Coactions of hope C-bimodules

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 étale 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