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

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

2011-05-17

arxiv.org/abs/1105.3406v1

Mathematics

Operator Algebras

25 pages

Scientific paper

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.

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