Amenable traces and Følner C∗C∗-algebras

In the present article we review an approximation procedure for amenable traces on unital and separable C∗C∗-algebras acting on a Hilbert space in terms of Følner sequences of non-zero finite rank projections. We apply this method to improve spectral approximation results due to Arveson and Bédos. We also present an abstract characterization in terms of unital completely positive maps of unital separable C∗C∗-algebras admitting a non-degenerate representation which has a Følner sequence or, equivalently, an amenable trace. This is analogous to Voiculescu’s abstract characterization of quasidiagonal C∗C∗-algebras. We define Følner C∗C∗-algebras as those unital separable C∗C∗-algebras that satisfy these equivalent conditions. Finally we also mention some permanence properties related to these algebras.