Bidualräume und Vervollständigungen von Banachmoduln

The paper proposes a unified approach to many key theorems proved in the last twenty years in different areas of abstract harmonic analysis. This approach is based on the so-called slowly oscillating functions which were introduced in coarse geometry. In addition to this method being the most natural and simple, it also leads to the generalisation of some of the results and to the achievement of some new results. Several of these results concern the topological centres of convolution algebras and semigroup compactifications.

Section: Some Prerequisitesmentioning

confidence: 85%

Slowly oscillating functions in semigroup compactifications and convolution algebras

Filali

Salmi

2007

“…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. Proposition 2.6.…”

Section: In Particular If a Has A Brai Or A Blai Then We Havementioning

confidence: 99%

Module maps on duals of Banach algebras and topological centre problems

Neufang

Ruan

2011

We study various spaces of module maps on the dual of a Banach algebra A, and relate them to topological centres. We introduce an auxiliary topological centre Z t ( A * A * ) ♦ for the left quotient Banach algebra A * A * of A * * . Our results indicate that Z t ( A * A * ) ♦ is indispensable for investigating properties of module maps over A * and for understanding some asymmetry phenomena in topological centre problems as well as the interrelationships between different Arens irregularity properties. For the class of Banach algebras of type (M) introduced recently by the authors, we show that strong Arens irregularity can be expressed both in terms of automatic normality of A * * -module maps on A * and through certain commutation relations. This in particular generalizes the earlier work on group algebras by Ghahramani and McClure (1992) [13] and by Ghahramani and Lau (1997) [12]. We link a module map property over A * to the space WAP(A) of weakly almost periodic functionals on A, generalizing a result by Lau and Ülger (1996) [34] for Banach algebras with a bounded approximate identity. We also show that for a locally compact quantum group G, the quotient strong Arens irregularity of L 1 (G) can be obtained from that of M(G) and can be characterized via the canonical C 0 (G)-module structure on LUC(G) * .

“…See Dales and Lau [6] for more information on topological centres Z t (X * , ) and Z t (X * , ♦). The reader is also referred to Grosser [12] 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.…”

Section: Preliminariesmentioning

confidence: 99%

On topological centre problems and SIN quantum groups

Neufang

Ruan

2009

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.