Three versions of categorical crossed-product duality

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Abstract

In this partly expository paper we compare three different categories of C*-algebras in which crossed-product duality can be for mulated, both for actions and for coactions of locally compact groups. In these categories, the isomorphisms correspond to C*-algebra isomor phisms, imprimitivity bimodules, and outer conjugacies, respectively. In each case, a variation of the fixed-point functor that arises from classical Landstad duality is used to obtain a quasi-inverse for a crossed product functor. To compare the various cases, we describe in a formal way our view of the fixed-point functor as an “inversion" of the process of forming a crossed product. In some cases, we obtain what we call \good" inversions, while in others we do not. For the outer-conjugacy categories, we generalize a theorem of Peder sen to obtain a fixed-point functor that is quasi-inverse to the reduced crossed-product functor for actions, and we show that this gives a good inversion. For coactions, we prove a partial version of Pedersen's the orem that allows us to define a fixed-point functor, but the question of whether it is a quasi-inverse for the crossed-product functor remains open.

Original languageEnglish (US)
Pages (from-to)293-339
Number of pages47
JournalNew York Journal of Mathematics
Volume22
StatePublished - 2016

Keywords

  • Action
  • Category equivalence
  • Coaction
  • Crossed-product duality
  • Exterior equivalence
  • Outer conjugacy

ASJC Scopus subject areas

  • Mathematics(all)

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