Module homomorphisms and multipliers on locally compact quantum groups

Abstract

For a Banach algebra $A$ with a bounded approximate identity, we investigate the $A$-module homomorphisms of certain introverted subspaces of $A^{\ast }$, and show that all $A$-module homomorphisms of $A^{\ast }$ are normal if and only if $A$ is an ideal of $A^{\ast \ast }$. 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.

The detailed proof of Lemma 4.1 is added in addendum. 11 pages, To appear in J. Math. Anal. Appl

Cited by in corpus (1)

• Reiter's properties for the actions of locally compact quantum groups on von Neumann algebras