C-pseudo-multiplicative unitaries and Hopf C-bimodules

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, a C∗-tensor category of representations, and in the regular case two reduced and two universal 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.