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

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 نشر من قبل Joan Bosa
 تاريخ النشر 2020
  مجال البحث
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In this paper we show that for an almost finite minimal ample groupoid $G$, its reduced $mathrm{C}^*$-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 ${rm Cu}(C_r^*(G))$ is almost divisible and ${rm Cu}(C_r^*(G))$ and ${rm Cu}(C_r^*(G)otimes mathcal{Z})$ are canonically order-isomorphic, where $mathcal{Z}$ denotes the Jiang-Su algebra.



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