Proper actions, fixed-point algebras and naturality in nonabelian duality

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13 Scopus citations

Abstract

Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on C0 (X). We consider a category in which the objects are C*-dynamical systems (A, G, α) for which there is an equivariant homomorphism of (C0 (X), γ) into the multiplier algebra M (A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra Aα which is Morita equivalent to A ×α, r G. We show that the assignment (A, α) {mapping} Aα is functorial, and that Rieffel's Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural.

Original languageEnglish (US)
Pages (from-to)2949-2968
Number of pages20
JournalJournal of Functional Analysis
Volume254
Issue number12
DOIs
StatePublished - Jun 15 2008

Keywords

  • Coaction
  • Comma category
  • Crossed product
  • Fixed-point algebra
  • Landstad duality
  • Proper actions

ASJC Scopus subject areas

  • Analysis

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