Duals of quantum semigroups with involution

Abstract

We define a category $\mathcal{QSI}$ of quantum semigroups with involution which carries a corepresentation-based duality map $M\mapsto \widehat M$. Objects in $\mathcal{QSI}$ are von Neumann algebras with comultiplication and coinvolution, we do not suppose the existence of a Haar weight or of a distinguished spatial realisation. In the case of a locally compact quantum group $\mathbb G$, the duality $\;\widehat{\ }\;$ in $\mathcal{QSI}$ recovers the universal duality of Kustermans: $\widehat{L^\infty(\mathbb G)} = C_0^u(\hat {\mathbb G})^{**}= \widehat{ C_0^u(\mathbb G)^{**}}$, and $\widehat{L^\infty(\hat{\mathbb G})} = C_0^u(\mathbb G)^{**} = \widehat{ C_0^u(\hat{\mathbb G})^{**}}$. Other various examples are given.