Multipliers of locally compact quantum groups via Hilbert C$^*$-modules

TL;DR

This paper extends Gilbert's classical result on multipliers of Fourier algebras to the setting of locally compact quantum groups, using Hilbert C*-modules and operator theory to analyze their structure and actions.

Contribution

It recasts multiplier theory for quantum groups in terms of Hilbert C*-modules and explores the actions of the antipode and dual quantum groups on these multipliers.

Findings

Representation of multipliers via adjointable operators on Hilbert C*-modules.

Extension of multiplier actions to the antipode and dual quantum groups.

Framework for understanding two-sided multipliers in quantum group theory.

Abstract

A result of Gilbert shows that every completely bounded multiplier $f$ of the Fourier algebra $A(G)$ arises from a pair of bounded continuous maps $\alpha,\beta:G \rightarrow K$, where $K$ is a Hilbert space, and $f(s^{-1}t) = (\beta(t)|\alpha(s))$ for all $s,t\in G$. We recast this in terms of adjointable operators acting between certain Hilbert C$^*$-modules, and show that an analogous construction works for completely bounded left multipliers of a locally compact quantum group. We find various ways to deal with right multipliers: one of these involves looking at the opposite quantum group, and this leads to a proof that the (unbounded) antipode acts on the space of completely bounded multipliers, in a way which interacts naturally with our representation result. The dual of the universal quantum group (in the sense of Kustermans) can be identified with a subalgebra of the completely bounded multipliers, and we show how this fits into our framework. Finally, this motivates a certain way to deal with two-sided multipliers.