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

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.