5_201_Gordji.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 3, 2009, (361-371) ISSN 1307-5543 – www.ejpam.com Module Extension Banach Algebras and (σ,τ)-amenability M. Eshaghi Gordji1∗ and A. Niyazi Motlagh2 1 Department of mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran. 2 Department of Mathematics, Ferdowsi University, P. O. Box 1159, Mashhad 91775, Iran Abstract. In this paper among other things we find some necessary and sufficient conditions for a Banach algebra A , to be (σ,τ)-amenable, where σ and τ are continuous homomor- phisms onA . 2000 Mathematics Subject Classifications: Primary 46H25; Secondary 47B47 Key Words and Phrases: (σ,τ)−derivation; Arens product; approximate identity 1. Introduction. Let A be a Banach algebra and X be a Banach A -bimodule, that X is both a Banach space and an algebraicA -bimodule, and the module operations (a, x) 7→ ax and (a, x) 7→ xa from A ×X into X are (jointly) continuous. Then X ∗ is also a BanachA -bimodule under the following module actions: (a · f )(x) = f (xa), ∗Corresponding author. Email addresses: madjid.eshaghi�gmail. om (M. Gordji), ab − ni40�stu-mail.um.a .ir andniazimotlagh�gmail. om (A. Motlagh) http://www.ejpam.com 361 c© 2009 EJPAM All rights reserved. M. Gordji and A. Motlagh / Eur. J. Pure Appl. Math, 2 (2009), (361-371) 362 ( f · a)(x) = f (ax), a ∈A , x ∈ X , f ∈X ∗. Let A be a Banach algebra. Given f ∈ A ∗ and F ∈ A ∗∗, then F f and f F are defined inA ∗ by the following formulae F f (a) = F( f · a), f F(a) = F(a · f ) (a ∈A ). Next, for F, G ∈A ∗∗, FG is defined inA ∗∗ by the formulae (FG)( f ) = F(G f ), this product is called first Arens product onA ∗∗ andA ∗∗ with the first Arens product is a Banach algebra. Let A be a Banach algebra and X be a Banach A -bimodule. The Banach space X ∗∗ is a BanachA ∗∗-bimodule under following actions F · G = w∗ − lim i lim j ai x j, G · F = w∗ − lim j lim i x jai where F = w∗ − limi ai, G = w∗ − lim j x j, (ai) is a net inA , (x j) and is a net in X . Suppose that ϕ : A → B is a Banach algebra homomorphism. The Banach algebraB is considered as a BanachA - bimodule by the following module actions a · b = ϕ(a)b, b · a = bϕ(a) (a ∈A , b ∈B) we denoteBϕ the aboveA -bimodule. Let A be a Banach algebra and σ,τ be continuous homomorphisms on A . Sup- pose that X is a Banach A -bimodule. A linear mapping d : A → X is called a (σ,τ)-derivation if d(ab) = d(a)σ(b) +τ(a)d(b) (a, b ∈ A). For example every ordinary derivation of an algebraA into an A -bimodule X is an (idA , idA )-derivation, where idA is the identity mapping on the algebraA . M. Gordji and A. Motlagh / Eur. J. Pure Appl. Math, 2 (2009), (361-371) 363 A linear mapping d : A −→ X is called (σ,τ)-inner derivation if there exists x ∈ X such that d(a) = τ(a)x − xσ(a) (a ∈A ). See also [3–6]. We denote the set of continuous (σ,τ)-derivations fromA intoX by Z1 (σ,τ) (A ,X ) and the set of inner (σ,τ)-derivations by B1 (σ,τ) (A ,X ). we define the space H1 (σ,τ) (A ,X ) as the quotient space Z1 (σ,τ) (A ,X )/B1 (σ,τ) (A ,X ). The space H1 (σ,τ) (A ,X ) is called the first (σ,τ)-cohomology group ofA with coefficients inX . A is called (σ,τ)-amenable if H1 (σ,τ) (A ,X ∗) = {0}, for each Banach A -bimodule X . Let A be a Banach algebra and let X be a Banach A -bimodule. Define A ⊕1X by actions: (a, x) + (b, y) = (a+ b, x + y) a(b, x) = (ab, ax) , (b, x)a = (ba, xa) (a, x)(b, y) = (ab, a y + x b), for every a, b ∈A and x , y ∈ X . It is clearA ⊕1X is a Banach algebra with the following norm: ‖(a, x)‖ = ‖a‖+ ‖x‖. This Banach algebra is called module extension Banach algebra. We use some ideas and terminology of [2] to investigate (σ,τ)-amenability of Banach algebras. 2. (σ,τ)-amenability of Banach Algebras. Let A be a Banach algebra and let σ,τ be continuous homomorphisms on A . Suppose that X is a Banach A -bimodule. Then X is a Banach A -bimodule by the following module actions: a · x = τ(a)b, x · a = bσ(a) (a ∈A , x ∈ X ). M. Gordji and A. Motlagh / Eur. J. Pure Appl. Math, 2 (2009), (361-371) 364 We denote X(σ,τ) for this A -bimodule. It is easy to check that (X(σ,τ)) ∗ = X ∗ (τ,σ) , and that every (σ,τ)-derivation from A into X is a derivation from A into X(σ,τ). Thus we can show that A is amenable, if and only if A is (σ,τ)-amenable, for each σ,τ ∈ Hom(A ). First we give the following examples for (σ,τ)-amenability of Banach algebras. Example 2.1. It is easy to see that ℓ1 is a Banach algebra equipped with the following product [7] a · b = a(1)b (a, b ∈ ℓ1), and ℓ1 has a left identity e defined by e(n) =    1 i f n = 1 0 i f n 6= 1. The dual space (ℓ1)∗ = ℓ∞ is a ℓ1-bimodule via the ordinary actions as follows a · f = f (a)e, f · a = a(1) f (a ∈ ℓ1, f ∈ ℓ∞), where e is regarded as an element of ℓ∞. Next let σ : ℓ1 −→ ℓ1 be a bounded homomorphism. We have a(1)σ(b) = σ(a · b) = σ(a) ·σ(b) = σ(a)(1)σ(b) and so σ(b)(a(1) −σ(a)(1)) = 0 for all a, b ∈ N. Since σ 6= 0, we have � σ(a) � (1) = a(1) (a ∈ ℓ1) (2.1) In [5] has been shown that ℓ1 is (σ,τ)-weakly amenable for all homomorphisms σ,τ but for some homomorphisms σ and τ it is not (σ,τ)-amenable. In the following we prove if the Banach algebra ℓ1 is (σ,τ)-amenable, then τ(a) = a(1)c where c(1) = 1. M. Gordji and A. Motlagh / Eur. J. Pure Appl. Math, 2 (2009), (361-371) 365 Let B = ℓ1 by product a • b = a(2)b. Then B is a Banach algebra and for each bounded homomorphism ψ : B −→ B we have � ψ(a) � (2) = a(2). Let a ∈ ℓ1 define a′ ∈ ℓ1 by a′ = � a(2), a(1), a(3), · · · � . Let ϕ : ℓ1 −→ B defined by ϕ(a) = a′. It is clear that ϕ is a homomorphism. Consider the Banach ℓ1-bimodule Bϕ under actions a ◦ b = ϕ(a) • b = a′ • b = a′(2)b = a(1)b and b ◦ a = b •ϕ(a) = b • a′ = b(2)a′ for each a ∈ ℓ1, b ∈Bϕ. Let D : ℓ1 −→B∗ ϕ be a bounded (σ,τ)-derivation. We have � D(a · b) � (c) = D(a)σ(b)(c) +τ(a)D(b)(c) a(1)D(b)(c) = D(a)(σ(b) ◦ c) + D(b)(c ◦ τ(a)) a(1)D(b)(c) = b(1)D(a)(c) + c(2)D(b)(τ(a)) for all a, b ∈ ℓ1 and c ∈ Bϕ . By taking a = b we obtain D(a)(τ(a)) = 0. Also by taking c ∈ Bϕ such that c(2) = 0 we can conclude a(1)D(b) = b(1)D(a). If ℓ1 is (σ,τ)-amenable, then there exists f ∈ B∗ ϕ such that D = D f is a (σ,τ)-inner derivation. So we have a(1)D f (b) = b(1)D f (a) a(1) f (b(1)c − c(2)τ(b)) = b(1) f (a(1)c − c(2)τ(a)) for all a, b ∈ ℓ1 and c ∈ Bϕ . Then f (b(1)c(2)τ(a)− a(1)c(2)τ(b)) = 0. Since f ∈ B∗ ϕ is arbitrary, immediately is conclude a(1)τ(b) = b(1)τ(a). By taking b = e we have τ(a) = a(1)τ(e), where τ(e)(1) = 1. So we have the following result. Corollary 2.1. Let σ,τ be two continuous homomorphisms on ℓ1 (by above product). If ℓ1 is (σ,τ)-amenable then there is c ∈ ℓ1 such that τ(a) = a(1)c, and c(1) = 1. M. Gordji and A. Motlagh / Eur. J. Pure Appl. Math, 2 (2009), (361-371) 366 Example 2.2. LetA be a Banach algebra. ThenA has a bounded approximate identity if and only ifA is (id, 0) and (0, id)-amenable. Corollary 2.2. LetA be a C∗−algebra orA = L1(G) for a locally compact topological group G. ThenA is (id, 0) and (0, id)-amenable. Let T : A → B be a continuous linear map between Banach algebras. Two continuous linear maps T ′ : B∗ → A ∗ and T ′′ : A ∗∗ → B∗∗ are known, that are defined by the following formula � T ′( f ) � (a) = f � T (a) � , � T ′′(G) � ( f ) = G � T ′( f ) � where a ∈A , f ∈B∗ and G ∈A ∗∗. Lemma 2.1. Let A be a Banach algebra, X be a Banach A -bimodule, and let σ and τ be two continuous homomorphisms on A . Suppose that D : A −→ X is (σ,τ)- derivation. Then D′′ :A ∗∗ −→X ∗∗ is a (σ′′,τ′′)-derivation. Proof. Let F, G ∈ A ∗∗ and let F = w∗ − limα aα, G = w∗ − limβ bβ in A ∗∗, where (aα), (bβ) are nets inA with ||aα|| ≤ ||F ||, ||bβ || ≤ ||G||. Then D′′(FG) = D′′ � w∗ − lim α w∗ − lim β aαbβ � = w∗ − lim α w∗ − lim β D′′(aαbβ) = w∗ − lim α w∗ − lim β � τ(aα)D(bβ) + D(aα)σ(bβ ) � = τ′′(F)D′′(G) + D′′(F)σ′′(G) and so D′′ is a (σ′′,τ′′)-derivation. Now we are ready to state some equivalent conditions by (σ,τ)-amenability of Banach algebras. M. Gordji and A. Motlagh / Eur. J. Pure Appl. Math, 2 (2009), (361-371) 367 Theorem 2.1. Let σ and τ be two continuous homomorphisms on Banach algebra A . The following statements are equivalent: 1. A is (σ,τ)-amenable. 2. For each Banach algebraB and every homomorphism ϕ :A −→B , H1 (σ,τ) (A ,B∗ ϕ ) = 0. 3. For each Banach algebra B and every injective homomorphism ϕ : A −→ B , H1 (σ,τ) (A ,B∗ ϕ ) = 0. 4. For each Banach algebra B and every injective homomorphism ϕ :A −→ B , if d :A −→Bϕ ∗ is a (σ,τ)-derivation satisfies (d(a))(ϕ(b)) + (d(b))(ϕ(a)) = 0 (a, b ∈A ), then d is (σ,τ)-inner derivation. Proof. Clearly (1)⇒ (2)⇒ (3)⇒ (4). It is sufficient to show that (4)⇒ (1). LetX be a BanachA -bimodule and D :A −→X ∗ be a (σ,τ)-derivation. SetB =A⊕1X and define injective homomorphism ϕ : A −→ B by ϕ(a) = (a, 0) and so we can assume that A is a subalgebra of B . Define d :A −→B∗ ϕ by d(a) = (0, D(a)). The map d is (σ,τ)-derivation, since d(ab) = (0, D(ab)) = (0, D(a)σ(b) +τ(a)D(b)) = (0, D(a))(0,σ(b)) + (0,τ(a))(0, D(b)) = d(a)ϕ(σ(b)) +ϕ(τ(a))d(b) = d(a) ·σ(b) +τ(a) · d(b) (a, b ∈A ). Since (d(a))(ϕ(b)) + (d(b))(ϕ(a)) = (0, D(a))((b, 0)) + (0, D(b))((a, 0)) = 0, we have (d(a))(ϕ(b)) + (d(b))(ϕ(a)) = 0. M. Gordji and A. Motlagh / Eur. J. Pure Appl. Math, 2 (2009), (361-371) 368 It follows from our assumption that d is a (σ,τ)-inner derivation. Hence there are f ∈A ∗ and g ∈ X ∗ such that (0, D(a)) = d(a) = (σ(a), 0)( f , g)− ( f , g)(τ(a), 0) = (σ(a) f − f τ(a),σ(a)g − gτ(a)). Thus D(a) = σ(a)g − gτ(a), hence D is (σ,τ)-inner derivation. Definition 2.1. LetA be a Banach algebra and σ be a continuous homomorphisms on A . The Banach algebra A is called approximately σ-contractible, if for each Banach A -bimodule X and σ-derivation D : A −→ X , there exists a bounded net (xα) ⊆ X such that D(a) = lim α � σ(a)xα − xασ(a) � (a ∈A ). In the following theorem we follow the structure of Proposition 2.8.59 [1]. Theorem 2.2. LetA be a Banach algebra and σ be a bounded homomorphism on A . Then the following assertion are equivalent: 1. A is σ-amenable. 2. For everyA -bimodule X , H1 (σ,σ) (A ,X ∗∗) = 0 3. A is approximately σ-contractible. Proof. (1)⇒ (2) is trivially. (2)⇒ (3): Let D :A −→ X be a σ-derivation from A into A -bimodule X and let JX : X −→ X ∗∗ be the canonical embedding, then for each a, b ∈A we have eD(ab) = (JX ◦ D)(ab) = JX � σ(a)D(b) + D(a)σ(b) � M. Gordji and A. Motlagh / Eur. J. Pure Appl. Math, 2 (2009), (361-371) 369 = σ(a)eD(b) + eD(a)σ(b). Thus eD is a σ-derivation. Then by (2) there exists Λ ∈ X ∗∗ such that eD(a) = σ(a)Λ − Λσ(a) (a ∈ A ). Set m = ||Λ||,U = X[m]. Then Λ ∈ JX (U ) w∗ . Let a1, a2, a3, . . . , an ∈A , then V = Πn j=1 � σ(a j)U −Uσ(a j) � is a convex subset of X (n) and (D(a1), D(a2), . . . , D(an)) ∈ V weak . Thus for each finite subset F ofA , and ǫ > 0, there exists x(F,ǫ) ∈U such that ||D(a)− (σ(a)x(F,ǫ) − x(F,ǫ)σ(a))|| < ǫ (a ∈ F). The family of such pairs (F,ǫ) is a directed if order ≤ given by (F1,ǫ1)≤ (F2,ǫ2)⇔ F1 ⊆ F2,ǫ1 ≤ ǫ2. Also we have D(a) = lim (F,ǫ) � σ(a)x(F,ǫ) − x(F,ǫ)σ(a) � . (3)⇒ (1): Let D : A −→ X ∗ be a σ-derivation. Then there exists a net (x ′ α ) ⊆ X ∗ such that D(a) = limα � σ(a)x ′ α − x ′ α σ(a) � (a ∈ A ). By passing to a subnet we may assume that w∗ − lim x ′ α = x ′ in X ∗ and then D(a) = σ(a)x ′ − x ′σ(a). Thus A is σ-amenable. Theorem 2.3. Let A be a Banach algebra and σ be a continuous homomorphism on A . IfA ∗∗ is σ′′-amenable, thenA is σ-amenable. Proof. Let X be a Banach A -bimodule, and D : A −→ X ∗∗ be a σ-derivation. Then by Lemma 2.1, D′′ : A ∗∗ −→ X ∗∗∗∗ is a σ′′-derivation. Since A ∗∗ is σ′′- amenable, then there exists x (4) ∈ X ∗∗∗∗ such that D′′(a′′) = σ′′(a′′)x (4)− x (4)σ′′(a′′), (a′′ ∈A ∗∗). We haveX ∗∗∗∗ =X ∗∗⊕(X ∗)⊥ (asA ∗∗-bimodules). Let P :X ∗∗∗∗ −→X ∗∗ be the natural projection. Then for each a ∈ A , we have D(a) = σ(a)P(x (4)) − P(x (4))σ(a), and so D ∈ N 1 (σ,σ) (A ,X ∗∗). Thus by above theorem,A is σ-amenable. REFERENCES 370 In the following we fined an easy equivalent condition for σ-amenability of a Banach algebra. Proposition 2.1. LetA be a Banach algebra and let σ be a continuous homomorphism on A . Then A is a σ-amenable if and only if for every Banach algebra B and every injective homomorphism ϕ :A −→B , H1 (σ,σ) (A , B∗∗ ϕ ) = 0. Proof. One side is clear, so we prove the other side. Let X be a Banach A - bimodule and D :A −→X ∗∗ be a σ-derivation. If φ :A −→A ⊕1X is defined by ϕ(a) = (a, 0). Then ϕ is injective and ϕ∗∗ :A ∗∗ −→ (A⊕1X ) ∗∗ the second transpose of ϕ is a Banach algebra homomorphism and ((A ⊕1 X )ϕ) ∗∗ ≃ (A ∗∗ ⊕1 X ∗∗)ϕ∗∗ as A ∗∗-bimodules. Then H1 (σ,σ) (A , (A ∗∗⊕1X ∗∗)ϕ∗∗) = H1 (σ,σ) (A , ((A ⊕1X )ϕ) ∗∗) = {0}. (2.2) Now we define D1 : A −→ A ∗∗ ⊕1 X ∗∗ by D1(a) = (0, D(a)). For a, b ∈ A we have D1(ab) = D1(a)ϕ ∗∗(bb) + ϕ∗∗(ba)D1(b). Thus D1 is a σ-derivation from A into (A ∗∗ ⊕1X ∗∗)ϕ∗∗ . By (2.2), D1 is σ-inner. Therefore there exist a′′ ∈ A ∗∗, x ′′ ∈ X ∗∗ such that (0, D(a)) = D1(a) = (a ′′, x ′′)(0,σ(a))− (0,σ(a))(a′′, x ′′), Thus D is σ-inner. Therefore H1 (σ,σ) (A ,X ∗∗) = 0, and by Theorem 2.2, A is σ- amenable. ACKNOWLEDGEMENTS The authors would like to thank Professor M. S. Moslehian for his useful comments. References [1] H. G. Dales, Banach algebra and Automatic continuity, Oxford university Press, 2001. REFERENCES 371 [2] M. Eshaghi Gordji, Homomorphisms, Amenability and weak amenability of Banach alge- bras, Vietnam J. Math. 36 (2008), no. 3, 253–260. [3] M. Mirzavaziri, M. S. Moslehian, σ-derivations in Banach algebras, Bull. Iranian Math. Soc. 32 (2006), no. 1, 65–78 [4] M. S. Moslehian, Approximate (σ− τ)-contractibility, Nonlinear Funct. Anal. Appl., 11 (2006), no. 5, 805–813. [5] M. S. Moslehian and A. N. Motlagh, (σ,τ)-amenability of Banach algebras, preprint. [6] M. Mirzavaziri and M. S. Moslehian, Automatic continuity of σ-derivations in C∗- algebras, Proc. Amer. Math. Soc., 11 (2006), no. 5, 805–813. [7] Yong Zhang, Weak Amenability of a Class of Banach Algebras, Canada. Math. Bull. Vol. 44(4), 2001 pp.504-508