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Induced Coactions of Discrete Groups on C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Siegfried Echterhoff  and
John Quigg
Siegfried Echterhoff
Affiliation:
Westfälishche Wilhelms-Universität Münster, SFB 478, Hittorfstrasse 27, D-48149 Münster, Germany email: echter@math.uni-muenster.de
John Quigg
Affiliation:
Department of Mathematics, Arizona State University, Tempe, Arizona 85287, U.S.A. email: quigg@math.la.asu.edu
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Abstract

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Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a ${{C}^{*}}$ -coaction $\delta :D\to D\otimes {{C}^{*}}\left( G/N \right)$ of a quotient group $G/N$ of a discrete group $G$ to a ${{C}^{*}}$-coaction $\text{Ind}\,\delta \text{:}\,\text{Ind}\,D\to \text{Ind}\,D\otimes {{C}^{*}}\left( G \right)\,\text{of }G$. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products $\text{Ind}\,D{{\times }_{\text{Ind}\,\delta }}\,G\,\text{and }D{{\times }_{\delta }}\,G/N$ are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

Keywords

46L55
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 4 , 01 August 1999 , pp. 745 - 770
Copyright
Copyright © Canadian Mathematical Society 1999

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