2008/10/15 by Astrid an Huef, Huef, Astrid an, S. Kaliszewski +5
Mathematics · #46L55 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.0810.2819
openalex publication_date 2008/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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\rtimesα,rG 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.