A Kurosh-Type Theorem for Type III Factors

TL;DR

This paper generalizes Ozawa's Kurosh-type theorem to free products of semiexact II_1 factors with non-tracial states, enabling the distinction of certain type III factors and showing their non-isomorphism.

Contribution

It extends Kurosh-type decomposition results to a broader class of type III factors with arbitrary states, providing new tools for their classification.

Findings

Different free product constructions yield mutually non-isomorphic type III factors.

The theorem applies to free products of II_1 factors with arbitrary faithful normal states.

The results help distinguish complex type III factors in operator algebra theory.

Abstract

We prove a generalization of N. Ozawa's Kurosh-type theorem to the setting of free products of semiexact II_1 factors with respect to arbitrary (non-tracial) faithful normal states. We are thus able to distinguish certain resulting type III factors. For example, if M = LF_n \otimes LF_m and {\phi_i} is any sequence of faithful normal states on M, then the l-various (M,\phi_1) * ... * (M,\phi_l) are all mutually non-isomorphic.