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\section*{Comments to Chapter 3}
\addcontentsline{toc}{section}{Comments to Chapter 3}
\markright{Comments to Chapter 3}

\noindent\textbf{Section 3.1.}

3.1.1, 3.1.2.
In the theory of representations of algebras, it was suggested \cite{89} to
regard a representation problem as wild, if it contains the classical
unsolved problem of representation theory:  to describe, up to a
similarity, a pair of matrices without relations. This unsolved problem of
representation theory contains in itself the problem to describe, up to
similarity, $n$ matrices without relations for any $n \in  \mathbb{N}$, and
therefore, it contains the problem of description, up to similarity, of
representations of any finitely generated algebra.

Numerous examples of wild problems of representation theory can be found,
e.g., in \cite{aus,gab_roi_book}, also see bibliography therein.

To define an analogue of wildness for $*$-algebras ($*$-wildness), it was
suggested in \cite{krusam} to choose, for a standard $*$-wild problem in
the theory of $*$-representations, the problem of description of a pair of
self-adjoint (or unitary) operators up to a unitary equivalence, or which
is the same, representations of the free $*$-algebra $\mathfrak{S}_2$ (or
$\mathfrak{U}_2$) generated by a pair of self-adjoint (or unitary)
generators. It was also suggested to regard as wild problems the ones
that 
contain a standard $*$-wild problem; it was proven that the standard
$*$-wild problem contains, as a sub-problem, the problem of description of
$*$-representations of any finitely or countably generated $*$-algebra.

A number of papers \cite{84,pirsam,85} etc. are devoted to
elaborating the meaning of the statement ``description of
$*$-representations of a $*$-algebra $\mathfrak{A}$ contains, as a
sub-problem, the description of $*$-representations of a $*$-algebra
$\mathfrak{B}$''. Used in the book approach to estimation of the
complexity of $*$-representations  based on
the concepts of majorization
relation for $*$-algebras (Definition~\ref{majoriz}), and $*$-wildness
(Definition~\ref{def:14}) is due to S.~Kruglyak and is exposed in
\cite{kru_sam98,kru_sam_ams}. Theorem~\ref{bound}
on majorization for
$C^*$-algebras and Corollary~\ref{cor:q-order}
establishing that the majorization of
$*$-algebras is a quasi-order relation are also outlined there. Given in
the book proofs are due to S.~Popovych.

In the book we do not discuss relations between the notions of
majorization and Morita equivalence (on Morita equivalence for
$*$-algebras, see \cite{rief74,43,lance95}, etc.)

For representations of finite-dimensional algebras (and for a wider class
of matrix problems as well), it was shown in \cite{drozd}, that these
problems can be subdivided into ``tame'' and ``wild'' (for accurate
definitions, see \cite{89}). We do not discuss here what it means that a
$*$-algebra is tame; however, if one chooses 
type I $*$-algebras (or even nuclear $*$-algebras) to be ``$*$-tame'', 
then there exists a large
set of intermediate $*$-algebras, which are neither $*$-tame, nor
$*$-wild (see, e.g., Section~\ref{sec:3.1.6}).

In Sections \ref{sec:3.1.3}--\ref{sec:3.1.6}, a number of examples of
$*$-wild problems are given. For more examples of $*$-wild problems, see
also \cite{86}, \cite{bagro_kru_2}, \cite{besp_mfat}, etc.

\smallskip{3.1.3.}
The exposition of topics on $*$-wildness of $*$-algebras generated by
orthogonal projections and idempotents essentially follows \cite{krusam}, 
\cite{kru_sam98,kru_sam_ams}. The proof of $*$-wildness of
$*$-algebras $\mathcal{R}_{5,2}$, and $\mathcal{R}_{5, \frac52}$ in
Subsection~5 of \ref{sec:3.1.3} is
given by S.~Kruglyak, Yu.~Samo\u\i{}lenko and A.~Piryatinskaya.

\smallskip{3.1.4.}
For facts on $*$-wildness of semi-linear relations (Propositions
\ref{pro:wild1},~\ref{pro:wild2})
see \cite{bss,sam_tur_sh}. The proof given here is
due to S.~Kruglyak.

\smallskip{3.1.5.}
$*$-Wildness of description of pairs of self-adjoint operators,
$A$, $B$, such that $B^2 = I$, up to a unitary equivalence, follows
directly from \cite{krusam}. We give a simple criterion of $*$-wildness
for pairs of self-adjoint operators connected by a quadratic relation
(A. Piryatinskaya), or by a cubic semi-linear relation
in terms of coefficients of the relation.



\smallskip{3.1.6.}
We give only the simplest examples of $*$-wild groups. The proof of
Theorem~\ref{th:kal} that periodic groups are not $*$-wild can be found
in \cite{kal_sam}. 

In Section~\ref{sec:3.1}, we listed a number of examples of $*$-algebras
and mappings $\psi \colon \mathfrak{A} \to M_n(\mathfrak{B})$ such that
the functor $F_\psi \colon \rep \mathfrak{B} \to \rep\mathfrak{A}$ is
full. However, we do not discuss methods of construction of such mappings.
This is a separate topic; it needs an advanced language of $*$-categories
\cite{roi_box}, and $*$-quivers \cite{84}, \cite{serg87}, etc.
We also do not discuss the question what is the minimal number $n$ for
which there exists a homomorphism $\psi \colon \mathfrak{A} \to
M_n(\mathfrak{B})$ such that the corresponding functor $F_\psi$ is full.
We only notice that in \cite{ols_zame}, it was shown that for the
$C^*$-algebra $\mathcal{A}$ with $m$ self-adjoint generators, $a_1$,
\dots, $a_m$, the algebra $M_n(\mathcal{A})$ for $n \ge -3+\sqrt{9+8m}$,
is singly generated, i.e., generated
by a pair of self-adjoint generators. It was shown in
\cite{rab_umz} that the estimate $n\ge \sqrt{m-1}$ holds, and that this
estimate is exact, i.e., there exists a commutative $C^*$-algebra $C(K)$
with $m$ self-adjoint generators such that $M_n(C(K))$ is not singly
generated for $n< \sqrt{m-1}$.


\medskip\noindent\textbf{Section 3.2.}
In Section~\ref{sec:3.2} we consider an application of the theory of
representations of $*$-algebras to a study of classes of operators that
are singled out algebraically.

The problem to describe the class of operators which satisfy  relations up
to a unitary equivalence is equivalent to the one of describing
representations of the corresponding $*$-algebra $\mathfrak{A}$. 

For such algebras we estimate the complexity of the corresponding problem
of $*$-representation theory, i.e., the  complexity of the unitary
description of the corresponding class of operators.

The contents of Section~\ref{sec:3.2} is directly related to the papers
\cite{91,brown,halm_mac,pear,wog_2,wog,benk,benk_ii,benk_iii,camp}
and the bibliography therein. In particular, it
was proven that
in any factor of infinite type there exists a generator, which is a
partial isometry, and which is, at the same time, a weakly 
centered,  or hyponormal,
or subnormal, etc.\  operator; this indicates that the description of such
operators is complicated. Moreover, the constructions used in the proofs,
in some (but not all) cases can be applied to the proof of $*$-wildness of
the corresponding class of operators (see Section~\ref{sec:3.2.3},
item~1).
Essentially, teh $*$-wildness of the corresponding class of operators
(for example, hyponormal) means that one can choose shuch operator
(hyponormal) to be a generator of the $*$-wild $C^*$-algebra.

\smallskip{3.2.1.}
Simple criteria  of $*$-wildness for classes of non self-adjoint
operators, for which $X$ and $X^*$ are related by a quadratic or cubic
semilinear relation, follow the results
exposed in Section \ref{sec:3.1.5}.

Quasi-normal operators have a rather simple structure \cite{91,halm2},
 etc. The study of the classes of operators considered in
items  4 and 5 of \ref{sec:3.2.1} is also not too complicated.

\smallskip{3.2.2.}
On the complexity of the unitary description of partial isometries, see
\cite{halm_mac,halm2}. The $*$-wildness of the classes of weakly
centered operators and weakly centered partial isometries are proved
in \cite{85,piryat}.

On algebraic operators, see \cite{bessam1}. In \cite{besp_mfat}, the proof
of $*$-wildness of the description of non self-adjoint operators, $X$,
$X^*$, such that $[X^j{X^*}^j, {X^*}^kX^k] =  [X^j{X^*}^j,
X^k{X^*}^k] = [{X^*}^jX^j,{X^*}^kX^k] =0$, $1\le j,k \le n$, is given 
 for a fixed
$n\ge1$. On the other hand, the class of centered operators (for which
these relations hold for all $j$, $k \ge1$), is not $*$-wild (see
\ref{sec:2.5.2}).

\smallskip{3.2.3.}
As A. Piryatinskaya noticed, the construction in \cite{wog} means
essentially that the class of hyponormal operators is $*$-wild.

The $*$-wildness of pairs of commuting partial isometries can be obtained
directly from \cite{berg_cob_leb}. The proof given in the book is due
to D.~Proskurin.

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