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Følner Sequences in Operator Theory and Operator Algebras

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Operator Theory, Operator Algebras and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 242))

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.

Mathematics Subject Classification (2010). Primary 43A07; Secondary 47B20, 47A65, 46L05.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Universitat Autònoma de Barcelona, E-08193, Bellaterra, (Barcelona), Spain

    Pere Ara

  2. Department of Mathematics, University Carlos III Madrid, Avda. de la Universidad 30, E-28911, Leganés (Madrid), Spain

    Fernando Lledó

  3. Instituto de Ciencias Matemáticas (CSIC – UAM – UC3M – UCM), Madrid, Spain

    Fernando Lledó & Dmitry V. Yakubovich

  4. Departamento de Matemáticas, Universidad Autonoma de Madrid, E-28049, Cantoblanco, (Madrid), Spain

    Dmitry V. Yakubovich

Authors
  1. Pere Ara
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  2. Fernando Lledó
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  3. Dmitry V. Yakubovich
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Corresponding author

Correspondence to Pere Ara .

Editor information

Editors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, Lisbon, Portugal

    M. Amélia Bastos

  2. Departamento de Matemática, Instituto Superior Técnico, Lisbon, Portugal

    Amarino Lebre

  3. Departamento de Matemática, Universidade do Algarve, Faro, Portugal

    Stefan Samko

  4. Department of Mathematics, College of William & Mary, Williamsburg, Virginia, USA

    Ilya M. Spitkovsky

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Ara, P., Lledó, F., Yakubovich, D.V. (2014). Følner Sequences in Operator Theory and Operator Algebras. In: Bastos, M., Lebre, A., Samko, S., Spitkovsky, I. (eds) Operator Theory, Operator Algebras and Applications. Operator Theory: Advances and Applications, vol 242. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0816-3_1

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