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Separable C*-algebras and weak* fixed point property

Fendler G., Leinert M.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

233--241

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