Structure and K-theory of ℓ^p uniform Roe algebras

2019/04/15 by Yeong Chyuan Chung, Kang Li, Chung, Yeong Chyuan +1
Mathematics · Physics and Astronomy · #46H20 #46L80 #54F45 #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Noncommutative and Quantum Gravity Theories

paper · pdf · doi:10.48550/arxiv.1904.07050

openalex publication_date 2019/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we characterize when the ℓ^p uniform Roe algebra of a metric space with bounded geometry is (stably) finite and when it is properly infinite in standard form for p∈ [1,∞). Moreover, we show that the ℓ^p uniform Roe algebra is a (non-sequential) spatial L^p AF algebra in the sense of Phillips and Viola if and only if the underlying metric space has asymptotic dimension zero. We also consider the ordered K₀ groups of ℓ^p uniform Roe algebras for metric spaces with low asymptotic dimension, showing that (1) the ordered K₀ group is trivial when the metric space is non-amenable and has asymptotic dimension at most one, and (2) when the metric space is a countable locally finite group, the (ordered) K₀ group is a complete invariant for the (bijective) coarse equivalence class of the underlying locally finite group. It happens that in both cases the ordered K₀ group does not depend on p∈ [1,∞).

Citations

• Crossed products of L^p operator algebras and the K-theory of Cuntz algebras on L^p spaces
• Quasi-Locality and Property A

Related

• Morita equivalence of two ℓ^p Roe-type algebras
• On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces
• Embeddings of uniform Roe algebras
• Maximal and reduced Roe algebras of coarsely embeddable spaces
• Two versions of the uniform Roe algebras for spaces without bounded geometry