Approximate Amenability for Banach Modules

U.P.B. Sci. Bull., Series A, Vol. 78, Iss. 2, 2016 ISSN 1223-7027 APPROXIMATE AMENABILITY FOR BANACH MODULES Fatemeh Anousheh1 , Davood Ebrahimi Bagha2 , Abasalt Bodaghi3 For a Banach algebra A, a Banach A-bimodule E and a bounded Banach A-bimodule homomorphism ∆ : E −→ A, the notions of approximate ∆amenability and ∆-contractibility for E are introduced. The general theory is developed and some hereditary properties are given. In analogy with approximate amenability and contractibility for Banach algebras, it is shown that under some mild conditions approximate ∆-amenability and approximate ∆-contractibility are the same properties. Keywords: Banach modules; Module amenability: Module contractibility. MSC2010: 43A07, 46H25. 1. Introduction The concept of amenability for a Banach algebra was introduced by Johnson [9] in 1972, and it has been proved to be of enormous importance in Banach algebra theory (for example, [1], [2], [5] and [10]). The main example in [9] asserts that the group algebra L1 (G) of a locally compact group G is amenable if and only if G is amenable. The definition of an amenable Banach algebra is strong enough to allow for the development of a rich general theory, but still weak enough to include a variety of interesting examples. For example, Johnson’s result fails to be true for discrete semigroups. This failure is partially due to the fact that l1 (S) is equipped with two algebraic structures. It is a Banach algebra and a Banach module over l1 (ES ), where S is a discrete inverse semigroug with the set of idempotents ES . There are many examples of Banach modules which do not have any natural algebra structure. One example is Lp (G) which is a left L1 (G) module, for a locally compact group G [4]. There is one thing in common in all of them, namely the existence of a module homomorphism, from the Banach module to the underlying Banach algebra. This consideration was the motivation to study the concept of module amenability (more precisely ∆-amenability) which is defined for a Banach module E over a Banach algebra A with a given module homomorphism ∆ : E −→ A. This notion was introduced by Ebrahimi Bagha and Amini in [6]. 1 Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran, e-mail: [email protected] 2 Corresponding Author, Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran, e-mail: [email protected] 3 Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran, e-mail: [email protected] 79 80 F. Anousheh, D. Ebrahimi Bagha, A. Bodaghi The concepts of approximate amenability, contractibility and some other related concepts, were introduced and studied in [7] and further developed in [8]. In [7], the authors showed that the corresponding class of approximately amenable (contractible) Banach algebras is larger than that for the classical amenable algebras introduced by Johnson; for the module approximate amenability case refer to [11]. In this paper, we introduce the concepts of approximate ∆-amenability and ∆contractibility, and indicate some basic properties of approximately ∆-amenable Banach modules. We also study the hereditary properties of approximate ∆-amenability for a Banach module. Finally, we provide some examples. 2. Approximate amenability for Banach modules Let A be a Banach algebra and X be a Banach A-bimodule. A derivation from A to X is a linear map D : A −→ X such that D(ab) = D(a) · b + a · D(b) for a, b ∈ A. Also, D is said to be inner if there exists x ∈ X such that D(a) = a·x−x·a for every a ∈ A. In this case, we denote D by adx . Moreover, D is said to be approximately inner if there exists a net (xi ) in X such that for any a ∈ A, D(a) = limi (a · xi − xi · a) = limi adxi (a). A Banach algebra A is called amenable if every continuous derivation D : A −→ X ∗ , where X ∗ is the dual module of X, is inner for every Banach A-bimodule X and A is called approximately amenable if every continuous derivation D : A −→ X ∗ is approximately inner, for all Banach Abimodule X. Throughout this paper, A is a Banach algebra, E is a Banach A-bimodule and ∆ : E −→ A is a bounded Banach A-bimodule homomorphism, that is, a bounded linear map such that for any a ∈ A and x ∈ E, ∆(a · x) = a · ∆(x) and ∆(x · a) = ∆(x) · a. Let X be a Banach A-bimodule. A bounded linear map D : A −→ X is called a module derivation (or more specifically ∆-derivation) if D(∆(a · x)) = a · D(∆(x)) + D(a) · ∆(x), D(∆(x · a)) = D(∆(x)) · a + ∆(x) · D(a), for all a ∈ A and x ∈ E. Also, D is called ∆-inner if there is f ∈ X such that for any x ∈ E D(∆(x)) = ∆(x) · f − f · ∆(x) = adf (∆(x)). An A-bimodule E is called module amenable (or more specifically ∆-amenable) if for each Banach A-bimodule X, all ∆-derivations from A to X ∗ are ∆-inner. It is clear that A is module amenable (with ∆ = id) if and only if it is amenable as a Banach algebra. A (weak) right approximate identity of E is a net (aα ) in A such that for each x ∈ E, ∆(x) · aα − ∆(x) → 0 [∆(x) · aα − ∆(x) → 0 in the weak topology]. The (weak) left and two sided approximate identities are defined similarly. Definition 2.1. An A-bimodule E is called approximately module amenable (approximately ∆-amenale as an A-bimodule) if for each Banach A-bimodule X, all ∆derivations from A to X ∗ are approximately ∆-inner. A ∆-derivation D : A −→ X ∗ Approximate amenability for Banach modules 81 is called approximately ∆-inner if there is a net (fα ) ⊆ X ∗ such that D(∆(x)) = limα (∆(x) · fα − fα · ∆(x)) (x ∈ E). A point ∆-derivation d at a character ϕ of an algebra A is a linear functional d satisfying d(∆(a · x)) = d(a · ∆(x)) = d(a)ϕ(∆(x)) + ϕ(a)d(∆(x)), d(∆(x · a)) = d(∆(x) · a) = d(∆(x))ϕ(a) + ϕ(∆(x))d(a) (x ∈ E, a ∈ A). Proposition 2.1. Suppose that A admits a nonzero continuous point ∆-derivation d at a character ϕ. If ϕ ◦ ∆ ̸= 0, then E is not approximately ∆-amenable. Proof. Let d be a non zero point ∆-derivation at a character ϕ, with ϕ ◦ ∆ ̸= 0. Then the map D : A −→ A∗ ; a 7→ d(a)ϕ is a ∆-derivation. We have D(∆(a · x)) = D(a · ∆(x)) = (d(a)ϕ(∆(x)) + ϕ(a)d(∆(x))ϕ = d(a)ϕ(∆(x))ϕ + ϕ(a)d(∆(x))ϕ = d(a)ϕ · ∆(x) + a · d(∆(x))ϕ for all a ∈ A and x ∈ E. The last equality holds, in fact, for each b ∈ A (d(a)ϕ · ∆(x) + a · d(∆(x))ϕ)(b) = (d(a)ϕ · ∆(x))(b) + (a · d(∆(x))ϕ)(b) = d(a)ϕ(∆(x)b) + d(∆(x))ϕ(ba) = d(a)ϕ(∆(x))ϕ(b) + d(∆(x))ϕ(b)ϕ(a) = d(a)ϕ(∆(x))ϕ(b) + ϕ(a)d(∆(x))ϕ(b) = (d(a)ϕ(∆(x))ϕ + ϕ(a)d(∆(x))ϕ)(b). Suppose the assertion of the proposition is false. Hence, there is a net (fα ) in A∗ such that (D ◦ ∆)(x) = limα (adfα ◦ ∆)(x) for all x ∈ E. Thus (D(∆(x)))(∆(x)) = d(∆(x))ϕ(∆(x)) = lim(adfα (∆(x)))∆(x) α = lim(∆(x) · fα − fα · ∆(x))(∆(x)) α = lim fα ((∆(x))2 − (∆(x))2 ) = 0 α for all x ∈ E. So, d(∆(x))ϕ(∆(x)) = 0. This shows that d ◦ ∆ vanishes off ker(ϕ ◦ ∆). On the other hand, if z ∈ / ker(ϕ ◦ ∆) and x ∈ ker(ϕ ◦ ∆), then 2x = (x + z) + (x − z) with x + z, x − z ∈ / ker(ϕ ◦ ∆). So d ◦ ∆(x) = 0. Thus d ◦ ∆ = 0. Now suppose that d(a) ̸= 0 for some a ∈ A. Hence, for any x ∈ E we have d(∆(a · x)) = d(a · ∆(x)) = d(a)ϕ(∆(x)) + ϕ(a)d(∆(x)). Therefore, d(a)ϕ(∆(x)) = 0. Since d(a) ̸= 0, we have ϕ ◦ ∆ = 0 which is a contradiction.  The proof of the following lemma is similar to the proof of [7, Lemma 2.1], so we do not include it. 82 F. Anousheh, D. Ebrahimi Bagha, A. Bodaghi Lemma 2.1. Suppose that E has a weak left (right) approximate identity, then E has a left (right) approximate identity. Proposition 2.2. Let E be approximately ∆-amenable. Then, E has left and right approximate identities. In particular, ∆(E) · A = A · ∆(E) = ∆(E). Proof. Take A∗∗ with usual left action and zero right action as an A-bimodule. Then, the natural injection A −→ A∗∗ ; a 7→ b a is a ∆-derivation. Thus, there is a ∗∗ [ net (fα ) ⊆ A with ∆(x).fα → ∆(x) for each x ∈ E. Choose the finite sets F ⊆ E, Φ ⊆ A∗ and ϵ > 0. Let H = {ϕ · ∆(x) : x ∈ F, ϕ ∈ Φ}, K = max{∥ψ∥, ∥ϕ∥ : ψ ∈ H, ϕ ∈ Φ}. Similar to the proof of [7, Lemma 2.2], we can show that E has a weak left approximate identity. Now, apply Lemma 2.1.  Proposition 2.3. Suppose that E is approximately ∆-amenable (as an A-bimodule) and ϕ : A −→ B is a continuous epimorphism such that E · ker ϕ = ker ϕ · E = {0}. If E is considered as a B-bimodule via b · x := a · x, x · b := x · a (b ∈ B, x ∈ E) where a ∈ A with b = ϕ(a), then approximate ∆-amenability of E (as an A-bimodule) implies approximate ϕ ◦ ∆-amenability of E (as a B-bimodule). Proof. Suppose that X is a B-bimodule and D : B −→ X ∗ is a ϕ ◦ ∆-derivation. Then X is naturally an A-bimodule via a · x = ϕ(a) · x, x · a = x · ϕ(a) (x ∈ X, a ∈ A). Thus D ◦ ϕ : A −→ X ∗ is a ∆-derivation, so there is a net (fα ) ⊆ X ∗ such that D ◦ ϕ(∆(x)) = lim(∆(x) · fα − fα · ∆(x)) α = lim(ϕ(∆(x)) · fα − fα · ϕ(∆(x))) α for all x ∈ E. This shows that D is approximately ϕ ◦ ∆-inner.  The next corollary is a direct consequece of Proposition 2.3. Corollary 2.1. Let E be approximately ∆-amenable (as an A-bimodule) and J be a closed two-sided ideal of A such that J · E = E · J = {0}. If π : A −→ A/J is the canonical map, then E is an A/J-bimodule which is approximately π ◦ ∆-amenable. Proposition 2.4. Let E and E ′ be Banach A-bimodules with corresponding module homomorphisms ∆ : E −→ A and ∆′ : E ′ −→ A, recpectively. If θ : E −→ E ′ is a bounded module epimorphism such that ∆′ ◦ θ = ∆, then approximate ∆-amenability of E implies approximate ∆′ -amenability of E ′ . Proof. Suppose that D : A −→ X ∗ is a ∆′ -derivation where X is a Banach Abimodule. So D : A −→ X ∗ is also a ∆-derivation because D(∆(a · x)) = D(a · ∆(x)) = D(a · (∆′ ◦ θ)(x)) = D(a · ∆′ (θ(x)) = D(a) · ∆′ (θ(x)) + a · D(∆′ (θ(x)) = D(a) · ∆(x) + a · D(∆(x)) Approximate amenability for Banach modules 83 for all a ∈ A and x ∈ E. Due to the approximate ∆-amenability of E, there is a net (fα ) ⊆ X ∗ such that D(∆(x)) = limα ∆(x) · fα − fα · ∆(x) (x ∈ E). Hence, D(∆′ (θ(x)) = D(∆(x)) = lim(∆(x) · fα − fα · ∆(x)) α ′ = lim(∆ (θ(x)) · fα − fα · ∆′ (θ(x)) α for all x ∈ E. Since θ is surjective, we conclude that D is approximately ∆′ -inner and so E ′ is approximately ∆′ -amenable.  Proposition 2.5. Let J be a closed submodule of E and I be the closed ideal of A generated by ∆(J), q : A −→ A/I and q̃ : E −→ E/J be the corresponding quotient maps. Then, E is approximately ∆-amenable whenever J is ∆|J -amenable ˜ ˜ : E/J −→ A/I is (∆|J : J −→ I) and E/J is approximately ∆-amenable whereas ∆ ˜ the unique A/I module map with ∆ ◦ q̃ = q ◦ ∆ Proof. Let X be a Banach A-bimodule and let D : A −→ X ∗ be a ∆-derivation. Then, D|I : I → X ∗ is a ∆|J -derivation. Since J is ∆|J -amenable, there exists λ1 ∈ X ∗ with D(∆(j)) = adλ1 (∆(j)) (j ∈ J). Replacing D by D − adλ1 , we may suppose that D|∆(J) = 0 so D|I = 0. Set F = I · X + X · I. Then F is a closed A-submodule of X and X/F is clearly a Banach (A/I)-bimodule (indeed X/F is an A-bimodule such that I(X/F ) = (X/F )I = 0). Also, (X/F )∗ ∼ = F⊥ = {f ∈ X ∗ : f |F = 0} is a dual Banach (A/I)-bimodule. For each a ∈ A and b ∈ I, we have a · D(b) = D(ab) = 0 and so D(a) · b = 0. Take x ∈ X. Then, ⟨b · x, D(a)⟩ = ⟨x, D(a) · b⟩ = 0 so D(a)|I·X = 0. Similarly, D(a)|X·I = 0 and so D(a)|F = 0. Thus D(A) ⊆ F ⊥ and the map DI : A/I → F ⊥ , DI (a + I) = D(a) is ˜ ˜ a continuous ∆-derivation. By hypothesis, E/J is approximately ∆-amenable. So, ⊥ ˜ ˜ there exists a net (fα ) ⊆ F with DI (∆(e + J)) = limα adfα (∆(e + J)). For each e ∈ E, we get ˜ + J) = ∆ ˜ ◦ q̃(e) = q ◦ ∆(e) = ∆(e) + I. ∆(e Therefore ˜ + J)) = lim adf (∆(e ˜ + J)) D(∆(e)) = DI (∆(e) + I) = DI (∆(e α α = lim adfα (∆(e) + I) = lim adfα (∆(e)). α α Consequently, D is the sum of a ∆-inner derivation adλ1 and approximately ∆-inner derivation D − adλ1 .  We have the following lemmas which are analogous to [7, Lemma 2.3] and [7, Lemma 2.4], respectively. Since the proofs are similar, we omit them. Lemma 2.2. Let A be a unital Banach algebra with identity e, X be an A-bimodule, and let D : A −→ X ∗ be a ∆-derivation. Then, there is a ∆-derivation D1 : A −→ e · X ∗ · e and η ∈ X ∗ such that (i) ∥η∥ ≤ 2C∥D∥ (where C is a constant depending on X); (ii) D(∆(x)) = D1 (∆(x)) + adη (∆(x)) (x ∈ E). 84 F. Anousheh, D. Ebrahimi Bagha, A. Bodaghi Lemma 2.3. Let A be a unital Banach algebra with identity e and E be approximately ∆-amenable, X be an A-bimodule and D : A −→ X ∗ be a ∆-derivation. Then, there are a net (fα ) ⊂ e · X ∗ · e and η ∈ X ∗ such that (i) ∥η∥ ≤ 2C∥D∥; (ii) D(∆(x)) = adη (∆(x))) + limα (adfα (∆(x)) (x ∈ E). Remark 2.1. In the previous lemma if E is ∆-amenable then there are f ∈ e · X ∗ · e and η ∈ X ∗ such that (1) ∥η∥ ≤ 2C∥D∥; (2) D(∆(x)) = adη (∆(x)) + adf (∆(x)) (x ∈ E). Let A be a non-unital Banach algebra. Then, A♯ = A ⊕ C, the unitization of A, is a unital Banach algebra which contains A as a closed ideal. If E is a Banach A-bimodule and ∆ : E −→ A is an A-bimodule homomorphism, then E is an A♯ -bimodule with the actions (a, λ) · x = a · x + λx, x · (a, λ) = x · a + λx (x ∈ E, λ ∈ C, a ∈ A). It is easy to check that ∆′ : E −→ A♯ is an A♯ -bimodule homomorphism, where for any x ∈ E, ∆′ (x) = ∆(x). Proposition 2.6. E is approximately ∆-amenable (as an A-bimodule) if and only if E is approximately ∆′ -amenable (as an A♯ -bimodule). Proof. Sufficient part: Let D : A♯ −→ X ∗ be a ∆′ -derivation, where X is an A♯ bimodule. Clearly, X is an A-bimoule and D|A : A −→ X ∗ is a ∆-derivation. Since E is approxmiately ∆-amenable as an A-bimodule, there is a net (fα ) ⊂ X ∗ such that D|A (∆(x)) = limα adfα (∆(x)). It follows from Im(∆′ ) = Im(∆) ⊆ A ⊆ A♯ that D(∆′ (x)) = D|A (∆(x)) = limα adfα (∆(x)) = limα adfα (∆′ (x)). So, E is approximately ∆′ -amenable as an A♯ -bimodule. Necessary part: Let D : A −→ X ∗ be a ∆-derivation where X is an Abimodule. Then X is an A♯ -bimodule with the usual actions. Now, D can be extended to D̃ : A♯ → X ∗ by D̃(a, λ) = D(a). Then D̃ is a ∆′ -derivation. In fact D̃(∆′ ((a, λ) · x))) = D̃((a, λ) · ∆′ (x)) = D̃((a · ∆(x) + λ∆(x)) = D̃((a · ∆(x)) + D̃(λ∆(x)) = D(a · ∆(x)) + λD(∆(x)) = D(a) · ∆(x) + a · D(∆(x)) + λD(∆(x)) = D̃(a, λ) · ∆′ (x) + (a, λ) · D̃(∆′ (x)). Then, D̃ is approximately ∆′ -inner, whence D is approximately ∆-inner.  Definition 2.2. A Banach A-bimodule X is called right ∆-essential if for x ∈ X, there are a ∈ ∆(E) and y ∈ X such that x = y · a. The left ∆-essential and (two sided ) ∆-essential modules are defined similarly. Theorem 2.1. Suppose that ∆ has a dense range and E has a bounded approximate identity. Then E is approximately ∆-amenable if and only if for each ∆-essential Banach A-bimodule X, all ∆-derivations from A to X ∗ are approximately ∆-inner. Approximate amenability for Banach modules 85 Proof. Let (aα ) ⊆ A be a bounded approximate identity for E. Let X be a Banach A-bimodule and D : A −→ X ∗ be a ∆-derivation. Consider Tα : X ∗ −→ X ∗ defined by Tα (f ) = aα · f , for all f ∈ X ∗ . Since (aα ) is bounded in A, {Tα } is bounded in B(X ∗ ). Hence, it has a w∗ -cluster point, say T . We may assume that Tα → T in w∗ -topology. For each e ∈ E, x ∈ X, f ∈ X ∗ we have ⟨x · ∆(e), T f ⟩ = lim⟨x · ∆(e), Tα f ⟩ = lim⟨x · ∆(e), aα · f ⟩ α α = lim⟨x · ∆(e)aα , f ⟩ = ⟨x · ∆(e), f ⟩. α X∗ Thus, T − I : −→ (X · ∆(E))⊥ is a bounded projection. Also, the following short exact sequence of Banach A-bimodules is admissible 0 −→ (X · ∆(E))⊥ −→ X ∗ −→ (X · ∆(E))∗ −→ 0. On the other hand, X (X·∆(E)) · ∆(E) = 0. We have X ∗ = (X · ∆(E))∗ ⊕ (X · ∆(E))⊥ . This implies that T D and (I − T )D are ∆-derivations, the latter being ∆-inner. X )∗ and · ∆(E) = 0 that ∆(E) · It follows from (X · ∆(E))⊥ ∼ = ( X (X·∆(E)) ( X )∗ (X·∆(E)) = 0. Since ∆ has dense range, A·( (X·∆(E)) X )∗ = (X·∆(E)) 0 and so A·(X ·∆(E))⊥ = 0. We now consider ∆(E) · (X · ∆(E)) and proceed as before to find that D is the sum of two ∆-inner derivations, plus a derivation mapping into the dual of the ∆-essential module ∆(E) · (X · ∆(E)).  Lemma 2.4. Let A be an approximately amenable Banach algebra. If B is another Banach algebra such that A is an ideal of B and ∆ : A −→ B is the inclusion map, then A is approximately ∆-amenable (as a B-bimodule). Proof. Suppose that X is a B-bimodule and D : B −→ X ∗ is a ∆-derivation. Then, X is also an A-bimodule and D|A : A → X ∗ is a derivation. Since A ia approximately amenable there exists a net (fα ) ⊂ X ∗ such that D|A (a) = limα adfα (a), we have D(∆(a)) = D(a) = D|A (a) = limα adfα (a).  It is shown in [6, Proposition 2.8] that if ∆ has a dense range, then ∆amenability of the A-module E is equivalent to amenability of the Banach algebra A. Also, it is known that the direct sum of two amenable Banach algebras is an amenable Banach algebra. Summing up: Lemma 2.5. Let E be a Banach A-bimodule and F be a Banach B-bimodule. If α : E −→ A and β : F −→ B are bounded Banach A-bimodule homomorphism and B-bimodule homomorphism with dense ranges, respectively, then E ⊕ F is a Banach A⊕B-bimodule with the natural action (x, y)·(a, b) = (xa, yb), (a, b)·(x, y) = (ax, by) for (a, b) ∈ A ⊕ B, (x, y) ∈ E ⊕ F. Also, (α ⊕ β) : E ⊕ F −→ A ⊕ B is defined by (α ⊕ β)(x, y) = (α(x), β(y)) is a bounded Banach A ⊕ B-bimodule homomorphism. In particular, if E is α-amenable (as an A-bimodule) and F is β-amenable (as a B-bimodule), then E ⊕ F is α ⊕ β-amenable as an A ⊕ B-bimodule. 86 F. Anousheh, D. Ebrahimi Bagha, A. Bodaghi Definition 2.3. A Banach A-bimodule E is called approximately ∆-contractible if for any Banach A-bimodule X every ∆-derivation D : A −→ X is approximately ∆-inner. Proposition 2.7. Let E be a Banach A-bimodule, and ∆ : E −→ A be a bounded Banach A-bimodule homomorphism. Let ∆ ⊕ ∆ : E ⊕ E −→ A ⊕ A be defined by (∆ ⊕ ∆)(x, y) = (∆(x), ∆(y)). If E ⊕ E is approximately ∆ ⊕ ∆ contractible (as an A ⊕ A bimodule), then A has an approximate identity. Proof. Let A be an A ⊕ A-bimodule with the following actions (a, b) · x = ax, x · (a, b) = xb (x ∈ A, a, b ∈ A). Define D : A ⊕ A → A by D(a, b) = a − b. Then, D is a derivation and hence a (∆ ⊕ ∆)-derivation. So, there is a net (ai ) ⊂ A such that D(∆(x), ∆(y)) = ∆(x) − ∆(y) = lim(∆(x), ∆(y))ai − ai (∆(x), ∆(y)) i = lim ∆(x)ai − ai ∆(y). i for all x, y ∈ E. Therefore, limi ∆(x)ai = ∆(x) and limi ai ∆(y) = ∆(y).  Theorem 2.2. Let E be a Banach A-bimodule and ∆ : E −→ A be a bounded Banach A-bimodule homomorphism. Suppose that ∆(E) is norm closed in A and ∆(E) has a bounded approximate identity. Then, E is approximately ∆-amenable if and only if it is approximately ∆-contractible. Proof. The sufficient part is clear. For the necessary part, assume that E is approximately ∆-amenable (as an A-bimodule). We claim that ∆(E) is an approximately amenable Banach algebra. So, by [8, Theorem 2.1] ∆(E) is an approximately contractible Banach algebra. Now, let D : A −→ X be a ∆-derivation for some Banach A-bimoule X. Then, D |∆(E) : ∆(E) −→ X is a derivation. Since ∆(E) is approximately contractible, D |∆(E) is approximately inner and hence D is approximately ∆-inner. Therefore E is approximately ∆-contractible. To prove the claim suppose that D : ∆(E) −→ X ∗ is a derivation for a Banach ∆(E)-bimodule X. Since ∆(E) is a norm closed ideal in A and ∆(E) has a bounded approximate identity, by [12, Proposition 2.1.6] we can extend D to a derivation D̄ : A −→ X ∗ . Due to the approximate ∆-amenability of E (as an A-bimodule), D̄ is approximately ∆-inner and thus D is approximately inner.  ⊗ For a Banach algebra A, let π : A c A −→ A be the canonical map, that is, π(a ⊗ b) = ab for any a, b ∈ A. Theorem 2.3. Let A be a unital Banach algebra with identity e, E be a Banach A-bimodule and ∆ : E −→ A be a bounded Banach A-bimodule homomorphism. Consider the following assertions: (i) E is approximately ∆-amenable as a Banach A-bimodule; ⊗ (ii) There is a net (Mv ) ⊆ (A c A)∗∗ such that for each x ∈ E, ∆(x) · Mv − Mv · ∆(x) −→ 0 and π ∗∗ (Mv ) −→ e; Approximate amenability for Banach modules 87 ⊗ (iii) There is a net (Mv′ ) ⊆ (A c A)∗∗ such that for each x ∈ E, ∆(x) · Mv′ − Mv′ · ∆(x) −→ 0 and π ∗∗ (Mv′ ) = e for every v. Then (i)⇒(iii)⇒ (ii), and in the case that ∆ has dense range, (ii) implies (i). ⊗ Proof. (i)⇒ (iii). Let D : A −→ (A c A)∗∗ be defined by D(a) = a · u − u · a for any a ∈ A, where u = e ⊗ e. Then, Im(D) ⊆ ker(π ∗∗ ) ∼ = (ker π)∗∗ . Since E is approximately ∆-amenable, there is a net (tv ) ⊂ ker(π ∗∗ ) such that for any x ∈ E, D(∆(x)) = limv ∆(x) · tv − tv · ∆(x). Let Mv′ = u − tv . So, π ∗∗ (Mv′ ) = e and ∆(x) · Mv′ − Mv′ · ∆(x) = ∆(x) · u − u · ∆(x) − (∆(x) · tv − tv · ∆(x)) −→ 0. (iii)⇒(ii) It is trivial. (ii)⇒(i). Let D : A −→ X ∗ be a ∆-derivation for some ∆-essential Banach ⊗ A-bimodule X. For each x ∈ X, let µx ∈ (A c A)∗ defined by µx (a ⊗ b) = ⟨aD(b), x⟩ for all a, b ∈ A. Now, for each v, put fv (x) = Mv (µx ) for any x ∈ X. We show that for any y ∈ E, D(∆(y)) = limv adfv (∆(y)) and hence by Theorem 2.1 E is ⊗ approximately ∆-amenable. It is easy to check that for x ∈ X, m ∈ A c A µx·∆(y)−∆(y)·x (m) = (µx · ∆(y) − ∆(y) · µx )(m) + (π(m)Da)(x). ⊗ There is a net (mαv ) in A c A such that Mv = w∗ − limα mαv . So, (∆(y) · fv − fv · ∆(y))(x) = fv (x · ∆(y) − ∆(y) · x) = Mv (µ∆(y)·x−x·∆(y) ) = lim(µ∆(y)·x−x·∆(y) )(mαv ) α = Mv (µx · ∆(y) − ∆(y) · µx ) + lim(π(mαv )D(∆(y))(x) α = (∆(y) · Mv − Mv · ∆(y))(µx ) + (π ∗∗ (Mv )D(∆(y)))(x). Thus ∥(∆(y) · fv − fv · ∆(y))(x) − D(∆(y))(x)∥ 6 ∥∆(y) · Mv − Mv · ∆(y)∥ · ∥D∥ · ∥x∥ + ∥x∥ · ∥π ∗∗ (Mv ) − e∥ · ∥D(∆(y))∥ Therefore, D(∆(y)) = limv adfv (∆(y)) as required.  Given a sequence {An } of Banach algebras, define their l∞ direct sum as l∞ (An ) = {(xn ) : xn ∈ An , ∥(xn )∥ = sup ∥xn ∥ < ∞} and c0 (An ) = {(xn ) ∈ l∞ (An ) : ∥xn ∥ → 0}. We finish the paper by four examples. Example 2.1. We present an approximately ∆-amenable module which is not a ∆-amenable Banach module. Consider the algebra Mn of n × n matrices with norm ∑ ∥aij ∥2 = ( i,j |aij |2 )1/2 . Then ∥AB∥2 ≤ ∥A∥2 ∥B∥2 for any A, B ∈ Mn . One should ∑ remember that the duality between Mn and Mn∗ is, ⟨A, E⟩ = i,j aij eij . Also, the 88 F. Anousheh, D. Ebrahimi Bagha, A. Bodaghi [ ] 0 −1 map Mn → Mn∗ : A 7−→ A, is isometric. Let as an element of M2∗ . 1 0 [ ] 0 −Pn Inductively, define Pn+1 = so that Pn ∈ M2n . Let An = M2♯n . Pn 0 By [7, Example 6.2], c0 (An ) is an approximately amenable Banach algebra which is not amenable, since c0 (An ) is an ideal of l∞ (An ), by Theorem 2.1 c0 (An ) is ∞ approximately ∆-amenable as an l∞ (An )-bimodule, where ∆ : c0 (An ) −→ ( l (An ))is ad Pn (xn ) the inclusion map. Now we define D : l∞ (An ) −→ l1 (A∗n ) by D((xn )) = . n2 As in [7, Example 6.2] D cannot be ∆-inner. Thus, c0 (An ) is not ∆-amenable as an l∞ (An )-bimodule. Example 2.2. Let A be an approximately amenable Banach algebra and let π : b → A be the canonical map. Then A⊗A b A⊗A is approximately π-amenable (as an A-bimodule). It is easy to see that π is a Banach A-bimodule homomorphism. Since A is an approximately amenable Banach algebra, A has left approximate identity. Therefore, π has a dense range. Let D : A −→ X ∗ be a π-derivation where X is a Banach A-bimodule. Since π has a dense range, D is a derivation. Due to the approximate amenability of A an Banach algebra, there exists a net (fα ) ⊂ X ∗ such b that D(a) = limα adfα (a), So D(π(x)) = limα adfα (π(x)) for all x ∈ A⊗A. Example 2.3. Let G be a locally compact group. We know that L1 (G) is a closed two sided ideal in M (G). We can consider L1 (G) as a Banach M (G)-bimodule. Let i : L1 (G) −→ M (G) be the inclusion map. If G is a non discrete amenable group then M (G) is not an approximately amenable Banach algebra [7]. Let D : M (G) −→ X ∗ be an i-derivation where X is a Banach M (G)-bimodule. Then, X is also an L1 (G)bimodule and D|L1 (G) : L1 (G) −→ X ∗ is a derivation. Since G is amenable, D|L1 (G) is inner and hence D|L1 (G) is an approximately inner derivation. Consequently, L1 (G) is an approximately i-amenable M (G)-bimodule. Example 2.4. Let G be an abelian compact group. Then, Lp (G) (1 < p < ∞) is a Banach L1 (G)-bimodule. If 1/p + 1/q = 1 and f ∈ Lq (G), then define ∆f : Lp (G) → L1 (G) by ∆f (g) = g ∗ f . Since G is an abelian compact group, ∆f has dense range. If G is amenable so L1 (G) is an amenable Banach algebra and so Lp (G) is ∆f -amenable. Therefore, Lp (G) is approximately ∆f -amenable. The idea of the next example is motivated by [7, Example 6.1]. Example 2.5. For each n ∈ N , let An be a unital Banach algebra with identity en . Let Mn be an An -bimodule such that there exists kn > 0 such that for each x ∈ Mn and a ∈ An , we have ∥a · x∥ ≤ kn ∥a∥∥x∥, ∥x · a∥ ≤ kn ∥x∥∥a∥. Let ∆n : Mn −→ An be a bounded Banach An -bimodule homomorphism with dense range. Suppose that Mn is ∆n amenable as an An -bimodule. Let M = c0 (Mn ) and A = c0 (An ). If sup{kn : n ∈ N } < ∞, then M is a Banach A-bimodule. Consider the mapping ∆ : M −→ A defined through (∆(mn )) = (∆n (mn )) and sup{∥∆n ∥ : n ∈ N } < ∞. Approximate amenability for Banach modules 89 Then ∆ is a bounded Banach A-bimodule homomorphism and M is approximately ∆-amenable (as an A-bimodule). It is easy to see that M is a Banach A-bimodule and ∆ is a bounded Banach A-bimodule homomorphism. Let X be an A-bimodule and D : A −→ X ∗ be a ∆-derivation. Put Bk = {(xn ) ∈ c0 (An ) : xn = 0 for n > k} and Ck = {(mn ) ∈ c0 (Mn ) : mn = 0 for n > k}. Set En = (e1 , e2 , e3 , ..., en , 0, ...). Then, (En ) is a central approximate identity for A. Restricting D to some Bn we have a ∆|Cn -derivation Dn : Bn −→ X ∗ . Since Bn is unital, by Lemma 2.5 and Remark 2.1, there exists fn ∈ En · X ∗ · En and ηn ∈ X ∗ (∥ηn ∥ 6 2C∥D∥) such that Dn (∆|Cn (x)) = adfn (∆|Cn (x)) + adηn (∆|Cn (x)), for any x ∈ Cn . Note that for each x ∈ M , ∥adηn (En ∆(x) − ∆(x))∥ → 0, because (En ) is an approximate identity and (ηn ) is bounded. Since fn ∈ En · X ∗ · En , we have aEn · fn = a · fn and fn · En a = fn · a for any a ∈ A. 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