Abstract: We show that the spectrum of a separable -algebra is discrete if and only if
, the Banach space dual of , has the weak fixed point property. We prove further that
these properties are equivalent among others to the uniform weak Kadec-Klee property of
and to the coincidence of the weak topology with the norm topology on the pure states
of . If one assumes the set-theoretic diamond axiom, then the separability is necessary.