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Naturality of symmetric imprimitivity theorems


Authors: Astrid an Huef, S. Kaliszewski, Iain Raeburn and Dana P. Williams
Journal: Proc. Amer. Math. Soc. 141 (2013), 2319-2327
MSC (2010): Primary 46L55
DOI: https://doi.org/10.1090/S0002-9939-2013-11712-0
Published electronically: February 26, 2013
MathSciNet review: 3043013
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Abstract: The first imprimitivity theorems identified the representations of groups or dynamical systems which are induced from representations of a subgroup. Symmetric imprimitivity theorems identify pairs of crossed products by different groups which are Morita equivalent and hence have the same representation theory. Here we consider commuting actions of groups $ H$ and $ K$ on a $ C^*$-algebra which are saturated and proper as defined by Rieffel in 1990. Our main result says that the resulting Morita equivalence of crossed products is natural in the sense that it is compatible with homomorphisms and induction processes.


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Additional Information

Astrid an Huef
Affiliation: Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand
Email: astrid@maths.otago.ac.nz

S. Kaliszewski
Affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287-1804
Email: kaliszewski@asu.edu

Iain Raeburn
Affiliation: Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand
Email: iraeburn@maths.otago.ac.nz

Dana P. Williams
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
Email: dana.williams@dartmouth.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11712-0
Keywords: Symmetric imprimitivity theorem, proper actions on $C^{*}$-algebras, naturality, fixed-point algebras, crossed products
Received by editor(s): March 18, 2011
Received by editor(s) in revised form: October 10, 2011
Published electronically: February 26, 2013
Additional Notes: This research was supported by the University of Otago and the Edward Shapiro Fund at Dartmouth College.
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society