Biexact von Neumann algebras

TL;DR

This paper introduces biexactness for von Neumann algebras, explores its implications for solidity, and distinguishes it from related properties through new examples and techniques involving nuclear embeddings.

Contribution

It defines the concept of biexactness for von Neumann algebras, connects it to solidity, and provides new examples and characterizations, advancing understanding of weak exactness.

Findings

Biexactness implies solidity for von Neumann algebras.

Many known solid von Neumann algebras are also biexact.

Examples of solid but not biexact algebras are provided.

Abstract

We introduce the notion of biexactness for general von Neumann algebras, naturally extending the notion from group theory. We show that biexactness implies solidity for von Neumann algebras, and that many of the examples of solid von Neumann algebras contained in the literature are, in fact, biexact. We also give examples of certain crossed products arising from Gaussian actions that are solid but not biexact, and we give examples of certain $q$-Gaussian von Neumann algebras that are strongly solid but not biexact. The techniques developed involve studying a certain weak form of nuclear embeddings, and we use this setting to give a new description of weak exactness for von Neumann algebras, which allows us to answer several open problems in the literature about weakly exact von Neumann algebras.