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

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 $C^{\ast }$-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.

29 pages

References in corpus (3)

• C*-pseudo-multiplicative unitaries
• $C^{\ast }$-pseudo-Kac systems and duality for coactions of concrete Hopf $C^{\ast }$-bimodules
• Finite-dimensional Hopf C*-bimodules and C*-pseudo-multiplicative unitaries