F{\o}lner sequences in operator theory and operator algebras

The present article is a review of recent developments concerning the notion of F{\o}lner sequences both in operator theory and operator algebras. We also give a new direct proof that any essentially normal operator has an increasing F{\o}lner sequence $\{P_n\}$ of non-zero finite rank projections that strongly converges to 1. The proof is based on Brown-Douglas-Fillmore theory. We use F{\o}lner sequences to analyze the class of finite operators introduced by Williams in 1970. In the second part of this article we examine a procedure of approximating any amenable trace on a unital and separable C*-algebra by tracial states $\mathrm{Tr}(\cdot P_n)/\mathrm{Tr}(P_n)$ corresponding to a F{\o}lner sequence and apply this method to improve spectral approximation results due to Arveson and B\'edos. The article concludes with the analysis of C*-algebras admitting a non-degenerate representation which has a F{\o}lner sequence or, equivalently, an amenable trace. We give an abstract characterization of these algebras in terms of unital completely positive maps and define F{\o}lner C*-algebras as those unital separable C*-algebras that satisfy these equivalent conditions. This is analogous to Voiculescu's abstract characterization of quasidiagonal C*-algebras.