Følner sequences in operator theory and operator algebras

Pere Ara, Fernando Lledó, Dmitry V. Yakubovich

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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 languageEnglish
Title of host publicationOperator Theory: Advances and Applications
Pages1-24
Number of pages23
Volume242
ISBN (Electronic)2296-4878
Publication statusPublished - 1 Jan 2014

Keywords

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

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    Ara, P., Lledó, F., & Yakubovich, D. V. (2014). Følner sequences in operator theory and operator algebras. In Operator Theory: Advances and Applications (Vol. 242, pp. 1-24)