Tracial state space with non-compact extreme boundary

Abstract

Let A be a unital simple separable C*-algebra. If A is nuclear and infinite-dimensional, it is known that strict comparison of positive elements is equivalent to Z-stability if the extreme boundary of its tracial state space is compact and of finite covering dimension. Here we provide the first proof of this result in the case of certain non-compact extreme boundaries. Additionally, if A has strict comparison of positive elements, it is known that the Cuntz semigroup of this C*-algebra is recovered functorially from the Murray–von Neumann semigroup and the tracial state space whenever the extreme boundary of the tracial state space is compact and of finite covering dimension. We extend this result to the case of a countable extreme boundary with finitely many cluster points.