Free products of hyperfinite von Neumann algebras and free dimension

TL;DR

This paper investigates the structure of free products of finite hyperfinite von Neumann algebras, revealing they decompose into a finite dimensional algebra and an interpolated free group factor, with implications for group von Neumann algebras of amenable groups.

Contribution

It determines the precise structure of free products of hyperfinite von Neumann algebras and introduces a method to compute the free dimension of the free group factor component.

Findings

The free product decomposes into a finite dimensional algebra plus an interpolated free group factor.

The free dimension r is explicitly computed using the original algebras' projections.

Group von Neumann algebras of certain amenable groups are identified as interpolated free group factors.

Abstract

The free product of an arbitrary pair of finite hyperfinite von Neumann algebras is examined, and the result is determined to be the direct sum of a finite dimensional algebra and an interpolated free group factor $L(\freeF_r)$. The finite dimensional part depends on the minimal projections of the original algebras and the "dimension", r, of the free group factor part is found using the notion of free dimension. For discrete amenable groups $G$ and $H$ this implies that the group von Neumann algebra $L(G*H)$ is an interpolated free group factor and depends only on the orders of $G$ and $H$.