A unique Cartan subalgebra result for Bernoulli actions of weakly amenable groups

2024/04/12 by Changying Ding, Ding, Changying
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2404.08182

openalex publication_date 2024/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if Γ\curvearrowright (X^Γ,μ^Γ) is a Bernoulli action of an i.c.c. nonamenable group Γ which is weakly amenable with Cowling-Haagerup constant 1, and Λ\curvearrowright(Y,ν) is a free ergodic p.m.p. algebraic action of a group Λ, then the isomorphism L^∞(X^Γ)\rtimesΓ≅ L^∞(Y)\rtimesΛ implies that L^∞(X^Γ) and L^∞(Y) are unitarily conjugate. This is obtained by showing a new rigidity result of non properly proximal groups and combining it with a rigidity result of properly proximal groups from \citeBIP21.

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