Simplicity of C^*-algebras of contracting self-similar groups

2025/01/20 by Eusebio Gardella, Gardella, Eusebio, Volodymyr Nekrashevych +5 · 4 citations
Mathematics · Physics and Astronomy · #20F10 #20F65 #46L99 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2501.11482

openalex publication_date 2025/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the C^*-algebra associated by Nekrashevych to a contracting self-similar group is simple if and only if the corresponding complex ∗-algebra is simple. We also improve on Steinberg and Szakać's algorithm to determine if the ∗-algebra is simple. This provides an interesting class of non-Hausdorff amenable, effective and minimal ample groupoids for which simplicity of the C^*-algebra and the complex ∗-algebra are equivalent.

Cited by

• On dense subalgebras of the singular ideal in groupoid C*-algebras
• Simplicity of algebras and C^*-algebras of self-similar groupoids
• Algebraic singular functions are not always dense in the ideal of C^*-singular functions
• Characterizations of zero singular ideal in étale groupoid C*-algebras via compressible maps

Related

• Simplicity of algebras associated to non-Hausdorff groupoids
• Simplicity of algebras and C^*-algebras of self-similar groupoids
• On the simplicity of Nekrashevych algebras of contracting self-similar groups
• Simplicity of *-algebras of non-Hausdorff ℤ₂-multispinal groupoids
• On simplicity of reduced C*-algebras of groups