Existence of tracial states on reduced group C*-algebras

2017/06/16 by Brian E. Forrest, Brian Forrest, Nico Spronk +4
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics #math.FA #math.OA

paper · pdf · doi:10.48550/arxiv.1706.05354

The proof of Remark 2.6 (ii) has been corrected, and a typo has been fixed

openalex publication_date 2017/06/16 · arxiv created 2017/07/19 · arxiv updated 2017/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a locally compact group. It is not always the case that its reduced C*-algebra C^*_r(G) admits a tracial state. We exhibit closely related necessary and sufficient conditions for the existence of such. We gain a complete answer when G compactly generated. In particular for G almost connected, or more generally when C^*_r(G) is nuclear, the existence of a trace is equivalent to amenability. We exhibit two examples of classes of totally disconnected groups for which C^*_r(G) does not admit a tracial state.

Citations

• A new look at C*-simplicity and the unique trace property of a group

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