\documentstyle[12pt]{article}
\textheight23cm
\textwidth15cm
\def\boldmat{\bf}
\def\Hom{{\rm Hom}}
\def\End{{\rm End}}
\def\dist{{\rm dist}}
\def\Span{{\rm Span}}
\def\Ker{{\rm Ker}\,}
\def\Im{{\rm Im}\,}
\def\Sp{{\rm Sp}\,}
\def\e{{\varepsilon}}
\def\la{{\langle}}
\def\ra{{\rangle}}
\def\v{{\vert}}
\def\V{{\Vert}}
\def\p{{\perp}}
\def\arr{{\longrightarrow}}
\def\b{{\beta}}
\def\c{{\cdot}}
\def\q{{\quad\bullet}}
\frenchspacing % for spaces after the end of proposition
\renewcommand{\theenumi}{\roman{enumi}}
\renewcommand{\labelenumi}{$\theenumi)$}
\sloppy
\addtolength{\topmargin}{-60pt}
%\addtolength{\oddsidemargin}{-1cm}
\date{}
\author{V.~M.~Manuilov}
\title{Adjointability of operators on Hilbert $C^{*}$-modules}
\begin{document}
\maketitle
\begin{abstract}
Can a functional $f\in H^{*}_{A} = \Hom_{A}(H_{A};A)$ on the non-self-dual
Hilbert module $H_{A}$ over a $C^{*}$-algebra $A$ be represented as an
operator of some inner product by an element of the module $H_{A}$, this
inner product being equivalent to the given one ? We discuss this question
and prove that for some classes of $C^{*}$-algebras
the closure with respect to the given norm of unification of
such functionals for all equivalent inner products coincides with the dual
module $H^{*}_{A}$.
We discuss the notion of compactness of operators in relation to
representability of functionals.
We also show
how an operator on $H_{A}$ in some situations (e.g. if it is Fredholm)
can be made adjointable by change of the inner product to an
equivalent one.
\end{abstract}
\section*{Introduction}
Let $A$ be a $C^{*}$-algebra with unity.
We consider Hilbert $A$-modules over $A$~\cite{pa1},
i.e. (right)
$A$-modules $M$ together with an $A$-valued inner product $\la\cdot,
\cdot\ra : M\times M\arr A$ satisfying the following
conditions:
\begin{enumerate}
\item
$\la x,x\ra\geq 0$ for every $x\in M$ and $\la x,x\ra =0$ iff $x=0$,
\item
$\la x,y\ra =\la y,x\ra^*$ for every $x,y\in M$,
\item
$\la\cdot,\cdot\ra$ is $A$-linear in the second argument,
\item
$M$ is complete with respect to the norm $\V x\V^2=\V\la x,x\ra\V_A$.
\end{enumerate}
By $M^*=\Hom_A(M;A)$ we denote $A$-module dual to $M$ consisting
of continuous $A$-valued functionals.
Let $H_A$ be the right Hilbert $A$-module
of sequences
$a=(a_k)$, $a_k\in A$, $k\in {\bf N}$ such that the series $\sum a^*_ka_k$
converges in $A$ in norm with the standard basis $\{e_k\}$ and
let $L_n(A)\subset H_A$ be the submodule generated by the elements
$e_1,\ldots,e_n$ of the basis. An inner $A$-valued product on the
module $H_A$ can be given by $\la x,y\ra=\sum x^*_ky_k$ for $x,y\in A$.
It is known~\cite{pa2} that in the case when $A$ is a $W^*$-algebra an
inner product can be naturally extended to the dual module $H^*_A$.
By an operator $S$ from a Hilbert $C^*$-module $M$ to another module $N$ we
mean a bounded $A$-homomorphism from $M$ to $N$ which possesses an
adjoint operator $S^*$ from $N$ to $M$ (of course there always exists
an adjoint operator $S^*$ from $N$ (or $N^*$) to $M^*$).
One of the essential distinctions between Hilbert
$C^{*}$-modules and Hilbert spaces is the existence of non-self-dual
modules (i.e. such that $M^{*}\neq M$). In other words there is no Riesz
representation theorem for Hilbert $C^{*}$-modules and not all
operators have an adjoint. In some
papers~\cite{2},\cite{8} it
is shown that it is useful sometimes to consider on $M$ other inner
products defining norms equivalent to the given one. We show how in
some situations one can make an operator have an adjoint by
changing an inner product to another one equivalent to the given one.
Partial result in this direction was announced in~\cite{m}.
\section{Representability of functionals on Hilbert $C^{*}$-modules}
In this section we study the question of
representability of functionals on a Hilbert $C^*$-module $M$ as
inner products by elements from $M$. Define $F$ to be the set of
functionals of the form
%\[
%F = \bigcup\limits_{\b\in {\boldmat B};\,y\in M}\la y,\cdot\ra_{\b},
%\]
$$
x\longmapsto\la y,x\ra_\b \ \ (x,y\in M,\b\in {\boldmat B})
$$
where ${\boldmat B}$ is the set of all inner products $\la\cdot,
\cdot\ra_{\b}$, defining norms equivalent to the given one.
We call a functional $f\in M^{*}$ representable if $f\in F$. Let $A^{**}$
be an enveloping $W^{*}$-algebra for $A$. By $H_{A}$ we denote the
standard Hilbert $A$-module $l_{2}(A)$ with the standard basis $\{ e_{i}\}$.
The extension of the given inner product from $H_{A^{**}}$ to
$H^{*}_{A^{**}}$~\cite{pa2} we also denote by $\la\c,\c\ra$. Obviously we have
$H^{*}_{A}\subset H^{*}_{A^{**}}$.
\medskip
{\bf Proposition 1.1.}\ {\it If $f\in H^{*}_{A}$ is representable then there
exists an element $z\in H_{A}$ such that the operator inequality
\begin{equation}
\alpha\la z,z\ra\leq\b\la f,f\ra\leq\la f,z\ra\leq\gamma\la
f,f\ra\leq\delta\la z,z\ra
\end{equation}
holds for some positive numbers $\alpha$, $\b$, $\gamma$, $\delta$.}
\medskip
{\bf Proof.}\ It was proved in~\cite{2} using results of L.~Brown~\cite{br}
that due to the fact that the module $H_A$ is countably generated,
any inner product equivalent to the given one is of the form
$\la x,y\ra_{\b} = \la Sx,Sy\ra$, where $S\in \End_{A}(H_{A})$ is an
invertible bounded operator. $S$ need not have an adjoint operator in
the module $H_A$ but it has an adjoint operator acting from $H_A$ to
$H^*_A$. If $f$ is representable then $f=S^{*}Sz$ for
some $S$ and some $z\in H_{A}$, the operator $\la f,z\ra\in A$ is
positive and we have
\[
\la f,z\ra = \la S^{*}Sz,z\ra = \la Sz,Sz\ra = \la z,z\ra_{\b}.
\]
Due to invertibility of $S$ we can find~\cite{2} positive numbers $a$ and
$b$ such that
\[
a\la z,z\ra\leq\la z,z\ra_{\b}\leq b\la z,z\ra
\]
and consequently
\begin{equation}
a\la z,z\ra\leq\la f,z\ra\leq b\la z,z\ra.
\end{equation}
We now estimate
$\la f,f\ra = \la S^{*}Sz,S^{*}Sz\ra$. Since $a^{2}\leq
S^{*}S\leq b^{2}$, so we have
\begin{equation}
a^{2}\la z,z\ra\leq\la f,f\ra\leq b^{2}\la z,z\ra.
\end{equation}
Gluing together (2) and (3) we obtain (1).$\quad\bullet$
\medskip
We call a functional $F\in M^{*}$ non-singular if there exists an element
$z\in M$ such that spectrum of the operator $\la f,z\ra\in A$ is separated
from zero. In this case without any loss of generality we can consider the
operator $\la f,z\ra$ as positive with $\la f,z\ra\geq c > 0$ for some
number $c$. The following example shows that there exist singular
functionals with the property $\la f,f\ra =1$.
\medskip
{\bf Example 1.2.}\ Let
$A = L^{\infty}\left([0;1]\right)$. Define $f\in H^{*}_{A}$
as a sequence of functions $f= (f_{k}(t))$, where
\[
f_{k}(t) = \left\lbrace
\begin{array}{cc}
1,& t\in \left[\frac{1}{2^{k}};\frac{1}{2^{k-1}}\right], \\
0,& \mbox{for other}\ t.
\end{array}
\right.
\]
It is obvious that $\la f,f\ra =1$. We show that the spectrum of $\la
f,z\ra$ is not separated from zero for all $z=(z_{k})\in H_{A}$. Since the
series $\sum^{\infty}_{k=1}z^{*}_{k}z_{k}$ is convergent in norm in $A$,
for any $\e > 0$ we can find a number $n$ such that
$\V\sum^{\infty}_{k=n+1}z^{*}_{k}z_{k}\V <\e$.
Then if $t<1/2^{n}$ we have
$\v f_{k}(t)z_{k}(t)\v <\e$, and consequently $\v\la f,z\ra (t)\v <\e$.
Hence $f$ is singular. As $\la f,z\ra$ is not invertible for all $z\in
H_{A}$ and $\la f,f\ra =1$, the inequality (1) is violated, hence $f$ is
not representable.
\medskip
{\bf Proposition 1.3.}\ {\it Let $f\in M^{*}$ be non-singular. Then it is
representable.}
\medskip
{\bf Proof.}\ The Cauchy--Schwarz inequality gives us
\[
00$
there
exists a free finitely generated $A$-module $N\cong A^{n}$; $N\subset
H_{A}$ such that $\dist (S,N)<\e$.
\medskip
{\bf Proposition 2.3.}\ {\it Let $T$ be an operator on $H_{A}$ (resp. an
operator having an adjoint). Then the following conditions are equivalent:
\\ i) $T\in BK(H_{A})$ (resp. $T$ is compact); \\
ii) the image $T(B_{1}(H_{A}))$ of the unit ball $B_{1}(H_{A})$ is
$A$-precompact.}
\medskip
{\bf Proof.}\ If the first statement is valid then it is sufficient to
prove that one can find an approximating module $N\cong A^{n}$ for a finite
set of elements of $H_{A}$. This can be done by the method
of~\cite{1}.
So suppose now that ($ii$) is valid. Then for any $\e >0$ we can find
elements $b_{1},\ldots,b_{k}\in H_{A}$ with $\la b_{i},b_{j}\ra =
\delta_{ij}$ which generate a module $N\subset H_{A}$ and $\dist
(T(B_{1}(H_{A})),N)<\e$. Denote by $P_{N}$ a projection onto $N$ and
consider the operator $P_{N}T$. It can be decomposed in the form
\begin{equation}
P_{N}Tx = b_{1}\la f_{1},x\ra + \cdots +b_{n}\la f_{n},x\ra
\end{equation}
with $f_{i}\in H^{*}_{A}$. As for any $x\in B_{1}(H_{A})$ we can find an
element $b\in N$ with $\V Tx - b\V <\e$, so
\begin{eqnarray*}
\V Tx - P_{N}Tx\V & = & \V Tx - b + b - P_{N}Tx\V \\
& = & \V Tx - b\V +\V P_{N}(b-Tx)\V\ \leq\ \e + \V P_{N}\V\,\e \ =\ 2\e,
\end{eqnarray*}
hence $\V T - P_{N}T\V\leq 2\e$ and $T$ belongs to the norm closure of
operators of the form (4). If $T$ is adjointable then $P_{N}T$ is also
adjointable, hence $f_{i}\in H_{A}$ and $T$ is compact.$\q$
\medskip
Remember that an operator is called Fredholm if it is invertible
modulo the ideal of compact operators. As in the case of Hilbert
modules one has two definitions of compactness (with and without
adjointness), so there are also two definitions of Fredholmness.
By a Banach-Fredholm operator we understand
an operator which is invertible modulo the ideal of the Banach-compact
operators.
Notice that this definition does not depend on a Hilbert structure.
Now we show that Banach-Fredholm
operators in Hilbert modules over arbitrary unital $C^{*}$-algebras
can be made adjontable by a change of inner product. Unfortunately the
case of $C^*$-algebras without unit is much more difficult and we
intend to consider it somewhere else.
The interest in the space of Banach-Fredholm operators lies in
the fact that due to contractibility of the general linear group of
all bounded operators of the module $H_A$~\cite{tr=k}
this space can be considered as a classifying space for
the topological $K$-theory with coefficients in the $C^*$-algebra $A$.
\medskip
{\bf Theorem 2.4.}\ {\it Let $M$, $N$ be Hilbert $C^{*}$-modules
isomorphic to $H_{A}$,
$T:M\arr N$ be a Banach-Fredholm operator having no adjoint. Then there exist
new inner products on these modules equivalent to the given ones so that
$T$ is adjointable with respect to these inner products.}
\medskip
{\bf Proof.}\ Consider a Banach-compact operator $K\in BK(M,N)$,
\begin{equation}
K=\sum_{i=1}^{\infty}y_{i}\la f_{i},\c\ra
\end{equation}
with $y_{i}\in N$, $f_{i}\in M^{*}$.
\medskip
{\bf Lemma 2.5.}\ {\it Let $S:M\arr N$ be an invertible operator, $S\in
\End_{A}(M;N)$. Then the operator $S+K$ is diagonal for some decompositions
$M=M_{1}\oplus M_{2}$; \, $N=N_{1}\oplus N_{2}$,\, where $M_{1}$, $N_{1}$
are finitely generated projective modules, direct sums are not necessarily
orthogonal
and $(S+K)\v_{M_{2}}$ is an
isomorphism.}
\medskip
{\bf Proof.}\ Without loss of generality we can consider the sum (5) to be
finite and the elements $y_{i}$ to be such that $\la y_{i},y_{i}\ra
=\delta_{ij}$. Denote by $N_{1}\cong A^{n}$ the $A$-module generated by
these $y_{i}$, $i=1,\ldots,n$.
Put $$M_{1}=S^{-1}(N_{1});\quad M_{2}=S^{-1}(N_{1}^{\p})$$ and define an
operator $$R: N_{1}^{\p}\arr N_{1}\ \ {\rm by}\ \ Ry=K(S^{-1}y),\ \
y\in N_{1}^{\p}.$$
Then the module $(S+K)(M_{2})$ is of the form
$$N_{2}=(S+K)(M_{2})=\{ y+Ry,\ y\in N_{1}^{\p}\}.$$
This module is obviously closed and $N=N_{1}\oplus N_{2}$
(not orthogonal direct sum). Indeed, if we denote by $P_{1}$ and $P_{2}$
the orthoprojections on $N_{1}$ and on $N_{1}^{\p}$ respectively,
then an element $z\in N$ can be decomposed:
$$z=(P_{1}z-RP_{2}z; P_{2}z+RP_{2}z).$$
As we have
$$(S+K)(M_{1})\subset N_{1},\quad (S+K)(M_{2})=N_{2},$$ so
the operator $S+K$ is diagonal with respect to the chosen decompositions of
$M$ and $N$.$\q$
If $T:M\arr N$ is Banach-Fredholm then there exists an operator $Q:N\arr M$
such
that $TQ-1$ and $QT-1$ are Banach-compact. Then by standard
methods~\cite{5},\cite{3}
one can find a decomposition
\[
T:M_{1}\oplus M_{2}\arr N_{1}\oplus N_{2}
\]
where $M_{1}$ and $N_{1}$ are projective and finitely generated,
$T\v_{M_{2}}:M_{2}\arr N_{2}$ is an isomorphism, but direct sums need not
be orthogonal.
By the proposition 1.4 we can make them orthogonal by changing inner
products on $M$ and $N$ to equivalent ones. Further, as $T\v_{M_{2}}$ is
an isomorphism, we can correct the inner product on $N_{2}$ so that this
isomorphism preserves Hilbert module structure. Then $T\v_{M_{2}}$ is
an identity, hence adjointable. But $T\v_{M_{2}}$ is an operator acting in
the auto-dual modules, hence it is also adjointable, so $T$ is
adjointable.$\q$
\medskip
Finally we show how the averaging theorem of~\cite{9} can be generalized from
compact to amenable groups in the case of $W^{*}$-algebras in order to find
an appropriate inner product. We state the following theorem for the group
${\boldmat Z}$ but the proof is valid for all amenable groups.
\medskip
{\bf Theorem 2.6.}\ {\it Let $A$ be a $W^{*}$-algebra and let $T:M\arr M$ be
an operator such that all its powers are uniformly bounded, $\V T^{n}\V\leq
C$, $n\in {\boldmat Z}$. Then there exists an inner product
$\la\c,\c\ra_{\b}$ equivalent to the given one so that $T$ is unitary with
respect to this inner product.}
\medskip
{\bf Proof.}\ Let $A_{*}$ be a predual Banach space for $A$. For any
$\phi\in A_{*}$ define a function $f_{x,y}$ on ${\boldmat Z}$ by
\[
f_{x,y}(n) = \phi(\la T^{n}x,T^{n}y\ra)
\]
for $x,y\in M$. By supposition this function is bounded. Put
\[
\phi_{x,y}= \lim\limits_{n\to\infty}\frac{1}{2n+1}\sum_{k=-n}^{n}f_{x,y}(k).
\]
Fixing $x$ and $y$ we obtain a linear bounded map
\[
a_{x,y}:A_{*}\arr {\boldmat C};\quad
\phi\longmapsto \phi_{x,y}.
\]
This map is an element of $(A_{*})^{*}=A$.
Define a new inner product on $M$ by $\la x,y\ra_{\b} = a_{x,y}\in A$. We
must check that $\la\c,\c\ra_{\b}$ is an
inner product. Its sesquilinearity is obvious. If $\phi\in A_{*}$ is a state
then $f_{x,x}(n)\geq 0$, hence $\phi(\la x,x\ra_{\b})=\phi_{x,x}\geq 0$.
Suppose that $\la x,x\ra_{\b} =0$ for some $x\in M$. Then we have
$\phi_{x,x}=0$. But as
\[
\la x,x\ra =\la T^{-k}(T^{k}x),T^{-k}(T^{k}x)\ra\leq C^{2}\la
T^{k}x,T^{k}x\ra,
\]
so $\frac{1}{C^{2}}f_{x,x}(0)\leq f_{x,x}(n)$ and
\[
\frac{1}{2n+1}\sum_{k=-n}^{n} f_{x,x}(k)\geq\frac{1}{C^{2}} f_{x,x}(0),
\]
hence $\phi_{x,x}\geq\frac{1}{C^{2}} f_{x,x}(0)$ and by supposition we must
have $f_{x,x}(0)=0$, i.e. $\phi(\la x,x\ra)=0$ for an arbitrary state $\phi$.
But then $\la x,x\ra =0$, hence $x=0$. So $\la\c,\c\ra_{\b}$ is an inner
product. The property $\la Tx,Ty\ra_{\b} =\la x,y\ra_{\b}$ is obvious, so
$T$ is unitary. Equivalence of $\la\c,\c\ra$ and $\la\c,\c\ra_{\b}$ follows
directly from the estimate
\[
\frac{1}{C^{2}}\la x,x\ra\leq\la T^{k}x,T^{k}x\ra\leq C^{2}\la x,x\ra
\]
being valid for all $k$.$\q$
\medskip
{\bf Acknowledgement.}\ The author is grateful to M.~Frank and to
E.~V.~Troitsky for submitting copies of their preprints and fruitful
discussions and to A.~S.~Mishchenko for helpful discussions.
This paper was partially supported by Russian Foundation for Fundamental
Research (grant \mbox{N 96-01-00182-a)}
and DAAD.
{\small
%\section*{References}
\begin{thebibliography}{99}
%$ $
\bibitem%
{br} {\sc Brown L.G.\/}: Close hereditary $C^*$-subalgebras and the
structure of quasimultipliers. MSRI preprint \# 11211-85, Purdue
University, 1985.
\bibitem%
{1} {\sc Dupr\'e M.J., Fillmore P.A.\/}: Triviality theorems for Hilbert
modules. In: Topics in modern operator theory, 5-th Internat.
conf. on operator theory. Timisoara and Herculane (Romania),
1980. Basel -- Boston -- Stuttgart: Birkh\"auser Verlag, 1981,
71 -- 79.
\bibitem%
{2} {\sc Frank M.\/}: Geometrical aspects of Hilbert $C^{*}$-modules.
K\o benhavns Universitet preprint series 22/1993.
\bibitem%
{3} {\sc Irmatov A.\/}: On a new topology in the space of Fredholm
operators. {\it Ann. Global Anal. and Geom.\/} \ {\bf 7\/} (1989), 93 --
106.
\bibitem%
{4} {\sc Kasparov G.G.\/}: Hilbert modules: theorems of Stinespring and
Voiculescu. {\it J. Operator Theory\/}\ {\bf 4\/} (1980), 133 -- 150.
\bibitem%
{m} {\sc Manuilov V.M.\/}: Representability of functionals on Hilbert
$C^*$-modules// {\it Func. Anal. i Pril.},\ to appear.
\bibitem%
{5} {\sc Mishchenko A.S., Fomenko A.T.\/}: The index of elliptic operators
over $C^{*}$-algebras. {\it Izv. Akad. Nauk SSSR. Ser. Mat.\/}
\ {\bf 43\/} (1979), 831 -- 859 (in Russian).
\bibitem%
{pa1}
{\sc Paschke W.L.\/}: Inner product modules over $B^{*}$-algebras.
{\it Trans. Amer. Math. Soc.\/}\ {\bf 182\/} (1973), 443 -- 468.
\bibitem%
{pa2} {\sc Paschke W.L.\/}: The double $B$-dual of an inner product module
over a $C^{*}$-algebra. {\it Canad. J. Math.\/}\ {\bf 26} (1974),
1272 -- 1280.
\bibitem%
{7} {\sc Putnam I.F.\/}: The invertible elements are dense in the
irrational rotation $C^{*}$-algebras. {\it J. Reine Angew. Math.\/}\ {\bf
410} (1990), 160 -- 166.
\bibitem%
{8} {\sc Takesaki M.\/}: Theory of operator algebras,1. New-York --
Heidelberg -- Berlin: Springer Verlag, 1979.
\bibitem%
{tr=k} {\sc Troitsky E.V.\/}: Contractibility of the full general linear
group of the $C^*$-Hilbert module $l_2(A)$. {\it Func. Anal. i
Pril.\/}\ {\bf 20} (1986), N 4, 58 -- 64 (in Russian)
\bibitem%
{9} {\sc Troitsky E.V.\/}: Some aspects of geometry of operators in
Hilbert modules. Ruhr-Universit\"at Bochum, preprint N 173, 1994.
\end{thebibliography}
}
\vspace{2cm}
V.~M.~Manuilov \\*
Dept. of Mech. and Math \\*
Moscow State University \\*
Moscow, 119899, Russia \\*
E-mail: manuilov@mech.math.msu.su
\end{document}