Limits of pure states

Abstract

In [7, Section 5], Glimm showed that if φ and ψ are inequivalent pure states of a liminal C*-algebra A such that the Gelfand-Naimark-Segal (GNS) representations πφ and πψ cannot be separated by disjoint open subsets of the spectrum  then ½ (φ+ψ) is a weak*-limit of pure states. We extend this to arbitrary C*-algebras (and more general convex combinations) by means of what we hope will be regarded as a transparent proof based on the notion of transition probabilities. As an application, we show that if J is a proper primal ideal of a separable C*-algebra A then there exists a state φ in (the pure state space) such that J=ker πφ (Theorem 3). The significance of this is discussed below after the introduction of further notation and terminology.