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{\bf Introduction to the Theory of Representations of Finitely Presented $*$-Algebras.\\ I. Representations by bounded operators}\\ V. Ostrovskyi, Yu. Samoilenko\vskip 5pt
\contentsline {chapter}{Preface}{3}
\contentsline {chapter}{\numberline {1}Pairs of self-adjoint operators connected by quadratic relations and some generalizations}{7}
\contentsline {section}{\numberline {1.1}Introduction to representations of $*$-algebras}{7}
\contentsline {subsection}{\numberline {1.1.1}$*$-Representations: key words}{7}
\contentsline {subsection}{\numberline {1.1.2}$C^*$-representable $*$-algebras}{11}
\contentsline {subsection}{\numberline {1.1.3}Enveloping $*$-algebras and $C^*$-algebras}{15}
\contentsline {subsection}{\numberline {1.1.4}$*$-Representations of generators and relations}{21}
\contentsline {subsection}{\numberline {1.1.5}Pairs of self-adjoint operators satisfying quad\discretionary {-}{}{}ratic relations}{24}
\contentsline {section}{\numberline {1.2}$F_n$-algebras and their representations}{27}
\contentsline {subsection}{\numberline {1.2.1}About $*$-representations of ${F}_n$-algebras}{27}
\contentsline {subsection}{\numberline {1.2.2}Examples of ${F}_n$-algebras generated by idempotents and their representations}{29}
\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}{39}
\contentsline {section}{\numberline {1.3}Representations of two-dimensional Lie algebras, their nonlinear transformations, and semilinear relations}{42}
\contentsline {subsection}{\numberline {1.3.1}Representations of two-dimensional real Lie algebras and their nonlinear transformations by bounded operators}{42}
\contentsline {subsection}{\numberline {1.3.2}Pairs of operators connected by semilinear relations}{44}
\contentsline {subsection}{\numberline {1.3.3}Kleinecke--Shirokov type theorems}{48}
\contentsline {subsection}{\numberline {1.3.4}Irreducible representations of semilinear relations}{60}
\contentsline {subsection}{\numberline {1.3.5}Representations of semilinear ${F}_4$-relations}{65}
\contentsline {section}{\numberline {1.4}Representations of $q$-relations}{67}
\contentsline {subsection}{\numberline {1.4.1}Finite-dimensional representations of $q$-re\discretionary {-}{}{}la\discretionary {-}{}{}ti\discretionary {-}{}{}ons}{67}
\contentsline {subsection}{\numberline {1.4.2}Hermitian $q$-plane and $q$-CCR}{70}
\contentsline {subsection}{\numberline {1.4.3}Real quantum plane and real quantum hyperboloid}{75}
\contentsline {section}{Comments to Chapter 1}{77}
\contentsline {chapter}{\numberline {2}Representations of dynamical $*$-algebras}{81}
\contentsline {section}{\numberline {2.1}Operator relations and one-dimensional dynamical\penalty -\@M systems}{81}
\contentsline {subsection}{\numberline {2.1.1}Operator relations connected with one-dimen\discretionary {-}{}{}sional dynamical systems}{81}
\contentsline {subsection}{\numberline {2.1.2}Finite-dimensional representations}{89}
\contentsline {subsection}{\numberline {2.1.3}Infinite-dimensional representations}{94}
\contentsline {section}{\numberline {2.2}Some classes of $*$-algebras with 3 and 4 generators}{104}
\contentsline {subsection}{\numberline {2.2.1}Representations of graded $so(3)$ and four-tup\discretionary {-}{}{}les of projections satisfying a linear relation}{104}
\contentsline {subsection}{\numberline {2.2.2}Representations of a class of quadratic algebras with three generators}{111}
\contentsline {subsection}{\numberline {2.2.3}Operator relations connected with a dynamical system on a plane}{113}
\contentsline {subsection}{\numberline {2.2.4}Representation of real forms of Witten's first deformation}{117}
\contentsline {subsection}{\numberline {2.2.5}Representations of the Sklyanin algebra and $U_q(sl(2))$}{120}
\contentsline {section}{\numberline {2.3}Representations of $q$-deformed $U(so(3,{\@mathbb C}))$}{130}
\contentsline {subsection}{\numberline {2.3.1}Real forms of $U_q(so(3, {\@mathbb C}))$}{130}
\contentsline {subsection}{\numberline {2.3.2}Representations of $U_q(so(3,\@mathbb C))$}{132}
\contentsline {section}{\numberline {2.4}Many-dimensional dynamical systems}{149}
\contentsline {subsection}{\numberline {2.4.1}``Direct products'' of one-dimensional dynamical systems}{149}
\contentsline {subsection}{\numberline {2.4.2}``Triangular'' dynamical systems.}{152}
\contentsline {subsection}{\numberline {2.4.3}{}\kern -1pt{}Operator relations connected with many\kern -1pt-dimen\discretionary {-}{}{}si\discretionary {-}{}{}onal dynamical systems}{159}
\contentsline {subsection}{\numberline {2.4.4}Representations of the non-standard real quantum sphere}{163}
\contentsline {subsection}{\numberline {2.4.5}Heisenberg relations for the quantum $E(2)$\penalty -\@M group}{166}
\contentsline {subsection}{\numberline {2.4.6}Wick algebras related to dynamical systems}{170}
\contentsline {section}{\numberline {2.5}On representations of some nuclear algebras}{177}
\contentsline {subsection}{\numberline {2.5.1}Commutative models}{177}
\contentsline {subsection}{\numberline {2.5.2}Centered operators}{182}
\contentsline {subsection}{\numberline {2.5.3}Representations of Cuntz algebras}{186}
\contentsline {section}{Comments to Chapter 2}{195}
\contentsline {chapter}{\numberline {3}On the complexity of the description of representations of \hbox {$*$-}algebras}{201}
\contentsline {section}{\numberline {3.1}$*$-Wild algebras and relations}{201}
\contentsline {subsection}{\numberline {3.1.1}Majorization of $*$-algebras with respect to the complexity of their representations}{201}
\contentsline {subsection}{\numberline {3.1.2}$*$-Wildness of $*$-algebras}{210}
\contentsline {subsection}{\numberline {3.1.3}$*$-Wild algebras generated by orthogonal projections and idempotents}{212}
\contentsline {subsection}{\numberline {3.1.4}$*$-Wild semilinear relations}{219}
\contentsline {subsection}{\numberline {3.1.5}$*$-Wild quadratic and cubic relations}{220}
\contentsline {subsection}{\numberline {3.1.6}$*$-Wild groups. Periodic groups are not $*$-wild}{225}
\contentsline {section}{\numberline {3.2}{}\kern -1pt{}On the complexity of the description of classes of non-self-adjoint operators}{227}
\contentsline {subsection}{\numberline {3.2.1}Classes of non-self-adjoint operators singled\penalty -\@M out by a quadratic or a cubic relation}{228}
\contentsline {subsection}{\numberline {3.2.2}Partial isometries, weakly centered operators and algebraic operators}{232}
\contentsline {subsection}{\numberline {3.2.3}Hyponormal operators and pairs of commuting completely non-unitary isometries}{234}
\contentsline {section}{Comments to Chapter 3}{236}
\contentsline {chapter}{Bibliography}{241}
\contentsline {chapter}{Index}{257}