A Class of Freely Complemented von Neumann Subalgebras of L\mathbbF_n

2024/11/07 by Nicholas Boschert, Boschert, Nicholas, Davis, Ethan +2
Computer Science · Mathematics · #Coding theory and cryptography #Finite Group Theory Research #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2411.05136

Abstract

We prove that if A₁, A₂, …, A_n are tracial abelian von Neumann algebras for 2≤ n ≤ ∞ and M = A₁ * ⋯ * A_n is their free product, then any subalgebra A ⊂ M of the form A = ∑_i=1ⁿ u_i A_i p_i u_i^*, for some projections p_i ∈ A_i and unitaries u_i ∈ U(M), for 1 ≤ i ≤ n, such that ∑_i u_i p_i u_i^* = 1, is freely complemented (FC) in M. Moreover, if A₁, A₂, …, A_n are purely non-separable abelian, and M = A₁ * ⋯ * A_n, then any purely non-separable singular MASA in M is FC. We also show that any of the known maximal amenable MASAs A⊂ L\mathbbF_n (notably the radial MASA), satisfies Popa's weak FC conjecture, i.e., there exist Haar unitaries u∈ L\mathbbF_n that are free independent to A.

Related

• A remark on amenable von Neumann subalgebras in a tracial free product
• Maximal amenable MASAs of radial type in the free group factors
• Free products and rescalings involving non-separable abelian von Neumann algebras
• Radial multipliers on arbitrary amalgamated free products of finite von Neumann algebras
• Some explicit computations and models of free products