Module homomorphisms and multipliers on locally compact quantum groups

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

Closed quantum subgroups of locally compact quantum groups

We investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group. Two definitions - one due to S.Vaes and one due to S.L.Woronowicz - are analyzed and relations between them discussed. Among many reformulations we prove that the former definition can be phrased in terms of quasi-equivalence of representations of quantum groups while the latter can be related to an old definition of Podle\'s from the theory of compact quantum groups. The cases of classical groups, duals of classical groups, compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context. A deep relationship with the quantum group generalization of Herz restriction theorem from classical harmonic analysis is also established, in particular, in the course of our analysis we give a new proof of Herz restriction theorem.Comment: 24 pages, v3 adds another reference. The paper will appear in Advances in Mathematic