The Radial Masa in a Free Group Factor is Maximal Injective

TL;DR

This paper proves that the radial masa in a free group factor is a maximal injective von Neumann algebra and explores tensor product stability of maximal injectivity under certain conditions.

Contribution

It establishes the maximal injectivity of the radial masa in free group factors and shows tensor product preservation of this property under specific conditions.

Findings

Radial masa is maximal injective in free group factors.

Tensor products of maximal injective algebras remain maximal injective under asymptotic orthogonality.

Finite tensor products of generator and radial masas are maximal injective.

Abstract

The radial (or Laplacian) masa in a free group factor is the abelian von Neumann algebra generated by the sum of the generators (of the free group) and their inverses. The main result of this paper is that the radial masa is a maximal injective von Neumann subalgebra of a free group factor. We also investigate tensor products of maximal injective algebras. Given two inclusions $B_i\subset M_i$ of type $\mathrm{I}$ von Neumann algebras in finite von Neumann algebras such that each $B_i$ is maximal injective in $M_i$, we show that the tensor product $B_1\ \bar{\otimes}\ B_2$ is maximal injective in $M_1\ \bar{\otimes}\ M_2$ provided at least one of the inclusions satisfies the asymptotic orthogonality property we establish for the radial masa. In particular it follows that finite tensor products of generator and radial masas will be maximal injective in the corresponding tensor product of free group factors.