C*-pseudo-multiplicative unitaries and Hopf C*-bimodules

TL;DR

This paper introduces C*-pseudo-multiplicative unitaries and Hopf C*-bimodules, extending quantum groupoid theory within C*-algebras, and illustrates their application to locally compact groupoids.

Contribution

It develops a new framework of C*-pseudo-multiplicative unitaries and Hopf C*-bimodules, generalizing existing structures for quantum groupoids.

Findings

Defined C*-pseudo-multiplicative unitaries and Hopf C*-bimodules

Established duality and representation theories for these structures

Applied the theory to locally compact Hausdorff groupoids

Abstract

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-bimod-ules 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.