Maximal amenable MASAs of the free group factor of two generators arising from the free products of hyperfinite factors

TL;DR

This paper constructs infinitely many maximal amenable subalgebras within the free group factor of two generators, using free products of hyperfinite factors and specific Haar unitaries, advancing understanding of their structure.

Contribution

It introduces a new class of maximal amenable MASAs in free group factors derived from free products of hyperfinite factors, with explicit construction and conjugacy properties.

Findings

Identifies a von Neumann subalgebra generated by a specific self-adjoint operator as maximal amenable.

Provides examples of non-unitarily conjugate maximal amenable MASAs.

Expands the known landscape of maximal amenable subalgebras in free group factors.

Abstract

In this paper, we give examples of maximal amenable subalgebras of the free group factor of two generators. More precisely, we consider two copies of the hyperfinite factor $R_i$ of type $\mathrm{II}_1$. From each $R_i$, we take a Haar unitary $u_i$ which generates a Cartan subalgebra of it. We show that the von Neumann subalgebra generated by the self-adjoint operator $u_1+u_1^{-1}+u_2+u_2^{-1}$ is maximal amenable in the free product. This provides infinitely many non-unitary conjugate maximal amenable MASAs.