Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers
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Published:2020-01-30
Issue:4
Volume:63
Page:825-836
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ISSN:0008-4395
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Container-title:Canadian Mathematical Bulletin
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language:en
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Short-container-title:Can. Math. Bull.
Author:
Nemati Mehdi,Rajaei Rizi Maryam
Abstract
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 $I^{\ast }$.
Publisher
Canadian Mathematical Society
Subject
General Mathematics