Properly Proximal von Neumann Algebras

TL;DR

This paper introduces proper proximality for finite von Neumann algebras, extending group notions, and applies it to derive new results on embeddings, ergodicity, and properties like Haagerup, with broad implications in operator algebras.

Contribution

It defines proper proximality for von Neumann algebras, provides new examples, and applies this to solve open problems in embedding, ergodicity, and approximation properties.

Findings

Group von Neumann algebra of a nonamenable inner amenable group cannot embed into a free group factor.

Proper proximality for actions yields new invariants for orbit equivalence.

Proves the equivalence between the Haagerup property and the compact approximation property for II$_1$ factors.

Abstract

We introduce the notion of proper proximality for finite von Neumann algebras, which naturally extends the notion of proper proximality for groups. Apart from the group von Neumann algebras of properly proximal groups, we provide a number of additional examples, including examples in the settings of free products, crossed products, and compact quantum groups. Using this notion, we answer a question of Popa by showing that the group von Neumann algebra of a nonamenable inner amenable group cannot embed into a free group factor. We also introduce a notion of proper proximality for probability measure preserving actions, which gives an invariant for the orbit equivalence relation. This gives a new approach for establishing strong ergodicity type properties, and we use this in the setting of Gaussian actions to expand on solid ergodicity results first established by Chifan and Ioana, and later generalized by Boutonnet. The techniques developed also allow us to answer a problem left open by Anantharaman-Delaroche in 1995, by showing the equivalence between the Haagerup property and the compact approximation property for II$_1$ factors.