We introduce C*-pseudo-multiplicative unitaries and concrete Hopf C*-bimodules for the study of quantum groupoids in the setting of C*-algebras. These unitaries and Hopf C*-bimodules generalize multiplicative unitaries and Hopf C*-algebras and are analogues of the pseudo-multiplicative unitaries and Hopf–von Neumann-bimodules studied by Enock, Lesieur and Vallin. To each C*-pseudo-multiplicative unitary, we associate two Fourier algebras with a duality pairing and in the regular case two Hopf C*-bimodules. The theory is illustrated by examples related to locally compact Hausdorff groupoids. In particular, we obtain a continuous Fourier algebra for a locally compact Hausdorff groupoid.

To any groupoid, equipped with a Haar system, Jean-Michel Vallin had associated several objects (pseudo-multiplicative unitary, Hopf-bimodule) in order to generalize, up to the groupoid case, the classical notions of multiplicative unitary and Hopf–von Neumann algebra, which were intensely used to construct quantum groups in the operator algebra setting. In two former articles (one in collaboration with Jean-Michel Vallin), starting from a depth-2 inclusion of von Neumann algebras, we have constructed such objects, which allowed us to study two ‘quantum groupoids’ dual to each other. We are now investigating in greater details the notion of pseudo-multiplicative unitary, following the general strategy developed by Baaj and Skandalis for multiplicative unitaries.

AMS 2000 Mathematics subject classification: Primary 46L89; 22A22; 81R50