Which multiplier algebras are $W^{\ast }$-algebras?

Abstract

We consider the question of when the multiplier algebra $M(\mathcal{A})$ of a $C^{\ast }$-algebra $\mathcal{A}$ is a $W^{\ast }$-algebra, and show that it holds for a stable $C^{\ast }$-algebra exactly when it is a $C^{\ast }$-algebra of compact operators. This implies that if for every Hilbert $C^{\ast }$-module $E$ over a $C^{\ast }$-algebra $\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $W^{\ast }$-algebra, then $\mathcal{A}$ is a $C^{\ast }$-algebra of compact operators. Also we show that a unital $C^{\ast }$-algebra $\mathcal{A}$ which is Morita equivalent to a $W^{\ast }$-algebra must be a $W^{\ast }$-algebra.