Naturality of symmetric imprimitivity theorems

an Huef, Astrid; Kaliszewski, S.; Raeburn, Iain; Williams, Dana

doi:10.1090/S0002-9939-2013-11712-0

Naturality of symmetric imprimitivity theorems
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by Astrid an Huef, S. Kaliszewski, Iain Raeburn and Dana P. Williams

Proc. Amer. Math. Soc. 141 (2013), 2319-2327

DOI: https://doi.org/10.1090/S0002-9939-2013-11712-0

Published electronically: February 26, 2013

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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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