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| Paper IPM / M / 18008 |
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| Abstract: | |
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Let ${\mathbb G}$ be a locally compact quantum group. We study the existence of certain (weakly) compact right and left multipliers of the Banach algebra ${\frak X}^*$, where ${\frak 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 ${\Bbb G}$ is co-amenable and $L^1({\mathbb G})$ is semisimple we give a characterization for compactness of ${\mathbb G}$ in terms of the existence of a non-zero compact right multiplier on ${\frak X}^*$. Using this, for a locally compact group ${\mathcal G}$ we prove that ${\mathbb G}_a$ is compact if and only if there is a non-zero (weakly) compact right multiplier on ${\frak X}^*$. Similar assertion holds for ${\mathbb G}_s$ when ${\mathcal G}$ is amenable.
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