Weakly compact multipliers for some quantum group algebras

Let \(\mathbb{G}\) be a locally compact quantum group. We study the existence of certain (weakly) compact right and left multipliers of the Banach al- gebra \(\mathfrak{X} ^{*} \), where \(\mathfrak{X} \) is an introverted subspace of \(L^\infty(\mathbb{G})\) with some conditions, and relate them with some properties of \(\mathbb{G}\) such as compactness and amenability. For example, when \(\mathbb{G}\) is co-amenable and \(L^1(\mathbb{G})\) is semisimple we give a characteri- zation for compactness of \(\mathbb{G}\) in terms of the existence of a nonzero compact right multiplier on \(\mathfrak{X} ^{*} \). Using this, for a locally compact group \({\mathcal G}\) we prove that \(\mathbb{G}_a\) is compact if and only if there is a nonzero (weakly) compact right multiplier on \(\mathfrak{X} ^{*} \). Similar assertion holds for \(\mathbb{G}_s\) when \({\mathcal G}\) is amenable.