Følner Sequences in Operator Theory and Operator Algebras

Abstract

The present article is a review of recent developments concerning the notion of Følner sequences both in operator theory and operator algebras. We also give a new direct proof that any essentially normal operator has an increasing Følner sequence \(\begin{array}{lll}\left\{P_{n}\right\}\end{array}\) of non-zero finite rank projections that strongly converges to 1. The proof is based on Brown–Douglas–Fillmore theory. We use Følner 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 \(Tr(.P_{n})/Tr(P_{n})\) corresponding to a Følner sequence and apply this method to improve spectral approximation results due to Arveson and Bédos. The article concludes with the analysis of C*-algebras admitting a non-degenerate representation which has a Følner sequence or, equivalently, an amenable trace. We give an abstract characterization of these algebras in terms of unital completely positive maps and define Følner 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.