The amalgamated free product of hyperfinite von Neumann algebras over finite dimensional subalgebras

TL;DR

This paper characterizes the structure of the amalgamated free product of hyperfinite von Neumann algebras over finite dimensional subalgebras, showing it results in a finite sum of interpolated free group factors and remains closed under such operations.

Contribution

It provides a detailed description of the amalgamated free product structure and proves closure properties for this class of von Neumann algebras.

Findings

The free product decomposes into a finite sum of interpolated free group factors and a hyperfinite algebra.

The class of these von Neumann algebras is closed under amalgamated free products over finite dimensional subalgebras.

The structure of the free product can be explicitly described in terms of known factors.

Abstract

In this paper we describe the amalgamated free product of two hyperfinite von Neumann algebras over a finite dimensional subalgebra. In general the free product is a finite direct sum of interpolated free group factors and a hyperfinite von Neumann algebra. We then show that the class of von Neumann algebras of this form is closed under taking amalgamated free products over finite dimensional subalgebras.