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\noindent \textcolor[rgb]{0.99,0.00,0.00}{This is a submission to one of journals of TMRG: BJMA or AFA}\\[.5in]
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\title[$C^*$-reflexivity doesn't pass to quotients]%
{$C^*$-reflexivity doesn't pass to quotients}

\author[V. Manuilov, J. Zhu]{Vladimir Manuilov$^1$$^{*}$ and Jingming Zhu$^2$}

\address{$^{1}$ Dept. of Math., Harbin Institute of Technology, 92 West Da-Zhi Str., Harbin, 150001, P.R.China;
\newline
Dept. of Mech. and Math., Moscow State University, Leninskie Gory,
Moscow, 119991, Russia.}

\email{\textcolor[rgb]{0.00,0.00,0.84}{manuilov@mech.math.msu.su}}

\address{$^{2}$ Dept. of Math., Harbin Institute of Technology, 92 West Da-Zhi Str., Harbin, 150001,
P.R.China.}

\email{\textcolor[rgb]{0.00,0.00,0.84}{jingmingzhu\underline{\phantom{a}}math@163.com}}

%\dedicatory{This paper is dedicated to Professor ABCD}

\subjclass[2010]{Primary 46L08, Secondary 54D99}

\keywords{$C^*$-reflexivity, $C^*$-algebra, Higson corona.}

\date{Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.
\newline \indent $^{*}$ Corresponding author}

\begin{abstract}
Using a recently obtained criterion of $C^*$-reflexivity for
commutative $C^*$-algebras, we show that the $C^*$-algebra of
continuous functions on the Higson corona is not $C^*$-reflexive.
This implies that $C^*$-reflexivity doesn't pass to quotient
$C^*$-algebras.
\end{abstract}
\maketitle

\section{Introduction and preliminaries}

A unital $C^*$-algebra $A$ is {\it $C^*$-reflexive} if any Hilbert
$C^*$-module $M$ over $A$ is reflexive, i.e. if the second dual
$M''$ of $M$ coincides with $M$. In \cite{Tro,Mis}
$C^*$-reflexivity of some classes of $C^*$-algebras was
established, and in \cite{FMT3} the following criterion of
$C^*$-reflexivity was obtained for commutative $C^*$-algebras.

\begin{theorem}[\cite{FMT3}]
A commutative unital $C^*$-algebra $A$ is not $C^*$-reflexive if
and only if there exists a sequence $\{I_i\}_{i\in\mathbb N}$ of
non-intersecting $C^*$-subalgebras $I_i\subset A$ such that the
canonical inclusion $\oplus_{i=1}^\infty I_i\subset A$ extends to
a $*$-homomorphism $\prod_{i=1}^\infty I_i\to A$.

\end{theorem}

Note that the $*$-homomorphism $\prod_{i=1}^\infty I_i\to A$, if
exists, has to be injective because $\oplus_{i=1}^\infty I_i$ is
an essential ideal of $\prod_{i=1}^\infty I_i$.

\bigskip


Let $A$ be the $C^*$-subalgebra of $l^\infty$ that consists of all
bounded sequences $\{a_n\}_{n\in\mathbb N}$, $a_n\in\mathbb C$,
such that $\lim_{n\to\infty}|a_{n+1}-a_n|=0$. This $C^*$-algebra
is the algebra of all continuous functions on the {\em Higson
compactification} $\nu\mathbb N$ of $\mathbb N$ \cite{Roe}. Note
that the subset $c_0=C_0(\mathbb N)\subset A$ of all sequences
vanishing at infinity (i.e. $\lim_{n\to\infty}a_n=0$) is an ideal
in $A$. The quotient $C^*$-algebra $B=C(\nu\mathbb N)/C_0(\mathbb
N)=C(\nu\mathbb N\setminus\mathbb N)$ is the algebra of continuous
functions on the {\em Higson corona}.


It was proved in \cite{FMT3} that $A=C(\nu\mathbb N)$ is
$C^*$-reflexive. Our aim is to study Hilbert $C^*$-modules over
the quotient $C^*$-algebra $B=C(\nu\mathbb N)/C_0(\mathbb
N)=C(\nu\mathbb N\setminus\mathbb N)$. Hilbert $C^*$-modules over
quotient $C^*$-algebras are interesting to study in view of the
theory of extensions for Hilbert $C^*$-modules recently developed
by D. Baki\'c and B. Gulja\u s \cite{Bakic-Guljas}. Higson corona
provides a simple but non-trivial example of quotient
$C^*$-algebras. In contrast with $A=C(\nu\mathbb N)$, it turns out
that $B=C(\nu\mathbb N)/C_0(\mathbb N)=C(\nu\mathbb
N\setminus\mathbb N)$ is not $C^*$-reflexive.


\section{Main results}

\begin{theorem}
The $C^*$-algebra $B=C(\nu\mathbb N\setminus\mathbb N)$ is not
$C^*$-reflexive.

\end{theorem}
\begin{proof}
Let $n_1<n_2<\ldots$ and $k_1\leq k_2\leq \ldots$ be two sequences
of positive integers such that
\begin{equation}\label{1}
\lim_{i\to\infty}k_i=\infty
\end{equation} and
\begin{equation}\label{2}
n_i+k_i<n_{i+1}\quad \mbox{for each}\ \  i\in\mathbb N.
\end{equation}
Let
$$
U_i=\{n_i,n_i+1,n_i+2,\ldots,n_i+k_i\},\quad i\in\mathbb N,
$$
be a sequence of disjoint (due to (\ref{2})) subsets in $\mathbb
N$. Fix a bijection $\alpha:\mathbb N\times\mathbb N\to\mathbb N$
and set $V_i^j=U_{\alpha(i,j)}$.


For each $i\in\mathbb N$, let $I_i\subset A$ be the
$C^*$-subalgebra of sequences $\{a_n\}_{n\in\mathbb N}$ such that
$a_n=0$ when $n\notin \cup_{j\in\mathbb N}V_i^j$. Let
$J_i=I_i/(c_0\cap I_i)$ be a $C^*$-subalgebra in $B$. $J_i$ is
non-zero for each $i\in\mathbb N$ due to (\ref{1}). Since $I_i\cap
I_k=\{0\}$ when $i\neq k$, the same holds for the quotients $J_i$
and $J_k$.

Let $a^{(i)}\in I_i$, and let $\dot a^{(i)}\in J_i$ denote the
class of $a^{(i)}$. Assume that $\sup_{i\in\mathbb N}\|\dot
a^{(i)}\|$ is bounded. Then the sequence $(\dot a^{(1)},\dot
a^{(2)},\ldots)$ represents an element of $\prod_{i\in\mathbb
N}J_i$.

For a sequence $a=\{a_n\}_{n\in\mathbb N}$ and for a set
$U=\{m,m+1,\ldots,m+k\}$ define $L(a|_U)$ by
$$
L(a|_U)=\max\nolimits_{n=m-1}^{m+k}|a_{n+1}-a_n|.
$$
As $a^{(i)}\in A$ for each $i\in\mathbb N$, so, by definition, we
have $\lim_{j\to\infty}L(a^{(i)}|_{V_i^j})=0$. So, for each
$i\in\mathbb N$, we can find an integer $m_i$ such that
$$
\sup_{j\geq m_i}L(a^{(i)}|_{V_i^j})\leq\frac{1}{i}
$$
and set
$b^{(i)}_n=\left\lbrace\begin{array}{cl}a^{(i)}_n,&\mbox{if\ \
}n\in\cup_{j=m_i}^\infty
V_i^j\\0,&\mbox{otherwise}.\end{array}\right.$

As $a^{(i)}-b^{(i)}$ is finite for each $i\in\mathbb N$, so $\dot
b^{(i)}=\dot a^{(i)}$.


Define a sequence $c=\{c_n\}_{n\in\mathbb N}$ by
$c_n=\sum_{i=1}^\infty b_n^{(i)}$ and note that this sum contains
no more than one non-trivial summand for each $n\in\mathbb N$,
since all $V_i^j$ are disjoint.

\begin{lemma}
\begin{enumerate}
\item
$c\in A$;
\item
$c_n=b^{(i)}_n$ when $n\in V^j_i$ for some $j\in\mathbb N$;
\item
$\dot c\in B$ depends only on $\dot a^{(i)}\in B$, $i\in\mathbb
N$, but not on their representatives in $A$.

\end{enumerate}

\end{lemma}
\begin{proof}
The claim directly follows from the construction of $c$.

\end{proof}

The above construction determines a $*$-homomorphism
$$
\bar\iota:\prod_{i=1}^\infty J_i\to B\quad \mbox{by}\quad
\bar\iota(\dot a^{(1)},\dot a^{(2)},\ldots)=\dot c.
$$
It is clear that $\bar\iota$ makes the diagram
$$
\begin{xymatrix}{
\oplus_{i=1}^\infty
J_i\ar@{^{(}->}[rr]\ar@{^{(}->}[dr]_-{\iota}&&\prod_{i=1}^\infty
J_i\ar@{^{(}->}[dl]^-{\bar\iota}\\
&B& }\end{xymatrix}
$$
commuting, where $\iota_i:J_i\subset B$ is the canonical inclusion
and $\iota=\oplus_{i=1}^\infty \iota_i$. Hence, by
$C^*$-reflexivity criterion, $B$ is not $C^*$-reflexive.

\end{proof}


As the Higson corona is a closed subspace of the Higson
compactification, so we get the following corollary.

\begin{corollary}
$C^*$-reflexivity is not a hereditary property.

\end{corollary}

This result suggests the following definition.

\begin{definition}
A compact Hausdorff space $X$ is called {\it hereditarily
$C^*$-reflexive} if $C(Y)$ is $C^*$-reflexive for any compact
subset $Y\subset X$.

\end{definition}

Let $\beta\mathbb N$ denote the Stone--\v Cech compactification of
integers.


\begin{theorem}
$X$ is hereditarily $C^*$-reflexive if and only if no closed
subspace of $X$ is homeomorphic to $\beta\mathbb N$.

\end{theorem}
\begin{proof}
If $X$ is not hereditarily $C^*$-reflexive then there exists a
closed subspace $Y\subset X$ such that $C(Y)$ is not
$C^*$-reflexive. Then, by Theorem 4.1 of \cite{FMT3},
$\beta\mathbb N$ can be embedded in $Y$ (hence, in $X$ too) as a
closed subspace. In the opposite direction, if $Y=\beta\mathbb
N\subset X$ is a closed subspace then $X$ cannot be hereditarily
$C^*$-reflexive since $C(\beta\mathbb N)=l^\infty$ is not
$C^*$-reflexive.

\end{proof}


{\bf Acknowledgement.} The first named author acknowledges partial
support from RFFI, grant No. 10-01-00257.

\bibliographystyle{amsplain}
\begin{thebibliography}{99}


\bibitem{Bakic-Guljas}
D. Baki\'c, B. Gulja\u s, \textit{Extensions of Hilbert
$C^*$-modules}, Houston J. Math. \textbf{30} (2004), 537--558.

\bibitem{FMT3}
M. Frank, V. Manuilov, E. Troitsky, \textit{A reflexivity
criterion for Hilbert $C^*$-modules over commutative
$C^*$-algebras}, New York J. Math. \textbf{16} (2010), 399--408.

\bibitem{Mis}
A. S. Mishchenko, \textit{Representations of compact groups in
Hilbert modules over $C\sp{\ast} $-algebras},  Proc. Steklov Inst.
Math. \textbf{166} (1986), 179--195.

\bibitem{Roe}
J. Roe, \textit{Coarse cohomology and index theory for complete
Riemannian manifolds}, Memoirs Amer. Math. Soc. \textbf{497}.
1993.

\bibitem{Tro}
V. A. Trofimov, \textit{Reflexivity of Hilbert modules over an
algebra of compact operators with associated identity}, Mosc.
Univ. Math. Bull. \textbf{41} (1986), No.5, 51--55.

\end{thebibliography}

\end{document}

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