Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers. Issue 4 (30th December 2020)

• Record Type: Journal Article
• Authors: Nemati, Mehdi; Rajaei Rizi, Maryam
• Is Part Of: Canadian mathematical bulletin =. Volume 63:Issue 4(2020)
• Page Start: 825
• Page End: 836
• Publication Date: 2020-12-30
• Subjects: 46H20, -- 46L89, -- 43A07, -- 43A22, -- 58B32; amenability, -- Arens regularity, -- locally compact quantum group, -- (weakly) compact multiplier, -- module homomorphism; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439520000077
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: 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 $I^{\ast }$ .