\documentclass[openany]{book} \usepackage{x} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amscd} \usepackage[mathcal]{euscript} \usepackage{makeidx} %\usepackage{showkeys} %debugging \allowdisplaybreaks \input{diagram} \makeindex \title{Introduction to the Theory of Representations of Finitely Presented $*$-Algebras. \\ I. Representations by bounded operators} \author{Vasyl Ostrovsky\u\i{} and Yuri\u\i{} Samo\u\i{}lenko} \address{}{Institute of Mathematics, Ukrainian National Academy of Sciences} \input{defs} \begin{document} \pagenumbering{roman} \addtocontents{toc}{{\bf Introduction to the Theory of Representations of Finitely Presented $*$-Algebras.\\ I. Representations by bounded operators}\\ V. Ostrovskyi, Yu. Samoilenko\vskip5pt} \markboth{Contents}{Contents} \tableofcontents \clearpage \markboth{}{} \cleardoublepage \pagenumbering{arabic} \maketitle \markboth{}{} \begin{abstract} This review gives fundamentals of representations of finitely presented $*$-algebras by bounded operators. The theory is illustrated with numerous examples of $*$-algebras. The examples, in particular, include $*$-algebras with two self-adjoint generators that satisfy a quadratic or a more general relation, $*$-algebras with three and four generators, $*$-algebras that arise from one- and many-dimensional discrete dynamical systems, Wick $*$-algebras, various $*$-wild algebras. This book is intended for graduate students as well as for the researchers who specialize in the theory of representations of $*$\nobreakdash-algeb\-ras and related areas. \end{abstract} \cleardoublepage \markboth{Preface}{Preface} \input{intro} \chapter{Pairs of self-adjoint operators connected by quadratic relations and some generalizations} \input{chapt21} \input{chapt22} \input{chapt23} \input{chapt24} \input{remark1} \cleardoublepage \chapter{Representations of dynamical $*$-algebras} \markboth{Chapter 2. Representations of dynamical $*$-algebras}{} \input{chapt25} \input{chapt26} \input{chapt27} \input{chapt28} \input{chapt29} \input{remark2} \cleardoublepage \chapter{On the complexity of the description of representations of \hbox{$*$-}algebras} \markboth{Chapter 3. On the complexity of representations}{} \input{chapt2a} \input{chapt2b} \input{remark3} \clearpage \markboth{}{} \cleardoublepage \addcontentsline{toc}{chapter}{Bibliography} \markboth{Bibliography}{Bibliography} \footnotesize \bibliography{ref,new,2} \bibliographystyle{amsplain} \clearpage \markboth{}{} \normalsize \cleardoublepage \addcontentsline{toc}{chapter}{Index} \printindex \end{document}

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