The amalgamated free product of semifinite hyperfinite von Neumann algebras over atomic type I subalgebras

TL;DR

This paper extends the theory of amalgamated free products to semifinite hyperfinite von Neumann algebras over atomic type I subalgebras, introducing new notions and classes that are closed under these operations.

Contribution

It introduces extended notions of free dimension and standard embeddings for semifinite cases and defines classes R3 and R4 closed under amalgamated free products.

Findings

Extended free dimension and standard embeddings for semifinite algebras.

Defined classes R3 and R4 closed under amalgamated free products.

Described the structure of free products over atomic type I subalgebras.

Abstract

In this paper we describe the amalgamated free product of finite and semifinite hyperfinite von Neumann algebras over atomic type I subalgebras. To do this we extend the notions of free dimension and standard embeddings used in the related results for finite von Neumann algebras to ones which work better for the semifinite case. We also define classes R3 (of finite von Neumann algebras) and R4 (of semifinite von Neumann algebras) which are closed under such amalgamated free products.