Completely positive multipliers of quantum groups

TL;DR

This paper establishes a canonical correspondence between completely positive multipliers of quantum group convolution algebras and unitary corepresentations, linking algebraic and operator-theoretic structures.

Contribution

It proves that all completely positive multipliers are induced by unitary corepresentations and establishes a bijection with positive functionals on the universal quantum group.

Findings

Every completely positive multiplier is induced by a unitary corepresentation.

There is an order bijection between multipliers of $L^1( ext{G})$ and positive functionals on $C_0^u( ext{G})$.

The representation map is weak*-weak* continuous.

Abstract

We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group $\G$ (either a locally compact quantum group, or a quantum group coming from a modular or manageable multiplicative unitary) is induced in a canonical fashion by a unitary corepresentation of $\G$. It follows that there is an order bijection between the completely positive multipliers of $L^1(\G)$ and the positive functionals on the universal quantum group $C_0^u(\G)$. We provide a direct link between the Junge, Neufang, Ruan representation result and the representing element of a multiplier, and use this to show that their representation map is always weak$^*$-weak$^*$-continuous.