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# Journal of Operator Theory

Volume 68, Issue 1, Summer 2012  pp. 19-66.

Coactions of Hopf $C^*$-bimodules

Authors Thomas Timmermann
Author institution: FB 10 Mathematisches Institut, Westfaelische Wilhelms-Universitaet, Einsteinstrasse 62, 48149 Muenster, Germany

Summary:  Coactions of Hopf $C^{*}$-bimodules simultaneously generalize coactions of Hopf $C^{*}$-algebras and actions of groupoids. Following an approach of Baaj and Skandalis, we construct reduced crossed products and establish a duality for fine coactions. Examples of coactions arise from Fell bundles on groupoids and actions of a groupoid on bundles of $C^{*}$-algebras. Continuous Fell bundles on an \'etale groupoid correspond to coactions of the reduced groupoid algebra, and actions of a groupoid on a continuous bundle of $C^{*}$-algebras correspond to coactions of the function algebra.

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