W*-correlations of II$_1$ factors and rigidity of tensor products and graph products

TL;DR

This paper introduces W*-correlations for II$_1$ factors, establishes rigidity results for tensor and graph products, and constructs uncountably many groups with non-isomorphic von Neumann algebras, advancing understanding of group von Neumann algebra classification.

Contribution

It develops the concept of W*-correlations for II$_1$ factors, proves rigidity results, and constructs uncountably many groups with distinct von Neumann algebra properties.

Findings

Rigidity results for tensor and graph products of II$_1$ factors.

Construction of uncountably many groups with non-isomorphic von Neumann algebras.

Demonstration that these groups are not measure equivalent or von Neumann equivalent.

Abstract

A variant of Gromov's notion of measure equivalence for groups has been introduced for II$_1$ factors under different names. We propose the terminology of W*-correlated II$_1$ factors. We prove rigidity results up to W*-correlations for tensor products and graph products of II$_1$ factors. As a consequence, we construct the first uncountable family of discrete groups $\Gamma$ that are not von Neumann equivalent, which means that their group von Neumann algebras $L(\Gamma)$ are not W*-correlated, and which implies that these groups are neither measure equivalent, nor have isomorphic or virtually isomorphic group von Neumann algebras.