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

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

2007-12-30

arxiv.org/abs/0801.0161v3

Mathematics

Operator Algebras

19 pages; minor revision

Scientific paper

Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let gamma be the induced action on C_0(X). We consider a category in which the objects are C*-dynamical systems (A, G, alpha) for which there is an equivariant homomorphism of (C_0(X), gamma) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra A^alpha which is Morita equivalent to A times_{alpha,r} G. We show that the assignment (A, alpha) maps to A^alpha 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.

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