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Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 4 / December 2020
- Published online by Cambridge University Press:
- 30 January 2020, pp. 825-836
- Print publication:
- December 2020
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- Article
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Let
$\mathbb{G}$ 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 
$I^{\ast }$.
