A Class of Distal Functions on Semitopological Semigroups

The norm closure of the algebra generated by the set {n→λ^nk : λ belongs T and k belongs N} of functions on (Z,+) was studied in [11] (and was named as the Weyl algebra). In this paper, by a fruitful result of Namioka, this algebra is generalized for a general semitopological semigroup and, among other things, it is shown that the elements of the involved algebra are distal. In particular, we examine this algebra for (Z,+) and (more generally) for the discrete (additive) group of any countable ring. Finally, our results are treated for a bicyclic semigroup

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