Rigidity of free product von Neumann algebras

TL;DR

This paper proves that free product von Neumann algebras preserve the number and structure of their nonamenable factors, extending rigidity results to type III factors and revealing new phenomena.

Contribution

It unifies and extends Kurosh-type rigidity results for free product von Neumann algebras, including type III factors, showing structural preservation under conjugacy.

Findings

Preservation of factor cardinality in free products

Structural rigidity up to conjugacy for nonamenable factors

Extension of rigidity results to type III von Neumann algebras

Abstract

Let $I$ be any nonempty set and $(M_i, \varphi_i)_{i \in I}$ any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class $\mathcal C_{\rm anti-free}$ of (possibly type III) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing a Cartan subalgebra. For the free product $(M, \varphi) = \ast_{i \in I} (M_i, \varphi_i)$, we show that the free product von Neumann algebra $M$ retains the cardinality $|I|$ and each nonamenable factor $M_i$ up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type II$_1$ factors and is new for free product type III factors. It moreover provides new rigidity phenomena for type III factors.