C*-algebras and the weak* fixed point property

Abstract: 

Let A be a separable C*-algebra. We show that the  discreteness of its spectrum sp(A) is equivalent to the fact that its Banach space dual A* has the weak* fixed point property, and some other properties. Among these are the uniform Kadec-Klee property for  A* and coincidence of the weak* and the norm topology on the pure  states of A. This is joint work with Gero Fendler (Vienna).