The Space of Traces of the Free Group and Free Products of Matrix Algebras

2023/08/21 by Joav Orovitz, Raz Slutsky, Orovitz, Joav +3 · 2 citations
Computer Science · Mathematics · #20E06 #20F65 #46L05 #46L09 #46L10 #81P45 #Advanced Algebra and Logic #Advanced Topics in Algebra #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2308.10955

openalex publication_date 2023/08/21 · openalex created_date 2023/08/24 · openalex updated_date 2026/07/28

Abstract

We show that the space of traces of the free group F_d on 2≤ d ≤ ∞ generators is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. This fails for many virtually free groups. The same result holds for free products of the form C(X₁)*C(X₂) where X₁ and X₂ are compact metrizable spaces without isolated points. Using a similar strategy, we show that the space of traces of the free product of matrix algebras M_n(ℂ) * M_n(ℂ) is a Poulsen simplex as well, answering a question of Musat and R\ordam for n ≥ 4. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.

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• Entanglement in C^*-algebras: tensor products of state spaces
• Hyperlinearity, stability and asymptotic spectral gap of higher rank lattices

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