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Compact ideals and rigidity of representations for amenable operator algebras

Volume 244 / 2019

Raphaël Clouâtre, Laurent W. Marcoux Studia Mathematica 244 (2019), 25-41 MSC: Primary 47L55, 47L75, 46L05; Secondary 46L07, 46H25. DOI: 10.4064/sm170328-12-10 Published online: 21 May 2018

Abstract

We examine rigidity phenomena for representations of amenable operator algebras which have an ideal of compact operators. We establish that a generalized version of Kadison’s conjecture on completely bounded homomorphisms holds for the algebra if the associated quotient is abelian. We also prove that injective completely bounded representations of the algebra are similar to complete isometries. The main motivating example for these investigations is the recent construction by Choi, Farah and Ozawa of an amenable operator algebra that is not similar to a $C^*$-algebra, and we show how it fits into our framework. All of our results hold in the presence of the total reduction property, a property weaker than amenability.

Authors

  • Raphaël ClouâtreDepartment of Mathematics
    University of Manitoba
    Winnipeg, Manitoba, Canada R3T 2N2
    e-mail
  • Laurent W. MarcouxDepartment of Pure Mathematics
    University of Waterloo
    Waterloo, Ontario, Canada N2L 3G1
    e-mail

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