Strict comparison for $C^{\ast }$-algebras arising from Almost finite groupoids

Abstract

In this paper we show that for an almost finite minimal ample groupoid $G$, its reduced ${\mathrm{C}}^{\ast }$-algebra $C_r^{\ast }(G)$ has real rank zero and strict comparison even though $C_r^{\ast }(G)$ may not be nuclear in general. Moreover, if we further assume $G$ being also second countable and non-elementary, then its Cuntz semigroup $\mathrm{Cu}(C_r^{\ast }(G))$ is almost divisible and $\mathrm{Cu}(C_r^{\ast }(G))$ and $\mathrm{Cu}(C_r^{\ast }(G)\otimes \mathcal{Z})$ are canonically order-isomorphic, where $\mathcal{Z}$ denotes the Jiang-Su algebra.

References in corpus (1)

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