A description of amalgamated free products of finite von Neumann algebras over finite dimensional subalgebras

TL;DR

This paper investigates the structure of free products of finite von Neumann algebras over finite dimensional subalgebras, showing they remain within the same class and providing an algorithm for their description.

Contribution

It proves that such free products are always II_1-factors and introduces an algorithm to describe these products in terms of known constructions.

Findings

Free products of a II_1-factor and a finite von Neumann algebra over a finite dimensional subalgebra are II_1-factors.

Provides an explicit algorithm for describing these free products.

The class of certain finite von Neumann algebras is closed under free products with amalgamation over finite dimensional subalgebras.

Abstract

We show that a free product of a II_1-factor and a finite von Neumann algebra with amalgamation over a finite dimensional subalgebra is always a II_1-factor, and provide an algorithm for describing it in terms of free products (with amalgamation over the scalars) and compression/dilation. As an application, we show that the class of direct sums of finitely many von Neumann algebras that are interpolated free group factors, hyperfinite II_1-factors, type I_n algebras for n finite, and finite dimensional algebras, is closed under taking free products with amalgamation over finite dimensional subalgebras.