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\chapter*{Preface}
\addcontentsline{toc}{chapter}{Preface}

\noindent\textbf{1.}
The word algebra in this book means an associative algebra
over the field of complex numbers $\mathbb{C}$. The terms
$*$-algebra,
Banach $*$\nobreakdash-algebra, $C^*$-algebra, $W^*$-algebra, as well as their
properties, are used in this book, as a rule, without special
references. 

This book is devoted to \emph{representations} of finitely presented
$*$\nobreakdash-al\-gebras (defined by a finite number of generators and relations)
by bounded operators.

\medskip\noindent
{\bf 2.}
There is  a whole domain of Algebra called ``Representation
theory of algebras''.  If one introduces an involution $*$ into an
algebra ${\cal A}$ and considers only those representations by
operators in a Hilbert space $H$ which preserve the involution
($*$-representations), then these representations make just an island
among all the representations of ${\cal A}$.  Moreover,
indecomposable $*$-representations, so dear to
the algebraist's heart, coincide, in this
case, with irreducible representations, and two
$*$-representations are equivalent if and only if they are unitarily
equivalent. And hence, the problem of describing
$*$\nobreakdash-representations of ${\cal A}$, up to a unitary equivalence, is
a subproblem (a particular case) of the problem of describing all
representations up to an equivalence.

        But:

\leftskip \parindent
1)      Mathematical problems related to $*$-representations could
        turn out to be pithy and interesting.

2)      Considering $*$-representations allows one to  change the
        accent sharply, from algebra to functional analysis, and to consider
        not only representations by bounded operators in an
        infinite-dimen\-sional space $H$, but also representations by
        unbounded operators. Representations of Lie algebras
        and their applications show how important and useful such
        representations are.

3)      Moreover, knowing only $*$-representations can sometimes be
        satisfactory to consumers of representation theory.
        The representation theory of $*$-algebras suggests some
        applications of the theory to:

\leftskip 2\parindent
        a) the construction and study of models of quantum \linebreak
physics,
in particular, by using Wick algebras and
        their representations;

        b) a study of representations of $*$-algebras which are
        generated by idempotents and the corresponding resolution
        of the identity;

        c) a study of operator Banach algebras containing a dense
        $*$-sub\-al\-geb\-ra, and construction of invertibility symbols
        for
        operators in the algebra, etc.;

        d) structure theorems for algebraically defined classes of
        not self-adjoint operators.;



        e) the theory of algebras  and
        their representations, since the island of
        $*$-representations could turn out to be an archi\-pelago, and
        the facts about it could be useful for studying both the
        algebra ${\cal A}$ itself and its representations but already
        without the involution in the algebra.
In particular, even such a traditional part of algebra as the
theory of groups (especially countable groups) has long ago included,
in its stock-in-trade, the methods of theory of $*$-representations; 

f)  other applications.

\leftskip 0pt

\medskip\noindent
{\bf 3.}
In the course of some time, the authors have accumulated
a fairly large number of examples, 
and developed techniques for calculating $*$-representations
of classes of finitely presented $*$-algebras.
These classes include certain curves in the real plane, $*$-algebras
generated by idempotents, Wick algebras, and others.

There came an idea to present these examples, classes of examples,
and methods used to describe their representations gradually, with an
increase of complexity of the problem. Actually, the choice of examples and
methods was determined by authors' taste  and their experience in the
subject.


Trying to carry out this idea systematically, we split it into
information about $*$-representations of algebras considered in the
examples by bounded operators (I) and  unbounded
operators
(II). This book is based on a sufficiently large ``zoo'' of
examples that illustrate the notions and methods that appear in
studying bounded $*$-representations. A more accurate title of this book
would possibly be  ``Representations of $*$-algebras by bounded
operators by examples'', but, a similar title has already been
taken (see \cite{34}).

\medskip\noindent\textbf{4.}
A starting point for the exposition in this review is representations of
$*$-algebras generated by two self-adjoint generators satisfying a
quadratic relation (a ``noncommutative curve of degree two in the
real plane'').  But we also give far reaching
generalizations of such ``noncommutative curves'': a theory of
representations of operators satisfying a semilinear relation
(Sections~\ref{sec:1.3.2}--\ref{sec:1.3.5}, \ref{sec:3.1.4}),  an
account of noncommutative
dynamical systems, one-dimensional (Section~\ref{sec:2.1})  and
many-dimensional
(Section~\ref{sec:2.4}), representations of algebras with three and four
generators, which appear in theoretical physics (Sections~\ref{sec:2.2},
\ref{sec:2.3}),
various $*$-wild problems (Sections~\ref{sec:3.1}, \ref{sec:3.2}).

\medskip\noindent
\textbf{5.} In order to read this book, it is enough to  be
familiar with a
basic university course of operator theory and involutive algebras
($*$-algebras). 
Of course, a part devoted to a description of
$*$-algebras and their
properties would be useful in an enlarged edition, where
finitely generated and finitely presented algebras and $*$-algebras, 
properties
of such algebras and examples could be presented.

We would like to give a  list of some related monographs, which are
close, in contents, to  this book.

1) Associative algebras (see, e.g.,
\cite{32,31,33},
and the bibliography
therein), countable groups (see, e.g., \cite{121,olsh,76}, 
and the bibliography
therein), 
and  representations of countable groups and associative
algebras (see, e.g., \cite{cur_re,aus,gab_roi_book},
and the bibliography
therein).

2) Dynamical systems, especially one-dimensional
(see, e.g., \cite{shmr,sh_kol_etal,sin}, etc.).


3) Functional analysis and operator theory, including spectral theory
(see, e.g., \cite{akh_glaz,halm2,reedsim,137,69,ber_us_sh}, and others),
unitary representations of groups (see, e.g., \cite{kiril,zhel,59}, etc.),
 operator $*$\nobreakdash-alge\-bras and their representations 
(see, e.g., \cite{78,26,douglas,arv76,take79,28,kad_rin,118,murphy,34}, 
and others), 
in particular, representations by unbounded operators
(see e.g., \cite{jor_moore,book,jorg_book,135}, etc.)

4) Quantum groups and homgeneous spaces, their representations (especially
$*$-representations), and their applications to the theory of integrable
models (see, e.g., 
\cite{kac,lus,rief_book,chari,mad1,jant,klim_sch,kor_soi}).

5) Applications of the theory of 
$*$\nobreakdash-representations to models of
mathematical physics
(see, e.g., \cite{7,70,sak,wielandt}, etc.),
non-commu\-tative geometry (see, e.g., \cite{conn,manin2}, etc.),
non-commu\-tative probability theory (\cite{holevo,partas,boz_sp91},
etc.), to the construction of invertibility symbols
(\cite{142,116}, etc.), to the theory of non self-adjoint operators
(see, e.g., \cite{halm2,ern,murphy}, etc.).
 


\medskip\noindent
\textbf{6.} References to the literature often contained 
in the comments to
chapters do not claim to be complete and, presumably,  do
not contain a full
bibliography on  books and articles directly related  to
the questions
touched upon in this book. Sometimes, the references to original 
sources are
replaced with the references to available monographs or reviews 
containing
additional bibliographical material; probably, the authors  
too often refer to sources in Russian and their translations. 

We also included some references
related to $*$-representations by unbounded operators,  keeping in
mind the future second volume of this book that will be devoted to 
representations by unbounded operators.

\medskip\noindent
\textbf{7.} The authors are sincerely grateful to many  mathematicians
who
contributed to this work:  their teacher, professor  
Yu.~M.~Berezansky, for
his kind attention and useful advice, all participants  of
the seminars on
algebraic problems of functional analysis in the Institute of  Mathematics
of the
Ukrainian National Academy of Sciences,  colleagues  Stanislav
Kruglyak, Konrad Schm\"udgen
and Victor Shul'man, students Lyudmyla Turowska, Alexandra  Piryatinskaya,
Eduard Vaysleb, Yury Chapovsky,  Stanislav
Popovych,
Daniil Proskurin, Slavik Rabano\-vich for their valuable contributions to
this book.

We also gratefully acknowlege
financial support from the joint grant from the CRDF and
Ukrainian Government no. UM1\nobreakdash-311.


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inoue,isma,jones89,jorg_book,jor_moore,jorg_s_w,%
jorg_wer,kiril2,koe,koor_sw,krugl_q,kru_r_s,kul,17,20,mis,murneu,%
nagy_nica2,nelson,niz_tur,olsh,lomi,umz95,non,ped,pow_i,pow,proskurin,%
pro_mfat,pro,pusz_anti,pw,36,rab_mfat,135,133,132,134,shwe_we,82,81,silv,%
sklyan,skl_2,138,take79,tam,11,thoma,144,130,vas,vasil,voi_dy_ni,wen,%
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