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

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.

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