Journal of Operator Theory

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

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

Authors: (1) Charles A. Akemann, (2) Massoud Amini (3) Mohammad B. Asadi
Author institution: (1) Department of Mathematics, University of\break California, Santa Barbara, CA 93106, U.S.A.
(2) School of Mathematics, Tarbiat Modares University, Tehran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran 14115134
(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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