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\chapter*{Preface}
\addcontentsline{toc}{section}{Preface}
{\bf 1.} The word algebra in this book means an associative algebra
over the field of complex numbers $\mathbb{C}$. The terms
$*$-algebra,
Banach $*$-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 representations of finitely presented
$*$-algebras (defined by a finite number of generators and relations)
by bounded operators (Chapter~\ref{I}) and unbounded
operators
(Chapter~\ref{II}).

\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, so dear to
the algebraist's heart indecomposable $*$-representations, in this
case, coincide with irreducible representations, and two
$*$-representations are equivalent if and only if they are unitarily
equivalent. And hence (see ???), the problem of describing
$*$-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 to sharply change the
	accent, from algebra to functional analysis, and to consider
	not only representations with bounded operators in an
	infinite dimensional space $H$, but representations with
	unbounded operators as well. 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.
	Even the authors' modest knowledge of the
	representation theory of $*$-algebras suggests some
	applications of the theory to:

{\leftskip 2\parindent
	a) construction and study of models of quantum physics
	\cite{???}, in particular, by using the Wick's algebras and
	their representations \cite{???};

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

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

	d) structure theorems for algebraically defined classes of
	not self-adjoint operators \cite{???}.

}


4)	A special consumer is the theory of algebras ${\cal A}$ and
	their representations, since the island of
	$*$-representations could turn out to be an archipelago, 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.

\leftskip 0pt\smallskip
In particular, even such a traditional part of algebra as the
theory of groups (especially countable groups) has long ago included
in its stockintrade the methods of theory of representations
(especially $*$-representations).

\medskip\noindent
{\bf 3.}
In the course of some time, a fairly large number of examples have
been accumulated by the authors,
and techniques for calculating $*$-representations
of classes of finitely presented $*$-algebras have been developed.
These classes include certain curves in real plane, $*$-agebras
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. Indeed, the choice of examples and
methods was determined by own test of the authors, and their experience of the
subject.


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

A starting point for the exposition in the book 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{1.3.2}, \ref{1.3.3}, \ref{1.11.1}), an account of noncommutative
dynamical systems, one-dimensional (Section~\ref{2.1}) and multi-dimensional
(Section~\ref{2.4}), representations of algebras, with three and four
generators, that appear in theoretical physics (Sections~\ref{2.2},
\ref{2.33}),
various $*$-wild problems (Sections~\ref{3.1},\ref{3.2}).

\medskip\noindent
\textbf{4.} 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 of this book, where
finitely generated and finitely presented algebras and $*$-algebras, 
properties
of such algebras and examples could be presented.

\medskip\noindent
\textbf{5.} References to the literature, containded often in the comments to
the chapters, do not claim to be complete and, presumably, do not contain much
bibliography on the books and the 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.

\medskip\noindent
\textbf{6.} The authors are sincerely grateful to many mathematicians for their
contribution to this work: to their teacher, professor Yu. M. Berezansky for
his kind attitude and patient attention, all the participats of the seminars on
algebraic problems of functional analysis in the Institute of Mathematics of
Ukrainian National Academy of Sciences, to their colleagues Stanislav Kruglyak
and Victor Shul'man, students Lyudmyla Turowska, Alexandra Piryatinskaya,
Eduard Vaisleb, Yury Chapovsky, post-graduate students Stanislav Popovych,
Daniil Proskurin, Slavik Rabanovich.

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