C*-algebraic partial compact quantum groups

TL;DR

This paper introduces C*-algebraic partial compact quantum groups, extending the theory of quantum groups to infinite object sets and connecting to dynamical quantum SU(2) groups.

Contribution

It generalizes Hayashi's compact face algebras to infinite object sets and develops the C*-algebraic framework for partial compact quantum groups.

Findings

Defined C*-algebraic partial compact quantum groups.

Connected the theory to dynamical quantum SU(2) groups.

Extended algebraic quantum group theory to the C*-algebraic setting.

Abstract

In this paper, we introduce C*-algebraic partial compact quantum groups, which are quantizations of topological groupoids with discrete object set and compact morphism spaces. These C*-algebraic partial compact quantum groups are generalisations of Hayashi's compact face algebras to the case where the object set can be infinite. They form the C*-algebraic counterpart of an algebraic theory of partial compact quantum groups developed in an earlier paper by the author and T. Timmermann, the correspondence between which will be dealt with in a separate paper. As an interesting example to illustrate the theory, we show how the dynamical quantum SU(2) group, as studied by Etingof-Varchenko and Koelink-Rosengren, fits into this framework.