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\date{2 April 1995}
\author{V.~M.~Manuilov}
\title{Local minima of commutator norms in finite factors}
\frenchspacing
\sloppy
\begin{document}
\maketitle
\begin{abstract}
Perturbations in norm of pairs of unitary operators in finite
factors are studied. Cases when local minimum of commutator norm is (or is
not) reached on a pair of unitaries are discussed.
\end{abstract}
\bigskip
Let $\A$ be a $W^*$-algebra and let $\U(\A)$ be its group of unitary
elements with the norm topology. Let $u$, $v$ be a pair of non-commuting
unitary operators.
\medskip
{\bf Definition}~\cite{ex2}. If one can find a neighborhood $V$ of a pair
$(u,v)$ in $\U(\A)\times\U(\A)$ such that for any pair $(u',v')\in V$ one
has
$$ \V u'v'-v'u'\V\geq\V uv-vu\V, $$
then we say that a local minimum of commutator norm is reached on the pair
$(u,v)$.
\medskip
Variational analysis of commutator norms in the finite-dimensional case,
when $\A$ is a type ${\rm I}_n$ factor, is given in~\cite{ex2}. In
particular it is shown that the Voiculescu matrices~\cite{voi} in the group
$U(n)$ give a local minimum of commutator norm. The present paper is
inspired by
\cite{ex2} and exposes similar result for the case of $\A$ being a type
${\rm II}_1$ factor.
\medskip
Let $l_2({\bf Z})$ be the Hilbert space of two-sided sequences and let $u$,
$v$ be operators given by
\begin{equation}\label{uv}
(u\xi)_n={\rm e}^{i\theta n}\xi_n;\qquad (v\xi)_n=\xi_{n-1},
\end{equation}
where $\theta/\pi\in (0;1)$ and $(\xi_n)\in
l_2({\bf Z})$. Then $uv={\rm e}^{-i\theta}vu$.
Notice that if $\th/\pi$ is irrational then the $C^*$-algebra $A_\theta$
generated by these two
operators is the irrational rotation algebra~\cite{con} and the $W^*$-algebra
generated by them is a type ${\rm II}_1$ factor~\cite{br}. Operators
(\ref{uv}) in the type ${\rm II}_1$ factor $\A$ play role similar to that of
the Voiculescu matrices in the finite-dimensional case. In particular
their multiplicative commutator
$uvu^*v^*={\rm e}^{i\theta}I$ is a scalar operator.
\medskip
{\bf Theorem 1.}\ {\it Let $u,v\in\U(\A)$ and let $uvu^*v^*={\rm
e}^{i\theta}I$ with $\theta\neq\pi$. Then local minimum of commutator norm
is reached on the pair $(u,v)$.}
\medskip
{\bf Proof}\ \, is based on the properties of determinants as it is in the
finite-dimensional case (cf. \cite{voi},\cite{ex2}).
Remember that if an operator $a\in\A$ has the
property
\begin{equation}\label{*}
\mbox{Spectrum is separated from the set}\ \ (-\infty;0]
\end{equation}
then its determinant can be defined (cf. \cite{f-ka},\cite{burg}) by $\ld a =
\tau(\log a)$
where $\log$ is the branch of logarithm with $\log 1=0$ (notice that this
definition slightly differs from~\cite{f-ka} where determinant is defined
to be self-adjoint). But in the
infinite-dimensional case multiplicativity of determinant is not longer
valid. Nevertheless we have the following
\medskip
{\bf Lemma 2.}\ {\it Let $u_t$, $v_s$ be continuous families of unitary
operators in $\A$ and let
\begin{equation}\label{norm2}
\V u_tv_su_t^*v_s^*-1\V <2.
\end{equation}
Then the value $\ld(u_tv_su_t^*v_s^*)$ is well defined and does not depend on
$t$ \mbox{and $s$.}}
\medskip
{\bf Proof.}\ It is sufficient to prove the lemma with fixed $v_s=v$. As we
have (\ref{norm2}) so (\ref{*}) is valid and $\ld$ is well defined. Notice
that if a family of operators $a_t$ has the property (\ref{*}) then by
\cite{burg} we have
$$ \frac{d}{dt}\ld a_t=\t\left(\left(\frac{d}{dt}
a_t\right)a_t^{-1}\right).$$
Hence
\begin{eqnarray*}
&&\fd\ld(u_tvu_t^*v^*)\\
&&=\t\left(\left(\fd(u_t)vu_t^*v^* -
u_tvu_t^*\fd(u_t)u_t^*v^*\right)(u_tvu_t^*v^*)^{-1}\right)\\
&&=\t\left(\fd(u_t)u_t^*\right)-\t\left(u_tvu_t^*\fd(u_t)v^*u_t^*\right) =0,
\end{eqnarray*}
so $\ld(u_tvu_t^*v^*)={\rm Const}$.\q
\medskip
As we have $uvu^*v^*={\rm e}^{i\th}I$, so the estimate (\ref{norm2}) is
fulfilled in some
neighborhood of the operators $u$ and $v$, hence in this neighborhood
$\ld(uvu^*v^*)=\ld({\rm e}^{i\th}I)=i\th$ holds.
\medskip
Consider now the so-called ``eigenvalues in continuous scale''. For
self-adjoint operators in finite factors this notion was introduced in
\cite{mn}. Let $w$ be a unitary in $\A$ satisfying (\ref{*}) and let
$E(\ph)$ be its spectral measure,
$$w=\int_{-\pi}^\pi {\rm e}^{i\ph}dE(\ph)$$
for $-\pi <\ph <\pi$.
As in~\cite{mn} define for $0<\alpha\leq 1$
$$ \e_w(\a)=i\inf_{\t(E(\ph))\geq\a}\ph.$$ Then
\begin{eqnarray*}
\ld w&=&\t(\log w)=\t\left(\int_{-\pi}^\pi i\ph\,dE(\ph)\right) =
i\int_{-\pi}^\pi\ph\,\t(dE(\ph)) \\
& =& i\int_0^1\left(\inf_{\t(E(\ph))\geq\a}\ph\right)\,d\a =
\int_0^1\e_w(\a)\,d\a.
\end{eqnarray*}
Obviously ``eigenvalues in continuous scale'' for the operator $uvu^*v^*$
give the constant function equal to $i\th$. Let a pair $(u',v')$ be close to
the pair $(u,v)$. Denote $\e_{u'v'u'^*v'^*}(\a)=\e(\a)$. Then by the lemma 2
we have
\begin{equation}\label{=}
\ld uvu^*v^* =\ld u'v'u'^*v'^*=i\th=\int_0^1\e(\a)\,d\a.
\end{equation}
If $\e(\a)$ is constant, then $u'v'u'^*v'^*={\rm e}^{i\th}$ and $\V
u'v'-v'u'\V=\v{\rm e}^{i\th}-1\v=\V uv-vu\V$. If $\e(\a)$ is not constant,
then it follows from (\ref{=}) and from definition of $\e(\a)$ that
$\frac{1}{i}\e(1)>\th$. Then by the spectral theorem we obtain
$$
\V u'v'-v'u'\V=\V u'v'u'^*v'^*-1\V\geq\v {\rm e}^{\e(1)}-1\v > \v {\rm
e}^{i\th}-1\v =\V uv-vu\V, $$
so we are done.\q
\medskip
For each $v\in {\cal U}({\cal A})$ define an open subspace $\U_v$ in
$\U(\A)$ by
$${\cal U}_v=\{ u\in {\cal
U}({\cal A}) : \V uv-vu\V <2\}. $$
\medskip
{\bf Corollary 3.}\ {\it Let $u_0$, $u_1$ and $v$ be such unitaries in
${\cal A}$ that $u_kv={\rm e}^{-i\theta_k}vu_k$, $k=0,1$ and
$\theta_k\neq\pi$. Then $u_0$ and $u_1$ cannot be connected by a continuous
path in ${\cal U}_v$.}
\bigskip
As in~\cite{ex2}, we call a pair $(u,v)$ irreducible if there is no proper
invariant subspace for both $u$ and $v$ when ${\cal A}$ is considered as
represented in a Hilbert space. In the next two statements we
follow~\cite{ex2}. Denote by $\gamma$ the map
$$\gamma : (u,v)\in {\cal U(\cal A})\times {\cal U(\cal A})\longmapsto
uvu^*v^*\in {\cal U(\cal A}).$$
Notice that unlike the finite-dimensional case the image of $\gamma$ is the
whole ${\cal U(\cal A})$~\cite{halm}. Denote by $U_{(u,v)}\in {\cal U
(\cal A)}$ the set
of unitaries satisfying (\ref{*}) and having the same {\it log\,det}\ as
$uvu^*v^*$ and let $T_{(u,v)}U$ denote its tangent space at the point
$uvu^*v^*$. Define
the inner product on the tangent space $i{\cal A}_h$ of ${\cal U(\cal A})$
at unity by $\langle x,y\rangle =\t(x^*y)$. If $D$ is a subspace of scalar
operators in $i{\cal A}_h$, then its orthogonal complement $D^\perp$ is
isomorphic to $T_{(u,v)}U$.
\medskip
{\bf Lemma 4.}\ {\it Let $\V uv-vu\V <2$. The map $$d\gamma_{(u,v)} :
i{\cal A}_h\times i{\cal A}_h\longrightarrow T_{(u,v)}U$$
is a submersion if and only if $(u,v)$ is an irreducible pair.}
\medskip
{\bf Proof}\ \, is similar to the finite-dimensional case~\cite{ex2}. If $h$,
$k$ lie in the tangent space $i{\cal A}_h$ of ${\cal U(\cal A})$ at unity,
then
$$
d\gamma_{(u,v)}(uh,vk)=uv(v^*hv-h+k-u^*ku)u^*v^*, $$
and we must prove that the map
$$L:(h,k)\in i{\cal A}_h\times i{\cal A}_h\longmapsto v^*hv-h+k-u^*ku\in
D^\perp$$
is onto. If $x$ is orthogonal to the image of $L$ then we must have
$$ 0=\t(v^*hvx-hx+kx-u^*kux)=\t(h(vxv^*-x)+k(x-uxu^*))$$ for any
$h,k\in i{\cal A}_h$,
hence $xu=ux$ and $xv=vx$. But if the pair $(u,v)$ is irreducible then $x$
must be scalar operator, so we have $L^\perp =D$ and the map
$d\gamma_{(u,v)}$ is onto.\q
\medskip
{\bf Proposition 5.}\ {\it If $(u,v)$ is an irreducible pair in $\U(\A)$
and at the same time a local minimum for the commutator norm then
the operator $uvu^*v^*$ is scalar.}
\medskip
{\bf Proof}\ (cf.~\cite{ex2}). As the exponential map is a local
isomorphism near unity
between the group $\U(\A)$ and its tangent space $i\A_h$, so the
neighborhood of the pair $(u,v)$ is mapped onto a neighborhood of its
image $\gamma(u,v)$, hence $\gamma(u,v)$ is the local minimum for the map
$$w\in U_{(u,v)}\arr\V w-1\V $$
and it is possible only if $w=\gamma(u,v)$ is a scalar.\q
\bigskip
There remains a problem of classification of reducible pairs of local
minima for the commutator norm. It was conjectured in~\cite{ex2} for the
finite-dimensional case that
direct sums $u=\oplus u_i$ and $v=\oplus v_i$ with $u_iv_iu_i^*v_i^*$ being
scalars also are local minima for the commutator norm. We give here a
counterexample to this conjecture.
\medskip
{\bf Example.}\ \,
Though operators $u_0$ and $u_1$ with $u_kv=
{\rm e}^{i\theta_k}vu_k$, $k=0,1$ cannot be connected by a continuous path
in the set ${\cal U}_v$ for the operator $v$ defined by (\ref{uv}),
there is no obstruction to connect the operators
$$
{\bf u}_0=\left(\begin{array}{cc}
u^2 & 0\\ 0 & 1
\end{array}\right) \quad\mbox{and}\quad
{\bf u}_1=\left(\begin{array}{cc}
u & 0\\ 0 & u
\end{array}\right)
$$
from the $2\times 2$ matrix algebra ${\cal A}\cong {\cal A}_2=M_2({\cal A})$\
\ (where $u$ is also defined \mbox{by (\ref{uv}))}
by a path in the set ${\cal U}_{\bf v}$ for ${\bf v}=\left(\begin{array}{cc}
v & 0 \\ 0 & v
\end{array}\right)$ because one has $\ld {\bf u}_0 =\ld {\bf u}_1$.
Let
$$
{\bf u}_t=\left(\begin{array}{cc}
u^2 \cos^2t+u\sin^2t & -u\sqrt{2-u-u^*}\sin t\cos t\\
u\sqrt{2-u-u^*}\sin t\cos t & \cos^2t+u\sin^2t
\end{array}\right)
$$
be such path. It is easily seen that the matrix entries of ${\bf u}_t
{\bf v}{\bf u}_t^*{\bf v}^*$ lie in the commutative algebra generated by
the operator $u$, i.e. they are operators of multiplication by functions
on $l_2({\bf Z})$. Let $\lambda_0(t)$ and $\lambda_1(t)$ be the eigenvalues
of ${\bf u}_t{\bf v}{\bf u}_t^*{\bf v}^*-1$ with $\lambda_0(0)=0$,
$\lambda_1(0)=
{\rm e}^{i\th}-1$. Notice that both $\lambda_k(t)$, $k=0,1$ are functions
from $l_\i({\bf Z})$ acting on $l_2({\bf Z})$, $(\lambda_k(t)\xi)_n=
\lambda_k(t)(n)\xi_n$ for $\xi=(\xi_n)\in l_2({\bf Z})$.
For $t$ small enough the commutator norm of ${\bf u}$
and ${\bf v}$ obviously coincides with $\V\lambda_1(t)\V$.
Suppose now that $\th$ also is small enough and expand the function
$\lambda_1(t)$:
$$\lambda_1(t)=2i\th+i(-2+\cos \th n)\th+{\rm o}(\th t^2),$$
hence
\begin{equation}\label{lam}
\V\lambda_1\V=\v 2\th-t^2\th\v+{\rm o}(\th t^2)
\end{equation}
One can easily see from (\ref{lam}) that for $\th$ small enough the function
$\V\lambda_1(t)\V$ decreases for small enough $t$, hence we have
$$\V{\bf u}_t{\bf v}-{\bf v}{\bf
u}_t\V <\V{\bf u}_0{\bf v}-{\bf v}{\bf u}_0\V$$ and the pair $({\bf u}_0,
{\bf v})$ is not a local minimum. Notice that this example is valid also in
the finite-dimensional case for big enough dimensions.
\medskip
{\bf Remark.}\ For a $C^*$-algebra $A$ put $$n(A)=\sup_{v\in\U(A)}(\#\pi_0
(\U_v)).$$ Then it follows from the proof of the theorem 5.1 of~\cite{ex2}
that for matrix algebras we have $n(M_n)=n$ for $n\geq 3$ and it is easy to
see that $n(M_2)=1$ as it is for commutative $C^*$-algebras.
The \mbox{corollary 3} states that for $A$ being a type
${\rm II}_1$ factor $n(A)$ is not less than continuum. It would be
interesting to calculate $n(A)$ for type ${\rm I}_\i$ factor.
\bigskip
{\bf Acknowledgement.}\
This work was partially
supported by the Russian Foundation for Fundamental Research (grant
\mbox{N 94-01-00108-a)} and the International Science Foundation
(grant N MGM300).
I am indebted to M.~Frank, A.~A.~Irmatov, A.~S.~Mishchenko
and E.~V.~Troitsky for helpful discussions.
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\end{thebibliography}
\vspace{2.5cm}
\noindent
V.~M.~Manuilov \\*
Moscow State University \\*
Russia - 119899 Moscow \\*
E-mail:manuilov@mech.math.msu.su
\end{document}