# Følner sequences in operator theory and operator algebras

Pere Ara, Fernando Lledó, Dmitry V. Yakubovich

Research output: Chapter in BookChapterResearchpeer-review

1 Citation (Scopus)

## Abstract

© 2014 Springer Basel. 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 {Pn } 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(.Pn)/Tr(Pn) corresponding to a Følner sequence and apply this method to improve spectral approximation results due to Arveson and Bedos. 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.
Original language English Operator Theory: Advances and Applications 1-24 23 242 2296-4878 Published - 1 Jan 2014

## Keywords

• Amenable trace
• C*-algebra
• Essentially normal operators
• Følner sequences
• Non-normal operators
• Spectral approximation