Amenable traces and Følner C*-algebras

In the present article we review an approximation procedure for amenable traces on unital andseparable C*-algebras acting on a Hilbert space in terms of Følner sequences of non-zero finite rankprojections. 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 ofunital separable C*-algebras admitting a non-degenerate representation which has a Følner sequenceor, equivalently, an amenable trace. This is analogous to Voiculescu's abstract characterization ofquasidiagonal C*-algebras. We define Følner C*-algebras as those unital separable C*-algebras thatsatisfy these equivalent conditions. Finally we also mention some permanence properties related tothese algebras.The first author was partially supported by DGI MICIIN MTM2011-28992-C02-01, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The second author was partially supported by projects DGI MICIIN MTM2012-36372-C03-01 and Severo Ochoa SEV-2011-0087 of the Spanish Ministry of Economy and Competition