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 $A^*$, 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

Completely bounded multipliers over locally compact quantum groups

“…The above proof shows that if G is co-amenable, then every bounded right centralizer of L 1 (G) is automatically completely bounded. It was shown in [44,Theorem 31] that if G is co-amenable, then M (G) is isometrically isomorphic to the Banach algebra of bounded right centralizers of L 1 (G).…”

“…The situation for the C*-algebra introduced by Salmi was studied in [48]. We point out that the fact that π is an isometric algebra homomorphism as shown below was also stated in [44,Lemma 4.1]. However, the proof given in [44] is not appropriately explained, missing the strictly continuous extension property for μ in M (G).…”

“…It was also shown in [22] that the compactness of G is equivalent to LUC(G) * = M (G). We note that this equivalence was given in [44,Proposition 4.2] under the condition that G is co-amenable.…”

Module maps over locally compact quantum groups

“…This generalizes the corresponding result by Lau and Losert [42] on A(G). We note that an isometric embedding of M (G) −→ LU C(G) * was considered in [52,Lemma 4.1], but missing in its proof the above strictly continuous extension property.…”

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

“…Some aspects of abstract harmonic analysis on locally compact groups are intensively extended by V. Runde, [15,16], to the framework of locally compact quantum groups. The same discipline continued by the authors in [14].…”