Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers

Nemati, Mehdi ; Rizi, Maryam Rajaei

Canadian mathematical bulletin, Tome 63 (2020) no. 4, pp. 825-836

Résumé

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 $I^{\ast }$.

Détail

DOI : 10.4153/S0008439520000077

Mots-clés : amenability, Arens regularity, locally compact quantum group, (weakly) compact multiplier, module homomorphism