Duals of quantum semigroups with involution

Abstract

We define a category $\mathcal{Q}\mathcal{S}\mathcal{I}$ of quantum semigroups with involution which carries a corepresentation-based duality map $M\mapsto \hat{M}$. Objects in $\mathcal{Q}\mathcal{S}\mathcal{I}$ 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 $\hat{}$ in $\mathcal{Q}\mathcal{S}\mathcal{I}$ recovers the universal duality of Kustermans: $\widehat{L^{\infty }(\mathbb{G})}=C_0^u(\hat{\mathbb{G}}{)}^{\ast \ast }=\widehat{C_0^u(\mathbb{G}{)}^{\ast \ast }}$, and $\widehat{L^{\infty }(\hat{\mathbb{G}})}=C_0^u(\mathbb{G}{)}^{\ast \ast }=\widehat{C_0^u(\hat{\mathbb{G}}{)}^{\ast \ast }}$. Other various examples are given.

Title and terminology changed

References in corpus (2)

• A duality of locally compact groups which does not involve the Haar measure
• A c*-algebraic framework for quantum groups

Cited by in corpus (1)

• Invariant means on subspaces of quantum weakly almost periodic functionals