Duals of quantum semigroups with involution

TL;DR

This paper introduces a category of quantum semigroups with involution, defining a duality map that generalizes existing dualities for locally compact quantum groups, without requiring Haar weights.

Contribution

It defines a new categorical framework for quantum semigroups with involution and establishes a duality map that extends known dualities in quantum group theory.

Findings

The duality map recovers the universal duality of Kustermans for quantum groups.

Examples demonstrate the applicability of the duality to various quantum structures.

The framework does not assume the existence of Haar weights or spatial realizations.

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.