Naturality of rieffel's morita equivalence for proper actions

Astrid An Huef, Steven Kaliszewski, Iain Raeburn, Dana P. Williams

Research output: Contribution to journalReview articlepeer-review

8 Scopus citations

Abstract

Suppose that a locally compact group G acts freely and properly on the right of a locally compact space T. Rieffel proved that if α is an action of G on a C*-algebra A and there is an equivariant embedding of C0(T) in M(A), then the action α of G on A is proper, and the crossed product A ⋊α,r G is Morita equivalent to a generalised fixed-point algebra Fix(A, α) in M(A)α. We show that the assignment (A, α) → Fix(A, α) extends to a functor Fix on a category of C*-dynamical systems in which the isomorphisms are Morita equivalences, and that Rieffel's Morita equivalence implements a natural isomorphism between a crossed-product functor and Fix. From this, we deduce naturality of Mansfield imprimitivity for crossed products by coactions, improving results of Echterhoff-Kaliszewski-Quigg-Raeburn and Kaliszewski-Quigg-Raeburn, and naturality of a Morita equivalence for graph algebras due to Kumjian and Pask.

Original languageEnglish (US)
Pages (from-to)515-543
Number of pages29
JournalAlgebras and Representation Theory
Volume14
Issue number3
DOIs
StatePublished - Jun 2011

Keywords

  • Coaction
  • Crossed product
  • Fixed-point algebra
  • Proper actions on C-algebras

ASJC Scopus subject areas

  • Mathematics(all)

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