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

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.

25 pages

References in corpus (4)

• A spectral sequence to compute L2-Betti numbers of groups and groupoids
• L^2-Homology for von Neumann Algebras
• L^2-Betti Numbers of Discrete Measured Groupoids
• Applications of Foelner's condition to quantum groups

Cited by in corpus (3)

• Measure continuous derivations on von Neumann algebras and applications to L^2-cohomology
• $L^2$-Betti numbers and costs in the framework of discrete groupoids
• A groupoid approach to Lück's amenability conjecture