  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.
