Unbounded quasitraces, stable finiteness and pure infiniteness

2017/05/03 by David Pask, Pask, David, Adam Sierakowski +3 · 1 citation
Mathematics · #46L05 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1705.01268

openalex publication_date 2017/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if A is a σ-unital exact C*-algebra of real rank zero, then every state on K₀(A) is induced by a 2-quasitrace on A. This yields a generalisation of Rainone's work on pure infiniteness and stable finiteness of crossed products to the non-unital case. It also applies to k-graph algebras associated to row-finite k-graphs with no sources. We show that for any k-graph whose C*-algebra is unital and simple, either every twisted C*-algebra associated to that k-graph is stably finite, or every twisted C*-algebra associated to that k-graph is purely infinite. Finally we provide sufficient and necessary conditions for a unital simple k-graph algebra to be purely infinite in terms of the underlying k-graph.

Cited by

• Nuclear dimension, pure infiniteness and real rank for higher rank graph C^*-algebras

Related

• C*-algebras of labelled graphs II - Simplicity results
• Pure infiniteness, stability and C*-algebras of graphs and dynamical systems
• Quasitraces on exact C*-algebras are traces
• Stability of C*-algebras associated to graphs
• Simplicity of twisted C*-algebras of higher-rank graphs and crossed\n products by quasifree actions