# Module Actions on Iterated Duals of Banach Algebras

Authors: Abbas Sahleh; Abbas Zivari-Kazempour
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

Let𝒜 be a Banach algebra and 𝒜′′ its second dual equipped with the first Arens product. We consider three 𝒜′′-bimodule structures on the fourth dual of 𝒜. This paper discusses the situation that makes these structures coincide.


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## Body

## 1. Introduction and Preliminaries

Let𝒜 be a Banach algebra. It is well known that, on the second dual space 𝒜′′ of 𝒜, there are two multiplications, called the first and second Arens products which make 𝒜′′ into a Banach algebra (see [1, 2]). By definition, the first Arens product □ on 𝒜′′ is induced by the left 𝒜-module structure on 𝒜. That is, for each Φ,Ψ∈𝒜′′, f∈𝒜′, and a,b∈𝒜, we have (1.1)〈Φ□Ψ,f〉=〈Φ,Ψ⋅f〉,〈Ψ⋅f,a〉=〈Ψ,f⋅a〉,〈f⋅a,b〉=〈f,ab〉.
Similarly, the second Arens product ◊ on 𝒜′′ 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 𝒜′′.For any fixedΦ∈𝒜′′, the map Ψ↦Ψ□Φ and Ψ↦Φ◊Ψ are w*-w* continuous on 𝒜′′. Thus, with the w*-topology, (𝒜′′,□) is a right topological semigroup, and (𝒜′′,◊) is a left topological semigroup. The following sets (1.2)Zt1(A′′)={Φ∈A′′:Ψ⟼Φ□Ψisw*-w*continuous onA′′},Zt2(A′′)={Φ∈A′′:Ψ⟼Ψ◊Φisw*-w*continuous onA′′},
are called the first and the second topological centres of 𝒜′′, respectively. One can verify that 𝒜 is Arens regular if and only if Zt1(𝒜′′)=Zt2(𝒜′′)=𝒜′′. 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 spaceX with its canonical image in X′′.


## 2.𝒜′′-Bimodule Structures on 𝒜(4)

Suppose that𝒜 is a Banach algebra and X is a Banach 𝒜-bimodule. According to [5], X′′ is a Banach 𝒜′′-bimodule where 𝒜′′ is equipped with the first Arens product. The module actions are defined by (2.1)Φ⋅ν=w*-lim⁡ilim⁡jai⋅xĵ,ν⋅Φ=w*-lim⁡jlim⁡ixj⋅aî,
where (ai) and (xj) are nets in 𝒜 and X that converge, in w*-topologies, to Φ and ν, respectively.There are two𝒜′′-bimodule structures on 𝒜′′′ which are not always equal [6]. In the first way 𝒜′′′ as the second dual of 𝒜′ can be made into 𝒜′′-bimodule by module actions defined as above (for X=𝒜′). We denote this module action by the symbol “*”. In the second way 𝒜′′′ as the dual space of 𝒜′′ is 𝒜′′-bimodule by the natural module structures.Now the Banach algebra𝒜(4) has three 𝒜′′-bimodule structures.(i)
If we consider𝒜(4) as the second dual of 𝒜′′, then 𝒜(4) can be an 𝒜′′-bimodule by module actions defined as above (for X=𝒜′′). We denote this module action by the symbol “•”.(ii)
If we consider𝒜(4) as the dual of 𝒜′′′ in which 𝒜′′′=(𝒜′)′′, then 𝒜(4) can be an 𝒜′′-bimodule by module structures as follows:
(2.2)〈Φ⋆α,μ〉=〈α,μ*Φ〉,〈α⋆Φ,μ〉=〈α,Φ*μ〉,
for all Φ∈𝒜′′, μ∈𝒜′′′, and α∈𝒜(4).(iii)
We consider𝒜(4) as the dual of 𝒜′′′ in which 𝒜′′′=(𝒜′′)′, so 𝒜(4) can be an 𝒜′′-bimodule by the following module actions:
(2.3)〈Φ⋅α,μ〉=〈α,μ⋅Φ〉,〈α⋅Φ,μ〉=〈α,Φ⋅μ〉,
where μ·Φ and Φ·μ are the natural module structures on 𝒜′′′.It is straightforward to check that three𝒜′′-bimodule structures on 𝒜(4) are not coincide, in general.Theorem 2.1.
Let𝒜 be a Banach algebra. Then
(2.4)α•Φ=α⋆Φ,Φ⋅α=Φ⋆α(Φ∈A′′,α∈A(4)).Proof.
Suppose that(ai), (fj), and (Fk) are bounded nets in 𝒜, 𝒜′ and 𝒜′′, respectively, such that aî→Φ, fĵ→μ and Fk̂→α in the w*-topology. Then, we have
(2.5)〈α⋆Φ,μ〉=〈α,Φ*μ〉=lim⁡k〈Fk̂,Φ*μ〉=lim⁡k〈Φ*μ,Fk〉=lim⁡klim⁡ilim⁡j〈ai⋅fĵ,Fk〉=lim⁡klim⁡ilim⁡j〈Fk,ai⋅fj〉=lim⁡klim⁡ilim⁡j〈fĵ,Fk⋅ai〉=lim⁡klim⁡i〈μ,Fk⋅ai〉=lim⁡klim⁡i〈Fk⋅aî,μ〉=〈α•Φ,μ〉.
Therefore, α•Φ=α⋆Φ. Similarly, we obtain
(2.6)〈Φ⋅α,μ〉=〈α,μ⋅Φ〉=lim⁡k〈Fk̂,μ⋅Φ〉=lim⁡k〈μ,Φ□Fk〉=lim⁡klim⁡j〈fĵ,Φ□Fk〉=lim⁡klim⁡j〈Φ,Fk⋅fj〉=lim⁡klim⁡jlim⁡i〈aî,Fk⋅fj〉=lim⁡klim⁡jlim⁡i〈Fk,fj⋅ai〉=lim⁡k〈μ*Φ,Fk〉=lim⁡k〈Fk̂,μ*Φ〉=〈α,μ*Φ〉=〈Φ⋆α,μ〉.
This shows that Φ·α=Φ⋆α, as required.Let𝒜′′ 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 𝒜′′-module structure on 𝒜′′.Similarly, we obtain two Arens product on𝒜(4) with respect to the second Arens product ◊ on 𝒜′′, 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.
Let𝒜 be a Banach algebra.(i)
If𝒜 is commutative, then α⊡β=β◊¯α and α⟐β=β□¯α.(ii)
If𝒜 is Arens regular, then α⊡β=α□¯β and α⟐β=α◊¯β.Theorem 2.3.
For each Banach algebra𝒜, the following assertions hold.(i)
Zt1(𝒜(4),⊡)=𝒜(4) if and only if Zt1(𝒜(4),□¯)=𝒜(4).(ii)
Zt1(𝒜(4),⟐)=𝒜(4) if and only if Zt1(𝒜(4),◊¯)=𝒜(4).Proof.
We prove (i); the proof of (ii) follows from similar argument.
Assume thatZt1(𝒜(4),⊡)=𝒜(4). Then α⊡β=α⟐β(α,β∈𝒜(4)). Since Arens regularity of 𝒜′′ implies that of 𝒜, so we have Φ□Ψ=Φ◊Ψ for each Φ,Ψ∈𝒜′′. Therefore by above proposition we obtain
(2.7)α□¯β=α⊡β=α⟐β=α◊¯β.
It follows that Zt1(𝒜(4),□¯)=𝒜(4). The converse is similar.It is well known that everyC*-algebra 𝒜 is Arens regular and 𝒜′′ is also a C*-algebra [2]. Therefore 𝒜′′ itself is Arens regular, but in general case for Arens regular Banach algebra 𝒜,𝒜′′ is not Arens regular (see [7, 8]).Theorem 2.4.
Let𝒜 be an Arens regular Banach algebra. If one of the following conditions hold:(i)
μ·Φ and Φ·μ∈𝒜′, for each μ∈𝒜′′′ and Φ∈𝒜′′,(ii)
the operatorsLΦ and RΦ are w*-w continuous on 𝒜′′,(iii)
𝒜′′□𝒜′′⊆𝒜,then 𝒜′′ is Arens regular.Proof.
This follows from Theorem 2.3 of [9].Remark 2.5.
Let𝒜 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, 𝒜′′ is not Arens regular unless 𝒜 is unital.Theorem 2.6.
Let𝒜 be an Arens regular Banach algebra. If one of the conditions of Theorem 2.4 hold, then all 𝒜′′-bimodule actions on 𝒜(4) coincide.Proof.
Let(ai) and (Fj) be bounded nets in 𝒜 and 𝒜′′ that converge, in w*-topologies, to Φ and α, respectively. Then for each μ∈𝒜′′′, we have
(2.8)〈α⋅Φ,μ〉=〈α,Φ⋅μ〉=lim⁡j〈Φ⋅μ,Fj〉=lim⁡jlim⁡i〈ai⋅μ,Fj〉=lim⁡jlim⁡i〈μ,Fj⋅ai〉=lim⁡jlim⁡i〈Fj⋅aî,μ〉=〈α•Φ,μ〉.
Consequently, α·Φ=α•Φ. By hypothesis 𝒜′′ is Arens regular and so
(2.9)lim⁡jlim⁡iai⋅Fĵ=lim⁡ilim⁡jai⋅Fĵ.
Thus, for each μ∈𝒜′′′,
(2.10)〈Φ⋅α,μ〉=〈α,μ⋅Φ〉=lim⁡j〈μ⋅Φ,Fj〉=lim⁡jlim⁡i〈μ⋅ai,Fj〉=lim⁡jlim⁡i〈μ,ai⋅Fj〉=lim⁡jlim⁡i〈ai⋅Fĵ,μ〉=lim⁡ilim⁡j〈ai⋅Fĵ,μ〉=〈Φ•α,μ〉.
Therefore Φ·α=Φ•α. Now by Theorem 2.1, three 𝒜′′-bimodule structure on 𝒜(4) coincide.Example 2.7.
Let𝒜=l1, with pointwise product. Then 𝒜 is an Arens regular Banach algebra which is not reflexive, but  𝒜′′□𝒜′′⊆𝒜 [4]. Therefore by above theorem three 𝒜′′-bimodule structures on 𝒜(4) coincide.



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