Illinois Journal of Mathematics

Fixed-point algebras for proper actions and crossed products by homogeneous spaces

Astrid an Huef, S. Kaliszewski, Iain Raeburn, and Dana P. Williams

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We consider a fixed free and proper action of a locally compact group G on a space T, and actions α : G → Aut A on C-algebras for which there is an equivariant embedding of (C0(T), rt) in (M(A), α). A recent theorem of Rieffel implies that α is proper and saturated with respect to the subalgebra Cc(T)ACc(T) of A, so that his general theory of proper actions gives a Morita equivalence between Aα,r G and a generalised fixed-point algebra Aα. Here we investigate the functor (A, α) ↦ Aα and the naturality of Rieffel’s Morita equivalence, focusing in particular on the relationship between the different functors associated to subgroups and quotients. We then use the results to study induced representations for crossed products by coactions of homogeneous spaces G/H of G, which were previously shown by an Huef and Raeburn to be fixed-point algebras for the dual action of H on the crossed product by G.

Article information

Illinois J. Math., Volume 55, Number 1 (2011), 205-236.

First available in Project Euclid: 19 December 2012

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Zentralblatt MATH identifier

Primary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]


an Huef, Astrid; Kaliszewski, S.; Raeburn, Iain; Williams, Dana P. Fixed-point algebras for proper actions and crossed products by homogeneous spaces. Illinois J. Math. 55 (2011), no. 1, 205--236. doi:10.1215/ijm/1355927034.

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