Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers
Keyword(s):
AbstractLet $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$. In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$. Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$-module homomorphisms on $I^{\ast \ast }$ and on $I^{\ast }$.
2017 ◽
Vol 60
(1)
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pp. 122-130
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2016 ◽
Vol 68
(2)
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pp. 309-333
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2003 ◽
Vol 14
(08)
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pp. 865-884
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2014 ◽
Vol 57
(3)
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pp. 546-550
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2020 ◽
Vol 56
(1)
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pp. 33-53
2012 ◽
Vol 23
(12)
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pp. 1250132
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2003 ◽
Vol 233
(2)
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pp. 231-296
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2001 ◽
Vol 180
(2)
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pp. 426-480
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Keyword(s):
2008 ◽
Vol 60
(4)
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pp. 648-662
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