MODULE STRUCTURES ON ITERATED DUALS OF BANACH ALGEBRAS
A. Bodaghi, M. Ettefagh, M. Eshaghi Gordji, A. R. Medghalchi
Abstract
LetAbe a Banach algebra and (A′′,) be its second dual with first Arens product. We consider three (A′′
,)-bimodule structures on forth dual and four (A′′
,)-bimodule structures on fifth dual of a Banach al- gebra. This paper determines the conditions that make these structures equal. Among other results we show that ifA′′is weakly amenable with some conditions, thenAis 3 -weakly amenable.
1 Introduction
LetAbe a Banach algebra and letX be a Banach A-module, that isX is a Banach space and anA-module such that the module operations (a, x)7−→a·x and (a, x)7−→x·afromA×X intoX are jointly continuous. The dual space X′ ofX is also a BanachA-module by the following module actions:
ha·f, xi=hf, x·ai, hf·a, xi=hf, a·xi, (a∈A, x∈X, f ∈X′).
We setX′′ = (X′)′, and so on, and we regard X as a subspace of X′′ in the standard way. AlsoX′′′= (X′′)′,...
LetX be a BanachA-module. Then a continuous linear map D:A−→X is called aderivation if
D(ab) =a·D(b) +D(a)·b (a, b∈A).
Key Words: Banach algebra, Derivation, Arens product, Arens regular, strong double limit property, 3-weak amenability.
Mathematics Subject Classification: 46H25 Received: May, 2009
Accepted: January, 2010
63
Forx∈X we defineδx:A−→X as follows:
δx(a) =a·x−x·a (a∈A), it is easy to show thatδx is a derivation. Such derivations are calledinner derivations. Ais called amenable, if every derivation D : A−→ X′ is inner, for each Banach A-moduleX. If every derivation fromAintoA′ is inner,A is called weakly amenable. Letn∈N. A Banach algebraAis called n-weakly amenable if every derivation fromAinto A(n) is inner [4], where A(n) is the n-th dual ofAthat is a BanachA-module. We regardAas a subspace ofA′′
by canonical embedding ˆ : A→ A′′;a 7→ ˆa. We write Ab as the image of A under this mapping.
LetX, Y andZ be normed spaces and letf :X×Y −→Zbe a continuous bilinear map. Then the adjiont off is defined by
f′ :Z′×X−→Y′, hf′(z′, x), yi=hz′, f(x, y)i (z′∈Z′, x∈X, y∈Y).
Sincef′is a continuous bilinear map, this process may be repeated to define f′′ = (f′)′ :Y′′×Z′ −→X′, and thenf′′′ = (f′′)′ : X′′×Y′′ −→Z′′. The map f′′′ is the unique extension off such that X′′−→Z′′;x′′7→f′′′(x′′, y′′) isweak∗−weak∗continuous for all y′′∈Y′′andY′′−→Z′′;y′′7→f′′′(x, y′′) isweak∗−weak∗continuous for allx∈X. Let nowft:Y ×X −→Z be the transpose off defined byft(y, x) =f(x, y) for allx∈X andy∈Y. Thenft is a continuous bilinear map fromY ×X toZ, and so it may be extended as above to (ft)′′′:Y′′×X′′−→Z′′. The bilinear mapf is calledArens regular iff′′′ = ((ft)′′′)t (see [1, 2, 7, 8] and [13]). Letx′′∈X′′ and y′′∈Y′′. Then there exist nets (xα)⊂X and (yβ)⊂Y withxbα
w∗
−→x′′ andybβ w∗
−→y′′. We have
f′′′(x′′, y′′) = lim
α lim
β
f(x\α, yβ),
((ft)′′′)t(x′′, y′′) = lim
β lim
α
f(x\α, yβ).
Let Abe a Banach algebra, and letπ:A×A−→Adenote the product ofA, so thatπ(a, b) =ab (a, b∈A). forF andGinA′′,we denote π′′′(F, G) and ((πt)′′′)t(F, G) by symbolsFGandF♦G, respectively. These are called first and second Arens products onA′′. These products are defined in stages as follows. For everyF, G∈A′′, f ∈A′anda, b∈A,we definef·a, a·f, G·f andf·F in A′;FGandF♦Gin A′′ by
hf ·a, bi=hf, abi, ha·f, bi=hf, bai, hG·f, ai=hG, f·ai, hf·F, ai=hF, a·fi, hFG, fi=hF, G·fi, hF♦G, fi=hG, f·Fi.
A′′ is a Banach algebra with (above) Arens products. In fact FG=w∗−lim
α w∗−lim
β
a[αbβ
F♦G=w∗−lim
β w∗−lim
α
a[αbβ,
where F =w∗−limαbaα and G=w∗−limβbbβ. The algebra Ais Arens regular whenever the map π is Arens regular that is, whenever the first and second Arens products ofA′′coincide. Recall that a Banach algebraAis said to bedualif there is a closed submoduleA0 ofA′ such thatA=A0′.
Definition 1.1.
The Banach algebraAhas strongly double limit property (SDLP) if for each bounded net (aα) inAand each bounded net (fβ) inA′, limαlimβhfβ, aαi= limβlimαhfβ, aαi, whenever both iterated limits exist.This definition has been introduced in [14]. Medghalchi and Yazdanpanah in [14] showed that every reflexive Banach algebra has (SDLP). We know that reflexivity is equivalent with double limit property [3, Theorem A.3.31], so the (SDLP) is equivalent with reflexivity. Now suppose that the Banach algebra A has (SDLP), then for each f ∈ A′ and bounded nets (aα),(bβ) in A, we have
limβ lim
α hbβ·f, aαi= lim
α lim
β hbβ·f, aαi,
which means that for eachf ∈A′, the map a7→a.f, A−→A′ is weakly compact by [3, Theorem 2.6.17], i.e., A is Arens regular. Hence (SDLP) is stronger than Arens regularity. On the other hand this two are not equivalent in general. We know C([0,1]) is an Arens regular Banach algebra. If we consider the sequence (fm) inC([0,1]) defined byfm(x) =m+m1
x
for 0< x≤1 andfm(0) = 0 for allm∈N, and assume that sequence (µn) is inM([0,1]) = C([0,1])∗ ( the set of all regular Borel measures on [0,1]), where µn is the point mass at n1, for alln∈N. Then, we easily see that
limm lim
n hµn, fmi= 06= 1 = lim
n lim
mhµn, fmi.
ThereforeC([0,1]) has not (SDLP). Also there are Arens regular Banach algebras which are not reflexive as Banach spaces. For example, the disc algebra A(D) is Arens regular [16] but not reflexive [15].
One may consider the question of howAinherits the amenability or weak amenability of A′′. For amenability the answer is positive (see [12]). So for weak amenability, this problem was considered by few authors and a positive answer has been given in each of the following cases:
• Ais a left ideal inA′′ [12].
• Ais a dual Banach algebra [11].
• Ais Arens regular and every derivation fromAintoA′is weakly compact [5].
• Ahas (SDLP) [14].
• Ais a right ideal inA′′ andA′′A=A′′ [9].
In section two of this paper, we put many module structures on forth dual A(4)and show that these module structures are not always equal, and we show when these module structures are equal. By using part two, we make four module structures onA(5). This is done in section three, where these module structures onA(5)are not always equal. In section four we show that with some module structures onA(5), weak amenabilityA′′ implies weak amenabilityA. This is a question that ifA′′ is 3 -weakly amenable, isA3-weakly amenable?
We show that the 3 -weak amenability ofA′′ implies the 3 -weak amenability of Aif D′′(A′′)·A(4) ⊆Ac′, for each derivation D : A−→ A′′′. It is known that every (n+ 2)-weakly amenable Banach algebra is n-weakly amenable for n ≥ 1 [4]. In particular the 3-weak amenability of A implies the weak amenability of A. Does weak amenability imply 3 -weak amenability? The answer is negative. Yong Zhang [19] gave an example of a weakly amenable Banach algebra that it is not 3-weakly amenable, but he had showed in [20]
that ifAis weakly amenable with a left (right) bounded approximate identity such that it is a left (right) ideal in A′′, then A is (2n+1)-weakly amenable forn≥1. A different proof are provided by Dales, Ghahramani and Grønbæk in [4] in which Ais an ideal inA′′. Finally we put some conditions onAand A′′such that ifAis weakly amenable, thenAis 3-weakly amenable. For the remainder of this paper, A′′ is regarded as a Banach algebra with respect to the first Arens product.
2 A
′′- bimodule structures on forth dual of a Banach al- gebra
A′′′ has two A′′-bimodule structures. First we regard A′′′, as the dual space of A′′, (A′′′ = (A′′)′) and so A′′′ can be made into an A′′-bimodule by the following actions
hλ·F, Gi=hλ, FGi, hF·λ, Gi=hλ, GFi, (λ∈A′′′;F, G∈A′′).
In the second way,A′′′, as the second dual ofA′, (A′′′ = (A′)′′), can be an A′′-bimodule by the following formula. Forλ∈A′′′ andF ∈A′′, we have
λ◦F =w∗−lim
i w∗−lim
α
f\i·aα, F◦λ=w∗−lim
α w∗−lim
i
a\α·fi,
where F = w∗−limαbaα in A′′and λ = w∗ −limifbi in A′′′, such that (aα) and (fi) are nets inAand A′ respectively. In factλ◦F andF ◦λare extensions of module actions (f, a)−→f ·a (A′×A−→A′) and (a, f)−→
a·f (A×A′−→A′).
These twoA′′-bimodule structures onA′′′ are considered in [10] and have been shown that two right A′′-bimodule actions on A′′′ always coincide but left A′′-bimodule structures on A′′′ are not always equal. Now the Banach algebra A(4) has threeA′′-bimodule structures.
(a)
We consider A(4) = (A′′′)′ in which A′′′ = (A′)′′, soA(4) can be an A′′-bimodule by following actionshF◦Λ, λi=hΛ, λ◦Fi, hΛ◦F, λi=hΛ, F◦λi where F∈A′′, λ∈A′′′ and Λ∈A(4).
(b)
We consider A(4) = (A′′′)′ in which A′′′ = (A′′)′, so A(4) can be an A′′-bimodule by following right and left module actionshF·Λ, λi=hΛ, λ·Fi, hΛ·F, λi=hΛ, F ·λi where F∈A′′, λ∈A′′′ and Λ∈A(4).
(c)
LetA(4) = (A′′)′′be as the second dual ofA′′. Take Λ∈A(4),F ∈A′′and bounded nets (Fα)⊂A′′,(aβ)⊂Awith Fbα w∗
−→Λ andbaβ w∗
−→F. Two module actions are defined by
F•Λ =w∗−lim
β lim
α
a\β·Fα Λ•F =w∗−lim
α lim
β
F\α·aβ.
Hence F •Λ and Λ•F are extensions of module actions (a, F) −→ a· F (A×A′′−→A′′) and (F, a)−→F·a (A′′×A−→A′′).
We show that these threeA′′-bimodule structures onA(4) are not always equal. Suppose that Λ ∈A(4) , λ∈A′′′, F ∈A′′ and bounded nets (Gα)⊂ A′′,(fγ)⊂A′,(aβ)⊂AbyGbα
w∗
−→Λ,fbγ w∗
−→λandbaβ w∗
−→F, then hΛ◦F, λi=hΛ, F◦λi
= lim
α hGbα, F ◦λi
= lim
α hF·λ, Gαi
= lim
α lim
β lim
γ hGα, aβ·fγi,
and
hΛ·F, λi=hΛ, F·λi
= lim
α hGbα, F ·λi
= lim
α hλ, GαFi
= lim
α lim
γ hfbγ, GαFi
= lim
α lim
γ hGα, F·fγi.
For structure (c), we have
hΛ•F, λi= lim
α lim
β hG\α·aβ, λi
= lim
α lim
β hλ, Gα·aβi
= lim
α lim
β lim
γ hfbγ, Gα·aβi
= lim
α lim
β lim
γ hGα, aβ·fγi.
We see two right actions in parts (a) and (c) are equal and different from the action of (b). For left actions, suppose that Λ∈A(4) , λ∈A′′′, F ∈A′′
and bounded nets (Gα)⊂A′′,(fγ)⊂A′,(aβ)⊂A withGbα w∗
−→Λ,fbγ w∗
−→λ andbaβ
w∗
−→F, then
hF◦Λ, λi=hΛ, λ◦Fi
= lim
α hGbα, λ◦Fi
= lim
α hλ◦F, Gαi
= lim
α lim
γ lim
β hGα, fγ·aβi,
and
hF·Λ, λi=hΛ, λ·Fi
= lim
α hGbα, λ·Fi
= lim
α hλ, FGαi
= lim
α lim
γ hfbγ, FGαi
= lim
α lim
γ hF, Gα·fγi
= lim
α lim
γ lim
β hbaβ, Gα·fγi
= lim
α lim
γ lim
β hGα, fγ·aβi.
For the structure (c), we have hF•Λ, λi= lim
β lim
α ha\β·Gα, b′′′i
= lim
β lim
α hλ, aβ·Gαi
= lim
β lim
α lim
γ hfbγ, aβ·Gαi
= lim
β lim
α lim
γ hGα, fγ·aβi.
We see that left actions in parts (a) and (b) are equal and different from the action of (c). We put some conditions onAand show that with this conditions allA′′-bimodule structures onA(4) are equal. First we bring some simple, but useful lemmas.
Lemma 2.1.
If A is Arens regular, then, for the bounded nets (Fα) and (Gβ)inA′′,(w∗−limαFα)(w∗−limβGβ) =w∗−limαw∗−limβ(FαGβ) =w∗−
limβw∗−limα(FαGβ).
Lemma 2.2.
Let the Banach algebra Awith one of the following condi- tions(i) The mapϕ:A′×A−→A′; ((f, a)−→f·a)is Arens regular,
(ii) The map ψ : A′′ −→ A′′; (G −→ GF) is weak-compact for every F ∈A′′,
(iii) The mapφ:A′′ −→A′′; (G−→GF) is w*-w-continuous for every F ∈A′′.
Then for each bounded net(aα)inAandλ∈A′′′,
hλ,(w∗−limαbaα)Fi= limαhλ,baαFi (1)
Proof.
(i) Let λ=w∗−limβfbβ, where (fβ) is a bounded net in A′, then we havehλ,(w∗−lim
α baα)Fi= lim
β h(w∗−lim
α baα)F, fβi
= lim
β lim
α hbaαF, fβi
= lim
β lim
α hF, fβ·aαi
=hw∗−lim
β w∗−lim
α
f\β·aα, Fi
=hw∗−lim
α w∗−lim
β
f\β·aα, Fi
= lim
α hw∗−lim
β
fbβ, aαFi
= lim
α hλ,baαFi.
(ii) From the double limit property of weak compact operatorψ, we see limβlimαhbaαF, fβi= limαlimβhbaαF, fβi, hence
hλ,(w∗−lim
α baα)Fi= lim
β h(w∗−lim
α baα)F, fβi
= lim
β lim
α hbaαF, fβi
= lim
α lim
β hbaαF, fβi
= lim
α hλ,baαFi.
(iii) Equation (1) is a consequence ofw∗−w−continuity ofφ.
Lemma 2.3.
If for every G∈A′′ the mapρ:A′′−→A′′; (F −→GF) is w*-w-continuous, then for every bounded net(Fj)inA′′hλ, G(w∗−limjFj)i= limjhλ, GFji, (λ∈ A′′′).
Proof.
It is similar to part (iii) of Lemma 2.2.Lemma 2.4.
Let A be an Arens regular Banach algebra. If the map ϕ: A′′−→A′′; (F −→GF)is weak-compact or w*-w-continuous for every G∈ A′′, thenhλ, w∗−limαw∗−limj(baαFj)i= limαlimjhλ,baαFji.(2) for all λ∈A′′′, bounded nets (aα)and(Fj)inAandA′′, respectively.
Proof.
Let (fβ) be a bounded net inA′ such thatfbβ−→w∗ λ. Then hλ, w∗−limα w∗−lim
j (baαFj)i= lim
β h(w∗−lim
α w∗−lim
j (baαFj), fβi
= lim
β lim
α lim
j hbaαFj, fβi
= lim
α lim
j lim
β hbaαFj, fβi
= lim
α lim
j hλ,baαFji.
Sinceϕisw∗−w−continuous, the equation (2) is obtained immediately.
Proposition 2.5.
Let A be a Banach algebra. If one of the following conditions holds, then the twoA′′-module actions in (a), (c) coincide.(i) The Banach algebra A and the map ϕ : A′×A −→ A′; ((f, a) −→
f ·a)are Arens regular and the map ψ :A′′ −→A′′; (F −→FG)is w*-w- continuous for every G∈A′′.
(ii) The Banach algebra A is Arens regular and the map φ : A′′ −→
A′′; (F −→FG)is weak-compact for everyG∈A′′.
(iii) For bounded nets(Gα), (fγ)and (aβ)in A′′,A′ and A, respectively, we have
limα lim
γ lim
β hGα, fγ·aβi= lim
β lim
α lim
γ hGα, fγ·aβi.
Proof.
We know that the two rightA′′-module actions onA′′′′ in (a) and (c) are equal tolimαlimβlimγhGα, aβ·fγi,
in which (Gα), (fγ) and (aβ) are bounded nets in A′′,A′ and A, respec- tively. For left A′′-module actions on A′′′′ it is enough to show the following equality
limα lim
γ lim
β hGα, fγ·aβi= lim
β lim
α lim
γ hGα, fγ·aβi.
(i) By Arens regularity of the mapϕwe have limγ lim
β hGα, fγ.aβi= lim
β lim
γ hGα, fγ.aβi.
Now suppose that λ =w∗−limβfbβ, by Lemma 2.1 and Lemma 2.4 we
have
limα lim
γ lim
β hGα, fγ·aβi= lim
α lim
β lim
γ hGα, fγ·aβi
= lim
α lim
β lim
γ hfbγ,baβGαi
= lim
α lim
β hλ,baβGαi
=hλ, w∗−lim
α w∗−lim
β (baβGα)i
=hλ, w∗−lim
β w∗−lim
α (baβGα)i
= lim
β lim
α lim
γ hGα, fγ·aβi.
(ii) By weak compactness ofφ, we have limγ lim
β hGα, fγ·aβi= lim
γ lim
β hfbγ,baβGαi= lim
β lim
γ hfbγ,baβGαi.
It is easy to check that the map φisw∗−w−continuous, and so the rest of the proof is the same as the proof of part (i).
(iii) It is clear.
3 A
′′- bimodule structures on fifth dual of a Banach al- gebra
Let A be a Banach algebra. We consider four A′′- bimodule structures on A(5).
(I)
We considerA(5)= (A(4))′in whichA(4)has anA′′-bimodule structure as in part(c)
in Section 2. Therefore A(5) is the dual space of A(4), by the following actionshF•Ψ,Λi=hΨ,Λ•Fi, hΨ•F,Λi=hΨ, F•Λi
whereF ∈A′′,Λ∈A(4) and Ψ∈A(5). In this case we haveA(5)= ((A′′)′′)′.
(II)
We considerA(5)= (A(4))′ in whichA(4) has anA′′-bimodule struc- ture as in part(b)
in Section 2, so the left actionA′′onA(5) = (((A′′)′)′)′ is defined byhF·Ψ,Λi=hΨ,Λ·Fi, hΛ·F, λi=hΛ, F·λi, hF·λ, Gi=hλ, GFi.
whereF, G∈A′′, λ∈A′′′,Λ∈A(4) and Ψ∈A(5). The right action is defined in a similar way.
(III)
Let A(5) = (A′′′)′′ be as the second dual of A′′′ in which A′′′ = ((A′)′)′ is anA-bimodule. Take Ψ∈A(5) ,F ∈A′′ and bounded nets (λα)⊂ A′′′,(aβ)⊂A′ withbλαw∗
−→Ψ andbaβ w∗
−→F. Two module actions is defined by
F◦Ψ =w∗−lim
β lim
α
a\β·λα Ψ◦F =w∗−lim
α lim
β
λ\α·aβ.
In fact F ◦Ψ and Ψ◦F are extension of module actions (a, λ) −→ a· λ (A×A′′′−→A′′′) and (λ, a)−→λ·a (A′′′×A−→A′′′).
(IV)
We considerA(5) = (A(4))′ in whichA(4)has anA′′-bimodule struc- ture as in part(a)
in Section 2, hence the A′′-module actions on A(5) are defined byhF ⋆Ψ,Λi=hΨ,Λ◦Fi, hΨ⋆ F,Λi=hΨ, F ◦Λi, whereF ∈A′′,Λ∈A(4) and Ψ∈A(5).
Suppose that Ψ ∈ A(5) , Λ ∈ A(4), F ∈ A′′ and bounded nets (λα) ⊂ A′′′,(Gγ)⊂A′′,(aβ)⊂A′ bybλα−→w∗ Ψ,Gbγ −→w∗ Λ andbaβ−→w∗ F, then
hF•Ψ,Λi=hΨ,Λ•Fi
= lim
α hbλα,Λ•Fi
= lim
α lim
γ lim
β hλα, Gγ·aβi, and
hF◦Ψ,Λi= lim
β lim
α ha\β·λα,Λi
= lim
β lim
α lim
γ hGbγ, aβ·λαi
= lim
β lim
α lim
γ hλα, Gγ·aβi,
so F◦Ψ andF•Ψ are not always equal. But hΨ◦F,Λi= lim
α lim
β hλ\α·aβ,Λi
= lim
α lim
β lim
γ hGbγ, λα·aβi
= lim
α lim
β lim
γ hλα, aβ·Gγi
= lim
α hF•Λ, λαi
=hΨ, F •Λi
=hΨ•F,Λi.
Hence the two rightA′′-bimodule structure parts (I) and (III) onA(5) always coincide. Also we can show that left (right)A′′-module action onA(5) in part (II) is
hF·Ψ,Λi= lim
α lim
γ hλα, GγFi (hΨ·F,Λi= lim
α lim
γ hλα, FGγi), hence the A′′-bimodule structure part (II) is different from (I) and (III).
For twoA′′-module action onA(5) in part (IV), we have
hF ⋆Ψ,Λi= lim
α lim
γ hF◦λα, Gγi, hΨ⋆ F,Λi= lim
α lim
γ hλα◦F, Gγi.
Lemma 3.1.
Let Abe a Banach algebra. Suppose that the map ϕ:A× A′′ −→ A′′; ((a, F) −→ a·F) is Arens regular and the map ψ : A′′ −→A′′; (G−→GF) is w*-w-continuous for everyF ∈A′′. Then the two right A′′-module actions onA(5)in (II) and (III) are equal.
Proof.
By w∗−w−continuity ofψ we must prove the following equality limα limβ lim
γ hλα, aβ·Gγi= lim
α lim
γ lim
β hλα, aβ·Gγi, (3) for bounded nets (λα),(Gγ) and (aβ) inA′′′,A′′ andA′, respectively. By Arens regularity ofϕ, we have
limβ lim
γ hλα, aβ·Gγi= lim
β lim
γ ha\β·Gγ, λαi= lim
γ lim
β ha\β·Gγ, λαi,
and so (3) is true.
Lemma 3.2.
LetAbe a Banach algebra. Assume that the Banach algebra Aand the map ϕ:A′′×A′′′ −→A′′′; ((F, λ)−→F ·λ)is Arens regular and the mapψ:A′′−→A′′; (G−→FG) is w*-w-continuous for everyF ∈A′′. Then two leftA′′-module actions onA(5)in (II) and (III) are equal.Proof.
By w∗−w−continuity of ψ, it is enough to prove the following equality for bounded nets (λα),(Gγ) and (aβ) inA′′′,A′′andA′, respectively.limα lim
γ lim
β hλα, Gγbaβi= lim
β lim
α lim
γ hλα, Gγbaβi. (4) By using Lemma 2.4 we can write
limγ lim
β hλα, Gγbaβi= lim
β lim
γ hλα, Gγbaβi.
Then, by Arens regularity ofϕ, we see limα lim
γ lim
β hλα, Gγbaβi= lim
α lim
β lim
γ hλα, Gγbaβi
= lim
α lim
β hλα,(w∗−lim
α Gγ)baβi
= lim
α lim
β hba\β·λα,(w∗−lim
α Gγ)bi
= lim
β lim
α hba\β·λα,(w∗−lim
α Gγ)bi
= lim
β lim
α lim
γ haβ·λα, Gγi
= lim
β lim
α lim
γ hλα, Gγbaβi.
4 3-weak amenability of the second dual
Let D : A −→ A′′′ be a derivation. Then D′′ : A′′ −→ A(5) = (A′′′)′′ the second transpose of D is a derivation (see [3] and [11]), that means that for every F, G∈A′′
D′′(FG) =D(F)◦G+F◦D(G).
But D′′ :A′′ −→A(5) = (A′′)′′′ is not always a derivation. In the following we put other conditions on Dsuch thatD′′ is a derivation (also see Theorem 4.3).
Proposition 4.1. Let A be a Banach algebra and let D : A −→ A′′′ be a derivation. Then D′′ : A′′ −→ A(5) = (A′′)′′′ is a derivation if and only if D′′(A′′)·A(4)⊆cA′.
Proof.
Let F, G ∈ A′′. Then there are nets (aα) and (bβ) in A which converge to F and G in the w∗- topology of A′′ respectively. Clearly D′′ is w∗- continuous. ThenD′′(FG) =w∗−lim
α w∗−lim
β D(aαbβ)
=w∗−lim
α w∗−lim
β D(aα)·bβ+w∗−lim
α w∗−lim
β aα·D(bβ)
=D′′(F)·G+ lim
α aα·D′′(G).
By (5), it is easy to see thatD′′is a derivation if and only if for everyF, G∈A′′
thatF =w∗−limαaα, the following equality holds
F·D′′(G) =w∗−limαaα·D′′(G) (6).
The relation (6) is true if and only if for every
Λ∈A(4),hF·D′′(G),Λi= limαhaα·D′′(G),Λi(7).
Also (7) holds if and only if
hD′′(G).Λ, Fi= limαhD′′(G)·Λ, aαi (8).
So (8) holds if and only ifD′′(G).Λ :A′′→Cisw∗−w∗−continuous. This
means thatD′′(G).Λ∈cA′.
Corollary 4.2LetAbe a Banach algebra such thatA′′is 3-weakly amenable.
IfD′′(A′′)·A(4)⊆cA′, for each derivationD:A−→A′′′. ThenAis 3 -weakly
amenable.
Let A be a Banach algebra and let ι : A′′ −→A(4) be an injective map (hι(F), λi = hλ, Fi) for F ∈ A′′ and λ ∈ A′′′. Then ι is an A-bimodule homomorphism . Also ι is an A′′-bimodule homomorphism with the mod- ule structures (a) and (b) on A(4), but it is not always an A′′-bimodule homomorphism with the module structures (c). Therefore the adjoint of ι (ι∗) is an A′′-bimodule homomorphism with the module structures (a) and (b). Let X be a Banach space. For n ∈ Z+, we denote X⊥, the subspace of X(2n+1) annihilating Xb, where X(2n+1) is the (2n+1)-th dual of X, i.e.
X⊥={λ∈X(2n+1);hλ, xi= 0, x∈X}. For the Banach algebraA, (A′′)⊥ is clearly w∗-closedA′′-submodule ofA(5). Now we get the main theorem of this paper.
Theorem 4.3. Let Abe a Banach algebra such that A′′ is weakly amenable.
Suppose that one the following conditions holds
(1) D′′(A′′)·A(4)⊆cA′,for each derivation D:A−→A′′′. (2) Conditions (i) of Lemma 3.1 and Lemma 3.2 are true.
(3) limαlimγlimβhλα, Fγ · aβi = limβlimαlimγhλα, Fγ · aβi, for every bounded nets (λα)inA′′′,(Fγ) inA′′ and every net (aβ) inA.
ThenAis 3 -weakly amenable.
Proof. Suppose that one of conditions (1) or (2) holds, so the two A′′- bimodule structures in parts (II) and (III) on A(5) are equal. We know (A′′)′′′ = (A′′)′ ⊕(A′′)⊥. In other words A(5) = (A′′)′′′ is a direct sum- mand of A′′−submodules of A(5). Let P : (A′′)′′′ −→ (A′′)′ be the projec- tion defined by the above direct sum. Suppose D : A −→ A′′′ is a deriva- tion. Then we can show that P is an A′′-module homomorphism. Thus P◦D′′:A′′−→(A′′′)′′= (A′′)′′′−→(A′′)′ is a derivation. SinceA′′is weakly amenable, there exists θ0∈(A′′)′ such thatP ◦D=δθ0.On the other hand D is the restriction ofP◦D′′to A.ThusD=δθ0.
Now assume that the condition (3) holds, then the twoA′′-bimodule struc- tures in parts (I) and (III) on A(5) are equal. SupposeD : A−→ A′′′ is a
derivation. Then ι∗◦D′′ : A′′ −→ (A′′′)′′ = (A(4))′ −→ (A′′)′ is a deriva- tion. Due to the weak amenability of A′′, there exists θ0 ∈(A′′)′ such that ι∗oD′′=δθ0.For everya∈AandF ∈A′′, we have
hι∗◦D′′(a), Fi=hD′′(a), ι(F)i=hι(F), D(a)i=hD(a), Fi.
SoD is the restriction of ι∗◦D′′ to A. Thus D =δθ0. Therefore Ais 3
-weakly amenable.
By applying Theorem 4.3 we have the following results.
Corollary 4.4. LetAbe a Banach algebra such that one of the conditions (1) up to (4) of Theorem 4.3 holds. If A′′ is weakly amenable, then A is weakly amenable.
Proof. It follows immediately from Theorem 4.3 and [4, Proposition 1.2].
Corollary 4.5. Let A be a Banach algebra such that one of the conditions (1) up to (3) of Theorem 4.3 holds. If A′′ is 3 -weakly amenable, thenAis 3 -weakly amenable.
Proof. By Theorem 4.3 and [4, Proposition 1.2], Ais 3 -weakly amenable.
The following Theorem has been proved in [10].
Theorem 4.6. Let Abe an Arens regular Banach algebra. Suppose that for every continuous derivation D:A′′−→A′′′ and everyF inA′′, D(F)andD are w∗-continuous. IfAis weakly amenable , then so isA′′.
Let one of the conditions of Theorem 4.3 holds. Then we have
Corollary 4.7. Under assumptions of Theorem 4.6, if Ais weakly amenable, then Ais 3-weakly amenable.
Proof. It is an immediate consequence of Theorem 4.3 and Theorem 4.6.
References
[1] R. Arens,The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848.
[2] N. Arikan, Arens regularity and reflexivity, Quart. J. Math. Oxford Ser.
32(1981), 383–388.
[3] H. G. Dales,Banach algebra and Automatic continuity, Oxford University Press, 2000.
[4] H. G. Dales, F. Ghahramani and N. Grønbæk, Derivations into iterated duals of Banach algebras, Studia Math.128(1998), 19–54.
[5] H. G. Dales, A. Rodriguez-Palacios and M. V. Velasco,The second trans- pose of a derivation, J. London Math. Soc. (2)64 (2001),707–721.
[6] J. Duncan and S. A. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A84(1979), 309–325.
[7] M. Eshaghi Gordji,Arens regularity of some bilinear maps, Proyecciones 28(2009), no. 1, 21–26.
[8] M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Math.181(2007), 237–254.
[9] M. Eshaghi Gordji and M. Filali, Weak amenability of the second dual of a Banach algebra, Studia Math.,182, no. 3, (2007), 205–213.
[10] M. Ettefagh, The third dual of a Banach algebra, Studia. Sci. Math.
Hung.,45(1), 1–11 (2008).
[11] F. Ghahramani and J. Laali,Amenability and topological center of the sec- ond duals of Banach algebras, Bull. Austral. Math. Soc.,65(2002),191–
197.
[12] F. Ghahramani, R. J. Loy and G. A. Willis, Amenability and weak amenability of second cojugate Banach algebras, Proc. Amer. Math. Soc.
124(1996), 1489–1497.
[13] A. T.-M. Lau and A. ¨Ulger, Topological centers of certain dual algebras, Trans. Amer. Math. Soc.,348(1996), 1191–1212.
[14] A. R. Medghalchi and T. Yazdanpanah, n-weak amenability and strong double limit property, Bull. Korean. Math. Soc., 42 (2005), no. 2, 359–
367.
[15] R. E. Megginson, An Introduction to Banach Space Theory , Springer- Verlag, New York Berlin Heidelberg, 1998.
[16] A. ¨Ulger, Weakly compact bilinear forms and Arens regularity, Proc.
Amer. Math. Soc.,101(1987), 697–704.
[17] A. ¨Ulger, Arens regularity sometimes implies the RNP, Pacific J. Math., 143(1990), 377–399.
[18] N. J. Young, The irregularity of multiplication in group algebras, Quart.
J. Math. Oxford,24(1973), 59-62.
[19] Yong Zhang, Weak amenability of module extension of Banach algebras, Trans. Amer. Math. Soc., 354, no.10(2002), 4131–4151.
[20] Yong Zhang, Weak amenability of a class of Banach algebras, Canad.
Math. Bull. Vol.,44(4) (2001), 504–508.
A. Bodaghi,
Islamic AZAD University, Garmsar Branch, Garmsar, Iran e-mail: abasalt [email protected] M. Ettefagh,
Islamic AZAD University, Tabriz Branch, Tabriz, Iran.
e-mail: mm [email protected] M. Eshaghi Gordji,
Semnan University,
Department of Mathematics, Semnan, Iran,
e-mail: [email protected] A. R. Medghalchi,
Tarbiat Moallem University,
Department of Mathematical Sciences, Tehran, Iran,
e-mail: a [email protected]
