A random matrix approach to the Peterson-Thom conjecture

2020/08/27 by Ben J. Hayes, Ben Hayes, Hayes, Ben · 9 citations
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Functional Analysis (math.FA) #Geometric and Algebraic Topology #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications #math.FA #math.OA #math.PR

paper · pdf · doi:10.48550/arxiv.2008.12287

46 pages. This is the final version, and will appear as such in the Indiana University Mathematics Journal

openalex publication_date 2020/08/27 · arxiv created 2022/03/14 · arxiv updated 2022/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by related results in Popa's deformation/rigidity theory and Peterson-Thom's results on L²-Betti numbers. We present an approach to this conjecture in terms of so-called strong convergence of random matrices by formulating a conjecture which is a natural generalization of the Haagerup-Thorbjornsen theorem whose validity would imply the Peterson-Thom conjecture. This random matrix conjecture is related to recent work of Collins-Guionnet-Parraud.

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