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