Separable $C^{\ast }$-Algebras and weak${}^{\ast }$-fixed point property

Abstract

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 $A$.

a proof revised, some results and references added, 7 pages

References in corpus (1)

• Consistency of a counterexample to Naimark's problem