Groupoids and C*-algebras for categories of paths

2011/11/29 by Jack Spielberg, Spielberg, Jack · 3 citations
Mathematics · #46L05 (Primary) 20L05 (Secondary) #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1111.6924

openalex publication_date 2011/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we describe a new method of defining C*-algebras from oriented combinatorial data, thereby generalizing the constructions of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the C*-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems.

Cited by

• On Diagonal bimodules of étale groupoid C^*-algebras
• Spectral Triples on a non-standard presentation of Effros-Shen AF algebras
• Groupoids of finitely aligned higher-rank graphs via filters and graph morphisms

Related

• Higher rank graph C*-algebras
• Groupoids and C^*-algebras for left cancellative small categories
• Categorising the operator algebras of groupoids and higher-rank graphs
• Uniqueness theorems for topological higher-rank graph C*-algebras
• Operator algebras associated with graphs and categories of paths: a Survey