From Hopf C*-families to concrete Hopf C*-bimodules

TL;DR

This paper bridges the gap between the theory of quantum groupoids in von Neumann algebras and $C^{*}$-algebras, showing how specific structures in the latter fit into a broader, more general framework.

Contribution

It demonstrates how the specialized theory of Hopf C*-families and bimodules embeds into a comprehensive approach for all locally compact quantum groupoids.

Findings

Embedding of the specialized theory into the general framework

Extension of quantum groupoid structures to all locally compact cases

Clarification of the relationship between different quantum groupoid models

Abstract

In the setting of von Neumann algebras, measurable quantum groupoids have successfully been axiomatized and studied by Enock, Vallin, and Lesieur, whereas in the setting of $C^{*}$-algebras, a similar theory of locally compact quantum groupoids could not yet be developed. Some basic building blocks for such a theory, like analogues of a Hopf-von Neumann bimodule and of a pseudo-multiplicative unitary, were introduced in the thesis and a recent article by the author. That approach, however, is restricted to decomposable quantum groupoids which generalize $r$-discrete groupoids. Recently, we developed a general approach that covers all locally compact groupoids. In this article, we explain how the special theory of our thesis embeds into the general one.