International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 439649 | 6 pages | https://doi.org/10.5402/2011/439649

Module Actions on Iterated Duals of Banach Algebras

Academic Editor: C. Zhu
Received23 Jun 2011
Accepted02 Aug 2011
Published18 Sep 2011

Abstract

Let 𝒜 be a Banach algebra and ğ’œî…žî…ž its second dual equipped with the first Arens product. We consider three ğ’œî…žî…ž-bimodule structures on the fourth dual of 𝒜. This paper discusses the situation that makes these structures coincide.

1. Introduction and Preliminaries

Let 𝒜 be a Banach algebra. It is well known that, on the second dual space ğ’œî…žî…ž of 𝒜, there are two multiplications, called the first and second Arens products which make ğ’œî…žî…ž into a Banach algebra (see [1, 2]). By definition, the first Arens product □ on ğ’œî…žî…ž is induced by the left 𝒜-module structure on 𝒜. That is, for each Φ,Î¨âˆˆğ’œî…žî…ž, ğ‘“âˆˆğ’œî…ž, and ğ‘Ž,𝑏∈𝒜, we have ⟨Φ□Ψ,𝑓⟩=⟨Φ,Ψ⋅𝑓⟩,⟨Ψ⋅𝑓,ğ‘ŽâŸ©=⟨Ψ,ğ‘“â‹…ğ‘ŽâŸ©,âŸ¨ğ‘“â‹…ğ‘Ž,𝑏⟩=⟨𝑓,ğ‘Žğ‘âŸ©.(1.1) Similarly, the second Arens product ◊ on ğ’œî…žî…ž is defined by considering 𝒜 as a right 𝒜-module. The Banach algebra 𝒜 is said to be Arens regular if Φ□Ψ=Φ◊Ψ on the whole of ğ’œî…žî…ž.

For any fixed Î¦âˆˆğ’œî…žî…ž, the map Ψ↦Ψ□Φ and Ψ↦Φ◊Ψ are 𝑤∗-𝑤∗ continuous on ğ’œî…žî…ž. Thus, with the 𝑤∗-topology, (ğ’œî…žî…ž,□) is a right topological semigroup, and (ğ’œî…žî…ž,◊) is a left topological semigroup. The following sets 𝑍1ğ‘¡î€·ğ’œî…žî…žî€¸=î€½Î¦âˆˆğ’œî…žî…žâˆ¶Î¨âŸ¼Î¦â–¡Î¨is𝑤∗-𝑤∗continuousonğ’œî…žî…žî€¾,𝑍2ğ‘¡î€·ğ’œî…žî…žî€¸=î€½Î¦âˆˆğ’œî…žî…žâˆ¶Î¨âŸ¼Î¨â—ŠÎ¦is𝑤∗-𝑤∗continuousonğ’œî…žî…žî€¾,(1.2) are called the first and the second topological centres of ğ’œî…žî…ž, respectively. One can verify that 𝒜 is Arens regular if and only if Z1𝑡(ğ’œî…žî…ž)=𝑍2𝑡(ğ’œî…žî…ž)=ğ’œî…žî…ž. For example, each 𝐶∗-algebra is Arens regular, and for locally compact group 𝐺, the group algebra 𝐿1(𝐺) is Arens regular if and only if 𝐺 is finite [3].

For a detailed account of Arens product and topological centres, we refer the reader to Memoire [4].

Throughout the paper we identify an element of a Banach space 𝑋 with its canonical image in ğ‘‹î…žî…ž.

2. ğ’œî…žî…ž-Bimodule Structures on 𝒜(4)

Suppose that 𝒜 is a Banach algebra and 𝑋 is a Banach 𝒜-bimodule. According to [5], ğ‘‹î…žî…ž is a Banach ğ’œî…žî…ž-bimodule where ğ’œî…žî…ž is equipped with the first Arens product. The module actions are defined by Φ⋅𝜈=𝑤∗−lim𝑖limğ‘—î„žğ‘Žğ‘–â‹…ğ‘¥ğ‘—,𝜈⋅Φ=𝑤∗−lim𝑗limğ‘–î„žğ‘¥ğ‘—â‹…ğ‘Žğ‘–,(2.1) where (ğ‘Žğ‘–) and (𝑥𝑗) are nets in 𝒜 and 𝑋 that converge, in 𝑤∗-topologies, to Φ and 𝜈, respectively.

There are two ğ’œî…žî…ž-bimodule structures on ğ’œî…žî…žî…ž which are not always equal [6]. In the first way ğ’œî…žî…žî…ž as the second dual of ğ’œî…ž can be made into ğ’œî…žî…ž-bimodule by module actions defined as above (for 𝑋=ğ’œî…ž). We denote this module action by the symbol “∗”. In the second way ğ’œî…žî…žî…ž as the dual space of ğ’œî…žî…ž is ğ’œî…žî…ž-bimodule by the natural module structures.

Now the Banach algebra 𝒜(4) has three ğ’œî…žî…ž-bimodule structures.(i)If we consider 𝒜(4) as the second dual of ğ’œî…žî…ž, then 𝒜(4) can be an ğ’œî…žî…ž-bimodule by module actions defined as above (for 𝑋=ğ’œî…žî…ž). We denote this module action by the symbol “•”.(ii)If we consider 𝒜(4) as the dual of ğ’œî…žî…žî…ž in which ğ’œî…žî…žî…ž=(ğ’œî…ž), then 𝒜(4) can be an ğ’œî…žî…ž-bimodule by module structures as follows: ⟨Φ⋆𝛼,𝜇⟩=⟨𝛼,𝜇∗Φ⟩,⟨𝛼⋆Φ,𝜇⟩=⟨𝛼,Φ∗𝜇⟩,(2.2) for all Î¦âˆˆğ’œî…žî…ž, ğœ‡âˆˆğ’œî…žî…žî…ž, and 𝛼∈𝒜(4).(iii)We consider 𝒜(4) as the dual of ğ’œî…žî…žî…ž in which ğ’œî…žî…žî…ž=(ğ’œî…žî…ž), so 𝒜(4) can be an ğ’œî…žî…ž-bimodule by the following module actions: ⟨Φ⋅𝛼,𝜇⟩=⟨𝛼,𝜇⋅Φ⟩,⟨𝛼⋅Φ,𝜇⟩=⟨𝛼,Φ⋅𝜇⟩,(2.3) where 𝜇⋅Φ and Φ⋅𝜇 are the natural module structures on ğ’œî…žî…žî…ž.

It is straightforward to check that three ğ’œî…žî…ž-bimodule structures on 𝒜(4) are not coincide, in general.

Theorem 2.1. Let 𝒜 be a Banach algebra. Then 𝛼•Φ=𝛼⋆Φ,Φ⋅𝛼=Î¦â‹†ğ›¼Î¦âˆˆğ’œî…žî…ž,𝛼∈𝒜(4).(2.4)

Proof. Suppose that (ğ‘Žğ‘–), (𝑓𝑗), and (𝐹𝑘) are bounded nets in 𝒜, ğ’œî…ž and ğ’œî…žî…ž, respectively, such that îğ‘Žğ‘–â†’Î¦, 𝑓𝑗→𝜇 and 𝐹𝑘→𝛼 in the 𝑤∗-topology. Then, we have ⟨𝛼⋆Φ,𝜇⟩=⟨𝛼,Φ∗𝜇⟩=lim𝑘𝐹𝑘,Φ∗𝜇=lim𝑘⟨Φ∗𝜇,𝐹𝑘⟩=lim𝑘lim𝑖limğ‘—î‚¬î„žğ‘Žğ‘–â‹…ğ‘“ğ‘—,𝐹𝑘=lim𝑘lim𝑖lim𝑗𝐹𝑘,ğ‘Žğ‘–â‹…ğ‘“ğ‘—î¬=lim𝑘lim𝑖lim𝑗𝑓𝑗,ğ¹ğ‘˜â‹…ğ‘Žğ‘–î‚­=lim𝑘lim𝑖⟨𝜇,ğ¹ğ‘˜â‹…ğ‘Žğ‘–âŸ©=lim𝑘limğ‘–î‚¬î„žğ¹ğ‘˜â‹…ğ‘Žğ‘–î‚­,𝜇=⟨𝛼•Φ,𝜇⟩.(2.5) Therefore, 𝛼•Φ=𝛼⋆Φ. Similarly, we obtain ⟨Φ⋅𝛼,𝜇⟩=⟨𝛼,𝜇⋅Φ⟩=lim𝑘𝐹𝑘,𝜇⋅Φ=lim𝑘⟨𝜇,Φ□𝐹𝑘⟩=lim𝑘lim𝑗𝑓𝑗,Φ□𝐹𝑘=lim𝑘lim𝑗Φ,𝐹𝑘⋅𝑓𝑗=lim𝑘lim𝑗limğ‘–î«îğ‘Žğ‘–,𝐹𝑘⋅𝑓𝑗=lim𝑘lim𝑗lim𝑖𝐹𝑘,ğ‘“ğ‘—â‹…ğ‘Žğ‘–î¬=lim𝑘⟨𝜇∗Φ,𝐹𝑘⟩=lim𝑘𝐹𝑘,𝜇∗Φ=⟨𝛼,𝜇∗Φ⟩=⟨Φ⋆𝛼,𝜇⟩.(2.6) This shows that Φ⋅𝛼=Φ⋆𝛼, as required.

Let ğ’œî…žî…ž have the first Arens product □. Then the first and second Arens products on 𝒜(4) which we denote by ⊡ and ⟐, respectively, are induced by the left and right ğ’œî…žî…ž-module structure on ğ’œî…žî…ž.

Similarly, we obtain two Arens product on 𝒜(4) with respect to the second Arens product ◊ on ğ’œî…žî…ž, which will denote by □ and ◊, respectively. We recall that the topological centres of 𝒜(4) with respect to each of Arens products can be defined analogously.

The proof of the following result is obvious and we omit it.

Proposition 2.2. Let 𝒜 be a Banach algebra. (i)If 𝒜 is commutative, then 𝛼⊡𝛽=𝛽◊𝛼 and 𝛼⟐𝛽=𝛽□𝛼.(ii)If 𝒜 is Arens regular, then 𝛼⊡𝛽=𝛼□𝛽 and 𝛼⟐𝛽=𝛼◊𝛽.

Theorem 2.3. For each Banach algebra 𝒜, the following assertions hold. (i)𝑍1𝑡(𝒜(4),⊡)=𝒜(4) if and only if 𝑍1𝑡(𝒜(4),□)=𝒜(4).(ii)𝑍1𝑡(𝒜(4),⟐)=𝒜(4) if and only if 𝑍1𝑡(𝒜(4),◊)=𝒜(4).

Proof. We prove (i); the proof of (ii) follows from similar argument.
Assume that 𝑍1𝑡(𝒜(4),⊡)=𝒜(4). Then 𝛼⊡𝛽=𝛼⟐𝛽(𝛼,𝛽∈𝒜(4)). Since Arens regularity of ğ’œî…žî…ž implies that of 𝒜, so we have Φ□Ψ=Φ◊Ψ for each Φ,Î¨âˆˆğ’œî…žî…ž. Therefore by above proposition we obtain 𝛼□𝛽=𝛼⊡𝛽=𝛼⟐𝛽=𝛼◊𝛽.(2.7) It follows that 𝑍1𝑡(𝒜(4),□)=𝒜(4). The converse is similar.

It is well known that every 𝐶∗-algebra 𝒜 is Arens regular and ğ’œî…žî…ž is also a 𝐶∗-algebra [2]. Therefore ğ’œî…žî…ž itself is Arens regular, but in general case for Arens regular Banach algebra 𝒜,ğ’œî…žî…ž is not Arens regular (see [7, 8]).

Theorem 2.4. Let 𝒜 be an Arens regular Banach algebra. If one of the following conditions hold: (i)𝜇⋅Φ and Î¦â‹…ğœ‡âˆˆğ’œî…ž, for each ğœ‡âˆˆğ’œî…žî…žî…ž and Î¦âˆˆğ’œî…žî…ž,(ii)the operators 𝐿Φ and 𝑅Φ are 𝑤∗-𝑤 continuous on ğ’œî…žî…ž,(iii)ğ’œî…žî…žâ–¡ğ’œî…žî…žâŠ†ğ’œ,then ğ’œî…žî…ž is Arens regular.

Proof. This follows from Theorem 2.3 of [9].

Remark 2.5. Let 𝒜 be a nonunital Banach algebra with a bounded approximate identity(=BAI). Then 𝒜 cannot be both WSC and Arens regular [10]. It follows that for every WSC Banach algebra 𝒜 with a BAI, ğ’œî…žî…ž is not Arens regular unless 𝒜 is unital.

Theorem 2.6. Let 𝒜 be an Arens regular Banach algebra. If one of the conditions of Theorem 2.4 hold, then all ğ’œî…žî…ž-bimodule actions on 𝒜(4) coincide.

Proof. Let (ğ‘Žğ‘–) and (𝐹𝑗) be bounded nets in 𝒜 and ğ’œî…žî…ž that converge, in 𝑤∗-topologies, to Φ and 𝛼, respectively. Then for each ğœ‡âˆˆğ’œî…žî…žî…ž, we have ⟨𝛼⋅Φ,𝜇⟩=⟨𝛼,Φ⋅𝜇⟩=lim𝑗Φ⋅𝜇,𝐹𝑗=lim𝑗limğ‘–î«ğ‘Žğ‘–â‹…ğœ‡,𝐹𝑗=lim𝑗lim𝑖𝜇,ğ¹ğ‘—â‹…ğ‘Žğ‘–î¬=lim𝑗limğ‘–î‚¬î„žğ¹ğ‘—â‹…ğ‘Žğ‘–î‚­,𝜇=⟨𝛼•Φ,𝜇⟩.(2.8) Consequently, 𝛼⋅Φ=𝛼•Φ. By hypothesis ğ’œî…žî…ž is Arens regular and so lim𝑗limğ‘–î„žğ‘Žğ‘–â‹…ğ¹ğ‘—=lim𝑖limğ‘—î„žğ‘Žğ‘–â‹…ğ¹ğ‘—.(2.9) Thus, for each ğœ‡âˆˆğ’œî…žî…žî…ž, ⟨Φ⋅𝛼,𝜇⟩=⟨𝛼,𝜇⋅Φ⟩=lim𝑗𝜇⋅Φ,𝐹𝑗=lim𝑗limğ‘–î«ğœ‡â‹…ğ‘Žğ‘–,𝐹𝑗=lim𝑗lim𝑖𝜇,ğ‘Žğ‘–â‹…ğ¹ğ‘—î¬=lim𝑗limğ‘–î‚¬î„žğ‘Žğ‘–â‹…ğ¹ğ‘—î‚­,𝜇=lim𝑖limğ‘—î‚¬î„žğ‘Žğ‘–â‹…ğ¹ğ‘—î‚­,𝜇=⟨Φ•𝛼,𝜇⟩.(2.10) Therefore Φ⋅𝛼=Φ•𝛼. Now by Theorem 2.1, three ğ’œî…žî…ž-bimodule structure on 𝒜(4) coincide.

Example 2.7. Let 𝒜=𝑙1, with pointwise product. Then 𝒜 is an Arens regular Banach algebra which is not reflexive, butâ€‰â€‰ğ’œî…žî…žâ–¡ğ’œî…žî…žâŠ†ğ’œ [4]. Therefore by above theorem three ğ’œî…žî…ž-bimodule structures on 𝒜(4) coincide.

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Copyright © 2011 Abbas Sahleh and Abbas Zivari-Kazempour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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