# Arens Regularity of Certain Class of  Banach Algebras

Authors: Abbas Sahleh; Abbas Zivari-Kazempour
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study Arens regularity of the left and right module actions ofA on A(n), where A(n) is the nth dual space of a Banach algebra A, and then
investigate (quotient) Arens regularity of U=A⊕A(n) as a module extension
of Banach algebras.


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## Body

## 1. Introduction and Preliminaries

In 1951, Arens showed that every bounded bilinear mapm:X×Y→Z on normed spaces has two natural but different extensions m′′′ and mt′′′t from X′′×Y′′ to Z′′ [1]. The first extension m′′′ of m is constructed by forming in turn the following bilinear maps: (1.1)m′:Z′×X⟶Y′,〈m′(z′,x),y〉=〈z′,m(x,y)〉,m′′:Y′′×Z′⟶X′,〈m′′(y′′,z′),x〉=〈y′′,m′(z′,x)〉,m′′′:X′′×Y′′⟶Z′′,〈m′′′(x′′,y′′),z′〉=〈x′′,m′′(y′′,z′)〉.
The bilinear map m′′′ is the unique extension of m which is w*-separately continuous on X×Y′′. The second extension mt′′′t of m can be made in the same way if we start by transpose map mt:Y×X→Z instead of m, which is defined by mt(y,x)=m(x,y). Similarly, it is the unique extension of m that is w*-separately continuous on X′′×Y. It is easy to check that (1.2)m′′′(x′′,y′′)=w*-lim⁡ilim⁡jm(xi,yj),mt′′′t(x′′,y′′)=w*-lim⁡jlim⁡im(xi,yj),
where (xi) and (yj) are nets in X and Y that converge, in w*-topologies, to x′′ and y′′, respectively. According to [1], m is said to be Arens regular if m′′′=mt′′′t.For the product mapπ:𝒜×𝒜→𝒜 of a Banach algebra 𝒜, we denote π′′′(Φ,Ψ) and πt′′′t(Φ,Ψ) by the symbols Φ□Ψ and Φ◊Ψ, respectively. These are called the  first and second Arens products on 𝒜′′. The Banach algebra 𝒜 is said to be Arens regular if Φ□Ψ=Φ◊Ψ on the whole of 𝒜′′. The higher extensions π(3n) and πt(3n)t of π and Arens products on 𝒜(2n) can be defined similarly. For any fixed Φ∈𝒜′′, the maps 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 (1.3)Zt1(A'')={Φ∈A'':Ψ⟼Φ□Ψisw*-w*continuousonA′′},Zt2(A'')={Φ∈A'':Ψ⟼Ψ◊Φisw*-w*continuousonA′′}
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, the group algebra L1(G) for locally compact group G is Arens regular if and only if G is finite [2]. The reader is referred to [3, 4] for more information on Arens products and topological centres.Throughout the paper we identify an element of a Banach spaceX with its canonical image in X′′. Also for closed linear subspace E of X we write E⊥={f∈X′:f∣E=0}.In [5], Eshaghi Gordji and Filali obtained significant results related to the topological centres of Banach module actions and regularity of bilinear maps. They showed that if 𝒜 enjoys a bounded approximate identity, then the left (right) module action of 𝒜 on 𝒜′ is regular if and only if 𝒜 is reflexive; see also [6].In this paper, under certain conditions we prove that the left and right module actions of𝒜 on 𝒜(n) are regular, where 𝒜 has not bounded approximate identity. Then we apply this fact to determine Arens regularity and quotient Arens regularity of certain class of Banach algebras.


## 2. Arens Regularity of Module Extension Banach Algebras

Suppose thatX is a Banach 𝒜-bimodule with the left and right module actions π1:𝒜×X→X and π2:X×𝒜→X, respectively. According to [7], 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.Now suppose that𝒰=𝒜⊕X. Then 𝒰 with norm ∥(a,x)∥=∥a∥+∥x∥ and product (2.2)(a,x)(b,y)=(ab,a⋅y+x⋅b)(a,b∈A,x,y∈X)
is a Banach algebra which is known as a module extension Banach algebra. The second dual 𝒰′′ of 𝒰 is identified with 𝒜′′⊕X′′, as a Banach space. Also the first Arens product □ on 𝒰′′ is specified by (2.3)(Φ,μ)□(Ψ,ν)=(Φ□Ψ,Φ⋅ν+μ⋅Ψ).
It is straightforward to check that (Φ,μ)∈Zt1(𝒰′′) if and only if (a)
Φ∈Zt1(𝒜′′),(b)
ν↦Φ·ν:X′′→X′′ is w*-w* continuous,(c)
Ψ↦μ·Ψ:𝒜′′→X′′ is w*-w* continuous,      (see [5, 8]).If𝒜′′ has the second Arens product ◊, then X′′ is an 𝒜′′-bimodule in the same way. We denote this module action by the symbol “•”. The second Arens product ◊ on 𝒰′′ and second topological centre Zt2(𝒰′′) of 𝒰′′ can be defined analogously. Thus, the Banach algebra 𝒰 is Arens regular if and only if 𝒜 is Arens regular and (2.4)Φ⋅ν=Φ•ν,ν⋅Φ=ν•Φ(Φ∈A′′,ν∈X′′).We consider𝒜 as a Banach 𝒜-bimodule equipped with its own multiplication. Then 𝒜=𝒜(0),𝒜′,𝒜′′,…,𝒜(n) can be made into a Banach 𝒜-bimodule in a natural fashion [4]. Clearly, regularity of 𝒰=𝒜⊕𝒜(n) implies that of 𝒜 but the converse is not true in general. For example, let 𝒜 be a nonreflexive Banach space and let φ be a nonzero element of 𝒜′ such that ∥φ∥≤1. Then the product a·b=φ(a)b turns 𝒜 into a Banach algebra [6], such that 𝒜(2n) is Arens regular for all n∈ℕ.Now we consider the bilinear mappings(2.5)π1:A×A(2n-1)⟶A(2n-1),π2:A(2n-1)×A⟶A(2n-1).
One can verify that π2 is Arens regular for all n∈ℕ but π1 is not regular for each n∈ℕ. This shows that 𝒰=𝒜⊕𝒜(2n-1) is not regular. However, 𝒰=𝒜⊕𝒜(2n) is Arens regular.We commence with the next result which studies Arens regularity of the left and right module actions of𝒜 on 𝒜(2n-1).Theorem 2.1.
Let𝒜 be a Banach algebra and n∈ℕ.(i)
Ifπ(3n)(𝒜(2n),𝒜(2n))⊆𝒜(2n-2), then the right module action of 𝒜 on 𝒜(2n-1) is Arens regular.(ii)
Ifπt(3n)t(𝒜(2n),𝒜(2n))⊆𝒜(2n-2), then the left module action of 𝒜 on 𝒜(2n-1) is Arens regular.Proof.
We prove (i) that the assertion (ii) can be proved similarly.
Since𝒜(n+2)=𝒜(n)⊕(𝒜(n-1))⊥ [7], as a direct sum of 𝒜-bimodules, it is enough to show that the result is valid for n=1, and it can be deduced for n≥2, analogously. To this end let Φ∈𝒜′′ and let (ai) be bounded net in 𝒜 that is w*-convergent to Φ. Since 𝒜′′′=𝒜′⊕𝒜⊥, for each μ∈𝒜′′′ there exist f∈𝒜′ and ρ∈𝒜⊥ such that μ=f̂+ρ. It follows that; for each Ψ∈𝒜′′,π′′′(ai,Ψ)→π′′′(Φ,Ψ) in the weak topology. So we have that(2.6)〈μ•Φ,Ψ〉=〈Φ,Ψ⋅μ〉=lim⁡i〈aî,Ψ⋅μ〉=lim⁡i〈μ,π′′′(ai,Ψ)〉=〈μ,π′′′(Φ,Ψ)〉=〈μ⋅Φ,Ψ〉.
Therefore the right module action of 𝒜 on 𝒜′ is regular, as required.The corollary below follows from Theorem 3.1 of [5] and Theorem 2.1.Corollary 2.2.
Letπ1 be the left module action of a Banach algebra 𝒜 on 𝒜′. If π1′′ is onto and 𝒜′′◊𝒜′′⊆𝒜, then 𝒜 is Arens regular.The following theorem, which is the main one in the paper, characterizes Arens regularity of𝒜(2n).Theorem 2.3.
Let𝒜 be an Arens regular Banach algebra. If 𝒜′′□𝒜′′⊆𝒜, then, for all n∈ℕ, 𝒜(2n) is Arens regular and
(2.7)π(3n+3)(A(2n+2),A(2n+2))⊆A(2n).Proof.
Since𝒜 is Arens regular, 𝒜′′ is a dual Banach algebra with predual space E=𝒜′ [4]. Let μ∈𝒜′′′ and Φ∈𝒜′′. Then the inclusion 𝒜′′□𝒜′′⊆𝒜 shows that μ·Φ is w*-continuous linear functional on 𝒜′′ and so it must be in 𝒜′. It follows that β·μ=0 for all β∈E⊥, and hence π(6)(α,β)=0 for each α∈E⊥. Similarly, we obtain πt(6)t(α,β)=0(α,β∈E⊥). Then by Proposition 2.16 of [4] 𝒜′′ is Arens regular and
(2.8)π(6)((Φ,α),(Ψ,β))=(Φ□Ψ,Φ⋅β+α⋅Ψ),(Φ,Ψ∈A′′,α,β∈E⊥).
One may verify that Φ·β=α·Ψ=0 and, since 𝒜(4)=𝒜′′⊕E⊥, that we have π(6)(𝒜(4),𝒜(4))⊆𝒜′′. Thus the result is established for n=1. An easy induction argument now finishes the proof.As a consequence of Theorems2.1 and 2.3, we have the next result.Corollary 2.4.
Let𝒜 be an Arens regular Banach algebra. If 𝒜′′□𝒜′′⊆𝒜, then the following assertions hold for all n∈ℕ.(i)
𝒰=𝒜⊕𝒜(n) is Arens regular.(ii)
𝒜(2n-1) is an 𝒜(2n)-submodule of 𝒜(2n+1).Let𝒜=l1, with pointwise product. Then 𝒜 is an Arens regular Banach algebra which is not reflexive but satisfies 𝒜′′□𝒜′′⊆𝒜 [4]. Therefore by the preceding corollary 𝒰=𝒜⊕𝒜(n) is Arens regular.It is easy to verify that regularity of the left and right module actions of𝒜 on 𝒜(2n-1) are equivalent for each Arens regular Banach algebra 𝒜 which is commutative.Remark 2.5.
It is well known that eachC*-algebra 𝒜 is Arens regular and 𝒜′′ is also a C*-algebra [3], and therefore 𝒜′′ itself are Arens regular. This shows that for each n∈ℕ, 𝒜(2n) and hence 𝒰=𝒜⊕𝒜(2n) is Arens regular. But in general, 𝒰=𝒜⊕𝒜(2n-1) is not Arens regular. Indeed, it is Arens regular if and only if 𝒜 is reflexive [5].


## 3. Quotient Arens Regularity of Module Extension Banach Algebras

Let𝒜 be a Banach algebra with a bounded approximate identity and let X=𝒜′·𝒜, the subspace of 𝒜′ consisting of the functionals of the form f·a, for all f∈𝒜′ and a∈𝒜. By Cohen’s factorization theorem [9], X is a closed 𝒜-submodule of 𝒜′. It is also left introverted in 𝒜′; that is, Φ·λ∈X for each λ∈X and Φ∈𝒜′′. Then X′ is a Banach algebra by the following (first Arens type) product: (3.1)〈Φ□Ψ,λ〉=〈Φ,Ψ⋅λ〉(Φ,Ψ∈X′,λ∈X).As in [10], the Banach algebra 𝒜 is said to be left quotient Arens regular if Zt(X′)=X′, where
(3.2)Zt(X′)={Φ∈X′:Ψ⟼Φ□Ψisw*-w*continuousonX′}.Similarly,X=𝒜·𝒜′ is an 𝒜-module and is right introverted in 𝒜′. As mentioned above, the second Arens product on 𝒜′′ induces naturally a Banach algebra product on X′, which is denoted by ◊. The topological centre Zt(X′) and right quotient Arens regularity can be defined analogously. Obviously, every Arens regular Banach algebra is quotient Arens regular but the converse does not hold; see example 38 of [10]. Also a direct proof shows that, if 𝒜 is an ideal in 𝒜′′, then 𝒜 is quotient Arens regular.Proposition 3.1.
Suppose that the Banach algebra𝒜 is a left ideal in 𝒜′′. Then 𝒰=𝒜⊕𝒜(2n) is a left ideal in 𝒰′′ for all n∈ℕ.Proof.
We first show that, if𝒜 is a left ideal in 𝒜′′, then it is also a left ideal in 𝒜(2n) for each n∈ℕ. So let a∈𝒜 and α∈𝒜(4). Then, for all μ∈𝒜′′′, there exist f∈𝒜′ and ρ∈𝒜⊥ such that μ=f̂+ρ. By assumption â·ρ=0, and therefore â·μ=â·f̂. This shows that â·μ is w*-continuous linear functional on 𝒜′′ and so â·μ∈𝒜′. Since 𝒜(4)=𝒜′′⊕(𝒜′)⊥, α=Φ+σ for some Φ∈𝒜′′ and σ∈(𝒜′)⊥. Then we have that
(3.3)〈α⋅â,μ〉=〈α,â⋅μ〉=〈Φ+σ,â⋅μ〉=〈Φ,â⋅μ〉=〈Φ⋅â,μ〉.
It follows that α·â=Φ·â, and thus 𝒜 is a left ideal in 𝒜(4). An easy induction argument now finishes our claim. Therefore by definition 𝒰 is a left ideal in 𝒰′′ for each n∈ℕ.In general, the above result is not valid if we replace2n with 2n-1. For example, let 𝒜 be the group algebra of an infinite compact group G. Then 𝒜 is an ideal in 𝒜′′, as is well known, but 𝒰=𝒜⊕𝒜(2n-1) is not ideal in 𝒰′′. By additional hypothesis we have the next result.Theorem 3.2.
If the Banach algebra𝒜 is a left ideal in 𝒜′′ and the right module action of 𝒜 on 𝒜(2n-2) is regular, then 𝒰=𝒜⊕𝒜(2n-1) is a left ideal in 𝒰′′.Proof.
The result is straightforward for the casen=1. So we give the proof for n=2. Let (a,μ)∈𝒰 and (Φ,Λ)∈𝒰′′. Then a similar argument to what has been used in the proof of the preceding proposition shows that Λ·â∈𝒜′′′. On the other hand, regularity of the right module action of 𝒜 on 𝒜′′ implies that Φ·μ̂ is w*-continuous linear functional on 𝒜(4) and so it must be in 𝒜′′′. Thus, Φ·μ̂+Λ·â∈𝒜′′′. Therefore by definition we have that (Φ,Λ)□(a,μ)∈𝒰, and hence 𝒰 is a left ideal in 𝒰′′. A similar discussion reveals that the result will be established for n>2.Recall that the right version of Proposition3.1 and Theorem 3.2 holds. Therefore, we have the following results.Corollary 3.3.
Let𝒜 be a Banach algebra such that 𝒜 is an ideal in 𝒜′′. Suppose that the left and right module actions of 𝒜 on 𝒜(2n-2) are regular. Then 𝒰=𝒜⊕𝒜(n) is quotient Arens regular.Corollary 3.4.
Let𝒜 be a C*-algebra and n∈ℕ. If 𝒜 is an ideal in 𝒜′′, then 𝒰=𝒜⊕𝒜(n) is quotient Arens regular.Example 3.5.
Let𝒜=c0, with pointwise product and 𝒰=𝒜⊕𝒜(n). Then 𝒜 is a commutative C*-algebra which is an ideal in 𝒜′′. Therefore by the above corollary 𝒰 is quotient Arens regular for all n∈ℕ. Note that, by Remark 2.5, 𝒰 is not Arens regular for the odd case n.



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*Source: 680952-2011-07-13.xml*