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

TL;DR

This paper characterizes when the dual of a separable $C^{}$-algebra has the weak$^{}$-fixed point property, linking it to discreteness, the Kadec-Klee property, and topological coincidences on pure states.

Contribution

It establishes equivalences between the discreteness of the dual, the weak$^{}$-fixed point property, and topological properties of pure states for separable $C^{}$-algebras.

Findings

Dual $\u0303A$ is discrete iff the Banach dual has the weak$^{}$-fixed point property.

Equivalence of these properties with the uniform weak$^{}$ Kadec-Klee property.

Coincidence of weak$^{}$ and norm topology on pure states.

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 $A$.