Strict comparison for $$C^*$$-algebras arising from almost finite groupoids

Abstract

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