Amenable traces and F{\o}lner C*-Algebras

TL;DR

This paper reviews approximation procedures for amenable traces on unital separable C*-algebras using F ext{l}ner sequences, improving spectral approximation results and characterizing F ext{l}ner C*-algebras through completely positive maps.

Contribution

It introduces F ext{l}ner C*-algebras and provides an abstract characterization similar to Voiculescu's quasidiagonality, with applications to spectral approximation.

Findings

Improved spectral approximation results for amenable traces.

Characterization of F ext{l}ner C*-algebras via unital completely positive maps.

Identification of permanence properties of F ext{l}ner C*-algebras.

Abstract

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.