  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.
