On topological centre problems and SIN quantum groups

Abstract

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 Z t ( A * A * ) of A * A * are obtained, and a characterization of Z t ( 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 Z t ( A * A * ) are also answered.

Module maps on duals of Banach algebras and topological centre problems

“…We start Section 2 with notation conventions followed by introducing an auxiliary topological centre Z t ( A * A * ) ♦ for ( A * A * , ), the canonical quotient Banach algebra of (A * * , ). We show that Z t ( A * A * ) ♦ happens to be the hidden piece that is responsible for some asymmetry phenomena occurring in topological centre problems as observed in [20,34]. Results in the paper indicate that Z t ( A * A * ) ♦ is indispensable for the comparison between the algebras B σ A (A * ), B A * * (A * ), and B A (A * ), and for the study of the interrelationships between different Arens irregularity (respectively, Arens regularity) properties.…”

“…The results obtained in the paper indicate that this Banach A-bimodule is exactly the missing piece in the study of Arens irregularity, without which some asymmetries occur in topological centre problems (cf. [20,Remark 28] and [34,Remark 5.2]). This A-bimodule shall be used to describe the pre-image of B A * * (A * ) under Φ, to compare the algebras B σ A (A * ), B A * * (A * ), and B A (A * ), and to study further interrelationships between various topological centre problems and properties of module maps on A * (see the results presented in this and the next sections).…”

“…We show that some of the automatic normality properties of module maps define new concepts between strong Arens irregularity and quotient strong Arens irregularity, the two notions introduced, respectively, by Dales and Lau [6] and by the authors [20]. More characterizations of module map properties over A * are obtained for Banach algebras A of type (M).…”

“…Most of these facts can be found in some form in the existing literature, but some of them were stated only for certain classes of Banach algebras. See, for example, [1,3,5,6,16,20,21,26,27,30,34]. The reader is also referred to [15] for a systematic study of left (respectively, right) Banach modules of the form V * A (respectively, AV * ), where V is a left (respectively, right) Banach A-module.…”

“…We introduced in [20] the Banach algebra A * A * R , which is a subspace of ( A * A * , ) but equipped with a distinct multiplication. In Section 5, we consider the corresponding Banach algebra of module maps on A * .…”