Multiplier completion of Banach algebras with application to quantum groups

Abstract

Let ${{\mathcal {A}}}$ be a Banach algebra and let $\varphi$ be a non-zero character on ${{\mathcal {A}}}$. Suppose that ${{\mathcal {A}}}_M$ is the closure of the faithful Banach algebra ${{\mathcal {A}}}$ in the multiplier norm. In this paper, topologically left invariant $\varphi$-means on ${{\mathcal {A}}}_M^*$ are defined and studied. Under some conditions on ${{\mathcal {A}}}$, we will show that the set of topologically left invariant $\varphi$-means on ${{\mathcal {A}}}^*$ and on ${{\mathcal {A}}}_M^*$ have the same cardinality. The main applications are concerned with the quantum group algebra $L^1({\mathbb {G}})$ of a locally compact quantum group ${\mathbb {G}}$. In particular, we obtain some characterizations of compactness of ${\mathbb {G}}$ in terms of the existence of a non-zero (weakly) compact left or right multiplier on $L^1_M({\mathbb {G}})$ or on its bidual in some senses.