Trace spaces of simple nuclear C*-algebras with finite-dimensional extreme boundary

TL;DR

This paper investigates the structure of trace spaces in simple nuclear C*-algebras, establishing conditions under which these algebras absorb the Jiang-Su algebra, with implications for their classification.

Contribution

It proves that simple nuclear C*-algebras with finite-dimensional extreme boundary trace spaces embed matrix algebras into their central sequence algebra, leading to Jiang-Su absorption under strict comparison.

Findings

Existence of unital embeddings of matrix algebras into the central sequence algebra

Finite-dimensional extreme boundary of trace space implies Jiang-Su absorption

Application to classification of simple nuclear C*-algebras

Abstract

Let A be a unital separable simple infinite-dimensional nuclear C*-algebra with at least one tracial state. We prove that if the trace space of A has compact finite-dimensional extreme boundary then there exist unital embeddings of matrix algebras into a certain central sequence algebra of A which is determined by the uniform topology on the trace space. As an application, it is shown that if furthermore A has strict comparison then A absorbs the Jiang-Su algebra tensorially.