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\contentsline {section}{Preface}{{\reset@font 4}} \contentsline {chapter}{\numberline {1}Pairs of self-adjoint operators connected by quadratic relations and some generalizations}{{\reset@font 8}} \contentsline {section}{\numberline {1.1}Introduction to representations of $*$-algebras}{{\reset@font 8}} \contentsline {subsection}{\numberline {1.1.1}$*$-Representations: key words}{{\reset@font 8}} \contentsline {subsection}{\numberline {1.1.2}$C^*$-representable $*$-algebras}{{\reset@font 12}} \contentsline {subsection}{\numberline {1.1.3}Enveloping $*$-algebras and $C^*$-algebras}{{\reset@font 16}} \contentsline {subsection}{\numberline {1.1.4}$*$-Representations of generators and relations}{{\reset@font 22}} \contentsline {subsection}{\numberline {1.1.5}Pairs of self-adjoint operators satisfying quad\discretionary {-}{}{}ratic relations}{{\reset@font 25}} \contentsline {section}{\numberline {1.2}$F_n$-algebras and their representations}{{\reset@font 28}} \contentsline {subsection}{\numberline {1.2.1}About $*$-representations of ${F}_n$-algebras}{{\reset@font 28}} \contentsline {subsection}{\numberline {1.2.2}Examples of ${F}_n$-algebras generated by idempotents and their representations}{{\reset@font 30}} \contentsline {subsection}{\numberline {1.2.3}Non-commutative ``circle'', ``pair of intersecting li\discretionary {-}{}{}nes'' and ``hyperbola''. More examples of $F_4$-alge\discretionary {-}{}{}bras}{{\reset@font 40}} \contentsline {section}{\numberline {1.3}Representations of two-dimensional Lie algebras, their nonlinear transformations, and semilinear relations}{{\reset@font 43}} \contentsline {subsection}{\numberline {1.3.1}Representations of two-dimensional real Lie algebras and their nonlinear transformations by bounded operators}{{\reset@font 43}} \contentsline {subsection}{\numberline {1.3.2}Pairs of operators connected by semilinear relations}{{\reset@font 45}} \contentsline {subsection}{\numberline {1.3.3}Kleinecke--Shirokov type theorems}{{\reset@font 49}} \contentsline {subsection}{\numberline {1.3.4}Irreducible representations of semilinear relations}{{\reset@font 61}} \contentsline {subsection}{\numberline {1.3.5}Representations of semilinear ${F}_4$-relations}{{\reset@font 66}} \contentsline {section}{\numberline {1.4}Representations of $q$-relations}{{\reset@font 68}} \contentsline {subsection}{\numberline {1.4.1}Finitely-dimensional representations of $q$-rela\discretionary {-}{}{}tions}{{\reset@font 68}} \contentsline {subsection}{\numberline {1.4.2}Hermitian $q$-plane and $q$-CCR}{{\reset@font 71}} \contentsline {subsection}{\numberline {1.4.3}Real quantum plane and real quantum hyperboloid}{{\reset@font 76}} \contentsline {section}{Comments to Chapter 1}{{\reset@font 78}} \contentsline {chapter}{\numberline {2}Representations of dynamical $*$-algebras}{{\reset@font 82}} \contentsline {section}{\numberline {2.1}Operator relations and one-dimensional dynamical\penalty -\@M systems}{{\reset@font 82}} \contentsline {subsection}{\numberline {2.1.1}Operator relations connected with one-dimen\discretionary {-}{}{}sional dynamical systems}{{\reset@font 82}} \contentsline {subsection}{\numberline {2.1.2}Finite-dimensional representations}{{\reset@font 90}} \contentsline {subsection}{\numberline {2.1.3}Infinite-dimensional representations}{{\reset@font 95}} \contentsline {section}{\numberline {2.2}Some classes of $*$-algebras with 3 and 4 generators}{{\reset@font 105}} \contentsline {subsection}{\numberline {2.2.1}Representations of graded $so(3)$ and four-tup\discretionary {-}{}{}les of projections satisfying a linear relation}{{\reset@font 105}} \contentsline {subsection}{\numberline {2.2.2}Representations of a class of quadratic algebras with three generators}{{\reset@font 112}} \contentsline {subsection}{\numberline {2.2.3}Operator relations connected with a dynamical system on a plane}{{\reset@font 115}} \contentsline {subsection}{\numberline {2.2.4}Representation of real forms of Witten's first deformation}{{\reset@font 118}} \contentsline {subsection}{\numberline {2.2.5}Representations of the Sklyanin algebra and $U_q(sl(2))$}{{\reset@font 122}} \contentsline {section}{\numberline {2.3}Representations of $q$-deformed $U(so(3,{\@mathbb C}))$}{{\reset@font 132}} \contentsline {subsection}{\numberline {2.3.1}Real forms of $U_q(so(3, {\@mathbb C}))$}{{\reset@font 132}} \contentsline {subsection}{\numberline {2.3.2}Representations of $U_q(so(3,\@mathbb C))$}{{\reset@font 134}} \contentsline {section}{\numberline {2.4}Many-dimensional dynamical systems}{{\reset@font 150}} \contentsline {subsection}{\numberline {2.4.1}``Direct products'' of one-dimensional dynamical systems}{{\reset@font 151}} \contentsline {subsection}{\numberline {2.4.2}``Triangular'' dynamical systems.}{{\reset@font 154}} \contentsline {subsection}{\numberline {2.4.3}Operator relations connected with many-dimen\discretionary {-}{}{}si\discretionary {-}{}{}onal dynamical systems}{{\reset@font 160}} \contentsline {subsection}{\numberline {2.4.4}Representations of the non-standard real quantum sphere}{{\reset@font 165}} \contentsline {subsection}{\numberline {2.4.5}Heisenberg relations for the quantum $E(2)$\penalty -\@M group}{{\reset@font 168}} \contentsline {subsection}{\numberline {2.4.6}Wick algebras related to dynamical systems}{{\reset@font 172}} \contentsline {section}{\numberline {2.5}On representations of some nuclear algebras}{{\reset@font 179}} \contentsline {subsection}{\numberline {2.5.1}Commutative models}{{\reset@font 179}} \contentsline {subsection}{\numberline {2.5.2}Centered operators}{{\reset@font 184}} \contentsline {subsection}{\numberline {2.5.3}Representations of Cuntz algebras}{{\reset@font 188}} \contentsline {section}{Comments to Chapter 2}{{\reset@font 197}} \contentsline {chapter}{\numberline {3}On the complexity of the description of representations of \hbox {$*$-}algebras}{{\reset@font 202}} \contentsline {section}{\numberline {3.1}$*$-Wild algebras and relations}{{\reset@font 202}} \contentsline {subsection}{\numberline {3.1.1}Majorization of $*$-algebras with respect to the complexity of their representations}{{\reset@font 202}} \contentsline {subsection}{\numberline {3.1.2}$*$-Wildness of $*$-algebras}{{\reset@font 211}} \contentsline {subsection}{\numberline {3.1.3}$*$-Wild algebras generated by orthogonal projections and idempotents}{{\reset@font 213}} \contentsline {subsection}{\numberline {3.1.4}$*$-Wild semilinear relations}{{\reset@font 220}} \contentsline {subsection}{\numberline {3.1.5}$*$-Wild quadratic and cubic relations}{{\reset@font 221}} \contentsline {subsection}{\numberline {3.1.6}$*$-Wild groups. Periodic groups are not $*$-wild}{{\reset@font 226}} \contentsline {section}{\numberline {3.2}On the complexity of the description of classes of non self-adjoint operators}{{\reset@font 228}} \contentsline {subsection}{\numberline {3.2.1}Classes of non self-adjoint operators singled out by a quadratic or a cubic relation}{{\reset@font 229}} \contentsline {subsection}{\numberline {3.2.2}Partial isometries, weakly centered operators and algebraic operators}{{\reset@font 233}} \contentsline {subsection}{\numberline {3.2.3}Hyponormal operators and pairs of commuting completely non-unitary isometries}{{\reset@font 235}} \contentsline {section}{Comments to Chapter 3}{{\reset@font 237}} \contentsline {section}{Bibliography}{{\reset@font 241}}