Amenability and vanishing of L^2-Betti numbers: an operator algebraic approach

TL;DR

This paper introduces an operator algebraic framework for understanding amenability and L^2-Betti numbers, unifying and extending previous results to new classes of algebras with vanishing L^2-Betti numbers.

Contribution

It recasts the Foelner condition in an operator algebraic setting, proving its implications for dimension flatness and injectivity of von Neumann algebras, and unifies various vanishing results.

Findings

Foelner condition implies dimension flatness.

Enveloping von Neumann algebra from a Foelner algebra is injective.

Provides new examples of algebras with vanishing L^2-Betti numbers.

Abstract

We recast the Foelner condition in an operator algebraic setting and prove that it implies a certain dimension flatness property. Furthermore, it is proven that the Foelner condition generalizes the existing notions of amenability and that the enveloping von Neumann algebra arising from a Foelner algebra is automatically injective. As an application we show how our techniques unify the previously known results concerning vanishing of L^2-Betti numbers for amenable groups, groupoids and quantum groups and moreover provides a large class of new examples of algebras with vanishing L^2-Betti numbers.