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%\date{15 April 1996}
\author{V.~M.~Manuilov}
\title{On almost commuting operators}
\sloppy
\parskip=0.5em%{\medskip}
\begin{document}
\begin{center}
{\Large\bf Spectrally stable relations
}
\bigskip
{\large\bf V.~M.~Manuilov
}
\bigskip
\end{center}
Let $\A$ be the class of $C^*$-algebras $A$ of topological rank one, of
real rank zero and such that for every projection $p\in A$ the unitary
group of $pAp$ is connected. Two main examples are the matrix
$C^*$-algebras and finite $W^*$-algebras. Results below do not depend on
a choice of an algebra from $\A$.
Let ${\cal R}=\{R_j\}$ be some relations on elements $a_i$,
$i=1,\ldots,n$, of an algebra $A$ from $\A$. Denote by $R_\e\in A^n$ the
space of all $n$-tuples $(a_1,\ldots,a_n)$, $a_i\in A$ such that
\begin{equation}\label{*}
\norm{R_j(a_1,\ldots,a_n)}\leq\e
\end{equation}
for every $j$.
We call the set ${\cal R}$ of relations {\em
topologically stable\/}
if for every $\e>0$ there exists $\d>0$ such that
whenever two $n$-tuples $(a_1,\ldots,a_n)$ and $(a'_1,\ldots,a'_n)$ lie in
$R_\d$ they can be connected by a path inside $R_\e$. Otherwise we call
such set of relations topologically non-stable.
Usually for non-stable sets of relations one can
construct some invariant which measures the
non-triviality of $n$-tuples (see~\cite{e-l},\cite{l2}). The notion of
topological stability is related to the strong stability of~\cite{l1}.
Remember that a set of relations ${\cal R}$ is called strongly stable if
for any $\e>0$ there exists $\d>0$ such that for any $n$-tuple
$(a_1,\ldots,a_n)\in R_\d$ one can find another $n$-tuple
$(a'_1,\ldots,a'_n)\in R_0$ close to the given one, i.e.
$\norm{a'_i-a_i}<\e$. Denote by $X_{\cal R}\in {\bf C}^n$ the set of all
complex numbers $(\l_1,\ldots,\l_n)$ satisfying relations $R_j$.
\begin{prop}
If the space $X_{\cal R}$ is connected then the strong
stability implies topological stability.
\end{prop}
Let $X_i$ be the closure of spectra of all possible $a_i$ satisfying
relations $R_j$.
We say that an operator $a_i$ has a spectrum with {\em defect} $L$ if
there exists a point $x_0\in X_i$ such that ${\rm dist}\, (x_0,{\rm
Sp}\,a_i)\geq L$.
Further we assume that the space $X_{\cal R}$ is connected.
We call a set of relations
{\em spectrally stable\/} if for any $L$ and
for every $\e>0$ there exists $d>0$ such that if
two $n$-tuples $(a_1,\ldots,a_n)$ and $(a'_1,\ldots,a'_n)$ lie in
$R_\d$
and if at least for one number $i$ operators $a_i$ and $a'_i$ have
spectra with defect $L$
(provided that the subspace $X_L=\{x=(x_1,\ldots,x_n)\in X_{\cal R}:
\v x_i-x_0\v\geq L\}$ is still connected)
then
these $n$-tuples can be connected by a path inside $R_\e$.
Of course spectral stability implies topological stability but not the
other way.
\begin{thm}
The following two sets of relations are spectrally stable:
\begin{enumerate}
\item
\underline{`almost commutative' two-dimensional torus}: \ $a_i$ are
unitaries, $i=1,2$;\ \ $ R(a_1,a_2)=[a_1,a_2]$,
\item
\underline{`almost commutative' sphere}: \ $a_i$ are selfadjoints;
%$i=1,2,3$,
$$ %\begin{eqnarray*}
R_{ik}(a_1,\ldots,a_n)=[a_i,a_k],
\quad R_0(a_1,\ldots,a_n)=a_1^2+\ldots+a_n^2-1.
$$ %\end{eqnarray*}
\end{enumerate}
\end{thm}
The first case is considered in~\cite{manFA}, section 3. Here we give
a sketch of proof for the second one when $n=3$. Suppose that the spectrum
of $a_1$ has a defect $L$. Then instead of initial relations we can deal
(see~\cite{l2}) with the set of relations
$$
\norm{[h,x]}\leq\e,\quad \norm{h^2+x^*x-1}\leq\e
$$
for positive $h$ (with defect $L$) and normal $x$.
It is sufficient to consider the case when ${\rm Sp}\,h\in [L,1]$ (in
the case when the lacuna of the length $L$ lies inside $[0,1]$ the space
$X_L$ is not connected). Divide the segment $[0,1]$ into small segments
of length $\e^{1/4}$. This division gives a decomposition of the
$C^*$-algebra $A$ into direct sum $\oplus p_kA$ with $p_k$ being the
spectral projections on these small segments. Operators $h$ and $x$ we
can now write as matrices with respect to this decomposition. Then as it
is shown in~\cite{manFA} changing $\e$ by $C_1\cdot\e^{1/4}$ for some
constant $C_1$ we can assume that $h$ is diagonal, $h=\diag\{h_k\}$, and
$x$ is three-diagonal,
\begin{equation}\label{3diag}
x=\left(\begin{array}{cccccc}
a_1&b_1&c_1&&&\\
&a_2&b_2&c_2&&\\
&&a_3&b_3&c_3&\\
&&&\ddots&\ddots&\ddots
\end{array}\right)
\end{equation}
with positive elements $a_k$
and $\norm{x^*x-xx^*}