# Journal of Operator Theory

Volume 74, Issue 1, Summer 2015 pp. 125-132.

Which multiplier algebras are $W^*$-algebras?

**Authors**:
Charles A. Akemann (1), Massoud Amini (2)
and Mohammad B. Asadi (3)

**Author institution:**(1) Department of Mathematics, University of
California, Santa Barbara, CA 93106, U.S.A.

(2) School of Mathematics, Tarbiat Modares University, Tehran
14115134, Iran and School of Mathematics, Institute for Research
in Fundamental Sciences (IPM), Tehran 19395-5746, Iran

(3) School of Mathematics, Statistics and Computer Science,
College of Science, University of Tehran, Enghelab Avenue, Tehran,
Iran and School of Mathematics, Institute for Research in
Fundamental Sciences (IPM),
Tehran 19395-5746, Iran

**Summary: **We consider the question of when the multiplier
algebra $M(\mathcal{A})$ of a $C^*$-algebra
$\mathcal{A}$ is a $ W^*$-algebra, and show that it holds for a stable
$C^*$-algebra exactly
when it is a $C^*$-algebra of compact operators. This implies that, if
for every Hilbert $C^*$-module $E$ over a $C^*$-algebra $\mathcal{A}$,
the algebra $B(E)$ of
adjointable operators on $E$ is a $ W^*$-algebra, then $\mathcal{A}$ is a
$C^*$-algebra of compact
operators.
Also we show that if unital operator algebras $\mathcal{A}$ and
$\mathcal{B}$ are strongly Morita equivalent, then $\mathcal{A}$ is a
dual
operator algebra if and only if $\mathcal{B}$ is a dual operator
algebra.

**DOI: **http://dx.doi.org/10.7900/jot.2014may07.2049

**Keywords: **Hilbert $C^*$-modules, strong Morita equivalence,
multiplier
algebras, operator algebras, $W^*$-algebras

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