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\newtheorem{cor}[thm]{Corollary}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{rmk}[thm]{Remark}


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\date{}
%\date{10 August 1997}
\author{V.~M.~Manuilov}
\title{Almost commutativity implies asymptotic commutativity}
\sloppy
\begin{document}
\maketitle

 \begin{abstract}
We show that there exist constants $\e_0>0$ and $C>0$ such that for any
finite set of
almost commuting unitaries
$u_1,\ldots,u_k\in U_n({\bf C})$ (with arbitrary $n$),
$\norm{[u_i,u_j]}<\e<\e_0$, there exists $N$ and unitary paths
$u_i(t)\in U_N({\bf C})$ such that
 $$
u_i(0)=u_i\oplus 1_{N-n},
\quad
\norm{[u_i(t),u_j(t)]}<C\e^{(1/4)^{k-1}}
\quad {\rm and} \quad
\norm{[u_i(1),u_j(1)]}<\e/2.
 $$
It means that almost commuting unitaries
generate asymptotically commuting unitary paths.

 \end{abstract}

\noindent
It is well known that asymptotic representations are closely related to
$K$-theoretical invariants~\cite{c-hig,dad-nem}. On the other hand in many
cases~\cite{voi,e-l,mish-noor} it is sufficient to have only an $\e$-almost
representation for some small enough $\e$.
The main purpose of the present paper is to show that almost commutative
unitaries generate asymptotically commuting unitary paths. Notice that
topological invariants (like the winding numbers) along these unitary
paths are constant. Our main result is the following

 \begin{thm}\label{alm->asym}
There exist constants $\e_0>0$ and $C>0$ such that for any $k$
almost commuting unitaries $u_1,\ldots,u_k\in U_n({\bf C})$,
$\norm{[u_i,u_j]}<\e<\e_0$ $($with arbitrary $n$$)$, there exists
a dimension $N$ and
unitary paths $u_i(t)\in U_N({\bf C})$ such that
 \begin{enumerate}
\item
$u_i(0)=u_i\oplus 1_{N-n}$,
\vspace{-\itemsep}
\item
$\norm{[u_i(t),u_j(t)]}<C\e^{(1/4)^{k-1}}$,
\vspace{-\itemsep}
\item
$\norm{[u_i(1),u_j(1)]}<\e/2$.
 \end{enumerate}
 \end{thm}

\noindent
{\bf Proof.}
We start with proof of the theorem \ref{alm->asym} in the case of $k=2$,
i.e. for two almost commuting unitaries $u=u_1$ and $v=u_2$.
%\medskip\noindent
Proof is based on several technical lemmas. We begin with these
lemmas in the first section and afterwards we will give an explicit
construction of the homotopy $u(t)=u_1(t)$ and $v(t)=u_2(t)$. The idea is
that if one of two almost commuting matrices is diagonal then the other
is almost of a block three-diagonal form (cf. \cite{man_f.a.97}) and
afterwards we use a variant of the Berg's technique (cf.~\cite{lor-berg}).
%\smallskip\noindent
The general case will easily follow from the case of two unitaries and we
deal with it at the end of the paper.


\medskip\noindent
We do not try to get the best possible estimates and the constants surely
can be chosen in a better way. On the other hand we believe that the power
in the asymptotic behaviour estimate $\e^{(1/4)^{k-1}}$ cannot be made
bigger.
%\smallskip\noindent
For convenience we assume that all
$\e$, $\d_1$, $\d_2$ etc. take values in the interval $(0,1)$.

 \begin{rmk}
{\rm
Let $A$ be a unital $C^*$-algebra of real rank zero~\cite{bp}.
If we change the complex numbers field ${\bf C}$ by $A$ and consider
almost commuting unitaries in matrix algebras over $A$ then the theorem
\ref{alm->asym} is valid too. The proof is the same since it is possible
to assume these unitaries to be linear combinations of projections.

}
 \end{rmk}



\section{Technical lemmas}\label{techn.lemmas}
%\setcounter{equation}{0}


We start with two unitaries $u,v\in U_n({\bf C})$ and fix a basis such that
the matrix $u$ is diagonal with diagonal entries $\l_i=e^{2\pi i\ph_i}$,
$i=1,\ldots,n$,
with ordered values: $\ph_i\in [0,1)$, $\ph_{i+1}\geq\ph_i$.
Divide the unit circle
into small arcs of the length $\sqrt[4]{\e}$ and
denote those arcs which contain at least one eigenvalue $\l_i$ by
$\D_k$. Then the number $l$ of such arcs is not bigger then
$2\pi/\sqrt[4]{\e}$ and the following properties hold:
\begin{enumerate}
\item
if $\l_i,\l_j\in\D_k$ then $\v \l_i-\l_j\v <\sqrt[4]{\e}$;
\vspace{-\itemsep}
\item
if $\l_i\in\D_{k-1}$ and $\l_j\in\D_{k+1}$ then $\v \l_i-\l_j\v
\geq\sqrt[4]{\e}$.
\end{enumerate}


\noindent
Denote the orthogonal spectral projections of $u$ corresponding to the
arcs $\D_k$ by $p_k$, $p_1\oplus\ldots\oplus p_l=1$. Then the matrix
algebra $M_n$ as a module over itself can be decomposed into a direct sum
corresponding to the above projections, $M_n=\oplus_{k=1}^lp_kM_n$ and we
can represent matrices from $M_n$ as smaller matrices of blocks with
regards to this decomposition: $u=\diag(\{u_k\})$ for $u_k=p_kup_k$ and
$v=(v_{km})$ for $v_{km}=p_kvp_m$.


\medskip\noindent
For a matrix $a=(a_{km})$ decomposed with respect to $\oplus_{k=1}^l
p_kM_n$ put
 \be\label{tridiag}
d(a)=\left(\begin{array}{cccc}
a_{11}&a_{12}&&a_{1l}\\
a_{21}&a_{22}&\ddots&\\
&\ddots&\ddots&a_{l-1,l}\\
a_{l1}&&a_{l,l-1}&a_{l,l}
\end{array}\right)
 \ee
where empty places are zeroes.
We call matrices of the form (\ref{tridiag}) block three-diagonal (we look
at the corner elements $a_{l1}$ and $a_{1l}$ as at continuations of the
diagonals $(a_{12},\ldots,a_{l-1,l})$ and $(a_{21},\ldots,a_{l,l-1})$
respectively).

\medskip\noindent
We begin by proving that the matrix
$v=(v_{km})$ is ``almost'' block three-diagonal.



 \begin{lem}\label{lemma01}
If
 \begin{equation}\label{ac1}
\norm{[u,v]}<\e
 \end{equation}
then
 $$
\norm{v-d(v)}<4\sqrt{\pi}\sqrt[4]{\e}.
 $$

 \end{lem}

\noindent
{\bf Proof.}
%\medskip\noindent
Divide once more the spectrum of the operator $u$ into smaller (than
$\D_k$) arcs $\ov{\D}_s$ of length $\e$. Let $\ov{\l}_s$ be the
middle points of the arcs $\ov{\D}_s$ and let $\ov{p}_s$ be the
spectral projections of $u$ corresponding to the arcs $\ov{\D}_s$.
Then put
 $$
\ov{u}=\sum_s \ov{\l}_s\ov{p}_s.
 $$
Obviously $\norm{u-\ov{u}}<\e/2$, hence in view of (\ref{ac1})
 \be\label{ac.2}
\norm{\ov{u}v-v\ov{u}}\leq \norm{\ov{u}v-uv}+
\norm{uv-vu}+\norm{vu-v\ov{u}}< 2\e.
 \ee
Let $M_n=\oplus_s\ov{p}_sM_n$ be the decomposition of $M_n$
corresponding to the spectral projections $\ov{p}_s$ of $u$. It is a
sub-decomposition of $\oplus_{k=1}^lp_kM_n$ and the matrix $v$ can be
written also as $v=(w_{ij})$,
$w_{ij}=\ov{p}_i v\ov{p}_j$. The matrix entries $v_{km}$
can be viewed as blocks of elements $w_{ij}$.
Denote by $M$ the number of columns of the matrix $(w_{ij})$. Then one has
 \be\label{N}
M<\frac{4\pi}{\e}.
 \ee


\medskip\noindent
%{\bf Proof.}
Consider the matrix $(\ov{u}v-v\ov{u})(\ov{u}v-v\ov{u})^*$.
From the inequality
(\ref{ac.2}) it follows that the norm of this matrix
is less then $4\e^2$,
hence the norm of any element of this matrix is also less than $4\e^2$.
So, as $\ov{\l}_j$ commute with $w_{ij}$, we obtain for every $i$
 \be\label{4}
\norm{\sum_{j=1}^M (\ov{\l}_i-\ov{\l}_j)^2 w_{ij}w^*_{ij}}<4\e^2.
 \ee
Let $\ov{\l}_i\in\D_{k_i}$, $\ov{\l}_j\in\D_{k_j}$.
As all the summands in (\ref{4}) are positive, so ignoring some
of them we will not increase the norm of the sum, hence
 $$
\norm{\sum\nolimits'_j (\ov{\l}_i-\ov{\l}_j)^2 w_{ij}w^*_{ij}}<4\e^2
 $$
where the sum $\sum'$ is taken for those $j$ for which
$\v k_j- k_i\v\geq 2$, i.e. we throw away those $w_{ij}$
for which $(v-d(v))_{ij}=0$.  As in the sum $\sum'$ we have
$\v\ov{\l}_k-\ov{\l}_j\v\geq\sqrt[4]{\e}$, so
 $$
4\e^2>\norm{\sum\nolimits'_j
(\ov{\l}_i-\ov{\l}_j)^2w_{ij}w^*_{ij}}\geq\sqrt{\e}\norm{\sum\nolimits'_j
w_{ij}w^*_{ij}},
 $$
hence
 \be\label{3}
\norm{\sum\nolimits'_j w_{ij}w^*_{ij}}<4\e^{3/2}.
 \ee
To finish the proof we need the following

 \begin{lem}\label{NN}
Let $A$ be a $C^*$-algebra and let
$a=(a_{ij})$, $a_{ij}\in A$ be a $M\times M$ matrix such that
for every $i$ one has $\norm{\sum_j a_{ij}a^*_{ij}}<\d_1^2$.
Then $\norm{a}<\d_1\sqrt{M}$.

 \end{lem}

\noindent
{\bf Proof.}
Take $\xi=(\xi_k)$, $k=1,\ldots, M$, $\xi_i\in M_n$.
Then using the generalized
Cauchi-Schwartz inequality~\cite{lance} we get
 \begin{eqnarray*}
\norm{a\xi}^2&=&\norm{\sum_{ijk}\xi^*_j a^*_{ij}a_{ik}\xi_k}
\leq\sum_i\norm{\sum_{kj}\xi_j^* a^*_{ij}a_{ik}\xi_k}\\
&\leq&\sum_i\norm{\sum_k a_{ik}a^*_{ik}}\cdot\norm{\sum_k \xi_k^*\xi_k}
< M\d_1^2 \norm{\xi}^2,
 \end{eqnarray*}
hence we have $\norm{a\xi}<\sqrt{M}\d_1\norm{\xi}$. \q

\medskip\noindent
Now in view of (\ref{N}) and (\ref{3}) from the lemma \ref{NN}
(for $\d_1=2\e^{3/4}$) we get
 $$
\norm{v-d(v)}<\sqrt{M}\ 2\e^{3/4}\leq
\sqrt{4\pi}\e^{-1/2}\,2\e^{3/4}= 4\sqrt{\pi}\sqrt[4]{\e}
 $$
which ends the proof of the lemma \ref{lemma01}.  \q

 \begin{rmk}\label{lacunas}
{\rm
If the spectrum of the matrix $u$ has a gap of length more
than $\sqrt[4]{\e}$ then without loss of generality we can assume that
this gap has zero as its central point. In that case norm of the elements
$v_{1l}$ and $v_{l1}$ is small too and we have
 $$
\norm{v-\left(\begin{array}{cccc}
v_{11}&v_{12}&&\\
v_{21}&v_{22}&\ddots&\\
&\ddots&\ddots&v_{l-1,l}\\
&&v_{l,l-1}&v_{l,l}
\end{array}\right)}<4\sqrt{\pi}\sqrt[4]{\e}.
 $$
If there exists another gap of length bigger than $\sqrt[4]{\e}$ in
the spectrum of $u$ (say between $\D_r$ and $\D_{r+1}$) then the matrix
$d(v)$ is close to a matrix decomposable into a direct sum of smaller
three-diagonal matrices because the norm of elements $v_{r+1,r}$ and
$v_{r,r+1}$ is small enough.
}
 \end{rmk}

%\medskip\noindent
Put
 $$
c=c(t)=e^{\pi it/2}\cos{\pi t/2},\qquad s=s(t)=-ie^{\pi it/2}\sin{\pi
t/2},\qquad t\in [0,1].
 $$
Then both paths
$\left(\begin{array}{cc}c&s\\s&c\end{array}\right)$ and
$\left(\begin{array}{cc}\ov{c}&\ov{s}\\\ov{s}&\ov{c}\end{array}\right)$
connect matrices $\left(\begin{array}{cc}1&0\\0&1\end{array}\right)$
and $\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$ in the group
$U_2({\bf C})$.
For a matrix
 \be\label{3diag}
w=\left(\begin{array}{cccc}
v_{11}&v_{12}&&v_{1l}\\
v_{21}&v_{22}&\ddots&\\
&\ddots&\ddots&v_{l-1,l}\\
v_{l1}&&v_{l,l-1}&v_{l,l}
\end{array}\right)
 \ee
and for an integer $m$
define a path $w(t)$ by formula %$w(t)=$
%{\small
 \be\label{bigmatr}
w(t){=}\!
\left(\!\!
\begin{array}{ccccccccccccc}
v_{11}&v_{12}&&cv_{1l}&&&&&&&&&sv_{1l}\\
v_{21}&v_{22}&\ddots&&&&&&&&&&\\
&\ddots&\ddots&\!\!v_{l{-}1,l}\!\!&&&&&&&&&\\
\ov{c}v_{l1}&&\!\!v_{l,l{-}1}\!\!&v_{l,l}&\ov{s}v_{l1}&&&&&&&&\\
&&&sv_{1l}&v_{11}&v_{12}&&cv_{1l}&&&&&\\
&&&&v_{21}&v_{22}&\ddots&&&&&&\\
&&&&&\ddots&\ddots&\!\!v_{l{-}1,l}\!\!&&&&&\\
&&&&\ov{c}v_{l1}&&\!\!v_{l,l{-}1}\!\!&v_{l,l}&\ov{s}v_{1l}&&&&\\
&&&&&&&sv_{1l}&\ddots&&&&\\
&&&&&&&&&v_{11}&v_{12}&&cv_{1l}\\
&&&&&&&&&v_{21}&v_{22}&\ddots&\\
&&&&&&&&&&\ddots&\ddots&\!\!v_{l{-}1,l}\!\!\\
\ov{s}v_{l1}&&&&&&&&&\ov{c}v_{1l}&&\!\!v_{l,l{-}1}\!\!&v_{l,l}\\
\end{array}\!\!
\right)
 \ee

%}
\noindent
where for $t=0$ the matrix $w$ is repeated $m$ times. Let $N=nm$ be the
dimension of $w(t)$. We show that the path $w(t)$ lies close to the
unitary group $U_N({\bf C})$ and that the corresponding estimate does not
depend on the number $m$.


 \begin{lem}\label{bigmatrix}
Suppose that $\norm{v-w}<\d_2$ and $\norm{[u,w]}<\d_3$. Let
$w(t)=\til{w}(t)h(t)$ be the polar decomposition with unitary $w(t)$.
Then
 $$
\norm{w(t)-\til{w}(t)}<100\d_2 \quad{\it and}\quad \norm{[u\oplus
1_{N-n},w(t)]}<\d_3+4\sqrt[4]{\e}.
 $$

 \end{lem}

\noindent
{\bf Proof.}
Notice that the inequality $\norm{v-w}<\d_2$ implies $\norm{1-v^*w}<\d_2$.
Let $w=\til{w}h$ be the polar decomposition. Then we have
 \be\label{ner-vo1}
\norm{1-v^*w}=\norm{\til{w}^*v-h}<\d_2.
 \ee
Passing to adjoints we obtain
 \be\label{ner-vo2}
\norm{\til{w}^*v-v^*\til{w}}<2\d_2,
 \ee
hence from (\ref{ner-vo1}) we get
 $$
\norm{\til{w}^*v-1}<2\d_2,
 $$
and in view of (\ref{ner-vo1})
 $$
\norm{h-1}<3\d_2.
 $$
Therefore we get $\norm{h^2-1}<(1+3\d_2)^2-1<15\d_2$, hence
 $$
\norm{w^*w-1}<15\d_2.
 $$
Notice that the matrix $w^*w-1$ is 9-diagonal (or 5-diagonal, if we
think of the upper right and lower left corners of this matrix as of
continuations of the diagonals close to the main diagonal), therefore the
norm of every diagonal is less than $15\d_2$.

\medskip\noindent
Now look at the matrix $w^*(t)w(t)-1$. It is a 13-diagonal matrix and it is
easily seen that every diagonal of this matrix is a direct sum of
diagonals of the matrix $w^*w-1$ (possibly with coefficients like $c(t)$
and $s(t)$), hence we get the estimate
 $$
\norm{w^*(t)w(t)-1}<13\cdot15\d_2=195\d_2,
 $$
i.e. $\norm{h^2(t)-1}<195\d_2$.
Then we have $\norm{h(t)-1}<100\d_2$ and finally
 $$
\norm{w(t)-\til{w}(t)}=\norm{\til{w}^*(t)w(t)-1}=\norm{h(t)-1}<100\d_2.
 $$

\medskip\noindent
Now we should estimate the commutator $[u\oplus 1_{N-n},w(t)]$. But it is
easy to see that
 \begin{eqnarray*}
\norm{[u\oplus 1_{N-n},w(t)]}&\leq&\norm{[u,w]}\\
&+&\norm{v_{l1}-v_{l1}u_l}+
\norm{u_1v_{l1}-v_{l1}}+\norm{v_{1l}-v_{1l}u_1}+\norm{u_lv_{1l}-v_{1l}}\\
&<&\d_3+2\norm{u_1-1}+2\norm{u_l-1}<\d_3+4\sqrt[4]{\e},
 \end{eqnarray*}
where $u_1$ and $u_l$ are the first and the last diagonal entries of the
matrix $u$.\q

 \begin{lem}\label{2matr.}
Let $w_1,w_2\in U_n({\bf C})$ be almost commuting block 3-diagonal
matrices of the form $($\ref{3diag}$)$, $\norm{[w_1,w_2]}<\d_4$. Define for
both these matrices paths $w_1(t)$ and $w_2(t)$ by $($\ref{bigmatr}$)$.
Then one has
 $$
\norm{[w_1(t),w_2(t)]}<13\d_4.
 $$

 \end{lem}

\noindent
{\bf Proof.}
It can be easily checked that the commutator $[w_1(t),w_2(t)]$ can be
decomposed into sum of the direct sum of smaller commutators $[w_1,w_2]$
and of 12-diagonal matrix with diagonals coinciding with direct sums of
certain diagonals of the matrix $[w_1,w_2]$.\q

 \begin{lem}
Let
 $$
w'=\left(\begin{array}{cccc}
w_{11}&w_{12}&&w_{1N}\\
w_{21}&w_{22}&\ddots&\\
&\ddots&\ddots&w_{N-1,N}\\
w_{N1}&&w_{N,N-1}&w_{N,N}
\end{array}\right)\in M_N({\bf C})
 $$
be a three-diagonal matrix and let
$w'=\til{w}'h'$ be the polar decomposition such that
 \be\label{otsen}
\norm{w'-\til{w}'}<\d_5.
 \ee
Then for any $\d_6>0$ there exist an integer
$r$ $($not depending on $N$$)$ and a decomposition
 $$
\til{w}'=w_0+w_1
 $$
such that the matrix $w_0$ is $(2r+3)$-diagonal and $\norm{w_1}<\d_6$.

 \end{lem}

\noindent
{\bf Proof.}
Consider the Taylor formula
 \begin{eqnarray*}
\til{w}'&=&w'((w')^*w')^{-1/2}=w'(1+((w')^*w'-1))^{-1/2}\\
&=&w'(1+a_1((w')^*w'-1)+a_2((w')^*w'-1)^2+\ldots+a_r((w')^*w'-1)^r+R_r),
 \end{eqnarray*}
where in view of (\ref{otsen}) we have
 $$
\norm{R_r}<\norm{(w')^*w'-1}^r=\norm{(h')^2-1}^r<(3\d_5)^r.
 $$
Take $r$ to satisfy
 \be\label{r}
(3\d_5)^r<\d_6
 \ee
and put
 \begin{eqnarray*}
&w_0=w'(1+a_1((w')^*w'-1)+a_2((w')^*w'-1)^2+\ldots+a_r((w')^*w'-1)^r),&\\
&w_1=w'R_r.&
 \end{eqnarray*}
Then as $w'$ and $((w')^*w'-1)$ are 3-diagonal and $(2r+1)$-diagonal
respectively, so the matrix $w_0$ is $(2r+3)$-diagonal.\q

 \begin{lem}\label{cont}
Let $v'(t)$ be a path in the matrix algebra $M_N({\bf C})$ such that
 $$
\dist(v'(t), U_N({\bf C}))<\d_7.
 $$
Then there exists a unitary path
$v(t)$ such that for every $t$ one has

 $$
\norm{v(t)-v'(t)}<3\d_7.
 $$

 \end{lem}

\noindent
{\bf Proof.}
Let $v'\in M_N({\bf C})$ be a matrix with $\dist(v',U_N({\bf C}))<\d_7$,
and let $v'=vh$ be its polar decomposition. By supposition there exists
a unitary $w$ such that $\norm{v'-w}<\d_7$. Then $\norm{vh-w}<\d_7$, hence
(as in proof of lemma \ref{bigmatrix}) we get the estimates
 \be\label{soprnorm1}
\norm{h-v^{-1}w}<\d_7,
 \ee
and
 \be\label{soprnorm2}
\norm{h-(v^{-1}w)^*}<\d_7.
 \ee
It follows from (\ref{soprnorm1}) and (\ref{soprnorm2}) that
 $$
\norm{v^{-1}w-(v^{-1}w)^*}<2\d_7,
 $$
hence
 \be\label{soprnorm3}
\norm{v^{-1}w-1}<2\d_7.
 \ee
Estimates (\ref{soprnorm1}) and (\ref{soprnorm3}) imply $\norm{h-1}<3\d_7$,
and finally
 $$
\norm{v'-v}=\norm{h-1}<3\d_7.
 $$
Now if we have a path $v'(t)$ with the polar decomposition $v'(t)=v(t)h(t)$
then the path $v(t)$ is continuous and $\norm{v(t)-v'(t)}<3\d_7$.\q

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Construction of homotopy. Case of two unitaries}\label{paths}
%\setcounter{equation}{0}

In this section we give an explicit construction of the paths $u(t)$ and
$v(t)$ satisfying the theorem \ref{alm->asym}. This construction consists
of four steps.

\medskip\noindent
We start with two unitaries $u,v\in U_n({\bf C})$ with $\norm{[u,v]}<\e$
and we assume that $u$ is already diagonal with ordered eigenvalues (cf.
section \ref{techn.lemmas}). Let $N=nm$ where the number $m$ will be
defined later. The homotopy which we are constructing should begin at the
pair of unitaries $u\oplus 1_{N-n}$ and $v\oplus 1_{N-n}$. In fact we will
construct paths $u(t)$ and $v'(t)$ with non-unitary $v'(t)$ but this
latter path will be close to the unitary group,
$\dist(v'(t),U_{N})<C\sqrt[4]{\e}$ for some constant $C$.
By the lemma \ref{cont} this would imply
existence of exactly unitary path $v(t)$ with the necessary properties.
At every step of the homotopy we should control two estimates --- the
commutator norm $\norm{[u(t),v'(t)]}$ and the distance from the unitary
group $\dist(v'(t),U_N({\bf C}))$.

\subsubsection*{Step 1}
Put $w=d(v)$ defined by (\ref{tridiag}).
While not changing the first matrix $u\oplus 1_{N-n}$, we connect the
matrix $v\oplus 1_{N-n}$ by the
linear path  with the block 3-diagonal matrix
$w\oplus 1_{N-n}$. At this step we have
 $$
\norm{[u(t),v'(t)]}<4\sqrt[4]{\e}
\quad{\rm and}\quad
\dist(v'(t),U_N({\bf C}))<\norm{v-w}<4\sqrt{\pi}\sqrt[4]{\e}.
 $$



\subsubsection*{Step 2}
Connect the unit matrix $1_{N-n}$ with the direct sum $w\oplus\ldots\oplus
w$ of $m-1$ copies of the matrix $w$. The matrix $u\oplus 1_{N-n}$ is
still fixed at the second step of homotopy and both the commutator norm
$\norm{[u(t),v'(t)]}$ and $\dist(v'(t),U_N({\bf C}))$ are
not being changed.

\subsubsection*{Step 3}
We still do not change the matrix $u\oplus 1_{N-n}$, and define the path
$v'(t)$ on the third step by $v'(t)=w(t)$, where $w(t)$ is given by
(\ref{bigmatr}). Then by lemma \ref{bigmatrix} (taking there
$\d_1=4\sqrt{\pi}\sqrt[4]{\e}$ and $\d_2=4\sqrt[4]{\e}$) we get the
estimates
 $$
\norm{u\oplus 1_{N-n},v'(t)}<8\sqrt[4]{\e}
\quad {\rm and}\quad
\dist(v'(t),U_N({\bf C}))\leq\norm{v'(t)-v(t)}<400\sqrt{\pi}\sqrt[4]{\e}
 $$
for $v(t)=v'(t)(v'^*(t)v'(t))^{-1/2}$.

\subsubsection*{Step 4}
At the last step we will not change the second matrix $v(1)$.
The matrix $v'(1)$ is 3-diagonal and
$\norm{v'(t)-v(t)}<400\sqrt{\pi}\sqrt[4]{\e}=\d_5$, so by lemma
\ref{bigmatrix} for $\d_6=\e/8$ we have a decomposition $v(t)=w_0+w_1$
with $\norm{w_1}<\e/8$ and with $w_0$ being $(2r+3)$-diagonal for $r$
satisfying the inequality (\ref{r}), i.e.
 $$
(1200\sqrt{\pi}\sqrt[4]{\e})^r<\e/8.
 $$
The choice of such $r$ is possible
if $\e<\e_0$, where $\e_0$ satisfies $1200\sqrt{\pi}\sqrt[4]{\e_0}\leq1$.

\medskip\noindent
Then $\norm{[u(t),v(1)]}\leq\norm{[u(t),w_0]}+\e/4$ and it will be
sufficient to find a path $u(t)$ starting from $u\oplus 1_{N-n}$ such
that
 $$
\norm{[u(1),w_0]}<\e/4.
 $$
For $t\in [0,1]$ define diagonal paths $u_j(t)\in U_n({\bf C})$ by
 $$
u^{(1)}(t)=\left(\begin{array}{ccc}
e^{2\pi i((1-t)\ph_1+t\ph_1/m)}&&\\
&\ddots&\\
&&e^{2\pi i((1-t)\ph_n+t\ph_n/m)}
\end{array}\right)
 $$
and
 $$
u^{(j)}(t)=\left(\begin{array}{ccc}
e^{2\pi it\ph_1j/m}&&\\
&\ddots&\\
&&e^{2\pi it\ph_nj/m}
\end{array}\right)
,\quad j=2,\ldots,m.
 $$
Put $u(t)=u^{(1)}(t)\oplus\ldots\oplus u^{(m)}(t)\in U_N({\bf C})$.
Then the diagonal matrix $u(1)$ will consist of all ordered power $m$
roots of eigenvalues $e^{2\pi i\ph_i}$ of the matrix $u$.

\medskip\noindent
Denote by $d_k(w_0)$ the diagonal of the matrix $w_0$ lying $k$ lines above
(or below if $k$ is negative) the main diagonal $d_0(w_0)$.
Then
 $$
w_0=\sum_{k=-r-1}^{r+1}d_k(w_0)
 $$
and $\norm{d_k(w_0)}<1+\e/8$.
It is easy to see (cf. remark \ref{lacunas} in the case of lacunas in
the spectrum of $u$) that
 $$
\norm{[u(t),d_k(w_0)]}<(1+\e/8)\cdot 2k\sqrt[4]{\e}
\cdot\left(1-t+\frac{t}{m}\right),
 $$
therefore
 $$
\norm{[u(t),w_0]}<
2\,(1+\e/8)\left(\sum_{k=1}^{r+1}k\right)\sqrt[4]{\e}\
\left(1-t+\frac{t}{m}\right)
<2\,(1+\e/8)\ (r+1)^2\ \sqrt[4]{\e}\ \left(1-t+\frac{t}{m}\right).
 $$
So along the whole path we have the necessary estimate
 $$
\norm{[u(t),v(t)]}<3(r+1)^2\sqrt[4]{\e}.
 $$
To finish the construction of homotopy we have to choose the number $m$ so
that it would satisfy the inequality
 $$
2\,(1+\e/8)\ (r+1)^2\ \sqrt[4]{\e}\ \frac{1}{m}<\e/4,
 $$
then we will have $\norm{[u(1),w_0]}<\e/4$, hence
$\norm{[u(1),v(1)]}<\e/2$. The case of two almost commuting unitaries is
proved.


%\bigskip\noindent
\section{Construction of homotopy. General case}

Now we can prove the general case of $k$ almost commuting unitaries
$u_1,\ldots,u_k$, $\norm{[u_i,u_j]}<\e$. As in the two matrices case
we diagonalize the first matrix $u_1$ and repeat the previous construction
for the other unitaries $u_2,\ldots,u_k$
(using lemma \ref{2matr.} we can control the commutator norm for these
unitaries). Then we will get the unitary
paths $u_i(t)$ such that $u_i(0)=u_i\oplus 1_{N-n}$, $u_i(1)=u_i^{(1)}$,
with properties
 \begin{enumerate}
\item
$\norm{[u_1^{(1)},u_j^{(1)}]}<\e'$,\quad $j\neq 1$,
\vspace{-\itemsep}
\item
$\norm{[u_i^{(1)},u_j^{(1)}]}<C\sqrt[4]{\e}$,
\vspace{-\itemsep}
\item
$\norm{[u_i(t),u_j(t)]}<C\sqrt[4]{\e}$
 \end{enumerate}
with small enough $\e'>0$ which we define below and with some constant
$C$. Then at the second step we diagonalize the matrix $u_2^{(1)}$ and
repeat the above construction to the matrices
$u_1^{(1)},u_3^{(1)},\ldots,u_k^{(1)}$. Then we will get the matrices
$u_1^{(2)},\ldots,u_k^{(2)}$ satisfying
 \begin{enumerate}
\item
$\norm{[u_2^{(2)},u_j^{(1)}]}<\e'$,\quad $j\neq 2$,
\vspace{-\itemsep}
\item
$\norm{[u_1^{(2)},u_j^{(2)}]}<C\sqrt[4]{\e'}$,\quad $j\neq 1,2$,
\vspace{-\itemsep}
\item
$\norm{[u_i^{(2)},u_j^{(2)}]}<C^2\sqrt[16]{\e}$,\quad $i,j\neq 1,2$,
\vspace{-\itemsep}
\item
$\norm{[u_i(t),u_j(t)]}<C^2\sqrt[16]{\e}$.
 \end{enumerate}
Repeating this procedure $k-1$ times we finally get the matrices
$u_1^{(k-1)},\ldots,u_k^{(k-1)}$ with properties
 \begin{enumerate}
\item
$\norm{[u_1^{(k-1)},u_j^{(k-1)}]}<C^{k-2}(\e')^{(1/4)^{k-2}}$,
\quad $j>1$,
\vspace{-\itemsep}
\item
$\norm{[u_2^{(k-1)},u_j^{(k-1)}]}<C^{k-3}(\e')^{(1/4)^{k-3}}$,
\quad $j>2$,\quad etc.,
\vspace{-\itemsep}
\item
$\norm{[u_{k-1}^{(k-1)},u_k^{(k-1)}]}<\e'$,
\vspace{-\itemsep}
\item
$\norm{[u_i(t),u_j(t)]}<C^{k-1}\e^{(1/4)^{k-1}}$.
 \end{enumerate}
So, we finally have $\norm{[u_i^{(k-1)},u_j^{(k-1)}]}<
C^{k-2}(\e')^{(1/4)^{k-2}}$ for all $i,j$.
Taking $\e'$ to satisfy the inequality
$C^{k-2}(\e')^{(1/4)^{k-2}}<\e/2$ we obtain the
statement of the theorem.\q



\bigskip\noindent
{\bf Acknowledgement.}
The present paper was prepared with the partial
support of RBRF (grant N 96-01-00276) and INTAS during my stay at the
Copenhagen University. I am grateful to G.~K.~Pedersen for his
hospitality and for helpful discussions. I am also grateful to
A.~S.~Mishchenko for useful discussions.


%\newpage

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\end{thebibliography}

%\end{document}

\vspace{2cm}
\noindent
V.~M.~Manuilov\\
Dept. of Mech. and Math.,\\
Moscow State University,\\
Moscow, 119899, RUSSIA\\
e-mail: manuilov@mech.math.msu.su

\end{document}
