Free products of finite dimensional and other von Neumann algebras with respect to non-tracial states

TL;DR

This paper investigates free products of finite-dimensional and other von Neumann algebras with non-tracial states, revealing their structure as type III factors and providing new methods to classify and construct such factors.

Contribution

It characterizes the structure of free products with non-tracial states as type III factors and introduces extremal almost periodic states for classification.

Findings

The free product is a type III factor or a sum with finite-dimensional algebra.

The free product state is an extremal almost periodic state.

Provides a new construction method for full type III_1 factors with arbitrary Connes Sd-invariant.

Abstract

The von Neumann algebra free product of arbitary finite dimensional von Neumann algebras with respect to arbitrary faithful states, at least one of which is not a trace, is found to be a type~III factor possibly direct sum a finite dimensional algebra. The free product state on the type~III factor is what we call an extremal almost periodic state, and has centralizer isomorphic to $L(\freeF_\infty)$. This allows further classification the type~III factor and provides another construction of full type~III$_1$ factors having arbitrary $\Sd$~invariant of Connes. The free products considered in this paper are not limited to free products of finite dimensional algebras, but can be of a quite general form.