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

2007/12/30 by S. Kaliszewski, John Quigg, Kaliszewski, S. +3 · 1 citation
Computer Science · Mathematics · #18A25 #46L55 #46M15 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Category Theory (math.CT) #FOS: Mathematics #Matrix Theory and Algorithms #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.0801.0161

openalex publication_date 2007/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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 C₀(X). We consider a category in which the objects are C*-dynamical systems (A, G, alpha) for which there is an equivariant homomorphism of (C₀(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.

Citations

• Categorical Landstad duality for actions

Cited by

• On groupoid-graded C*-algebras and equivalent subcategories linked via monads and comonads

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