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\section*{Comments to Chapter 1}
\addcontentsline{toc}{section}{Comments to Chapter 1}
\markright{Comments to Chapter 1}
\noindent\textbf{Section 1.1.} 

1.1.1.
We give basic definitions and properties of $*$-representa\-tions of
$*$-algebras by bounded operators, i.e., $*$-homomorphisms 
of $*$\nobreakdash-alge\-bras
into the $*$-algebra of operators on a Hilbert space $H$.

Notice  that in the book we do not consider $*$-homomorphisms into the
$*$-algebra of operators on spaces with indefinite metric. For such
representations, see, e.g., \cite{goh_etal,azi_io,goh_rei,serg90,%
serg92,rodm2,kis_sh}, as well as
the bibliography cited there.

\smallskip
1.1.2. Since in this book, our main concern is the representation theory of
$*$-algebras, the problems of $C^*$-representability of $*$-algebras is
very important in what follows. But it seems that there have not been
much work on the development of these problems (see \cite{10} and the
bibliography therein).

The fact that  in Proposition~\ref{prop:cstar} the implications $(i)
\Rightarrow (ii) \Rightarrow (iii) \Rightarrow (iv)$ hold is easy.
Counterexamples to the implications $(ii) \Rightarrow(i)$
and $(iii) \Rightarrow
(ii)$ are constructed by 
S.~Popovych. For a counterexample to $(iv) \Rightarrow  (iii)$, see
 \cite{wich,dor_bel} and  the bibliography therein.

The exposition of questions about residual family of finite-dimen\-sional
representations of a group $C^*$-algebra follows \cite{29}.

For Remark~\ref{rem:resid}, see the related bibliography: \cite{21,%
30,77} and the bibliography given there.

\smallskip
1.1.3.
In some cases, the representation theory of $*$-algebras can be reduced to that of
one of its enveloping $*$-algebra, $\sigma$-$C^*$-algebra, or
$C^*$\nobreakdash-algebra
(and vise versa). For a finitely generated $*$-algebra
$\mathfrak{A}$, its enveloping $C^*$-algebra exists and is unique if and
only if $\mathfrak{A}$ is $*$\nobreakdash-bounded
(see, e.g., \cite{78,118}).
The $*$\nobreakdash-algebra
generated by a pair of
orthogonal projections, which was considered in Example~\ref{ex:proj}, is $*$-bounded. For an investigation of its enveloping $C^*$-algebra,
see \cite{114,115} and others.

If a finitely generated $*$-algebra
is not $*$-bounded, then one can construct an enveloping
pro-$C^*$-algebra
(see \cite{56,13,15} and the references therein) which
possesses many useful properties of the enveloping $C^*$-algebra. The
exposition of Theorem~\ref{env} follows \cite{pop_mfat}.

\smallskip
1.1.4.
For finitely generated $*$-algebras, we set a natural language of
representation of generators and relations.

The spectral theorem for a single self-adjoint operator $A = A^* =
\int_{-\|A\|}^{\|A\|} \lambda \, dE(\lambda)$ gives a decomposition of the
operator $A$ into irreducible operators of multiplication by $\lambda
\in\mathbb{R}$. It is a standard part of a university course and is
exposed in details in textbooks on spectral theory \cite{akh_glaz,
137,reedsim,69,ber_us_sh}, etc.

Pairs of self-adjoint operators $A$ and $B$ have a  much more complicated
structure. We discuss the complexity of their unitary description in
Section~\ref{sec:3.1.2}. For a  unitary reduction of a pair of Hermitian
matrices, see \cite{serg84,shapiro} and the bibliography therein.

The spectral theorem for a pair of \emph{commuting} self-adjoint
operators (see, e.g., \cite{69,ber_us_sh},
etc.) gives a decomposition of a
pair of operators, $A_1 = \int_{\mathbb{R}^2} \lambda_1 \, dE_{(A_1,
A_2)}(\lambda_1, \lambda_2)$, $A_2
= \int_{\mathbb{R}^2} \lambda_2 \, dE_{(A_1,
A_2)}(\lambda_1, \lambda_2)$, into irreducible pairs of operators  of
multiplication by $\lambda_1$ and $\lambda_2$ in a  one-dimensional space.

The proof that there are no bounded self-adjoint operators satisfying CCR
follows  \cite{wielandt} (see also \cite{halm2,reedsim},
etc.).

\smallskip
1.1.5.
For a reduction of algebras (without an involution) defined by a pair of
generators that are related by a quadratic relation to a canonical 
form, see,
e.g., \cite{ver}.
Canonical forms for pairs of self-adjoint operators
($*$-algebras with a pair of self-adjoint generators) which satisfy a
quadratic relation (``non-commutative conics'' on a real plane) can be found in,
e.g.,
\cite{adv,romp2}, etc.
Newton's classification \cite{newt} (see also
\cite{smst} etc.) of the third-degree curves discouraged 
the authors to investigate the
problem of classification of $*$-algebras with two generators and one
cubic relation.

\medskip
The remaining part of this section is devoted to representations of
``non-commutative curves of the second degree on the real plane'' by bounded
operators, and representations of $*$-algebras which are more general than
these ``curves''. 

\medskip\noindent\textbf{Section 1.2} 

1.2.1.
We expose some known facts about algebras satisfying the standard
polynomial identity (see, e.g., \cite{31,33}
and the bibliography therein), and
their representations (see \cite{142} and others).

Exposition of Theorem \ref{th:r.f} follows \cite{rab_sam_mfat,
rab_sam_ieot}.

\smallskip
1.2.2.
For a number of examples of algebras and $*$-algebras generated by
idempotents and orthogonal projections, their representations are studied.

Representations of the algebra $Q_2$ (without an involution)
generated by a
pair of idempotents were studied, e.g., in \cite{yzette}. All irreducible
representation of $Q_2$ are either one- or two-dimensional.
The description of indecomposable
representations can be derived from \cite{naz,127}. The problem of
description of  representations of the $*$\nobreakdash-algebra
generated by a pair of
orthogonal projections on a finite-dimensional space is reduced to a description of
Jordan's angles between subspaces \cite{110}; in the case of a separable Hilbert
space, see \cite{ped2,halm1},
etc.\ (see, e.g., a detailed bibliography in \cite{116}).
For finite-dimensional representations of the $*$-algebra generated by
an idempotent and its adjoint, see \cite{112,113}, etc.; for
infinite-dimensional (and even unbounded) representations, see
\cite{pop_mfat}.

The algebra $Q_2$ generated by a pair of idempotents is the group
algebra of the simplest infinite group $\mathbb{Z}_2 * \mathbb{Z}_2 =
\mathbb{Z} \rtimes \mathbb{Z}_2$. For the group $G  = \mathbb{Z}^k \rtimes
G_f$, $k>1$, where $G_f$ is a finite group, $\mathbb{C}[G]$ is an
$F_{2|G_f|}$-algebra, and its irreducible $*$-representations
can be obtained using
Mackey's formalism of induced representations \cite{mackey}.
However, the description of all
indecomposable representations of $G$ is a very complicated problem
\cite{bon_dr}. Examples of $F_n$-algebras
generated by idempotents, which are not
group algebras, and their representations are discussed in \cite{124,
finck,125}.

The exposition of Section \ref{sec:1.2.2} and the proof of
Theorem~\ref{th:q2} follows \cite{rab_sam_mfat,rab_sam_ieot}.

\smallskip
1.2.3.
Non-commutative ``circle'', ``hyperbola'', ``pair of intersecting lines'', are
also natural examples of $F_4$-algebras. For their representations, see,
e.g., \cite{romp2,adv}.

For algebras that are generated by idempotents satisfying linear
relations, see \cite{besp_umz}.
The fact that $Q_{4,2}$ is an $F_4$-algebra
follows from the description of its $*$-representations \cite{gal_mfat}
(see also
Section~\ref{sec:2.2.1} below).

Given in Section \ref{sec:1.2.3}, items 4 and 5, examples of algebras
generated by idempotents are not $F_n$-algebras for any $n \in
\mathbb{N}$. This follows from the structure of their irreducible
representations \cite{besp_umz,
gal_mfat,gal_mur} (see also Section~
\ref{sec:2.2.1}).

\medskip\noindent\textbf{Section 1.3}

1.3.1.
Bounded representations of two-dimensional real Lie algebras and their
non-linear transformations can be easily described using the
Kleinecke--Shirokov theorem \cite{shirokov,kleinecke} 
(see also \cite{halm2}, etc.).

\smallskip
1.3.2--1.3.5. In these sections, a more general class of semilinear
relations is studied. The proof of the Kleinecke--Shirokov type theorems
and the study of their irreducible representations follow \cite{bss,
sam_tur_sh}.

\medskip\noindent\textbf{Section 1.4}

1.4.1.
Irreducible representations
of the $q$-relations by finite-dimen\-sional and compact
operators are one-dimensional. Their description using Propositions
\ref{prop:ssh}
and \ref{prop:ssh1} follows \cite{sam_sh_umz}.

\smallskip
1.4.2.
It was noticed in \cite{ver}, etc., that the Hermitian $q$-plane does
not have nontrivial representations by bounded operators. We follow
\cite{adv}. Bounded representations of one-dimensional $q$-CCR, $q \in
\mathbb{R}$, were studied in numerous papers (see, e.g.,
\cite{biede,macf}; for detailed references, see, e.g.,
\cite{klim_sch}). The exposition follows \cite{lomi,adv}. For unbounded
representations of $q$-CCR, $q \in \mathbb R$, see
\cite{koor_sw,adv,bukl2,heb_etal,romp2}, etc.

\smallskip
1.4.3. Real quantum plane and real quantum hyperboloid do not have
non-trivial representations by bounded operators. In \cite{lomi,adv},
this is proved by using the
Fuglede--Putnam--Rosenblum theorem (see, e.g., \cite{halm2,137}). For
unbounded representations of the real quantum hyperboloid, see \cite{133,132}.


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