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

2020/02/27 by Pere Ara, Christian Bönicke, Ara, Pere +5
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2002.12221

openalex publication_date 2020/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \rm Cu(C_r^*(G)) is almost divisible and \rm Cu(C_r^*(G)) and \rm Cu(C_r^*(G)⊗ Z) are canonically order-isomorphic, where Z denotes the Jiang-Su algebra.

Citations

• Trace spaces of simple nuclear C*-algebras with finite-dimensional extreme boundary
• Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed Algorithms
• The type semigroup, comparison and almost finiteness for ample groupoids

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