### 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 |
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Title of host publication | Operator Theory: Advances and Applications |

Pages | 1-24 |

Number of pages | 23 |

Volume | 242 |

ISBN (Electronic) | 2296-4878 |

Publication status | Published - 1 Jan 2014 |

### Keywords

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

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## Cite this

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)