On topological centre problems and SIN quantum groups

Let A be a Banach algebra with a faithful multiplication and ∗〈A∗A〉 be the quotient Banach algebra of A∗∗ with the left Arens product. We introduce a natural Banach algebra, which is a closed subspace of ∗〈A∗A〉 but equipped with a distinct multiplication. With the help of this Banach algebra, new characterizations of the topological centre Zt(∗〈A∗A〉) of ∗〈A∗A〉 are obtained, and a characterization of Zt(∗〈A∗A〉) by Lau and Ülger for A having a bounded approximate identity is extended to all Banach algebras. The study of this Banach algebra motivates us to introduce the notion of SIN locally compact quantum groups and the concept of quotient strong Arens irregularity. We give characterizations of co-amenable SIN quantum groups, which are even new for locally compact groups. Our study shows that the SIN property is intrinsically related to topological centre problems. We also give characterizations of quotient strong Arens irregularity for all quantum group algebras. Within the class of Banach algebras introduced recently by the authors, we characterize the unital ones, generalizing the corresponding result of Lau and Ülger. We study the interrelationships between strong Arens irregularity and quotient strong Arens irregularity, revealing the complex nature of topological centre problems. Some open questions by Lau and Ülger on Zt(∗〈A∗A〉) are also answered.