  It is shown that the dual $\hat{A}$ of a separable $C^{\ast}$-algebra $A$ is
discrete if and only if its Banach space dual has the weak$^{\ast}$-fixed point
property. We prove further that these properties are equivalent to the uniform
weak$^{\ast}$ Kadec-Klee property of $A^{\ast}$ and to the coincidence of the
weak$^{\ast}$ topology with the norm topology on the pure states of $A$.
