Von Neumann Orbit Equivalence

TL;DR

This paper introduces von Neumann orbit equivalence (vNOE), a new non-commutative group equivalence, and demonstrates its stability under free and graph product operations, linking group properties with von Neumann algebra structures.

Contribution

It defines vNOE for groups and von Neumann algebras, establishing its stability under free and graph products, and characterizing group vNOE via their von Neumann algebras.

Findings

vNOE is equivalent for groups and their von Neumann algebras.

vNOE is stable under free products.

vNOE is stable under graph products.

Abstract

We generalize the notion of orbit equivalence to the non-commutative setting by introducing a new equivalence relation on groups, which we call von Neumann orbit equivalence (vNOE). We prove the stability of this equivalence relation under taking free products and graph products of groups. To achieve this, we introduce von Neumann orbit equivalence of tracial von Neumann algebras, show that two countable discrete groups $\Gamma$ and $\Lambda$ are vNOE if and only if the corresponding group von Neumann algebras $L\Gamma$ and $L\Lambda$ are vNOE, and that vNOE of tracial von Neumann algebras is stable under taking free products and graph products of tracial von Neumann algebras.