordan,)i Fi(Essai)i(sur)g(la)f(ge)l(ometrie)g(\022)-36 b(a)22 b Fq(n)f Fi(dimensions)p FE(,)f(Bull.)e(So)r(c.)h(Math.)g(F)-6 b(rance)305 574 y Fh(3)23 b FE(\(1875\),)i(103{174.)118 658 y([123])42 b(P)-6 b(.)28 b(E.)g(T.)g(J\034rgensen,)j Fi(Op)l(er)l(ators)h(and)f(r)l(epr)l(esentation)g(the)l(ory)p FE(,)f(North-Holland)305 737 y(\(Elsevier\),)23 b(Amsterdam,)e(1988.) 118 821 y([124])42 b(P)-6 b(.)23 b(E.)g(T.)g(J\034rgensen)h(and)h(R.)d (T.)h(Mo)r(ore,)h Fi(Op)l(er)l(ator)i(c)l(ommutation)h(r)l(elations)p FE(,)d(D.)305 900 y(Reidel)f(Publ.)g(Comp.,)f(Dordrec)n(h)n(t,)i(1984.) 118 984 y([125])42 b(P)-6 b(.E.T.)38 b(J\034rgensen,)44 b(D.P)-6 b(.)38 b(Proskurin,)k(and)e(Y)-6 b(u.S.)39 b(Samoilenk)n(o,)j Fi(Gener)l(alize)l(d)305 1063 y(c)l(anonic)l(al)d(c)l(ommutation)h(r)l (elations:)59 b(r)l(epr)l(esentations)39 b(and)g(stability)e(of)h(uni-) 305 1142 y(versal)33 b(enveloping)g Fq(C)929 1118 y Fg(\003)965 1142 y Fi(-algebr)l(a)p FE(,)g(Pro)r(c)f(Ins.)f(Math.)h(NAS)f(Ukraine)g Fh(43)g FE(\(2002\),)305 1221 y(456{460.)118 1305 y([126])42 b(P)-6 b(.)26 b(E.)h(T.)f(J\034rgensen,)j(L.)d(M.)g(Sc)n(hmitt,)i(and)f (R.)f(F.)h(W)-6 b(erner,)28 b Fq(q)r Fi(-Canonic)l(al)h(c)l(om-)305 1384 y(mutation)23 b(r)l(elations)g(and)g(stability)e(of)i(the)f(Cuntz) g(algebr)l(a)p FE(,)f(P)n(aci\014c)g(J.)f(Math.)g Fh(165)305 1462 y FE(\(1994\),)25 b(131{151.)118 1546 y([127])p 305 1546 212 4 v 265 w(,)19 b Fi(Positive)h(r)l(epr)l(esentation)h(of)g (gener)l(al)g(c)l(ommutation)g(r)l(elations)h(al)t(lowing)305 1625 y(Wick)j(or)l(dering)p FE(,)f(J.)f(F)-6 b(unct.)25 b(Anal.)e Fh(134)f FE(\(1995\),)j(33{99.)118 1709 y([128])42 b(P)-6 b(.)25 b(E.)h(T.)g(J\034rgensen)h(and)g(R.)e(F.)h(W)-6 b(erner,)27 b Fi(Coher)l(ent)h(states)g(of)g(the)g Fq(q)r Fi(-c)l(anonic)l(al)305 1788 y(c)l(ommutation)f(r)l(elations)p FE(,)d(Comm)n(un.)d(Math.)j(Ph)n(ys.)g Fh(164)e FE(\(1994\),)j (455{471.)118 1872 y([129])42 b(V.)34 b(Kac,)j Fi(R)l(o)l(ot)h (systems,)g(r)l(epr)l(esentations)f(of)f(gr)l(aphs)i(and)f(invariant)f (the)l(ory)p FE(,)305 1951 y(Lect.)24 b(Notes)g(Math.,)g(v)n(ol.)f (996,)h(pp.)f(74{108,)i(Springer,)e(Berlin,)g(1983.)118 2035 y([130])42 b(R.)20 b(V.)g(Kadison)h(and)h(J.)e(R.)g(Ringrose,)h Fi(F)-5 b(undamentals)25 b(of)e(the)g(the)l(ory)h(of)f(op)l(er)l(ator) 305 2114 y(algebr)l(as,)j(I,)g(II)p FE(,)e(Acad.)g(Press,)e(1983,)j (1986.)118 2198 y([131])42 b(S.)22 b(A.)g(Kalutsky)i(and)f(Y)-6 b(u.)22 b(S.)h(Samo)-8 b(\025)-27 b(\020lenk)n(o,)22 b Fi(Perio)l(dic)k(gr)l(oups)g(ar)l(e)f(not)g(wild)p FE(,)f(Ukr.)305 2277 y(Mat.)f(Zh.)h Fh(49)e FE(\(1997\),)k(no.)d(5,)h (729{730,)h(\(Russian\).)118 2361 y([132])42 b(D.)27 b(Kazhdan,)i Fi(Conne)l(ction)h(of)f(the)g(dual)i(sp)l(ac)l(e)f(of)g(a) g(gr)l(oup)g(with)g(the)f(structur)l(e)305 2440 y(of)c(its)h(close)l(d) g(sub)l(gr)l(oups)p FE(,)g(F)-6 b(unct.)24 b(Anal.)f(Appl.)g Fh(1)g FE(\(1957\),)i(63{65.)118 2524 y([133])42 b(A.)31 b(Y)-6 b(a.)32 b(Khelemski)-8 b(\025)-27 b(\020,)33 b Fi(Banach)i(algebr)l(as)f(and)h(p)l(oly-norme)l(d)g(algebr)l(as:)50 b(gener)l(al)305 2603 y(the)l(ory,)25 b(r)l(epr)l(esentations,)i (homolo)l(gies)p FE(,)e(Nauk)l(a,)f(Mosco)n(w,)g(1989,)g(\(Russian\).) 118 2687 y([134])42 b(A.)22 b(A.)g(Kirillo)n(v,)f Fi(Dynamic)l(al)26 b(systems,)e(factors)i(and)g(r)l(epr)l(esentations)g(of)f(gr)l(oups)p FE(,)305 2765 y(Usp)r(ekhi)e(Mat.)h(Nauk)g Fh(22)f FE(\(1967\),)i(no.)f (5,)f(67{80,)i(\(Russian\).)118 2849 y([135])p 305 2849 V 265 w(,)d Fi(Elements)j(of)f(the)g(the)l(ory)h(of)f(r)l(epr)l (esentations)p FE(,)f(Springer,)f(Berlin,)f(1970.)118 2933 y([136])42 b(E.)24 b(Kissin)g(and)i(V.)e(Sh)n(ul'man,)g Fi(R)l(epr)l(esentations)k(on)f(Kr)l(ein)g(sp)l(ac)l(es)h(and)g (deriva-)305 3012 y(tions)c(of)h Fq(C)618 2989 y Fg(\003)653 3012 y Fi(-algebr)l(as)p FE(,)e(Pitman)e(Monographs)i(and)g(Surv.)f (Pure)g(Applied)g(Math.,)305 3091 y(v)n(ol.)h(89,)h(Addison)f(W)-6 b(esley)g(,)25 b(Longman,)e(1997.)118 3175 y([137])42 b(D.)18 b(Kleinec)n(k)n(e,)i Fi(On)g(op)l(er)l(ator)j(c)l(ommutators)p FE(,)e(Pro)r(c.)e(Amer.)d(Math.)j(So)r(c.)g Fh(8)f FE(\(1957\),)305 3254 y(535{536.)118 3338 y([138])42 b(S.)31 b(Klimek)e(and)j(A.)f (Lesniewski,)i Fi(Quantum)g(Riemann)h(surfac)l(es.)f(I.)g(The)g(unit) 305 3417 y(disc)p FE(,)23 b(Comm)n(un.)e(Math.)j(Ph)n(ys.)g Fh(146)e FE(\(1992\),)j(103{122.)118 3501 y([139])p 305 3501 V 265 w(,)30 b Fi(A)h(two-p)l(ar)l(ameter)i(quantum)e(deformation) i(of)e(the)g(unit)f(disc)p FE(,)h(Journ.)305 3580 y(F)-6 b(unct.)24 b(Anal.)f Fh(115)g FE(\(1993\),)i(no.)f(1,)f(1{23.)118 3664 y([140])42 b(A.)30 b(U.)f(Klim)n(yk)g(and)i(K.)f(Sc)n(hm)r(\177) -37 b(udgen,)33 b Fi(Quantum)f(gr)l(oups)i(and)f(their)e(r)l(epr)l (esen-)305 3743 y(tations)p FE(,)f(T)-6 b(exts)30 b(and)f(Monographs)h (in)e(Ph)n(ysics,)i(Springer,)g(Berlin,)f(Heidelb)r(erg,)305 3822 y(1997.)118 3906 y([141])42 b(H.)21 b(T.)g(Ko)r(elink,)h Fi(On)h FD(\003)p Fi(-r)l(epr)l(esentations)i(of)f(the)g(Hopf)g FD(\003)p Fi(-algebr)l(a)h(asso)l(ciate)l(d)g(with)305 3984 y(the)g(quantum)h(gr)l(oup)h Fq(U)960 3992 y Ff(q)994 3984 y FE(\()p Fq(N)7 b FE(\),)24 b(Comp)r(ositio)f(Math.)h Fh(77)f FE(\(1991\),)i(199{231.)118 4068 y([142])42 b(T.)25 b(H.)g(Ko)r(orn)n(winder)g(and)i(R.)e(F.)g(Sw)n(arttou)n(w,)i Fi(On)g Fq(q)r Fi(-analo)l(gues)i(of)f(the)f(Fourier)305 4147 y(and)f(Hankel)g(tr)l(ansforms)p FE(,)f(T)-6 b(rans.)23 b(Amer.)e(Math.)j(So)r(c.)g Fh(333)e FE(\(1992\),)k(445{461.)p eop %%Page: 250 254 250 253 bop 118 100 a FP(250)1866 b FK(Bibliograph)n(y)118 333 y FE([143])42 b(L.)24 b(I.)h(Korogo)r(dski)g(and)h(Y.)e(S.)h(Soib)r (elman,)f Fi(A)n(lgebr)l(as)k(of)f(functions)g(on)g(quantum)305 412 y(gr)l(oups.)g(Part)f(1)p FE(,)d(Amer.)f(Math.)h(So)r(c.,)h(Pro)n (vidence,)g(R.I.,)f(1998.)118 496 y([144])42 b(S.)17 b(A.)f(Krugly)n(ak,)j Fi(R)l(epr)l(esentations)i(of)f(fr)l(e)l(e)g (involutive)f(quivers)p FE(,)f(Represen)n(tations)305 575 y(and)g(quadratic)g(forms,)e(Inst.)i(Math.)f(Acad.)h(Sci.)f(Ukr.)f (SSR,)h(Kiev,)h(1979,)h(pp.)f(149{)305 654 y(151,)24 b(\(Russian\).)118 738 y([145])p 305 738 212 4 v 265 w(,)44 b Fi(R)l(epr)l(esentations)e(of)f(involutive)f(quivers)p FE(,)j(VINITI)e(7266-84,)k(1984,)305 817 y(\(Russian\).)118 901 y([146])d(S.)28 b(A.)g(Krugly)n(ak)h(and)g(A.)f(Y)-6 b(u.)29 b(Piry)n(atinsk)l(a)n(y)n(a,)h Fi(On)g(\\wild")h FD(\003)p Fi(-algebr)l(as)g(and)g(the)305 980 y(unitary)d(classi\014c)l (ation)i(of)e(we)l(akly)h(c)l(enter)l(e)l(d)g(op)l(er)l(ators)p FE(,)g(Prepr.)d(ser.)g(of)h(Mittag-)305 1059 y(Le\017er)c(Inst.)h(no.)g (11,)g(1995/96.)118 1143 y([147])42 b(S.)37 b(Krugljak,)i(S.)e(P)n(op)r (o)n(vyc)n(h,)42 b(and)c(Y)-6 b(u.)37 b(Samoilenk)n(o,)j Fi(R)l(epr)l(esentations)g(of)e FD(\003)p Fi(-)305 1222 y(algebr)l(as)j(asso)l(ciate)l(d)i(with)e(Dynkin)f(gr)l(aphs)j(and)e (Horn)-7 b('s)41 b(pr)l(oblem)p FE(,)k(Pro)r(c.)40 b(of)305 1301 y(T)-6 b(a)n(vric)n(h.)23 b(Univ.)g(\(2004\),)i(to)g(app)r(ear.) 118 1385 y([148])42 b(S.A.)20 b(Krugljak,)h(V.I.)f(Rabano)n(vic)n(h,)j (and)e(Y)-6 b(u.S.)21 b(Samoilenk)n(o,)g Fi(On)i(sum)h(of)f(pr)l(oje)l (c-)305 1464 y(tions)p FE(,)g(F)-6 b(unct.)25 b(Anal.)e(Prilozh.)f Fh(36)h FE(\(2002\),)i(no.)f(3,)f(20{35.)118 1548 y([149])p 305 1548 V 265 w(,)18 b Fi(De)l(c)l(omp)l(osition)j(of)e(a)h(sc)l(alar) g(matrix)f(into)g(a)h(sum)g(of)f(pr)l(oje)l(ctions)p FE(,)g(Linear)305 1627 y(Algebra)k(Appl.)g Fh(370)g FE(\(2003\),)i (217{225.)118 1711 y([150])42 b(S.)25 b(A.)h(Krugly)n(ak)g(and)g(Y)-6 b(u.)26 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)26 b Fi(On)h(unitary)h(e)l(quivalenc)l(e)g(of)g(c)l(ol)t(le)l(c-)305 1790 y(tions)e(of)h(self-adjoint)f(op)l(er)l(ators)p FE(,)g(F)-6 b(unct.)25 b(Anal.)f(i)g(Prilozhen.)g Fh(14)g FE(\(1980\),)i(no.)e(1,)305 1869 y(60{62,)g(\(Russian\).)118 1953 y([151])p 305 1953 V 265 w(,)e Fi(Structur)l(e)h(the)l(or)l(ems)i (for)f(families)g(of)g(idemp)l(otents)p FE(,)e(Ukr.)f(Mat.)g(Zh)n(urn.) 305 2032 y Fh(50)i FE(\(1998\),)i(no.)e(4,)h(523{533,)h(\(Russian\).) 118 2116 y([152])p 305 2116 V 265 w(,)19 b Fi(On)i(c)l(omplexity)g(of)g (description)h(of)f(r)l(epr)l(esentations)h(of)f FD(\003)p Fi(-algebr)l(as)h(gen-)305 2195 y(er)l(ate)l(d)k(by)f(idemp)l(otents)p FE(,)f(Pro)r(c.)f(Amer.)f(Math.)i(So)r(c.)g Fh(128)e FE(\(2000\),)j(no.)f(1.)118 2279 y([153])42 b(N.)22 b(Krupnik,)g Fi(Banach)j(algebr)l(as)h(with)f(symb)l(ol)g(and)h(singular)f(inte)l (gr)l(al)g(op)l(er)l(ators)p FE(,)305 2358 y(Op)r(er.)e(Theory)h(Adv.)f (Appl.,)g(v)n(ol.)g(90,)h(Birkh\177)-35 b(auser)23 b(V)-6 b(erlag,)23 b(Basel,)g(1987.)118 2443 y([154])42 b(N.)30 b(Krupnik,)i(S.)e(Ro)r(c)n(h,)j(and)f(B.)e(Silb)r(ermann,)h Fi(On)h Fq(C)1846 2419 y Fg(\003)1882 2443 y Fi(-algebr)l(as)h(gener)l (ate)l(d)f(by)305 2521 y(idemp)l(otents)p FE(,)24 b(J.)f(F)-6 b(unc.)24 b(Anal.)f Fh(137)g FE(\(1996\),)i(303{319.)118 2606 y([155])42 b(N.)21 b(Krupnik)h(and)h(E.)e(Spigel,)h Fi(Invertibility)h(symb)l(ols)i(for)g(a)f(Banach)i(algebr)l(a)f(gen-) 305 2685 y(er)l(ate)l(d)c(by)e(two)i(idemp)l(otents)g(and)h(a)e(shift)p FE(,)f(In)n(t.)g(Equat.)f(Op)r(er.)g(Theory)h Fh(17)e FE(\(1993\),)305 2763 y(567{578.)118 2848 y([156])42 b(P)-6 b(.)15 b(Kruszy)r(\023)-37 b(nski)16 b(and)g(S.)g(L.)f(W)-6 b(orono)n(wicz,)19 b Fi(A)f(nonc)l(ommutative)i(Gelfand{Naimark)305 2927 y(the)l(or)l(em)p FE(,)k(J.)f(Op)r(er.)g(Theory)j Fh(8)d FE(\(1982\),)i(361{389.)118 3011 y([157])42 b(P)-6 b(.)34 b(P)-6 b(.)34 b(Kulish,)h Fi(Contr)l(action)i(of)f(quantum)g (algebr)l(as)g(and)h Fq(q)r Fi(-oscil)t(lators)p FE(,)g(T)-6 b(eor.)305 3090 y(Math.)23 b(Ph)n(ys.)h Fh(86)f FE(\(1991\),)i (108{110.)118 3174 y([158])42 b(P)-6 b(.)21 b(P)-6 b(.)21 b(Kulish)f(and)i(N.)f(Y)-6 b(u.)21 b(Resh)n(tikhin,)h Fi(Quantum)i(line)l(ar)g(pr)l(oblem)h(for)f(the)f(sine-)305 3253 y(Gor)l(don)f(e)l(quation)g(and)g(higher)f(r)l(epr)l(esentations)p FE(,)f(Zap.)f(Nauc)n(h.)g(Sem.)e(LOMI)i Fh(101)305 3332 y FE(\(1981\),)25 b(101{110,)g(\(Russian\).)118 3416 y([159])42 b(M.)26 b(Laca,)i Fi(Endomorphisms)k(of)d Fe(B)p FE(\()p Fe(H)q FE(\))f Fi(and)h(Cuntz)g(algebr)l(as)p FE(,)f(J.)f(Op)r(er.)f(Theory)305 3495 y Fh(30)d FE(\(1993\),)i (85{101.)118 3579 y([160])42 b(E.)16 b(C.)h(Lance,)i Fi(Hilb)l(ert)g Fq(C)979 3556 y Fg(\003)1015 3579 y Fi(-mo)l(dules:)31 b(a)20 b(to)l(olkit)g(for)f(op)l(er)l(ator)j(algebr)l(aists)p FE(,)d(London)305 3658 y(Math.)k(So)r(c.)h(Lect.)h(Notes)f(Ser.,)f(v)n (ol.)g(210,)h(CUP)-6 b(,)23 b(1995.)118 3742 y([161])p 305 3742 V 265 w(,)28 b Fi(Finitely)g(pr)l(esente)l(d)i Fq(C)1215 3719 y Fg(\003)1251 3742 y Fi(-algebr)l(as)p FE(,)f(Op)r(erator)e(Algebras)g(and)h(Applica-)305 3821 y(tions)e(\(A.)f(Kata)n(v)n(olos,)i(ed.\),)f(Nato)g(ASI)g(Series,)g (Ser.)f(C:)g(Math.)h(and)g(Ph)n(ys.)f(Sci.,)305 3900 y(v)n(ol.)e(495,)h(Klu)n(w)n(er)f(Acad.)h(Publ.,)f(1997,)h(pp.)f (255{266.)118 3984 y([162])42 b(T.)29 b(A.)h(Loring,)h Fq(C)815 3961 y Fg(\003)851 3984 y Fi(-algebr)l(as)h(gener)l(ate)l(d)g (by)f(stable)h(r)l(elations)p FE(,)g(J.)e(F)-6 b(unct.)31 b(Anal.)305 4063 y Fh(112)22 b FE(\(1993\),)j(no.)f(1,)f(159{203.)118 4147 y([163])42 b(G.)23 b(Lusztig,)h Fi(Intr)l(o)l(duction)k(to)d (quantum)h(gr)l(oups)p FE(,)f(Birkh\177)-35 b(auser,)23 b(Boston,)h(1993.)p eop %%Page: 251 255 251 254 bop 118 100 a FK(Bibliograph)n(y)1865 b FP(251)118 333 y FE([164])42 b(A.)19 b(J.)h(Macfarlane,)h Fi(On)h Fq(q)r Fi(-analo)l(gues)i(of)f(the)f(quantum)i(harmonic)f(oscil)t (lator)h(and)305 412 y(the)h(quantum)h(gr)l(oup)h Fq(su)p FE(\(2\),)d(J.)f(Ph)n(ys.)h(A)f Fh(22)g FE(\(1989\),)i(4581{4586.)118 492 y([165])42 b(G.)17 b(W.)g(Mac)n(k)n(ey)-6 b(,)19 b Fi(Imprimitivity)g(for)h(r)l(epr)l(esentations)g(of)g(lo)l(c)l(al)t (ly)h(c)l(omp)l(act)g(gr)l(oups)p FE(,)305 571 y(Pro)r(c.)i(Nat.)h (Acad.)f(Sci.)h(USA)f Fh(35)g FE(\(1949\),)i(no.)f(9,)f(537{545.)118 651 y([166])p 305 651 212 4 v 265 w(,)d Fi(Induc)l(e)l(d)j(r)l(epr)l (esentations)g(of)e(lo)l(c)l(al)t(ly)i(c)l(omp)l(act)g(gr)l(oups)p FE(,)e(Ann.)e(Math.)g Fh(55)305 730 y FE(\(1952\),)25 b(no.)e(1,)h(101{139.)118 810 y([167])42 b(T.)26 b(Maistrenk)n(o,)i(S.) f(P)n(op)r(o)n(vyc)n(h,)i(and)f(Y)-6 b(u.)26 b(Samoilenk)n(o,)i Fi(On)g(the)g(algebr)l(a)i(of)f(un-)305 889 y(harmonic)f(quantum)g (oscil)t(lator)p FE(,)e(Pro)r(c.)e(of)h(Inst.)h(Of)e(Math.)h(of)g(NAS)g (of)g(Ukraine)305 968 y(\(2004\),)g(to)f(app)r(ear.)118 1048 y([168])42 b(S.)25 b(Ma)t(jid,)f Fi(F)-5 b(oundations)29 b(of)f(quantum)f(gr)l(oup)i(the)l(ory)p FE(,)c(Cam)n(bridge)g(Univ.)g (Press,)305 1127 y(Cam)n(bridge,)d(1995.)118 1207 y([169])42 b(Y)-6 b(u.)16 b(I.)g(Manin,)h Fi(T)-5 b(opics)19 b(in)g(non-c)l (ommutative)h(ge)l(ometry)p FE(,)d(Princeton)g(Univ.)f(Press,)305 1286 y(Princeton,)24 b(N.J.,)e(1991.)118 1366 y([170])42 b(T)18 b(Masuda,)h(K.)e(Mimac)n(hi,)g(Y.)g(Mak)l(agami,)h(M.)f(Noumi,)h (Y.)f(Saburi,)i(and)f(K.)f(Ueno,)305 1445 y Fi(Unitary)h(r)l(epr)l (esentations)i(of)f(the)f(quantum)i(gr)l(oup)g Fq(S)t(U)1809 1453 y Ff(q)1842 1445 y FE(\(1)p Fq(;)13 b FE(1\),)18 b(Lett.)f(Math.)f(Ph)n(ys.)305 1524 y Fh(19)23 b FE(\(1990\),)i(no.)e (3,)h(187{204.)118 1604 y([171])42 b(K.)26 b(McClanahan,)j Fq(C)899 1580 y Fg(\003)935 1604 y Fi(-algebr)l(as)g(gener)l(ate)l(d)g (by)f(elements)h(of)g(a)g(unitary)g(matrix)p FE(,)305 1683 y(J.)23 b(F)-6 b(unct.)25 b(Anal.)e Fh(107)f FE(\(1992\),)j(no.)f (2,)f(439{457.)118 1763 y([172])42 b(S.)29 b(A.)g(McCullough)h(and)h (L.)e(Ro)r(dman,)i Fi(Two)h(self-adjoint)f(op)l(er)l(ators)j(in)d(Kr)l (ein)305 1842 y(sp)l(ac)l(es)p FE(,)24 b(In)n(t.)h(Equat.)f(Op)r(er.)f (Theory)i Fh(26)e FE(\(1996\),)i(202{209.)118 1922 y([173])42 b(A.S.)27 b(Mellit,)i Fi(When)h(a)g(sum)h(of)f(thr)l(e)l(e)h(p)l (artial)g(r)l(e\015e)l(ctions)f(is)g(zer)l(o)p FE(,)g(Ukr.)e(Math.)305 2001 y(Zh.)23 b Fh(55)g FE(\(2003\),)i(no.)f(9.)118 2081 y([174])42 b(A.S.)15 b(Mellit,)i(V.I.)e(Rabano)n(vic)n(h,)k(and)e(Y)-6 b(u.S.)16 b(Samoilenk)n(o,)h Fi(When)j(a)f(sum)g(of)g(p)l(artial)305 2160 y(r)l(e\015e)l(ctions)25 b(is)h(a)g(sc)l(alar)h(op)l(er)l(ator)p FE(,)e(F)-6 b(unct.)25 b(Anal.)e(Prilozh.)f Fh(37)h FE(\(2003\),)i(no.) f(4.)118 2240 y([175])42 b(R.)25 b(Menik)n(o\013)h(and)g(D.)f(H.)g (Sharp,)h Fi(R)l(epr)l(esentations)j(of)e(a)h(lo)l(c)l(al)i(curr)l(ent) d(algebr)l(a:)305 2319 y(their)i(dynamic)l(al)j(determination)p FE(,)e(J.)e(Math.)g(Ph)n(ys.)g Fh(16)g FE(\(1975\),)j(no.)e(12,)g (2341{)305 2398 y(2360.)118 2478 y([176])42 b(M.)31 b(Misiurewicz,)i Fi(A)n(bsolutely)h(c)l(oninuous)h(me)l(asur)l(es)g(for)f(c)l(ertain)g (maps)g(of)g(an)305 2557 y(interval)p FE(,)23 b(Publ.)g(Math.)h(Inst.)g (Hautes)g(Etud.)g(Sci.)f Fh(53)g FE(\(1981\),)i(17{51.)118 2637 y([177])42 b(B.)33 b(Morrel)g(and)i(P)-6 b(.)33 b(Muhly)-6 b(,)37 b Fi(Center)l(e)l(d)e(op)l(er)l(ators)p FE(,)k(Studia)c(Math.)f Fh(51)f FE(\(1974\),)305 2716 y(251{263.)118 2796 y([178])42 b(G.)17 b(J.)g(Murph)n(y)-6 b(,)18 b Fq(C)802 2773 y Fg(\003)838 2796 y Fi(-algebr)l(as)j(and)f(op) l(er)l(ator)i(the)l(ory)p FE(,)d(Acad.)e(Press,)h(Boston,)h(1990.)118 2876 y([179])42 b(F.)24 b(Murra)n(y)h(and)g(J.)g(v)n(on)h(Neumann,)f Fi(On)i(rings)f(of)i(op)l(er)l(ators.)h(IV.)p FE(,)c(Ann.)g(Math.)305 2955 y Fh(44)e FE(\(1943\),)i(716{808.)118 3035 y([180])42 b(G.)35 b(Nagy)g(and)h(A.)e(Nica,)j Fi(On)f(the)g(\\quantum)h(disk")e (and)i(\\non-c)l(ommutative)305 3114 y(cir)l(cle")p FE(,)26 b(Algebraic)g(metho)r(ds)h(in)f(op)r(erator)h(theory)h(\(R.)e(E.)g (Curto)h(and)g(P)-6 b(.)26 b(E.)g(T.)305 3193 y(J\034rgensen,)e (eds.\),)g(Birkh\023)-35 b(auser)23 b(V)-6 b(elag,)24 b(Boston,)g(1994,)g(pp.)f(276{290.)118 3273 y([181])42 b(L.)19 b(A.)f(Nazaro)n(v)l(a,)j Fi(R)l(epr)l(esentations)i(of)f(a)g (quadruple)p FE(,)f(Izv.)f(AN.)f(SSSR)h Fh(31)e FE(\(1967\),)305 3352 y(no.)23 b(6,)h(1361{1377,)h(\(Russian\).)118 3432 y([182])42 b(E.)23 b(Nelson,)g Fi(A)n(nalytic)i(ve)l(ctors)p FE(,)f(Ann.)f(of)g(Math.)h Fh(70)f FE(\(1959\),)i(no.)f(2,)f(572{615.) 118 3512 y([183])42 b(I.)23 b(Newton,)h Fi(Enumer)l(atio)k(line)l(arum) e(p)l(ortii)h(or)l(dinis)p FE(,)d(Optics)g(\(1704\),)h(138{162.)118 3593 y([184])42 b(L.)22 b(P)-6 b(.)22 b(Nizhnik)h(and)g(L.)g(B.)f(T)-6 b(uro)n(wsk)l(a,)23 b Fi(R)l(epr)l(esentations)i(of)g(double)h(c)l (ommutator)305 3671 y(by)32 b(matrix-di\013er)l(ential)g(op)l(er)l (ators)p FE(,)j(Metho)r(ds)d(F)-6 b(unct.)32 b(Anal.)e(T)-6 b(op)r(ol.)31 b Fh(3)g FE(\(1997\),)305 3750 y(no.)23 b(3,)h(75{80.)118 3830 y([185])42 b(M.)17 b(Noumi)g(and)i(K.)f(Mimac)n (hi,)f Fi(Big)j Fq(q)r Fi(-Jac)l(obi)h(p)l(olynomials,)j Fq(q)r Fi(-Hahn)d(p)l(olynomials)305 3909 y(and)30 b(a)h(family)f(of)g (quantum)g FE(3)p Fi(-spher)l(es)p FE(,)g(Lett.)f(Math.)f(Ph)n(ys.)g Fh(19)g FE(\(1990\),)j(no.)d(4,)305 3988 y(299{305.)118 4068 y([186])42 b(A.)18 b(V.)h(Odesski,)g Fi(On)i(an)h(analo)l(gue)h (of)f(the)f(Sklyanin)g(algebr)l(a)p FE(,)g(F)-6 b(unct.)20 b(Anal.)e(Appl.)305 4147 y Fh(20)23 b FE(\(1986\),)i(152{154.)p eop %%Page: 252 256 252 255 bop 118 100 a FP(252)1866 b FK(Bibliograph)n(y)118 333 y FE([187])42 b(A.)27 b(V.)g(Odesski)h(and)h(B.)e(L.)h(F)-6 b(eigin,)29 b Fi(El)t(liptic)g(Sklyanin)h(algebr)l(as)p FE(,)g(F)-6 b(unkt.)28 b(Anal.)305 412 y(Prilozh.)22 b Fh(23)h FE(\(1989\),)i(no.)f(3,)f(45{54.)118 496 y([188])42 b(C.)22 b(L.)g(Olsen)h(and)g(W.)g(R.)f(Zame,)g Fi(Singly)i(gener)l(ate) l(d)i Fq(C)1819 472 y Fg(\003)1855 496 y Fi(-algebr)l(as)p FE(,)d(T)-6 b(rans.)22 b(Amer.)305 574 y(Math.)h(So)r(c.)h Fh(215)f FE(\(1976\),)i(205{215.)118 658 y([189])42 b(A.)16 b(Y)-6 b(u.)17 b(Ol'shanski)-8 b(\025)-27 b(\020,)17 b Fi(Ge)l(ometry)j(of)g(de\014ning)f(r)l(elations)i(in)e(gr)l(oups)p FE(,)h(Klu)n(w)n(er)c(Acad.)305 737 y(Publ.,)22 b(Dordrec)n(h)n(t,)i (1991,)h(T)-6 b(ransl)23 b(from)e(Russian)j(edn.:)31 b(Nauk)l(a,)24 b(Mosco)n(w,)g(1989.)118 821 y([190])42 b(V.)27 b(L.)g(Ostro)n(vsky)-8 b(\025)-27 b(\020,)29 b Fi(R)l(epr)l(esentations)h(of)g(a)g(family)g(of)f(quadr)l(atic)i (algebr)l(as)f(with)305 900 y(thr)l(e)l(e)c(gener)l(ators)p FE(,)d(Selecta)j(Math.)d(So)n(v.)h Fh(12)f FE(\(1993\),)i(119{127.)118 984 y([191])p 305 984 212 4 v 265 w(,)32 b Fi(On)f(op)l(er)l(ator)j(r)l (elations,)h(c)l(enter)l(e)l(d)d(op)l(er)l(ators,)j(and)e(nonbije)l (ctive)e(dy-)305 1063 y(namic)l(al)25 b(systems)p FE(,)c(Metho)r(ds)h (F)-6 b(unct.)22 b(Anal.)f(T)-6 b(op)r(ol.)21 b Fh(2)g FE(\(1996\),)i(no.)f(3-4,)f(114{121.)118 1147 y([192])42 b(V.)33 b(L.)h(Ostro)n(vsky)-8 b(\025)-27 b(\020)34 b(and)h(Y)-6 b(u.)34 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)36 b Fi(Applic)l(ation)h(of)e(the)h(pr)l(oje)l(ction)305 1226 y(sp)l(e)l(ctr)l(al)30 b(the)l(or)l(em)g(to)f(nonc)l(ommuting)h (families)g(of)f(op)l(er)l(ators)p FE(,)g(Ukr.)e(Math.)g(Zh.)305 1305 y Fh(40)c FE(\(1988\),)i(no.)e(4,)h(469{481,)h(\(Russian\).)118 1389 y([193])p 305 1389 V 265 w(,)d Fi(F)-5 b(amilies)25 b(of)g(unb)l(ounde)l(d)h(selfadjoint)f(op)l(er)l(ators,)i(which)e(ar)l (e)g(c)l(onne)l(cte)l(d)305 1468 y(with)20 b(non-Lie)f(r)l(elations)p FE(,)g(F)-6 b(unct.)18 b(Anal.)f(Prilozh.)f Fh(23)h FE(\(1989\),)j(no.) d(2,)h(67{68,)i(\(Rus-)305 1546 y(sian\).)118 1630 y([194])p 305 1630 V 265 w(,)g Fi(R)l(epr)l(esentations)i(of)g FD(\003)p Fi(-algebr)l(as)g(with)f(two)h(gener)l(ators)g(and)g(p)l (olynomial)305 1709 y(r)l(elations)p FE(,)f(Zap.)f(Nauc)n(hn.)h(Semin.) e(LOMI)h Fh(172)f FE(\(1989\),)k(no.)d(121{129,)i(\(Russian\).)118 1793 y([195])p 305 1793 V 265 w(,)30 b Fi(Unb)l(ounde)l(d)j(op)l(er)l (ators)g(satisfying)d(non-Lie)g(c)l(ommutation)i(r)l(elations)p FE(,)305 1872 y(Repts.)24 b(math.)e(ph)n(ys.)i Fh(28)f FE(\(1989\),)i(no.)f(1,)f(91{103.)118 1956 y([196])p 305 1956 V 265 w(,)i Fi(Structur)l(e)h(the)l(or)l(ems)i(for)f(a)g(p)l (air)h(of)e(unb)l(ounde)l(d)j(selfadjoint)f(op)l(er)l(ators)305 2035 y(satisfying)d(a)h(quadr)l(atic)g(r)l(elation)p FE(,)e(Adv.)g(So)n(v.)g(Math.)f Fh(9)h FE(\(1992\),)h(131{149.)118 2119 y([197])p 305 2119 V 265 w(,)g Fi(On)h(p)l(airs)i(of)f (self-adjoint)g(op)l(er)l(ators)p FE(,)g(Seminar)c(Soph)n(us)j(Lie)f Fh(3)f FE(\(1993\),)305 2198 y(no.)f(2,)h(185{218.)118 2282 y([198])p 305 2282 V 265 w(,)29 b Fi(On)h(r)l(epr)l(esentations)h (of)f(the)g(Heisenb)l(er)l(g)f(r)l(elations)i(for)f(the)g(quantum)305 2361 y Fq(E)t FE(\(2\))c Fi(gr)l(oup)p FE(,)e(Ukr.)f(Mat.)g(Zh.)h Fh(47)f FE(\(1995\),)i(no.)f(5,)f(689{692.)118 2445 y([199])p 305 2445 V 265 w(,)j Fi(R)l(epr)l(esentations)j(of)f FD(\003)p Fi(-algebr)l(as)h(and)g(dynamic)l(al)g(systems)p FE(,)d(Nonlinear)305 2524 y(Math.)d(Ph)n(ys.)h Fh(2)f FE(\(1995\),)i(no.)f(2,)f(133{150.)118 2608 y([200])p 305 2608 V 265 w(,)48 b Fi(R)l(epr)l(esentations)e(of)e(quadr)l(atic)h FD(\003)p Fi(-algebr)l(as)g(by)f(b)l(ounde)l(d)i(and)f(un-)305 2687 y(b)l(ounde)l(d)27 b(op)l(er)l(ators)p FE(,)e(Repts.)f(Math.)g(Ph) n(ys.)f Fh(35)g FE(\(1995\),)j(no.)d(2/3,)h(283{301.)118 2771 y([201])42 b(V.)19 b(L.)h(Ostro)n(vsky)-8 b(\025)-27 b(\020)21 b(and)f(S.)g(D.)g(Silv)n(estro)n(v,)h Fi(R)l(epr)l (esentations)j(of)e(the)h(r)l(e)l(al)g(forms)h(of)305 2849 y(a)g(gr)l(ade)l(d)h(analo)l(gue)g(of)f(the)f(Lie)g(algebr)l(a)h Fq(sl)q FE(\(2)p Fq(;)12 b Fb(C)d FE(\),)28 b(Ukr.)20 b(Mat.)h(Zh)n(urn.)h Fh(44)e FE(\(1992\),)305 2928 y(no.)j(11,)h (1518{1524,)i(\(Russian\).)118 3012 y([202])42 b(V.)21 b(L.)g(Ostro)n(vsky)-8 b(\025)-27 b(\020)22 b(and)g(L.)f(B.)g(T)-6 b(uro)n(vsk)l(a)n(y)n(a,)23 b Fi(R)l(epr)l(esentations)i(of)f FD(\003)p Fi(-algebr)l(as)h(and)305 3091 y(multidimensional)k(dynamic)l (al)g(systems)p FE(,)c(Ukr.)g(Mat.)g(Zh)n(urn.)h Fh(47)f FE(\(1995\),)i(no.)f(4,)305 3170 y(488{497.)118 3254 y([203])42 b(K.)c(R.)g(P)n(arthasarath)n(y)-6 b(,)44 b Fi(A)n(n)c(intr)l(o)l(duction)i(to)e(quantum)g(sto)l(chastic)h(c)l (alculus)p FE(,)305 3333 y(Birkh\177)-35 b(auser{V)-6 b(erlag,)23 b(Basel,)g(1992.)118 3417 y([204])42 b(C.)21 b(P)n(earcy)-6 b(,)23 b Fi(On)h(c)l(ertain)g(von)h(Neumann)g(algebr)l (as)g(which)g(ar)l(e)g(gener)l(ate)l(d)f(by)g(p)l(ar-)305 3496 y(tial)h(isometries)p FE(,)f(Pro)r(c.)f(Amer.)e(Math.)j(So)r(c.)g Fh(15)f FE(\(1964\),)i(393{395.)118 3580 y([205])42 b(G.)21 b(K.)h(P)n(edersen,)g Fi(Me)l(asur)l(e)j(the)l(ory)f(for)g Fq(C)1474 3556 y Fg(\003)1510 3580 y Fi(-algebr)l(as)p FE(,)f(Math.)e(Scand.)i Fh(22)e FE(\(1968\),)305 3659 y(63{74.)118 3743 y([206])p 305 3743 V 265 w(,)j Fq(C)627 3719 y Fg(\003)663 3743 y Fi(-algebr)l(as)j(and)g(their)f(automorphism) i(gr)l(oups)p FE(,)e(London)f(Math.)f(So)r(c.)305 3822 y(Monographs,)g(v)n(ol.)f(14,)h(Acad.)f(Press,)g(London,)h(1979.)118 3906 y([207])42 b(S.)c(P)n(edersen,)43 b Fi(A)n(ntic)l(ommuting)d (selfadjoint)h(op)l(er)l(ators)p FE(,)j(J.)39 b(F)-6 b(unct.)39 b(Anal.)f Fh(89)305 3984 y FE(\(1990\),)25 b(no.)e(2,)h(428{443.)118 4068 y([208])42 b(N.)25 b(C.)g(Phillips,)f Fi(Inverse)k(limits)f(of)h Fq(C)1371 4045 y Fg(\003)1407 4068 y Fi(-algebr)l(as)p FE(,)e(J.)f(Op)r(er.)g(Theory)i Fh(19)e FE(\(1988\),)305 4147 y(153{195.)p eop %%Page: 253 257 253 256 bop 118 100 a FK(Bibliograph)n(y)1865 b FP(253)118 333 y FE([209])42 b(R.)e(S.)g(Pierce,)45 b Fi(Asso)l(ciative)c(algebr)l (as)p FE(,)46 b(Graduate)c(texts)g(in)e(math.,)k(v)n(ol.)d(88,)305 412 y(Springer-V)-6 b(erlag,)22 b(New)i(Y)-6 b(ork{Heidelb)r (erg{Berlin,)23 b(1982.)118 492 y([210])42 b(A.)27 b(Piry)n(atinsk)l(a) n(y)n(a,)j Fi(On)f(unitary)h(classi\014c)l(ation)h(of)f(we)l(akly)g(c)l (enter)l(e)l(d)g(op)l(er)l(ators)p FE(,)305 571 y(V)-6 b(estnik)24 b(T)-6 b(am)n(b)r(o)n(v)23 b(Univ.)g Fh(3)h FE(\(1998\),)h(no.)e(1,)h(79{83.)118 651 y([211])42 b(A.)19 b(Y)-6 b(u.)20 b(Piry)n(atinsk)l(a)n(y)n(a)g(and)h(Y)-6 b(u.)19 b(S.)h(Samo)-8 b(\025)-27 b(\020lenk)n(o,)20 b Fi(Wild)j(pr)l(oblems)g(in)f(r)l(epr)l(esenta-)305 730 y(tion)k(the)l(ory)g(of)h FD(\003)p Fi(-algebr)l(as)g(with)f(gener) l(ators)h(and)g(r)l(elations)p FE(,)e(Ukr.)f(Mat.)g(Zh)n(urn.)305 809 y Fh(47)f FE(\(1995\),)i(no.)e(1,)h(70{78,)g(\(Russian\).)118 889 y([212])42 b(S.)23 b(P)n(op)r(o)n(vyc)n(h,)i Fi(R)l(epr)l (esentations)i(of)f(r)l(e)l(al)h(forms)f(of)g(Witten)-7 b('s)24 b(\014rst)i(deformation)p FE(,)305 968 y(Symmetry)c(Nonlin.)h (Math.)g(Ph)n(ys.)h Fh(2)f FE(\(1997\),)i(393{396.)118 1048 y([213])p 305 1048 212 4 v 265 w(,)i Fi(Unb)l(ounde)l(d)k(idemp)l (otents)p FE(,)d(Metho)r(ds)g(F)-6 b(unct.)28 b(Anal.)e(T)-6 b(op)r(ol.)27 b Fh(5)f FE(\(1999\),)305 1127 y(no.)d(1,)h(95{103.)118 1207 y([214])42 b(S.V.)53 b(P)n(op)r(o)n(vyc)n(h)k(and)e(T.Y)-6 b(u.)53 b(Maistrenk)n(o,)63 b Fq(C)1723 1184 y Fg(\003)1759 1207 y Fi(-algebr)l(as)54 b(asso)l(ciate)l(d)i(with)305 1286 y(quadr)l(atic)29 b(dynamic)l(al)h(systems)p FE(,)d(Pro)r(c.)f (Inst.)h(Math.)f(NAS)h(Ukraine)f Fh(30)g FE(\(2000\),)305 1365 y(no.)d(2,)h(364{370.)118 1445 y([215])42 b(S.V.)31 b(P)n(op)r(o)n(vyc)n(h)k(and)e(T.Y)-6 b(u.)32 b(Maistrenk)n(o,)i Fq(C)1607 1422 y Fg(\003)1643 1445 y Fi(-algebr)l(as)h(asso)l(ciate)l (d)g(with)g Fe(F)2502 1454 y Fc(2)2532 1440 y Fw(n)305 1524 y Fi(dynamic)l(al)27 b(systems)p FE(,)c(Ukr.)f(Mat.)i(Zh.)f Fh(53)g FE(\(2001\),)i(no.)f(7,)f(929{938.)118 1604 y([216])42 b(S.V.)35 b(P)n(op)r(o)n(vyc)n(h)i(and)f(Y)-6 b(u.S.)36 b(Samoilenk)n(o,)i Fi(On)e(homomorphisms)41 b(of)c(algebr)l(as)305 1683 y(gener)l(ate)l(d)26 b(by)f(pr)l(oje)l(ctions)j(and)f(c)l(oxeter)e (functors)p FE(,)f(Ukr.)f(Mat.Zurn.)h Fh(55)f FE(\(2003\),)305 1762 y(no.)g(9,)h(1224{1237.)118 1842 y([217])42 b(R.)30 b(T.)g(P)n(o)n(w)n(ers,)h Fi(Selfadjoint)i(algebr)l(as)g(of)f(unb)l (ounde)l(d)i(op)l(er)l(ators.)g(I)p FE(,)d(Comm)n(un.)305 1921 y(Math.)23 b(Ph)n(ys.)h Fh(21)f FE(\(1971\),)i(85{124.)118 2001 y([218])p 305 2001 V 265 w(,)43 b Fi(Selfadjoint)e(algebr)l(as)h (of)e(unb)l(ounde)l(d)j(op)l(er)l(ators.)f(II)p FE(,)e(T)-6 b(rans)40 b(Amer.)305 2080 y(Math.)23 b(So)r(c.)h Fh(187)f FE(\(1974\),)i(261{293.)118 2160 y([219])p 305 2160 V 265 w(,)20 b Fi(Simplicity)i(of)g(the)g Fq(C)1150 2137 y Fg(\003)1186 2160 y Fi(-algebr)l(a)g(asso)l(ciate)l(d)i(with)e(the)g (fr)l(e)l(e)g(gr)l(oup)h(on)f(two)305 2239 y(gener)l(ators)p FE(,)h(Duk)n(e)i(Math.)e(J.)h Fh(42)f FE(\(1975\),)i(151{156.)118 2320 y([220])42 b(D.)27 b(Proskurin,)h Fi(Homo)l(gene)l(ous)k(ide)l (als)f(in)e(Wick)g(algebr)l(as)p FE(,)h(Pro)r(c.)e(Amer.)e(Math.)305 2398 y(So)r(c.)e Fh(126)e FE(\(1998\),)j(no.)f(11,)g(3371{3376.)118 2479 y([221])42 b(D.)22 b(P)-6 b(.)23 b(Proskurin,)f Fi(A)n(b)l(out)k(p)l(ositivity)e(of)i(Fo)l(ck)g(inner)f(pr)l(o)l(duct)h (of)g(a)f(c)l(ertain)g(Wick)305 2557 y(algebr)l(as)p FE(,)f(Metho)r(ds)g(F)-6 b(unct.)25 b(Anal.)e(T)-6 b(op)r(ol.)23 b Fh(5)h FE(\(1999\),)h(no.)e(1,)h(88{94.)118 2638 y([222])42 b(D.)31 b(P)-6 b(.)32 b(Proskurin)f(and)i(Y)-6 b(u.)31 b(S.)h(Samo)-8 b(\025)-27 b(\020lenk)n(o,)34 b Fi(R)l(epr)l (esentations)g(of)g(Wick)f(CCR)305 2717 y(algebr)l(a)p FE(,)28 b(Sp)r(ectral)h(and)e(ev)n(olutionary)i(problems,)e(v)n(ol.)g (8)g(\(Simferop)r(ol\),)g(T)-6 b(a)n(vria,)305 2795 y(1998,)24 b(pp.)f(43{45.)118 2876 y([223])p 305 2876 V 265 w(,)g Fi(Stability)i(of)h(the)g Fq(C)1108 2852 y Fg(\003)1144 2876 y Fi(-algebr)l(a)g(asso)l(ciate)l(d)h(with)f(twiste)l(d)g(CCR)p FE(,)e(Algebra)305 2955 y(Repr.)f(Theory)i Fh(5)e FE(\(2002\),)i (433{444.)118 3035 y([224])42 b(W.)e(Pusz,)k Fi(Twiste)l(d)e(c)l (anonic)l(al)h(antic)l(ommutation)g(r)l(elations)p FE(,)i(Repts.)c (Math.)305 3114 y(Ph)n(ys.)23 b Fh(27)g FE(\(1989\),)i(349{360.)118 3194 y([225])42 b(W.)17 b(Pusz)h(and)g(S.)g(L.)f(W)-6 b(orono)n(wicz,)20 b Fi(Twiste)l(d)g(se)l(c)l(ond)i(quantization)p FE(,)d(Repts.)f(Math.)305 3273 y(Ph)n(ys.)23 b Fh(27)g FE(\(1989\),)i(231{257.)118 3353 y([226])42 b(I.)17 b(F.)g(Putnam,)h Fq(C)791 3330 y Fg(\003)827 3353 y Fi(-algebr)l(as)j(arising)f(fr)l(om) h(interval)e(exchange)h(tr)l(ansformations)p FE(,)305 3432 y(J.)j(Op)r(er.)g(Theory)i Fh(27)e FE(\(1992\),)i(231{250.)118 3512 y([227])42 b(V.)22 b(I.)h(Rabano)n(vic)n(h,)h Fi(Banach)i(algebr)l (as)g(gener)l(ate)l(d)f(by)g(thr)l(e)l(e)g(idemp)l(otents)p FE(,)f(Meth-)305 3591 y(o)r(ds)f(F)-6 b(unct.)25 b(Anal.)e(T)-6 b(op)r(ol.)24 b Fh(4)f FE(\(1998\),)i(no.)f(1,)f(65{67.)118 3671 y([228])p 305 3671 V 265 w(,)35 b Fi(Singly)e(gener)l(ate)l(d)i Fq(C)1184 3648 y Fg(\003)1220 3671 y Fi(-algebr)l(as)p FE(,)g(Ukr.)d(Mat.)g(Zh.)h Fh(51)f FE(\(1999\),)k(no.)d(8,)305 3750 y(\(Russian\).)118 3830 y([229])42 b(V.)d(I.)i(Rabano)n(vic)n(h)h (and)f(Y)-6 b(u.)40 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)44 b Fi(On)d(r)l(epr)l(esentations)h(of)g Fe(F)2509 3838 y Ff(n)2549 3830 y Fi(-)305 3909 y(algebr)l(as)26 b(and)h (invertibility)d(symb)l(ols)p FE(,)g(Metho)r(ds)g(F)-6 b(unct.)25 b(Anal.)e(T)-6 b(op)r(ol.)24 b Fh(4)f FE(\(1998\),)305 3988 y(no.)g(4,)h(86{96.)118 4068 y([230])p 305 4068 V 265 w(,)k Fi(On)h(r)l(epr)l(esentations)h(of)f Fq(F)1329 4076 y Ff(n)1371 4068 y Fi(-algebr)l(as)h(and)g(their)e(applic)l (ations)p FE(,)i(Op)r(er.)305 4147 y(Theory)24 b(Adv.)f(Appl.,)g(v)n (ol.)g(94,)h(Birkh\177)-35 b(auser)23 b(V)-6 b(erlag,)23 b(Basel,)g(1999.)p eop %%Page: 254 258 254 257 bop 118 100 a FP(254)1866 b FK(Bibliograph)n(y)118 333 y FE([231])p 305 333 212 4 v 265 w(,)31 b Fi(When)g(a)g(sum)h(of)f (idemp)l(otents)h(or)f(pr)l(oje)l(ctions)i(is)e(a)g(multiple)h(of)f (the)305 412 y(identity)p FE(,)22 b(F)-6 b(unct.)24 b(Anal.)f(Prilozh.) g Fh(34)g FE(\(2000\),)i(no.)f(4,)f(91{93.)118 496 y([232])p 305 496 V 265 w(,)31 b Fi(Cases)h(in)g(which)g(a)g(sc)l(alar)h(op)l(er) l(ator)h(is)d(a)h(sum)g(of)g(pr)l(oje)l(ctions)p FE(,)h(Ukr.)305 575 y(Math.)23 b(Zh.)h Fh(53)f FE(\(2001\),)i(no.)e(7,)h(1116{1133.)118 659 y([233])42 b(V.I.)34 b(Rabano)n(vic)n(h,)39 b(Y)-6 b(u.S.)34 b(Samoilenk)n(o,)j(and)f(A.V.Strelets,)h Fi(On)f(identities)f (in)305 738 y(algebr)l(as)29 b(gener)l(ate)l(d)g(by)e(line)l(arly)i(c)l (onne)l(cte)l(d)g(idemp)l(otents)p FE(,)f(Ukr.)e(Math.)g(Zh.)g Fh(56)305 817 y FE(\(2004\),)f(to)f(app)r(ear.)118 902 y([234])42 b(I.)27 b(Raeburn)i(and)f(A.)f(M.)g(Sinclair,)h Fi(The)h Fq(C)1506 878 y Fg(\003)1542 902 y Fi(-algebr)l(a)h(gener)l (ate)l(d)g(by)e(two)i(pr)l(oje)l(c-)305 981 y(tions)p FE(,)23 b(Math.)h(Scand.)g Fh(65)f FE(\(1989\),)i(278{290.)118 1065 y([235])42 b(M.)23 b(Reed)j(and)f(B.)f(Simon,)f Fi(Metho)l(ds)k(of)g(mo)l(dern)h(mathematic)l(al)g(physics)p FE(,)d(v)n(ol.)f(1,)305 1144 y(Acad.)g(Press,)e(New)i(Y)-6 b(ork,)23 b(1972.)118 1229 y([236])42 b(J.)24 b(Renault,)i Fi(A)h(gr)l(oup)l(oid)i(appr)l(o)l(ach)h(to)d Fq(C)1485 1205 y Fg(\003)1521 1229 y Fi(-algebr)l(as)p FE(,)e(Lect.)h(Notes.)f (Math.,)g(v)n(ol.)305 1307 y(793,)f(Springer{V)-6 b(erlag,)23 b(1980.)118 1392 y([237])42 b(M.)24 b(Rie\013el,)g Fi(Quantum)j (deformations)h(for)f(actions)g(of)g Fb(R)1894 1369 y Ff(d)1930 1392 y FE(,)e(Mem.)e(Amer.)g(Math.)305 1471 y(So)r(c.,)g(v)n(ol.)h(506,)g(Amer.)d(Math.)j(So)r(c.,)f(Pro)n (vidence,)i(RI,)e(1993.)118 1555 y([238])p 305 1555 V 265 w(,)39 b Fi(Morita)f(e)l(quivalenc)l(e)g(for)g Fq(C)1400 1532 y Fg(\003)1436 1555 y Fi(-algebr)l(as)g(and)g Fq(W)1971 1532 y Fg(\003)2007 1555 y Fi(-algebr)l(as)p FE(,)h(J.)e(Pure)305 1634 y(Appl.)23 b(Algebra)g Fh(5)h FE(\(1974\),)h(51{96.)118 1719 y([239])42 b(S.)31 b(Ro)r(c)n(h)i(and)g(B.)e(Sib)r(ermann,)i Fi(A)n(lgebr)l(as)h(gener)l(ate)l(d)g(by)e(idemp)l(otents)j(and)f(the) 305 1798 y(symb)l(ol)28 b(c)l(alculus)i(for)e(singular)g(inte)l(gr)l (al)h(op)l(er)l(ators)p FE(,)f(In)n(t.)f(Equat.)g(Op)r(er.)e(Theory)305 1877 y Fh(11)e FE(\(1988\),)i(385{419.)118 1961 y([240])42 b(A.)32 b(V.)f(Roiter,)k Fi(Boxes)f(with)g(an)g(involution)p FE(,)h(Represen)n(tations)g(and)e(quadratic)305 2040 y(forms,)27 b(Inst.)i(Math.)g(Acad.)g(Sci.)g(Ukr.)f(SSR,)g(Kiev,)i (1979,)h(pp.)d(124{126,)k(\(Rus-)305 2119 y(sian\).)118 2203 y([241])42 b(W.)23 b(Rudin,)h Fi(F)-5 b(unctional)26 b(analysis)p FE(,)e(McGra)n(w-Hill,)d(New)j(Y)-6 b(ork,)23 b(1973.)118 2288 y([242])42 b(S.)16 b(Sak)l(ai,)i Fi(Op)l(er)l(ator)i (algebr)l(as)g(in)f(dynamic)l(al)h(systems.)f(The)g(the)l(ory)g(of)h (unb)l(ounde)l(d)305 2367 y(derivations)26 b(in)f Fq(C)814 2343 y Fg(\003)850 2367 y Fi(-algebr)l(as)p FE(,)f(Cam)n(bridge)f (Univ.)g(Press,)f(Cam)n(brige,)h(1991.)118 2451 y([243])42 b(Y)-6 b(u.)32 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)34 b Fi(Sp)l(e)l(ctr)l(al)h(the)l(ory)g(of)f(families)g(of)g(self-adjoint) g(op)l(er)l(ators)p FE(,)305 2530 y(Klu)n(w)n(er)c(Academic)h (Publisher,)h(1991,)i(T)-6 b(ransl.)30 b(from)f(Russian)i(edn.:)46 b(Nauk)n(o)n(v)l(a)305 2609 y(Dumk)l(a,)22 b(Kiev,)h(1984.)118 2694 y([244])42 b(Y)-6 b(u.)22 b(S.)h(Samo)-8 b(\025)-27 b(\020lenk)n(o)23 b(and)g(V.)f(S.)h(Sh)n(ul'man,)f Fi(On)i(r)l(epr)l (esentations)i(of)g(r)l(elations)g(of)305 2772 y(the)f(form)g Fq(i)p FE([)p Fq(A;)11 b(B)s FE(])20 b(=)f Fq(f)7 b FE(\()p Fq(A)p FE(\))16 b(+)e Fq(g)r FE(\()p Fq(B)s FE(\),)24 b(Ukr.)e(Mat.)h(Zh.)g Fh(43)g FE(\(1991\),)h(no.)f(1,)g(110{114,)305 2851 y(\(Russian\).)118 2936 y([245])42 b(Y)-6 b(u.)26 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o)27 b(and)g(L.)f(B.)g(T)-6 b(uro)n(wsk)l(a,)27 b Fi(On)h(r)l(epr)l(esentations)h(of)g FD(\003)p Fi(-algebr)l(as)305 3015 y(by)j(unb)l(ounde)l(d)j(op)l(er)l (ators)p FE(,)g(F)-6 b(unkt.)33 b(Anal.)e(Prilozh.)f Fh(31)h FE(\(1997\),)k(no.)c(4,)j(80{83,)305 3094 y(\(Russian\).)118 3178 y([246])p 305 3178 V 265 w(,)d Fi(R)l(epr)l(esentations)h(of)f (cubic)g(semiline)l(ar)h(r)l(elations)g(and)g(r)l(e)l(al)h(forms)f(of) 305 3257 y(the)24 b(Fairlie)g(algebr)l(a)p FE(,)f(Quan)n(tum)f(groups)h (and)f(quan)n(tum)h(spaces,)g(Banac)n(h)g(Cen)n(ter)305 3336 y(Publ.,)f(v)n(ol.)i(40,)f(Inst.)h(Math.)g(P)n(olish)f(Acad.)h (Sci.,)f(W)-6 b(arsza)n(w)n(a,)24 b(1997,)h(pp.)e(21{40.)118 3420 y([247])42 b(Y)-6 b(u.)33 b(S.)h(Samo)-8 b(\025)-27 b(\020lenk)n(o,)35 b(L.)f(B.)f(T)-6 b(uro)n(wsk)l(a,)36 b(and)f(S.)e(P)n(op)r(o)n(vyc)n(h,)38 b Fi(R)l(epr)l(esentations)305 3499 y(of)e(a)g(cubic)g(deformation)h(of)g Fq(su)p FE(\(2\))f Fi(and)h(p)l(ar)l(asup)l(ersymmetric)i(c)l(ommutation)305 3578 y(r)l(elations)p FE(,)24 b(Symmetry)e(in)h(Nonlin.)g(Math.)h(Ph)n (ys.)f Fh(2)h FE(\(1997\),)h(272{383.)118 3663 y([248])42 b(Y)-6 b(u.)24 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)24 b(L.)g(B.)g(T)-6 b(uro)n(wsk)l(a,)25 b(and)g(V.)f(S.)g(Sh)n(ul'man,)f Fi(Semiline)l(ar)k(r)l(ela-)305 3742 y(tions)g(and)g(their)g FD(\003)p Fi(-r)l(epr)l(esentations)p FE(,)f(Metho)r(ds)g(F)-6 b(unct.)25 b(Anal.)g(T)-6 b(op)r(ol.)25 b Fh(2)f FE(\(1996\),)305 3820 y(no.)f(1,)h(55{111.)118 3905 y([249])42 b(K.)30 b(Sc)n(hm)r(\177)-37 b(udgen,)33 b Fi(Unb)l(ounde)l(d)h(op)l(er)l(ator) g(algebr)l(as)f(and)h(r)l(epr)l(esentation)f(the)l(ory)p FE(,)305 3984 y(Birkh\177)-35 b(auser,)23 b(Basel,)g(1990.)118 4068 y([250])p 305 4068 V 265 w(,)36 b Fi(Op)l(er)l(ator)h(r)l(epr)l (esentations)g(of)e Fb(R)1538 4076 y Ff(q)1572 4068 y FE(,)i(Publ)d(RIMS)g Fh(28)f FE(\(1992\),)39 b(no.)34 b(6,)305 4147 y(1029{1061.)p eop %%Page: 255 259 255 258 bop 118 100 a FK(Bibliograph)n(y)1865 b FP(255)118 333 y FE([251])p 305 333 212 4 v 265 w(,)38 b Fi(Inte)l(gr)l(able)g(op) l(er)l(ator)g(r)l(epr)l(esentations)g(of)f Fb(R)1862 309 y Fc(2)1862 349 y Ff(q)1897 333 y Fi(,)i Fq(X)2016 341 y Ff(q)r(;\015)2141 333 y Fi(and)f Fq(S)t(L)p FE(\(2)p Fq(;)12 b Fb(R)p FE(\),)305 412 y(Comm)n(un.)21 b(Math.)j(Ph)n(ys)g Fh(159)e FE(\(1994\),)j(217{237.)118 500 y([252])p 305 500 V 265 w(,)38 b Fi(Op)l(er)l(ator)g(r)l(epr)l(esentations)g(of)e Fq(U)1543 508 y Ff(q)1577 500 y FE(\()p Fq(sl)1658 509 y Fc(2)1693 500 y FE(\()p Fb(R)p FE(\)\),)j(Lett.)e(Math.)e(Ph)n(ys.)g Fh(37)305 579 y FE(\(1996\),)25 b(211{222.)118 667 y([253])42 b(J.)30 b(Sc)n(h)n(w)n(enk)i(and)g(J.)e(W)-6 b(ess,)33 b Fi(A)f Fq(q)r Fi(-deforme)l(d)h(quantum)g(me)l(chanic)l(al)h(toy)e (mo)l(del)p FE(,)305 746 y(Ph)n(ys.)23 b(Lett.)i(B.)e Fh(291)f FE(\(1992\),)j(273{277.)118 834 y([254])42 b(V.)19 b(V.)g(Sergeic)n(h)n(uk,)j Fi(Classi\014c)l(ation)h(of)g(line)l(ar)f (op)l(er)l(ators)j(in)d(a)g(\014nite-dimensional)305 913 y(unitary)j(sp)l(ac)l(e)p FE(,)g(F)-6 b(unct.)24 b(Anal.)f(Appl.)g Fh(18)g FE(\(1984\),)i(no.)f(3,)f(224{230.)118 1001 y([255])p 305 1001 V 265 w(,)c Fi(Classi\014c)l(ation)j(pr)l (oblems)g(for)f(systems)g(of)f(forms)i(and)f(line)l(ar)h(mappings)p FE(,)305 1080 y(Math.)h(USSR)h(Izv)n(estiy)n(a)h Fh(31)e FE(\(1988\),)j(no.)d(3,)h(481{501.)118 1168 y([256])p 305 1168 V 265 w(,)f Fi(Classi\014c)l(ation)k(of)e(p)l(airs)i(of)e (subsp)l(ac)l(es)i(in)e(sp)l(ac)l(es)i(with)f(sc)l(alar)g(pr)l(o)l (duct)p FE(,)305 1247 y(Ukrain.)d(Math.)g(J.)h Fh(42)e FE(\(1990\),)k(no.)d(4,)h(478{491.)118 1335 y([257])p 305 1335 V 265 w(,)41 b Fi(Symmetric)e(r)l(epr)l(esentations)h(of)f (algebr)l(as)h(with)g(involution)p FE(,)i(Math.)305 1414 y(Notes)24 b Fh(50)f FE(\(1992\),)i(no.)f(3{4,)g(1058{1061.)118 1502 y([258])p 305 1502 V 265 w(,)17 b Fi(Unitary)i(and)g(Euclide)l(an) i(r)l(epr)l(esentations)f(of)f(a)g(quiver)p FE(,)d(Linear)h(Algebra)305 1581 y(Appl.)23 b Fh(278)f FE(\(1998\),)j(37{62.)118 1669 y([259])42 b(H.)22 b(Shapiro,)i Fi(A)h(survey)g(of)h(c)l(anonic)l (al)h(forms)f(and)g(invariants)g(for)g(unitary)f(simi-)305 1748 y(larity)p FE(,)e(Linear)g(Algebra)h(Appl.)f Fh(147)f FE(\(1991\),)k(101{167.)118 1836 y([260])42 b(A.)23 b(N.)h(Shark)n(o)n (vski)-8 b(\025)-27 b(\020,)24 b(Y)-6 b(u.)23 b(L.)h(Maistrenk)n(o,)g (and)h(E.)e(Y)-6 b(u.)24 b(Romanenk)n(o,)g Fi(Di\013er)l(enc)l(e)305 1915 y(e)l(quations)34 b(and)g(their)f(applic)l(ations)p FE(,)j(Klu)n(w)n(er)c(Acad.)g(Publ.,)i(Dordrec)n(h)n(t,)g(1993,)305 1994 y(T)-6 b(ransl.)22 b(from)g(Russian)h(edn.:)32 b(Nauk)n(o)n(v)l(a) 25 b(Dumk)l(a,)d(Kiev,)i(1986.)118 2082 y([261])42 b(A.)24 b(N.)g(Shark)n(o)n(vsky)-6 b(,)27 b(S.)d(F.)h(Koly)n(ada,)g(A.)g(G.)g (Siv)l(ak,)g(and)h(V.)e(V.)g(F)-6 b(edorenk)n(o,)27 b Fi(Dy-)305 2161 y(namics)j(of)g(one-dimensional)i(maps)p FE(,)e(Klu)n(w)n(er)e(Acad.)h(Publ.,)f(Dordrec)n(h)n(t,)i(1997,)305 2240 y(T)-6 b(ransl.)22 b(from)g(Russian)h(edn.:)32 b(Nauk)n(o)n(v)l(a) 25 b(Dumk)l(a,)d(Kiev,)i(1989.)118 2328 y([262])42 b(F.)20 b(V.)g(Shirok)n(o)n(v,)h Fi(Pr)l(o)l(of)j(of)f(the)g(Kaplansky)g(hyp)l (othesis)p FE(,)f(Usp)r(ekhi)f(Mat.)f(Nauk)h Fh(11)305 2407 y FE(\(1956\),)k(167{168,)g(\(Russian\).)118 2495 y([263])42 b(V.)d(S.)h(Sh)n(ulman,)k Fi(Multiplic)l(ation)e(op)l(er)l (ators)h(and)f(sp)l(e)l(ctr)l(al)g(synthesis)p FE(,)i(Dokl.)305 2574 y(Ak)l(ad.)23 b(Nauk)i(SSSR)f Fh(313)e FE(\(1990\),)j(no.)f(5,)f (1047{1051,)j(\(Russian\).)118 2662 y([264])42 b(S.)29 b(D.)g(Silv)n(estro)n(v,)i Fi(Hilb)l(ert)g(sp)l(ac)l(e)h(r)l(epr)l (esentations)g(of)g(the)f(gr)l(ade)l(d)i(analo)l(gue)g(of)305 2741 y(the)28 b(Lie)f(algebr)l(a)i(of)f(the)g(gr)l(oup)h(of)f(plane)h (motions)p FE(,)f(Studia)f(Math.)f Fh(117)f FE(\(1996\),)305 2820 y(no.)e(2,)h(195{203.)118 2908 y([265])p 305 2908 V 265 w(,)h Fi(R)l(epr)l(esentations)j(of)g(c)l(ommutation)g(r)l (elations.)g(A)f(dynamic)l(al)i(systems)305 2987 y(appr)l(o)l(ach)p FE(,)d(Hadronic)e(Journ.)f(Suppl.)h Fh(11)f FE(\(1996\),)i(no.)f(1,)f (1{116.)118 3075 y([266])42 b(S.)29 b(D.)f(Silv)n(estro)n(v)i(and)g(L.) f(B.)f(T)-6 b(uro)n(wsk)l(a,)31 b Fi(R)l(epr)l(esentations)h(of)f(the)g Fq(q)r Fi(-deforme)l(d)305 3154 y(Lie)20 b(algebr)l(a)j(of)f(the)f(gr)l (oup)i(of)f(motions)g(of)g(the)f(Euclide)l(an)i(plane)p FE(,)e(J.)e(F)-6 b(unct.)20 b(Anal.)305 3233 y Fh(160)i FE(\(1998\),)j(79{114.)118 3321 y([267])42 b(S.)19 b(D.)g(Silv)n(estro) n(v)g(and)h(H.)f(W)-6 b(allin,)20 b Fi(R)l(epr)l(esentations)i(of)g (algebr)l(as)h(asso)l(ciate)l(d)g(with)305 3400 y(M\177)-36 b(obius)22 b(tr)l(ansformation)p FE(,)g(J.)d(Nonlin.)g(Math.)g(Ph)n (ys.)h Fh(3)f FE(\(1996\),)j(no.)d(1-2,)h(202{213.)118 3488 y([268])42 b(Y)-6 b(a.)33 b(G.)h(Sinai,)h Fi(Mo)l(dern)h(pr)l (oblems)h(of)e(er)l(go)l(dic)h(the)l(ory)p FE(,)g(Mo)r(dern)e(Problems) e(of)305 3567 y(Mathematics,)23 b(Fizik)n(o-Matematic)n(hesk)l(a)n(y)n (a)i(Literatura,)f(Mosco)n(w,)g(1995.)118 3655 y([269])42 b(E.)16 b(K.)h(Skly)n(anin,)h Fi(On)h(some)h(algebr)l(aic)g(structur)l (es)g(r)l(elate)l(d)g(to)g(Yang{Baxter)g(e)l(qua-)305 3734 y(tion)p FE(,)j(F)-6 b(unkt.)24 b(Anal.)f(i)g(Prilozhen.)h Fh(16)f FE(\(1982\),)i(no.)e(4,)h(27{34,)g(\(Russian\).)118 3822 y([270])p 305 3822 V 265 w(,)31 b Fi(On)f(some)i(algebr)l(aic)f (structur)l(es)g(r)l(elate)l(d)h(to)f(the)g(Yang{Baxter)h(e)l(qua-)305 3901 y(tion.)j(II.)g(R)l(epr)l(esentations)h(of)g(quantum)g(algebr)l(a) p FE(,)g(F)-6 b(unct.)35 b(Anal.)f(Prilozh.)e Fh(17)305 3980 y FE(\(1983\),)25 b(no.)e(4,)h(34{48,)g(\(Russian\).)118 4068 y([271])42 b(A.)25 b(S.)g(Smogorzhevski)-8 b(\025)-27 b(\020)25 b(and)i(E.)e(S.)g(Stolo)n(v)l(a,)i Fi(Handb)l(o)l(ok)j(in)d (the)g(the)l(ory)h(of)g(plane)305 4147 y(curves)d(of)h(the)f(thir)l(d)i (or)l(der)p FE(,)d(Fizmatgiz,)f(Mosco)n(w,)g(1961,)i(\(Russian\).)p eop %%Page: 256 260 256 259 bop 118 100 a FP(256)1866 b FK(Bibliograph)n(y)118 333 y FE([272])42 b(Y)-6 b(a.)32 b(S.)h(Soib)r(elman)f(and)h(L.)f(L.)h (V)-6 b(aksman,)34 b Fi(The)g(algebr)l(a)h(of)f(functions)g(on)h(the) 305 412 y(quantum)23 b(gr)l(oup)h Fq(S)t(U)7 b FE(\()p Fq(n)i FE(+)g(1\))24 b Fi(and)f(o)l(dd-dimensional)j(quantum)d(spher)l (es)p FE(,)f(Algebra)305 490 y(i)h(Analiz)g Fh(2)g FE(\(1990\),)j(no.)d (5,)h(101{120,)h(\(Russian\).)118 579 y([273])42 b(S.)25 b(Stratila)i(and)f(D.)g(V)-6 b(oiculescu,)27 b Fi(R)l(epr)l (esentations)h(of)g(AF-algebr)l(as)h(and)g(of)f(the)305 658 y(gr)l(oup)f Fq(U)7 b FE(\()p FD(1)p FE(\),)23 b(Lect.)h(Notes.)g (Math.,)f(v)n(ol.)h(486,)g(Springer,)f(1975.)118 747 y([274])42 b(M.)22 b(T)-6 b(ak)n(esaki,)25 b Fi(The)l(ory)h(of)g(op)l (er)l(ator)h(algebr)l(as)p FE(,)d(Springer,)f(1979.)118 835 y([275])42 b(P)-6 b(.)27 b(K.)g(T)-6 b(am,)27 b Fi(On)i(the)h (unitary)f(e)l(quivalenc)l(e)h(of)f(c)l(ertain)h(classes)g(of)f (non-normal)305 914 y(op)l(er)l(ators.)e(I)p FE(,)d(Canad.)g(J.)f (Math.)h Fh(23)f FE(\(1971\),)i(no.)f(5,)f(849{856.)118 1003 y([276])42 b(P)-6 b(.)22 b(T)-6 b(app)r(er,)22 b Fi(Emb)l(e)l(dding)k FD(\003)p Fi(-algebr)l(as)f(into)g Fq(C)1539 979 y Fg(\003)1575 1003 y Fi(-algebr)l(as)p FE(,)d(Ph.)g(D.)g(thesis,)g(Univ.)g(of)305 1082 y(Leeds,)i(1996.)118 1170 y([277])42 b(E.)29 b(Thoma,)685 1154 y Fi(\177)672 1170 y(Ub)l(er)i(unit\177)-36 b(ar)l(e)32 b(Darstel)t(lungen)f(abz\177) -36 b(ahlb)l(ar)l(er,)35 b(diskr)l(etten)c(Grupp)l(en)p FE(,)305 1249 y(Math.)23 b(Ann.)h Fh(153)e FE(\(1964\),)j(111{138.)118 1338 y([278])42 b(J.)d(T)-6 b(omijama,)42 b Fi(Invitation)f(to)f(the)h Fq(C)1408 1315 y Fg(\003)1444 1338 y Fi(-algebtas)f(and)i(top)l(olo)l (gic)l(al)i(dynamics)p FE(,)305 1417 y(W)-6 b(orld)23 b(Sci.)h(Publ.,)e(Singap)r(ore,)i(1987.)118 1506 y([279])42 b(D.)25 b(M.)g(T)-6 b(opping,)27 b Fi(L)l(e)l(ctur)l(es)g(on)h(von)g (Neumann)h(algebr)l(as)p FE(,)e(V)-6 b(an)26 b(Nostrand)g(Rein-)305 1584 y(hold)e(Comp.,)d(London,)k(1971.)118 1673 y([280])42 b(L.)18 b(T)-6 b(uro)n(wsk)l(a)n(y)n(a,)21 b Fi(R)l(epr)l(esentations)h (of)g(some)g(r)l(e)l(al)g(forms)h(of)e Fq(U)2013 1681 y Ff(q)2047 1673 y FE(\()p Fq(sl)q FE(\(3\)\),)g(Algebras,)305 1752 y(Groups)j(and)g(Geometries)f Fh(12)g FE(\(1995\),)i(321{338.)118 1841 y([281])42 b(E.)24 b(Twietmey)n(er,)h Fi(R)l(e)l(al)j(forms)f(of)g Fq(U)1301 1849 y Ff(q)1335 1841 y FE(\()p Fa(J)p FE(\),)f(Lett.)g (Math.)f(Ph)n(ys.)g Fh(24)f FE(\(1992\),)j(no.)e(1,)305 1920 y(49{59.)118 2008 y([282])42 b(L.)19 b(L.)h(V)-6 b(aksman)19 b(and)i(L.)e(I.)h(Korogo)r(dskii,)g Fi(The)j(algebr)l(a)g (of)f(b)l(ounde)l(d)i(functions)e(on)305 2087 y(the)28 b(quantum)h(gr)l(oup)h(of)f(motions)g(of)g(the)f(plane)i(and)f Fq(q)r Fi(-analo)l(g)h(of)f(Bessel)f(func-)305 2166 y(tions)p FE(,)23 b(Dokl.)g(Ak)l(ad.)h(Nauk)g(USSR)g Fh(304)e FE(\(1989\),)j(no.) f(5,)g(1036{1040,)h(\(Russian\).)118 2255 y([283])42 b(F.-H.)23 b(V)-6 b(asilescu,)26 b Fi(A)n(ntic)l(ommuting)h (selfadjoint)h(op)l(er)l(ators)p FE(,)f(Rev.)f(Roum.)e(Math.)305 2334 y(Pures)f(Appl.)g Fh(28)g FE(\(1983\),)i(77{91.)118 2422 y([284])42 b(A.)29 b(N.)g(V)-6 b(asil'ev,)30 b Fi(The)l(ory)i(of)g (r)l(epr)l(esentations)g(of)f(a)h(top)l(olo)l(gic)l(al)i (\(non-Banach\))305 2501 y(involutary)26 b(algebr)l(a)p FE(,)e(T)-6 b(eor.)23 b(Math.)h(Ph)n(ys.)f Fh(2)g FE(\(1970\),)j (113{123.)118 2590 y([285])42 b(N.)20 b(V)-6 b(asilevski,)20 b Fq(C)796 2566 y Fg(\003)832 2590 y Fi(-algebr)l(as)j(gener)l(ate)l(d) h(by)e(pr)l(oje)l(ctions)i(and)f(their)g(applic)l(ations)p FE(,)305 2669 y(In)n(tegr.)h(Equat.)g(Op)r(er.)f(Theory)i Fh(31)e FE(\(1998\),)j(113{132.)118 2757 y([286])42 b(N.)25 b(V)-6 b(asilevski)26 b(and)h(I.)f(Spitk)n(o)n(vski,)h Fi(On)h(algebr)l(a)h(gener)l(ate)l(d)f(by)f(two)i(pr)l(oje)l(ctions)p FE(,)305 2836 y(Dokl.)23 b(Ak)l(ad.)h(Nauk)g(Ukr.)e(SSR,)i(Ser.)f(A)g Fh(8)h FE(\(1981\),)h(10{13,)f(\(Russian\).)118 2925 y([287])42 b(E.)21 b(Y)-6 b(e.)21 b(V)-6 b(a)n(ysleb,)22 b Fi(In\014nite-dimensional)j FD(\003)p Fi(-r)l(epr)l(esentations)f(of) g(Sklyanin)g(algebr)l(a)305 3004 y(in)19 b(de)l(gener)l(ate)h(c)l(ase)h (\(the)e(quantum)i(algebr)l(a)f Fq(U)1589 3012 y Ff(q)1623 3004 y FE(\()p Fq(sl)q FE(\(2\)\))p Fi(\))p FE(,)g(Metho)r(ds)e(of)f(F) -6 b(unctional)305 3083 y(Analysis)18 b(in)h(problems)f(of)h (Mathematical)g(Ph)n(ysics,)h(Inst.)f(Math.)g(Acad.)h(Sci.)f(Ukr.)305 3162 y(SSR,)k(Kiev,)g(1990,)i(pp.)e(50{62,)i(\(Russian\).)118 3250 y([288])p 305 3250 212 4 v 265 w(,)19 b Fi(R)l(epr)l(esentations)i (of)g(r)l(elations)h(which)f(c)l(onne)l(ct)g(a)g(family)f(of)h(c)l (ommuting)305 3329 y(op)l(er)l(ators)35 b(with)e(non-sefadjoint)h(one)p FE(,)f(Ukrain.)e(Math.)h(Zh.)f Fh(42)g FE(\(1990\),)k(1258{)305 3408 y(1262,)24 b(\(Russian\).)118 3497 y([289])42 b(E.)19 b(Y)-6 b(e.)20 b(V)-6 b(a)n(ysleb)21 b(and)g(V.)e(V.)h(F)-6 b(edorenk)n(o,)21 b Fi(R)l(epr)l(esentations)j(of)e(op)l(er)l(ator)j(r) l(elations)305 3576 y(and)e(one-dimensional)h(dynamic)l(al)g(systems)p FE(,)d(Application)g(of)f(Metho)r(ds)h(of)f(F)-6 b(unc-)305 3654 y(tional)18 b(Analysis)g(in)f(Mathematical)i(Ph)n(ysics,)g(Inst.)f (Math.)g(Ak)l(ad.)h(Nauk)f(UkrSSR,)305 3733 y(Kiev,)23 b(1989,)h(\(Russian\),)g(pp.)g(12{20.)118 3822 y([290])42 b(E.)27 b(Y)-6 b(e.)27 b(V)-6 b(a)n(ysleb)28 b(and)g(Y)-6 b(u.)27 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)28 b Fi(R)l(epr)l(esentations)i(of)f(op)l(er)l(ator)i(r)l(ela-)305 3901 y(tions)e(by)g(unb)l(ounde)l(d)i(op)l(er)l(ators)h(and)e (multi-dimensional)h(dynamic)l(al)f(systems)p FE(,)305 3980 y(Ukrain.)23 b(Math.)g(Zh.)h Fh(42)f FE(\(1990\),)i(no.)e(9,)h (1011{1019,)h(\(Russian\).)118 4068 y([291])p 305 4068 V 265 w(,)c Fi(R)l(epr)l(esentations)j(of)g(the)f(r)l(elations)h Fq(AU)i FE(=)19 b Fq(U)7 b(F)j FE(\()p Fq(A)p FE(\))24 b Fi(by)e(unb)l(ounde)l(d)j(self-)305 4147 y(adjoint)f(and)g(unitary)f (op)l(er)l(ators)p FE(,)g(Selecta)g(Math.)e(So)n(v.)g Fh(13)f FE(\(1994\),)j(no.)e(1,)h(35{54.)p eop %%Page: 257 261 257 260 bop 118 100 a FK(Bibliograph)n(y)1865 b FP(257)118 333 y FE([292])42 b(A.)28 b(M.)h(V)-6 b(ershik,)30 b Fi(A)n(lgebr)l(as)h(with)g(quadr)l(atic)h(r)l(elations)p FE(,)f(Sp)r(ectral)g(theory)f(of)f(op-)305 412 y(erators)e(and)i (in\014nite-dimensional)e(analysis,)h(Inst.)g(Math)g(Acad.)g(Sci.)f (Ukraine,)305 490 y(Kiev,)c(1984,)h(\(Russian\),)g(pp.)g(32)g({)f(56.) 118 575 y([293])42 b(A.)30 b(M.)h(V)-6 b(ershik,)32 b(I.)f(M.)g (Gelfand,)i(and)f(M.)e(I.)i(Graev,)h Fi(R)l(epr)l(esentations)h(of)f (the)305 654 y(gr)l(oup)24 b Fq(S)t(L)p FE(\(2)p Fq(;)12 b(R)p FE(\))p Fi(,)25 b(wher)l(e)f Fq(R)g Fi(is)g(a)f(function)h(ring)p FE(,)d(Usp)r(ehi)g(Mat.)h(Nauk)g Fh(28)e FE(\(1973\),)305 733 y(no.)j(5,)h(83{128,)h(\(Russian\).)118 817 y([294])p 305 817 212 4 v 265 w(,)34 b Fi(Commutative)f(mo)l(del)i(of)e(the)g(r)l (epr)l(esentation)i(of)e(the)g(curr)l(ent)g(gr)l(oup)305 896 y Fq(S)t(L)p FE(\(2)p Fq(;)11 b Fb(R)p FE(\))568 873 y Ff(X)660 896 y Fi(r)l(elate)l(d)34 b(to)g(the)f(unip)l(otent)h (sub)l(gr)l(oup)p FE(,)h(F)-6 b(unct.)33 b(Anal.)e(Priolozh.)g Fh(17)305 975 y FE(\(1983\),)25 b(no.)e(2,)h(70{72,)g(\(Russian\).)118 1059 y([295])42 b(M.)31 b(Vlasenk)n(o,)j(A.)d(Mellit,)i(and)g(Y)-6 b(u.)32 b(Samoilenk)n(o,)h Fi(On)g(algebr)l(as)i(gener)l(ate)l(d)e(by) 305 1138 y(line)l(arly)24 b(c)l(onne)l(cte)l(d)h(elements)f(with)h (given)e(sp)l(e)l(ctrum)p FE(,)g(F)-6 b(unct.)23 b(Anal.)e(Prilozh.)g Fh(38)305 1217 y FE(\(2004\),)k(to)f(app)r(ear.)118 1302 y([296])42 b(D.)16 b(V.)h(V)-6 b(oiculescu,)19 b(K.)d(J.)h(Dyk)n(ema,)h (and)g(A.)e(Nica,)i Fi(F)-5 b(r)l(e)l(e)20 b(r)l(andom)i(variables)p FE(,)c(CRM)305 1381 y(Monograph)i(Ser.,)g(Cen)n(tre)g(de)g(Rec)n(herc)n (hes)h(Math.)e(Univ.)g(Motr)n(\023)-33 b(eal,)20 b(v)n(ol.)g(1,)g (Amer.)305 1460 y(Math.)j(So)r(c.,)h(Pro)n(vidence,)g(R.)f(I.,)g(1992.) 118 1544 y([297])42 b(Y.)29 b(W)-6 b(eiss,)32 b Fi(On)f(algebr)l(as)i (gener)l(ate)l(d)f(by)f(two)h(idemp)l(otents)p FE(,)h(Seminar)c (Analysis:)305 1623 y(Op)r(erator)18 b(Equations)h(and)g(Numer.)e (Anal.)g(1987/88)j(\(Berlin\),)f(Karl-W)-6 b(eierstrass-)305 1702 y(Institut)25 b(f)r(\177)-37 b(ur)23 b(Mathematik,)g(1988,)h(pp.)g (139{145.)118 1786 y([298])42 b(H.)21 b(W)-6 b(enzl,)22 b Fi(R)l(epr)l(esentations)j(of)f(br)l(aid)h(gr)l(oups)g(and)g(the)f (quantum)g(Yang{Baxter)305 1865 y(e)l(quation)p FE(,)g(P)n(acif.)e(J.)i (Math.)g Fh(145)e FE(\(1990\),)j(153{180.)118 1950 y([299])42 b(J.)25 b(Wic)n(hmann,)g Fi(Hermitian)i FD(\003)p Fi(-algebr)l(as)h (which)g(ar)l(e)g(not)g(symmetric)p FE(,)c(J.)i(London)305 2029 y(Math.)d(So)r(c.)h Fh(8)g FE(\(1974\),)h(109{112.)118 2113 y([300])42 b(H.)35 b(Wielandt,)768 2096 y Fi(\177)755 2113 y(Ub)l(er)i(die)g(Unb)l(eschr\023)-36 b(anktheit)37 b(der)g(Op)l(er)l(ator)l(en)h(des)f(Quanten-)305 2192 y(me)l(chanik)p FE(,)24 b(Math.)f(Ann.)h Fh(121)e FE(\(1949\),)j(21.) 118 2277 y([301])42 b(E.)30 b(Witten,)35 b Fi(Gauge)e(the)l(ories,)i (vertex)c(mo)l(dels,)36 b(and)e(quantum)f(gr)l(oups)p FE(,)h(Repts.)305 2355 y(Nuclear)23 b(Ph)n(ys.)h Fh(b330)f FE(\(1990\),)i(285{346.)118 2440 y([302])42 b(W.)35 b(R.)g(W)-6 b(ogen,)39 b Fi(On)e(gener)l(ators)g(for)f(von)h(Neumann)h(algebr)l(as) p FE(,)g(Bull.)d(Amer.)305 2519 y(Math.)23 b(So)r(c.)h Fh(75)f FE(\(1969\),)i(95{99.)118 2603 y([303])p 305 2603 V 265 w(,)19 b Fi(On)h(sp)l(e)l(cial)i(gener)l(ators)f(for)f(pr)l (op)l(erly)j(in\014nite)c(von)i(Neumann)g(algebr)l(as)p FE(,)305 2682 y(Pro)r(c.)i(Amer.)f(Math.)h(So)r(c.)h Fh(28)f FE(\(1971\),)i(no.)f(1,)f(107{113.)118 2767 y([304])42 b(S.)24 b(L.)h(W)-6 b(orono)n(wicz,)26 b Fi(Comp)l(act)i(matrix)e (pseudo)l(gr)l(oups)p FE(,)i(Comm)n(un.)23 b(Math.)i(Ph)n(ys.)305 2846 y Fh(111)d FE(\(1987\),)j(613{665.)118 2930 y([305])p 305 2930 V 265 w(,)d Fi(Quantum)j Fq(E)t FE(\(2\))g Fi(gr)l(oup)g(and)h (its)e(Pontryagin)g(dual)p FE(,)g(Lett.)f(Math.)f(Ph)n(ys.)305 3009 y Fh(23)h FE(\(1991\),)i(251{263.)118 3094 y([306])p 305 3094 V 265 w(,)c Fi(Unb)l(ounde)l(d)k(elements)e(a\016liate)l(d)h (with)g Fq(C)1718 3070 y Fg(\003)1754 3094 y Fi(-algebr)l(as)f(and)h (non-c)l(omp)l(act)305 3172 y(quantum)i(gr)l(oups)p FE(,)f(Comm)n(un.)c (Math.)j(Ph)n(ys.)f Fh(136)g FE(\(1991\),)i(399{432.)118 3257 y([307])p 305 3257 V 265 w(,)f Fq(C)627 3234 y Fg(\003)663 3257 y Fi(-algebr)l(as)i(gener)l(ate)l(d)h(by)e(unb)l(ounde)l(d)j (elements)p FE(,)c(Rev.)g(Math.)g(Ph)n(ys.)305 3336 y Fh(7)f FE(\(1995\),)i(no.)f(3,)f(481{521.)118 3420 y([308])42 b(Sh.)30 b(Y)-6 b(amagami,)30 b Fi(On)h(unitary)h(r)l(epr)l(esentation) h(the)l(ories)f(of)h(c)l(omp)l(act)g(quantum)305 3499 y(gr)l(oups)p FE(,)24 b(Comm)n(un.)e(Math.)h(Ph)n(ys.)h Fh(167)e FE(\(1995\),)k(509{529.)118 3584 y([309])42 b(C.)24 b(Zac)n(hos,)j Fi(Elementary)g(p)l(ar)l(adigms)j(of)d(quantum)h (algebr)l(as)p FE(,)e(Con)n(t.)f(Math.)h Fh(134)305 3663 y FE(\(1992\),)f(351{377.)118 3747 y([310])42 b(S.)24 b(Zakrzewski,)h Fi(R)l(e)l(ali\014c)l(ations)j(of)f(c)l(omplex)h (quantum)f(gr)l(oups)p FE(,)f(Groups)f(and)g(re-)305 3826 y(lated)30 b(topics)h(\(R.)f(I.)g(Gielerak)g(et)h(al.,)f(ed.\),)i (Klu)n(w)n(er)d(Acad.)i(Publ.,)f(Dordrec)n(h)n(t,)305 3905 y(1992,)24 b(pp.)f(83{100.)118 3990 y([311])42 b(A.)19 b(S.)g(Zhedano)n(v,)j Fq(Q)p Fi(-r)l(otations)g(and)h(other)f Fq(Q)p Fi(-tr)l(ansformations)h(as)f(unitary)g(non-)305 4068 y(line)l(ar)37 b(automorphisms)i(of)e(quantum)g(algebr)l(as)p FE(,)h(J.)d(Math.)h(Ph)n(ys.)f Fh(35)f FE(\(1994\),)305 4147 y(3756{3764.)p eop %%Page: 258 262 258 261 bop 118 100 a FP(258)1866 b FK(Bibliograph)n(y)118 333 y FE([312])42 b(D.)19 b(P)-6 b(.)21 b(Zhelob)r(enk)n(o,)h Fi(Comp)l(act)i(Lie)e(gr)l(oups)i(and)f(their)g(r)l(epr)l(esentations)p FE(,)e(T)-6 b(ransla-)305 412 y(tions)24 b(of)g(Math.)h(Monographs.,)f (v)n(ol.)g(40,)g(Amer.)f(Math.)h(So)r(c.,)h(Pro)n(vidence,)g(R.I.,)305 490 y(1973,)f(T)-6 b(rans.)23 b(from)f(Russian)h(edn.:)31 b(Nauk)l(a,)25 b(Mosco)n(w,)e(1970.)p eop %%Page: 259 263 259 262 bop 118 748 a FS(Index)118 1163 y FP(algebra)284 1263 y FO(F)337 1275 y FM(n)383 1263 y FP(-algebra,)25 b(27)284 1363 y(en)n(v)n(eloping)h(pro-)p FO(C)906 1333 y FN(\003)944 1363 y FP(,)h(20)284 1463 y(en)n(v)n(eloping)f FQ(\003)p FP(-algebra,)f(15)284 1563 y(en)n(v)n(eloping)h FO(\033)s FP(-)p FO(C)835 1533 y FN(\003)874 1563 y FP(,)i(17)284 1663 y(en)n(v)n(eloping)e FO(C)757 1633 y FN(\003)796 1663 y FP(,)i(16)284 1764 y(residually)18 b(\014nite)g(dimensional,)450 1863 y(13)118 1963 y FQ(\003)p FP(-algebra)284 2063 y FQ(\003)p FP(-b)r(ounded,)27 b(16)284 2164 y FQ(\003)p FP(-wild,)g(214)284 2264 y FO(C)349 2234 y FN(\003)388 2264 y FP(-represen)n(tatble,)f(11)284 2364 y FO(\033)s FP(-)p FO(C)427 2334 y FN(\003)466 2364 y FP(-represen)n(tatble,)g(20) 284 2464 y(\014nitely)i(generated,)f(21)284 2564 y(generated)17 b(b)n(y)h(idemp)r(oten)n(ts,)450 2664 y(29)284 2764 y(group,)27 b(11)284 2864 y(lo)r(cally)g FO(C)612 2834 y FN(\003)651 2864 y FP(,)g(20)284 2964 y(wild,)h(212)284 3064 y(with)h(generators)d (and)j(rela-)450 3164 y(tions,)f(16)118 3355 y(Can)n(tor)e(set,)i(106) 118 3455 y FO(\026)p FP(-CAR)g(algebra,)e(158)118 3555 y FO(\026)p FP(-CCR)i(algebra,)d(156)118 3656 y(comm)n(utativ)n(e)i(mo) r(del,)h(180)118 3756 y(con)n(tin)n(ued)f(fractions,)g(84,)g(95)118 3856 y(Cun)n(tz)h(algebra,)e(189)118 4047 y(de\014ciency)i(indices,)f (73)118 4147 y(dynamical)g(system,)g(89)1658 1163 y(cycle,)g(89)1658 1262 y(measurable)f(section,)h(90)1658 1362 y(non-bijectiv)n(e,)g(91) 1658 1462 y(orbit,)g(89)1658 1561 y(p)r(erio)r(dic)g(p)r(oin)n(t,)h(89) 1658 1661 y(triangular,)e(155)1492 1839 y(F)-7 b(airlie)27 b(algebra,)f(133)1492 2018 y(graded)g FO(so)p FP(\(3\),)i(107)1492 2118 y(group)1658 2217 y FQ(\003)p FP(-wild,)f(227)1658 2317 y(amenable,)g(229)1658 2416 y(Burnside,)g(229)1658 2516 y(Co)n(xeter,)f(32)1741 2616 y(not)h(a\016ne,)h(228)1658 2715 y(h)n(yp)r(erb)r(olic,)f(228)1658 2815 y(p)r(erio)r(dic,)g(229) 1658 2915 y(residually)f(\014nite,)j(13)1492 3093 y(Heisen)n(b)r(erg)d (relations,)h(169)1492 3271 y(idemp)r(oten)n(ts)1658 3371 y(comm)n(uting)g(pairs,)g(220)1658 3471 y(pairwise)f(orthogonal,)g (220)1492 3570 y(in)n(v)n(olution)1658 3670 y(completely)h(prop)r(er,)g (12)1658 3770 y(prop)r(er,)g(12)1492 3869 y(isomertries)1658 3969 y(comm)n(uting)g(pair,)g(237)1492 4147 y(Jacobi)f(matrix,)h(73) 1284 4357 y(259)p eop %%Page: 260 264 260 263 bop 118 100 a FP(260)118 333 y(Lie)28 b(algebras)284 432 y(nonlinear)18 b(transformations,)450 532 y(26)118 713 y(ma)5 b(jorization,)26 b(203,)g(206)118 813 y(marginally)g(n)n (ull)i(subset,)g(61)118 913 y(measurable)e(section,)i(65)118 1012 y(measure)284 1112 y(ergo)r(dic,)f(62,)f(90)284 1212 y(pro)r(duct,)i(194)284 1311 y(quasi-in)n(v)-5 b(arian)n(t,)26 b(62,)g(89)118 1493 y(non-comm)n(utativ)n(e)g(curv)n(es)284 1592 y(circle,)h(39)284 1692 y(h)n(yp)r(erb)r(ola,)g(39)284 1792 y(pair)34 b(of)g(in)n(tersecting)g(lines,)450 1891 y(39)118 2073 y(op)r(erator)284 2172 y(algebraic,)26 b(236)284 2272 y(cen)n(tered,)h(86,)g(185)367 2372 y(partial)g (isometry)-7 b(,)27 b(97)284 2471 y(h)n(yp)r(onormal,)f(236)284 2571 y(in)n(tert)n(wining,)h(8)284 2671 y(non-self-adjoin)n(t,)g(230) 284 2770 y(partial)g(isometry)-7 b(,)27 b(234)284 2870 y(pseudo-in)n(tegral,)f(59)284 2969 y(quasi-normal,)g(233)284 3069 y(supp)r(orting)h(set,)h(49)284 3169 y(w)n(eakly)f(cen)n(tered,)g (234)118 3268 y(op)r(erators)284 3368 y(b)r(ounded)20 b(self-adjoin)n(t)g(pair,)450 3468 y(22)284 3567 y(irreducible)27 b(family)-7 b(,)28 b(21)118 3749 y(p)r(olynomial)284 3848 y(standard,)f(27)118 3948 y(pro)5 b(jections)284 4048 y(all)18 b(but)h(one)f(orthogonal,)g(217)284 4147 y(four-tuples,)27 b(107,)g(111,)f(217)1492 333 y(quan)n(tum)h(disk,)h (84,)f(102,)f(105)1492 433 y(quan)n(tum)h(sphere,)g(166)1492 632 y(real)f(quan)n(tum)i(h)n(yp)r(erb)r(oloid,)f(75)1492 733 y(real)f(quan)n(tum)i(plane,)f(75)1492 833 y(relations)1658 934 y FO(F)1711 946 y FL(4)1748 934 y FP(-relations,)g(26)1658 1035 y FO(q)s FP(-relations,)f(26)1658 1135 y(quadratic,)g(24)1741 1236 y(homogeneous,)g(24)1658 1336 y(represen)n(tation,)g(21)1658 1437 y(semilinear,)h(44)1741 1537 y(c)n(haracteristic)16 b(binary)i(re-)1824 1637 y(lation,)27 b(47)1741 1738 y(c)n(haracteristic)h(function,)1824 1837 y(47)1741 1938 y(graph,)e(47)1658 2038 y FQ(\003)p FP(-wild,)h(26)1741 2139 y(cubic,)h(222)1741 2239 y(quadratic,)e(222)1741 2340 y(semilinear,)h(221)1492 2441 y(represen)n(tation,)f(7)1658 2541 y FQ(\003)p FP(-complete,)h(62)1658 2642 y(an)n(ti-F)-7 b(o)r(c)n(k,)26 b(100)1658 2742 y(F)-7 b(o)r(c)n(k,)27 b(74,)g(100)1658 2843 y(indecomp)r(osable,)g(9)1658 2943 y(irreducible,)g(9)1492 3044 y(represen)n(tations)1658 3145 y(category)e FQ(\003)p FP(-Rep)14 b FA(A)p FP(,)27 b(8)1658 3245 y(equaiv)-5 b(alen)n(t,)27 b(7)1658 3346 y(residual)g(family)-7 b(,)27 b(11)1658 3446 y(unitarily)g(equiv)-5 b(alen)n(t,)27 b(8)1492 3547 y(resolution)f(of)i(the)g(iden)n(tit)n(y) 1658 3647 y(non-orthogonal,)d(110)1492 3847 y(second-degree)16 b(mapping,)k(83,)f(94,)1824 3946 y(100,)26 b(105,)g(106)1492 4047 y(Skly)n(anin)h(algebra,)f(123)1492 4147 y(standard)g FQ(\003)p FP(-wild)h(problem,)h(213)p eop %%Page: 261 265 261 264 bop 2450 100 a FP(261)118 333 y(theorem)284 432 y(Amitsur{Levitski,)27 b(27)284 532 y(F)-7 b(uglede{Putnam{Rosen)n (blum,)450 632 y(75)284 731 y(Jacobson,)26 b(44)284 831 y(Kleinec)n(k)n(e{Shirok)n(o)n(v,)e(43)284 930 y(Kleinec)n(k)n (e{Shirok)n(o)n(v)32 b(t)n(yp)r(e,)450 1030 y(44,)27 b(48)284 1130 y(Shark)n(o)n(vsky)-7 b(,)25 b(93)118 1229 y(t)n(wisted)j(CAR,)g(155)118 1329 y(t)n(wisted)g(CCR,)g(155)118 1512 y(Wic)n(k)g(algebra,)d(173)118 1611 y(Wic)n(k)j(ideal,)f(174)118 1711 y(Wic)n(k)h(ordered)e(monomials,)g(174)118 1810 y FQ(\003)p FP(-wildness,)h(203,)f(212)118 1910 y(Witten's)19 b(deformations)f(of)g FO(so)p FP(\(3\),)450 2010 y(119)118 2109 y(W)-7 b(old)28 b(decomp)r(osition,)f(75)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF