Compact C*-quantum groupoids

TL;DR

This paper introduces a new framework for compact quantum groupoids within C*-algebras, constructs associated unitaries and dual structures, and explores examples related to classical groupoids.

Contribution

It defines compact quantum groupoids in C*-algebras, constructs their duals, and connects to measurable quantum groupoids and classical examples.

Findings

Established a definition of compact quantum groupoids in C*-algebras.

Constructed regular C*-pseudo-multiplicative unitaries for these structures.

Explored examples related to compact and étale groupoids.

Abstract

We propose a definition of compact quantum groupoids in the setting of C*-algebras, associate to such a quantum groupoid a regular C*-pseudo-multiplicative unitary, and use this unitary to construct a dual Hopf C*-bimodule and to pass to a measurable quantum groupoid in the sense of Enock and Lesieur. Moreover, we discuss examples related to compact and to \'etale groupoids and study principal compact C*-quantum groupoids.