Free products of finite-dimensional and other von Neumann algebras in terms of free Araki-Woods factors

TL;DR

This paper demonstrates that free products of finite-dimensional von Neumann algebras with non-tracial states are isomorphic to free Araki-Woods factors, extending the classification to certain infinite-dimensional cases with almost periodic states.

Contribution

It provides a complete classification of free products of finite-dimensional von Neumann algebras in terms of free Araki-Woods factors, answering longstanding open questions.

Findings

Free products of finite-dimensional von Neumann algebras are isomorphic to free Araki-Woods factors.

Extension of classification to infinite-dimensional von Neumann algebras with almost periodic states.

Resolution of questions posed by Dykema and Shlyakhtenko.

Abstract

We show that any free product of finite-dimensional von Neumann algebras equipped with non-tracial states is isomorphic to a free Araki-Woods factor with its free quasi-free state possibly direct sum a finite-dimensional von Neumann algebra. This gives a complete answer to questions posed by Dykema and Shlyakhtenko, which had been partially answered by work of Houdayer and work of Ueda. We also extend this to suitable infinite-dimensional von Neumann algebras with almost periodic states.