Module Actions on Iterated Duals of Banach Algebras

Abstract

Let 𝒜 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.

1. Introduction and Preliminaries

Let 𝒜 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 ⟨Φ□Ψ,𝑓⟩=⟨Φ,Ψ⋅𝑓⟩,⟨Ψ⋅𝑓,𝑎⟩=⟨Ψ,𝑓⋅𝑎⟩,⟨𝑓⋅𝑎,𝑏⟩=⟨𝑓,𝑎𝑏⟩.(1.1) 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 fixed Φ∈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 𝑍𝑍^1𝑡_2𝑡𝒜^=Φ∈𝒜^∶Ψ⟼Φ□Ψis𝑤^∗-𝑤^∗continuouson𝒜^,𝒜^=Φ∈𝒜^∶Ψ⟼Ψ◊Φis𝑤^∗-𝑤^∗continuouson𝒜^,(1.2) are called the first and the second topological centres of 𝒜^, respectively. One can verify that 𝒜 is Arens regular if and only if Z^1𝑡(𝒜^)=𝑍^2𝑡(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 space 𝑋 with its canonical image in 𝑋^.

2. 𝒜^-Bimodule Structures on 𝒜⁽⁴⁾

Suppose that 𝒜 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 Φ⋅𝜈=𝑤^∗−lim_𝑖lim_𝑗𝑎_𝑖⋅𝑥_𝑗,𝜈⋅Φ=𝑤^∗−lim_𝑗lim_𝑖𝑥_𝑗⋅𝑎_𝑖,(2.1) where (𝑎_𝑖) and (𝑥_𝑗) are nets in 𝒜 and 𝑋 that converge, in 𝑤^∗-topologies, to Φ and 𝜈, respectively.

There are two 𝒜^-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 algebra 𝒜⁽⁴⁾ has three 𝒜^-bimodule structures.(i)If we consider 𝒜⁽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 consider 𝒜⁽⁴⁾ as the dual of 𝒜^ in which 𝒜^=(𝒜^)^, then 𝒜⁽⁴⁾ can be an 𝒜^-bimodule by module structures as follows: ⟨Φ⋆𝛼,𝜇⟩=⟨𝛼,𝜇∗Φ⟩,⟨𝛼⋆Φ,𝜇⟩=⟨𝛼,Φ∗𝜇⟩,(2.2) for all Φ∈𝒜^, 𝜇∈𝒜^, and 𝛼∈𝒜⁽⁴⁾.(iii)We consider 𝒜⁽⁴⁾ as the dual of 𝒜^ in which 𝒜^=(𝒜^)^, so 𝒜⁽⁴⁾ can be an 𝒜^-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 𝒜^.

It is straightforward to check that three 𝒜^-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=Φ⋆𝛼Φ∈𝒜^,𝛼∈𝒜⁽⁴⁾.(2.4)

Proof. Suppose that (𝑎_𝑖), (𝑓_𝑗), and (𝐹_𝑘) are bounded nets in 𝒜, 𝒜^ and 𝒜^, respectively, such that 𝑎_𝑖→Φ, 𝑓_𝑗→𝜇 and 𝐹_𝑘→𝛼 in the 𝑤^∗-topology. Then, we have ⟨𝛼⋆Φ,𝜇⟩=⟨𝛼,Φ∗𝜇⟩==lliimm_𝑘𝑘𝐹^𝑘,Φ∗𝜇=lim_𝑘⟨Φ∗𝜇,𝐹_𝑘⟩=lim_𝑘lim_𝑖lim_𝑗𝑎^𝑖⋅𝑓_𝑗,𝐹_𝑘=lim_𝑘lim_𝑖lim_𝑗𝐹_𝑘,𝑎_𝑖⋅𝑓_𝑗=lim_𝑘lim_𝑖lim_𝑗𝑓^𝑗,𝐹_𝑘⋅𝑎_𝑖=lim_𝑘lim_𝑖⟨𝜇,𝐹_𝑘⋅𝑎_𝑖⟩=⟨𝛼•lΦim_𝑖,𝜇⟩𝐹._𝑘⋅𝑎_𝑖,𝜇(2.5) Therefore, 𝛼•Φ=𝛼⋆Φ. Similarly, we obtain ⟨Φ⋅𝛼,𝜇⟩=⟨𝛼,𝜇⋅Φ⟩==lliimm_𝑘𝑘𝐹^𝑘,𝜇⋅Φ=lim_𝑘⟨𝜇,Φ□𝐹_𝑘⟩=lim_𝑘lim_𝑗𝑓^𝑗,Φ□𝐹_𝑘=lim_𝑘lim_𝑗Φ,𝐹_𝑘⋅𝑓_𝑗=lim_𝑘lim_𝑗lim_𝑖𝑎_𝑖,𝐹_𝑘⋅𝑓_𝑗=lim_𝑘lim_𝑗lim_𝑖𝐹_𝑘,𝑓_𝑗⋅𝑎_𝑖=lim_𝑘⟨𝜇∗Φ,𝐹_𝑘⟩==⟨⟨𝛼Φ,𝜇⋆∗𝐹𝛼,_𝑘Φ𝜇,⟩𝜇⟩.∗Φ(2.6) This shows that Φ⋅𝛼=Φ⋆𝛼, as required.

Let 𝒜^ 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 on 𝒜⁽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. 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)𝑍^1𝑡(𝒜⁽⁴⁾,⊡)=𝒜⁽⁴⁾ if and only if 𝑍^1𝑡(𝒜⁽⁴⁾,□)=𝒜⁽⁴⁾.(ii)𝑍^1𝑡(𝒜⁽⁴⁾,⟐)=𝒜⁽⁴⁾ if and only if 𝑍^1𝑡(𝒜⁽⁴⁾,◊)=𝒜⁽⁴⁾.

Proof. We prove (i); the proof of (ii) follows from similar argument.
Assume that 𝑍^1𝑡(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 𝛼□𝛽=𝛼⊡𝛽=𝛼⟐𝛽=𝛼◊𝛽.(2.7) It follows that 𝑍^1𝑡(𝒜⁽⁴⁾,□)=𝒜⁽⁴⁾. The converse is similar.

It is well known that every 𝐶^∗-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. Let 𝒜 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 operators 𝐿_Φ and 𝑅_Φ 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. 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, 𝒜^ 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 𝒜^-bimodule actions on 𝒜⁽⁴⁾ coincide.

Proof. Let (𝑎_𝑖) and (𝐹_𝑗) be bounded nets in 𝒜 and 𝒜^ that converge, in 𝑤^∗-topologies, to Φ and 𝛼, respectively. Then for each 𝜇∈𝒜^, we have ⟨𝛼⋅Φ,𝜇⟩=⟨𝛼,Φ⋅𝜇⟩==lliimm_𝑗𝑗Φ⋅𝜇,𝐹_𝑗=lim_𝑗lim_𝑖𝑎_𝑖⋅𝜇,𝐹_𝑗=lim_𝑗lim_𝑖𝜇,𝐹_𝑗⋅𝑎_𝑖=⟨𝛼•lΦim_𝑖,𝜇⟩𝐹._𝑗⋅𝑎_𝑖,𝜇(2.8) Consequently, 𝛼⋅Φ=𝛼•Φ. By hypothesis 𝒜^ is Arens regular and so lim_𝑗lim_𝑖𝑎_𝑖⋅𝐹_𝑗=lim_𝑖lim_𝑗𝑎_𝑖⋅𝐹_𝑗.(2.9) Thus, for each 𝜇∈𝒜^, ⟨Φ⋅𝛼,𝜇⟩=⟨𝛼,𝜇⋅Φ⟩==lliimm_𝑗𝑗𝜇⋅Φ,𝐹_𝑗=lim_𝑗lim_𝑖𝜇⋅𝑎_𝑖,𝐹_𝑗=lim_𝑗lim_𝑖𝜇,𝑎_𝑖⋅𝐹_𝑗=lim_𝑖lim_𝑖𝑎^𝑖⋅𝐹_𝑗,𝜇=⟨Φ•li𝛼m_𝑗,𝜇⟩𝑎._𝑖⋅𝐹_𝑗,𝜇(2.10) Therefore Φ⋅𝛼=Φ•𝛼. Now by Theorem 2.1, three 𝒜^-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, but  𝒜^□𝒜^⊆𝒜 [4]. Therefore by above theorem three 𝒜^-bimodule structures on 𝒜⁽⁴⁾ coincide.