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\section*{Comments to Chapter2}
\addcontentsline{toc}{section}{Comments to Chapter 2}
\markright{Comments to Chapter 2}
\noindent\textbf{Section 2.1.}

2.1.1. The study of a class of the relations $XX^* = F(X^*X)$ was
motivated by numerous examples arising in mathematical physics, including
the Hermitian $q$-plane, $q$-CCR, quantum disk, etc.
(see \cite{macf,biede,damku,kul,klles,klles2} etc.).
 In \cite{greek}, such relations are treated as a  general
deformation of CCR. Note that these relations are ``singly generated''
(connect a single operator $X$ and its adjoint).

In \cite{vai}, representations of such relations were
studied. 
The study of the relation is based  on the study of relations of the form
$AU = UF(A)$ with a self-adjoint (or normal) operator $A$. Properties and
representations of such relations were discussed in
\cite{romp,vai,vaisam1,vai_sam_sel}, etc.

For a one-to-one continuous mappings $F(\cdot)$  of a compact set
$\Delta$, the mapping $F(\cdot)$ defines an automorphism of $C(\Delta)$,
and representations of the relation are representations of the
corresponding $C^*$-algebra which is the crossed product $C(\Lambda) \rtimes
\mathbb{Z}$. In this case, methods of study of representation go back to
\cite{mackey1}, see also \cite{efr,tomi} and the bibliography
therein.

In our exposition, we do not assume that the mapping $F(\cdot)$ is
continuous and one-to-one; this fact makes the problem of introducing and
study of the corresponding $*$-algebra more complicated 
(see, e.g., \cite{renau,arz_ver2}, etc.). 

We show in Proposition~\ref{xx-center} 
that the considered class of operators is a 
subset of the class of centered
operators studied in \cite{mormu}. In Section \ref{sec:2.4.2} we
discuss centered operators again.

In the case of the crossed product, the relationships between ergodic measures and
irreducible representations were discussed by many  authors (see, e.g.,
\cite{tomi} and the bibliography therein). In the non-bijective case, the
dynamical system admits a standard infinite-dimen\-sional one-to-one
realization (see, e.g., \cite{vaisam1}); different conditions for a non-bijective
dynamical system to be simple are discussed in \cite{vai_sam_sel},
\cite{vai_fed} and the cited
there bibliography.

\smallskip
2.1.2.
For the relationship between cycles of the dynamical system and
finite-dimensional representations of the relation, see \cite{efr}.
The connection with the Sharkovsky theorem is discussed in \cite{vai_sam_sel}.
Basic facts about cycles 
of second order and more general continuous mapping of an interval
and their detailed investigation can be found, e.g.,  in \cite{shmr},
\cite{sh_kol_etal}; representations of algebraic relations arising from
continuous fraction mapping were discussed in \cite{83}.

\smallskip
2.1.3.
The partition of all irreducible representations into two classes, degenerate
(with $U$ or $U^*$ having nonzero kernel) and non-degene\-rate (with
unitary $U$), is similar to the Wold decomposition for isometries. In fact,
one can  show that any representation space can be decomposed into the
orthogonal direct sum of two subspaces, $H = H_0 \oplus H_1$, such that
the operator $U$ is a completely non-unitary centered partial
isometry on
$H_0$ and unitary on $H_1$. The description of
centered partial isometries is essentially contained in \cite{mormu}; the
application of this description to the study of operator relations is a
 modified version of \cite{vo_mfat}. 

The description of the anti-Fock representations for a second order mapping
relies on the formalism of symbolic dynamics, see, e.g.,~\cite{sh_kol_etal}.

The exposition for a unitary $U$ follows \cite{vaisam1}.

\medskip\noindent\textbf{Section 2.2.}
In this section we give a number of examples of $*$-algebras
known from  papers on mathematical physics (see, e.g., \cite{zachos} and the
bibliography therein). Our purpose here is
to illustrate how the methods developed in Section~\ref{sec:2.1} can be
generalized to a wider class of relations connecting several operators.

2.2.1.
Triples of operators $A_1$, $A_2$, $A_3$, satisfying the relations $\{A_1,
A_2\} = A_3$,  $\{A_2, A_3\} = A_1$, $\{A_3,A_1\} = A_2$ arise as
representations of a  natural graded analogue of the Lie algebra $so(3)$ (on
the definition of graded Lie $*$-algebras, see, e.g., 
\cite{book}). The irreducible
representations of this algebra were studied in \cite{gorpod};
representations of these relations with other involutions were studied
in \cite{ossilv}. 

 Algebras generated by families of
projections satisfying linear relations were considered in
\cite{besp_umz}. In \cite{gal_mfat}, following the lines of
\cite{besp_umz},
the results of \cite{gorpod}  are applied to
the description of four-tuples of projections such that $\sum_{i=1}^4 P_i
= \alpha I$ (see also \cite{gal_mur}).
Notice that  solutions
of the latter relation exist only for a countable number of
the parameter $\alpha$; also see  \cite{rab_sam_fa}, where the authors studied the  sets
$\alpha \in \mathbb{R}$  for which there are
orthogonal projections $P_1$, \dots, $P_n$ satisfying 
$\sum_{i=1}^n P_i =\alpha I$.


\smallskip
2.2.2.
The exposition follows \cite{three}.

\smallskip
2.2.3.
The class of representations described in Section~\ref{sec:2.2.2} can be
easily extended to more general relations which arise from 
dynamical systems on the plane; the methods and ideas used here are 
the same as in Section~\ref{sec:2.1}. Notice that some facts from the
one-dimensional dynamics fail to hold in the
two-dimensional case. The exposition essentially follows \cite{vaisam1}.


\smallskip
2.2.4.
The exposition follows \cite{pop_snmp}.

\smallskip
2.2.5. 
The Sklyanin algebra was introduced and studied  in \cite{sklyan,skl_2}; 
these papers also
contain classes of 
representations of its real forms.

For representations of the quantum algebra $U_q(sl(2))$, see, e.g.,
 \cite{klim_sch,kor_soi}, and the bibliography
therein. 
The exposition follows \cite{vai3}. 
A more general class
of algebras was introduced in \cite{odes_fei}.
 

\medskip
\noindent\textbf{Section 2.3.}
The nonstandard $q$-deformed algebras $U_q(so(3,{\Bbb C}))$ defined 
in terms 
of trilinear relations for generating elements were introduced by 
D.~Fairlie
\cite{fai}. 
An algebra which can be reduced to $U_q(so(3,{\Bbb C}))$ was 
defined in \cite{odes}.
It is known that Lie algebras of the Lie 
groups $SL(2,{\Bbb C})$ and $SO(3)$  
are isomorphic, but the $q$-deformed  algebras
$U_q(so(3,{\Bbb C}))$ differ from
the quantum algebras $U_q(sl(2,{\Bbb C}))$ introduced by V.~Drinfeld and 
M.~Jimbo. Moreover, they have non-coinciding sets of irreducible finite and
infinite dimensional representations.
Finite-dimensional irreducible representations which, for 
$q\rightarrow 1$, yield the well-known finite-dimensional irreducible 
representations of the Lie algebra $so(3,{\Bbb C})$ were described in
\cite{fai}.
The complete classification of finite-dimensional 
representations when $q$ is not a root of unity have been obtained in
\cite{108}. 
This paper  also contains a description of 
some classes of irreducible representations when $q$ is a root of unity
(see also \cite{107}).

Irreducible representations of the $*$-algebras $U_q(so(3))$ ($q\in\mathbb R$ and
$|q|=1$) determined by the involution $I_1^*=-I_1$, $I_2^*=-I_2$ were 
described in \cite{100}. It was shown there that all such 
representations  are analogues of irreducible 
$*$-representations of the real compact form of $so(3,{\Bbb C})$ for $q>0$ and
its graded analogue for $q<0$. The later were classified in
 \cite{gorpod}. The classification of 
irreducible $*$-representations of $U_q(so(3))$, when $q$ is a root of unity,
was also obtained independently in
\cite{bagro_kru}.
It was shown in \cite{bagro_kru,107,108,100} that the algebras 
$U_q(so(3,{\Bbb C}))$
have irreducible finite-dimensional representations which have no analogue for
the Lie algebra $so(3,{\Bbb C})$ or its graded analogue, that is, 
which do not admit the limits as  $q\rightarrow 1$ or $q\rightarrow -1$. 
Some classes of irreducible representations of the $*$-algebra $U_q(so(2,1))$ 
corresponding to the involution $I_1^*=-I_1$, $I_2^*=I_2$ are given in
\cite{139}. Note that all such nontrivial
representations are unbounded. Unbounded representations of the $*$-algebra
defined by the involution $I_1^*=-I_1$, $I_2^*=-I_2$ were studied in
\cite{100}.
 

\medskip\noindent\textbf{Section 2.4}

2.4.1. Following  
 \cite{jorg}  we
consider the Wick analogue of the ``direct product'' of $q$-CCR which is
an example of $q_{ij}$-CCR constructed in  \cite{102}. All
 representations of the ``direct product'' are described in terms of the
 one-dimensional $q$-CCR.


\smallskip
2.4.2. We consider Wick analogues of the twisted canonical
commutation relations introduced and studied by Pusz and Woronowicz
\cite{pw}, and twisted canonical anticommutation relations studied
by Pusz \cite{pusz_anti}. We denote this algebras by 
$\mu$-CCR and $\mu$-CAR,
respectively. The proof of Proposition~\ref{ccr-klein} is given according to
\cite{jorg}. The same proposition for $\mu-CAR$ (Theorem~\ref{car-klein}) 
was proved in \cite{proskurin}.

\smallskip
2.4.3.
The exposition follows
\cite{ostur}.


\smallskip
2.4.4.
 Algebra of functions on the non-standard three-dimen\-sional
real quantum sphere was introduced by M. Noumi and K. Mimachi
\cite{noumi}. 
The description of irreducible representations of this algebra 
follows  \cite{ostur}.

\smallskip
2.4.5.  The Heisenberg relations for the quantum $E_2$ group was
introduced by  Woronowicz \cite{wor}. The classification of
irreducible representations of this algebra is due to \cite{umz95}.
Notice that we do not assume the natural spectral condition of
\cite{wor}, thus obtaining a wider class of representations.

\smallskip
2.4.6.
The notions of a Wick algebra and Wick ideal were introduced 
 in \cite{jorg}. 
Proposition~\ref{wick-proj} was proved for $n=2$ in \cite{jorg},
and for the general $n$ in \cite{proskurin}.


\medskip\noindent\textbf{Section 2.5.}

2.5.1.
The idea of decomposition of the representation space with respect to some
commutative family was used by G.~Mackey in a general framework of
imprimitivity systems resulted in his method of description of
representations of semi-direct products of locally compact groups
\cite{mackey1,mackey}.
The non-group examples of such decomposition are the
G\aa{}rding--Wightman decomposition of CCR and CAR with infinite number of
degrees of freedom \cite{gar_w_car,gar_w_ccr}; also we
mention  \cite{araki60,gel,gol,heg_mel} on commutative models for CAR
and CCR, \cite{str_voi}
for AF-algebras, \cite{ver_gel_g2,men_shar,goldin,isma}, and the
bibliography therin, for current
and local current algebras, etc. The term
``commutative model'' was introduced in \cite{ver_gel_g}. As a rule,
the general commutative model does not give a unitary description of
all representations of the corresponding operator relations; however, the
existence of a commutative model  is a property of
the relation reflecting the structure of its representations. General
operator relations for which there exists a commutative model in terms of
multiplication and weighted operator-valued shift were studied in
\cite{bos,umz88,romp} (see also \cite{berkon}).

2.5.2. The commutative model for centered operators was constructed in
\cite{mormu}.  We rewrite the relations which define the class of
centered operators in the form that enables us to
apply the general theorem from the previous section.

2.5.3. Representations of Cuntz algebras occupy a special place due to their
relation to endomorphisms of $L(H)$ preserving the identity
\cite{arv89,lac}, and other applications (see, e.g., \cite{jor}).
Their representations were studied in numerous papers
(see, e.g., \cite{bra_jor1,bra_jor2,brat-jorg} etc.).
We choose appropriate elements in the Cuntz algebra, which enables us
to apply the results of Section~\ref{sec:2.5.1}, and construct a
commutative model for its representations; we apply this model to the
construction and study of representations of the Cuntz algebras.

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