Module homomorphisms and multipliers on locally compact quantum groups

TL;DR

This paper explores module homomorphisms and multipliers on locally compact quantum groups, providing characterizations of compactness and discreteness, and linking multiplier algebras to the structure of quantum groups.

Contribution

It offers new characterizations of compactness and discreteness for quantum groups and relates multiplier algebras to the structure of these groups, partially answering a known open problem.

Findings

All $A$-module homomorphisms of $A^*$ are normal iff $A$ is an ideal of $A^{**}$.

Characterizations of compactness and discreteness for quantum groups are established.

In the co-amenable case, the multiplier algebra of $ ext{L}^1( extbf{G})$ is identified with $ ext{M}( extbf{G})$.

Abstract

For a Banach algebra $A$ with a bounded approximate identity, we investigate the $A$-module homomorphisms of certain introverted subspaces of $A^*$, and show that all $A$-module homomorphisms of $A^*$ are normal if and only if $A$ is an ideal of $A^{**}$. We obtain some characterizations of compactness and discreteness for a locally compact quantum group $\G$. Furthermore, in the co-amenable case we prove that the multiplier algebra of $\LL$ can be identified with $\MG.$ As a consequence, we prove that $\G$ is compact if and only if $\LUC={\rm WAP}(\G)$ and $\MG\cong\mathcal{Z}({\rm LUC}(\G)^*)$; which partially answer a problem raised by Volker Runde.