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.

让\({{\mathcal {A}}}\)是一个巴拿赫代数，让\(\varphi \)是\({{\mathcal {A}}}\)上的一个非零字符。假设\({{\mathcal {A}}}_M\)是乘子范数中忠实Banach 代数\({{\mathcal {A}}}\)的闭包。在本文中，拓扑左不变\(\varphi \) - 对\({{\mathcal {A}}}_M^*\)的均值进行了定义和研究。在\({{\mathcal {A}}}\) 的某些条件下，我们将证明拓扑左不变的集合\(\varphi \) -means on \({{\mathcal {A}}}^* \)和\({{\mathcal {A}}}_M^*\)具有相同的基数。主要应用涉及局部紧量子群\({\mathbb {G}}\)的量子群代数\(L^1({\mathbb {G}}) \)。特别地，我们根据\(L^1_M({\mathbb { G}}\)上非零（弱）紧致左或右乘子的存在性获得了\({\mathbb {G}}\)紧致性的一些特征G}})\)或从某种意义上说是双标的。