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

TL;DR

This paper characterizes when the multiplier algebra of a $C^*$-algebra is a $W^*$-algebra, showing it occurs precisely for stable $C^*$-algebras that are compact operators, and explores related Morita equivalence conditions.

Contribution

It provides a complete characterization of $C^*$-algebras whose multiplier algebras are $W^*$-algebras, linking stability, compact operators, and Morita equivalence.

Findings

Multiplier algebra of a stable $C^*$-algebra is a $W^*$-algebra iff it is a compact operator algebra.

If $B(E)$ is a $W^*$-algebra for all Hilbert $C^*$-modules $E$, then $ ext{A}$ is a compact operator algebra.

A unital $C^*$-algebra Morita equivalent to a $W^*$-algebra must itself be a $W^*$-algebra.

Abstract

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 a unital $C^*$-algebra $\mathcal{A}$ which is Morita equivalent to a $ W^*$-algebra must be a $ W^*$-algebra.