() CUBO A Mathematical Journal Vol.13, No¯ 02, (127–137). June 2011 Module amenability for Banach modules D. Ebrahimi Bagha Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran. email: d.ebrahimibagha@iauctb.ac.ir and M. Amini Department of Mathematics, Tarbiat Modares University, P.O.Box 14115-175, Tehran, Iran. email: mamini@modares.ac.ir ABSTRACT We study the module amenability of Banach modules. This is a natural generalization of Johnson’s amenability of Banach algebras. As an example we show that for a discrete abelian group G, ℓp(G) is amenable as an ℓ1(G)-module if and only if G is amenable, where ℓ1(G) is a Banach algebra with pointwise multiplication. RESUMEN Se estudia el módulo de receptividad de los módulos de Banach. Esta es una general- ización natural de la receptividad de Johnson de las álgebras de Banach. Como ejemplo se muestra que para un grupo abeliano discreto G ℓp(G) es receptivo como un G ℓp(G)- módulo, si y sólo si G es receptivo, donde ℓ1(G) es un álgebra de Banach con producto punto. 128 D. Ebrahimi Bagha, M. Amini CUBO 13, 2 (2011) Keywords and phrases: Banach modules, module amenability, weak module amenability, semi- group algebra, inverse semigroup. Mathematics Subject Classification: 43A07, 46H25 1. Introduction The concept of amenability for Banach algebras was introduced by B.E. Johnson in [J]. The main example in [J] asserts that the group algebra L1(G) of a locally compact group G is amenable if and only if G is amenable. This is far from true for semigroups. If S is a discrete inverse semigroup with the set of idempotents ES, ℓ 1(S) is amenable if and only if ES is finite and all the maximal subgroups of S are amenable [DN]. For an arbitrary discrete semigroup S, ℓ1(S) is amenable if and only if the minimum ideal of S exists and is an amenable group and S has a principal series whose corresponding quotients are regular Rees matrix semigroups of special form [DLS, 10.12].This failure is partly due to the fact that ℓ1(S) is equipped with two (related) algebraic structures. It is a Banach algebra and a Banach module over ℓ1(ES). This consideration was the motivation of the second named author to study the concept of module amenability for Banach algebras which have an extra Banach module structure (with compatible actions) in [A]. In particular it is shown in [A] that for an inverse semigroup S, ℓ1(S) is module amenable as a Banach module over ℓ1(ES) if and only if S is amenable. The authors introduced the concept of weak module amenability in [AE] and showed that for a commutative inverse semigroup S, ℓ1(S) is always weak module amenable as a Banach module over ℓ1(ES). The present paper investigates module amenability from a different angle. There are many examples of Banach modules which do not have any natural algebra structure. One example is Lp(G) which is a left Banach L1(G)-module, for a locally compact group G [D, 3.3.19]. As another example of this sort, one may consider a Banach algebraic bundle over a locally compact group G [FD]. Then the fibers on elements of G are Banach modules over the fiber on the identity. Crossed products of Banach algebra by groups are special cases of Banach algebraic bundles. The theory of module amenability developed in [A] does not cover these examples. There is one thing in common in these examples and that is the existence of a module homomorphism from the Banach module to the underlying Banach algebra. For instance in the case of crossed products, X is a Banach algebra, G is a topological group, and Xg = X × {g}, for g ∈ G, and {Xg} is a Banach algebraic bundle over G. In this case we have a module homomorphism ∆g : Xg → Xe which sends (x,g) to (x,e), where e is the identity of G. Also if G is a compact group and f ∈ Lq(G), then one has the module homomorphism ∆f : L p(G) → L1(G) which sends g to f ∗ g. In this paper, the concept of module amenability (more precisely ∆-amenability) is defined for a Banach module E over a Banach algebra A with a given module homomorphism ∆ : E → A. The next section gives the basic properties of module amenability and in particular establishes the equivalence of this concept with the existence of module virtual (approximate) diagonals in an appropriate sense. Section 3 covers the weak ∆-amenability. A few examples are discussed in the CUBO 13, 2 (2011) Module amenability for Banach modules 129 last section. 2. Module Amenability Let A be a Banach algebra and E be a Banach space with a left A-module structure such that, for some M > 0, ‖a.x‖ ≤ M‖a‖ ‖x‖ (a ∈ A,x ∈ E), then E is called a left Banach A-module. Right and two-sided Banach A-modules are defined similarly. Throughout this section E is a Banach A-bimodule and ∆ : E → A is a bounded Banach A-bimodule homomorphism. Definition 2.1. Let X be a Banach A-bimodule. A bounded linear map D : A → X is called a module derivation (or more specifically a ∆-derivation) if D(∆(a.x)) = a.D(∆(x)) + D(a).∆(x), D(∆(x.a)) = D(∆(x)).a + ∆(x).D(a), For each a ∈ A and x ∈ E. Also D is called inner (or ∆-inner) if there is f ∈ X such that D(∆(x)) = f.∆(x) − ∆(x).f =: Df(∆(x)) (x ∈ E). When ∆ has a dense range, Df extends uniquely to a ∆-derivation from A to X. Definition 2.2. A bimodule E is called module amenable (or more specifically ∆-amenable as a A-bimodule) if for each Banach A-bimodule X, all ∆-derivations from A to X∗ are ∆-inner. It is clear that A is A-module amenable (with ∆ = id) if and only if it is amenable as a Banach algebra. A right bounded approximate identity of E is a bounded net {aα} in A such that for each x ∈ E, ‖∆(x).aα −∆(x)‖ → 0, as α → ∞. The left and two-sided approximate identities are defined similarly. Proposition 2.3. If E is module amenable, then E has a bounded approximate identity. Proof Consider the double conjugate space A∗∗ as a Banach A-module with 〈F.a,f〉 = 〈F,a.f〉,〈a.f,b〉 = f(ba),a.F = 0 (a,b ∈ A,f ∈ A∗,F ∈ A∗∗). Then the canonical embedding D : A → A∗∗ is a module derivation, hence D = DF on ∆(E), for some F ∈ A∗∗. Choose a net {aα} in A which is w ∗-convergent to F in A∗∗. Clearly {aα} is a left bounded approximate identity of E. Right and two sided approximate identities now could be constructed similar to the classical case [D]. Definition 2.4. A Banach A-module X is called right ∆-essential if for each x ∈ X there is a ∈ ∆(E) and y ∈ X such that x = y.a. The left ∆-essential and (two sided) ∆-essential modules are defined similarly. 130 D. Ebrahimi Bagha, M. Amini CUBO 13, 2 (2011) The following two results are proved as in the classical case [J]. We just include the proof of Lemma 2.5(i), as it involves a variation of the Cohen factorization theorem. Lemma 2.5. (i) If ∆ has a dense range and E has a (right) bounded approximate identity, then E is module amenable iff for each (right) ∆-essential Banach A-bimodule X, all ∆-derivations from A to X∗ are ∆-inner. (ii) If E and E ′ are Banach A-modules with module homomorphisms ∆ and ∆ ′ and θ : E → E ′ is a bounded module map with dense range such that ∆ ′ ◦ θ = ∆, then ∆-amenability of E implies ∆ ′ -amenability of E ′ . (iii) If J is a closed submodule of E and J∆ is the closed ideal of A generated by ∆(J), and q : A → A/J∆ and q̃ : E → E/J are the corresponding quotient maps, then E is ∆ amenable whenever J is ∆|J-amenable and E/J is ∆̃-amenable, where ∆̃ : E/J → A/J∆ is the unique A/J∆- module map with ∆̃ ◦ q̃ = q ◦ ∆. Proof We prove part (i) as promised. We just need to check the necessity. Let {aα} ⊆ A be a right bounded approximate identity for E. Let X be a Banach A-bimodule. Consider Tα : X ∗ → X∗ defined by T(f) = aα.f, for f ∈ X ∗, where a.f(x) = f(x.a), for a ∈ A,x ∈ X. Since {aα} is bounded in A, {Tα} is bounded in B(X ∗). Hence it has a w∗-cluster point T. We may assume that Tα → T in w∗-topology. For each e ∈ E,x ∈ X,f ∈ X∗, we have 〈x,∆(e),Tf〉 = ĺım α 〈x,∆(e),Tαf〉 = ĺım α 〈x,∆(e),aα.f〉 = ĺım α 〈x,∆(e)aα,f〉 = 〈x,∆(e),f〉. Hence T −I : X∗ → (X.∆(E))⊥ is a bounded projection and we have the admissible short exact sequence 0 → (X.∆(E))⊥ → X∗ → (X.∆(E))∗ → 0 of Banach A-bimodules. But ∆(E).(X/(X.∆(E))) = 0 and ∆ has a dense range, hence each bounded ∆-derivation D1 : A → (X.∆(E)) ⊥ = (X/(X.∆(E)))∗ is zero. On the other hand, each bounded ∆-derivation D2 : A → (X.∆(E)) ∗ is ∆-inner, by assumption. Therefore each bounded ∆-derivation D : A → (X.∆(E))∗ is ∆-inner, and we are done. Lemma 2.6. Assume that A and B are Banach algebras, J is a closed ideal of A, E is a Banach A-module, and ∆ : E → A is an A-module homomorphism. (i) If F is a Banach A-module and Φ : E → F is an A-module homomorphism with dense range, then ∆-amenability of E implies ∆ ◦ Φ-amenability of F. (ii) If Ψ : A → B is a Banach algebra epimorphism with E.Ker(Ψ) = Ker(Ψ).E = {0}, and E is considered as a B-module via b.x := a.x, x.b := x.a (b ∈ B,x ∈ E), CUBO 13, 2 (2011) Module amenability for Banach modules 131 where a ∈ A on the right hand side is any element with b = Ψ(a). Then ∆-amenability of E, as an A-module, implies Ψ ◦ ∆-amenability of E as a B-module. (iii) In (ii), if B = A/J, Ψ : A → A/J is the quotient map, and E.J = J.E = {0}, then ∆-amenability of E, as an A-module, implies Ψ ◦ ∆-amenability of E as a A/J-module. (iv) If I is a closed ideal of A, E ′ is the closed submodule of E generated by IE, and ∆ ′ : E ′ → I is the restriction of ∆ : E → A, then E ′ is ∆ ′ -amenable whenever E is ∆-amenable and E ′ has a bounded approximate identity. Proposition 2.7. If I is a closed ideal of A which contains a bounded approximate identity (of itself), E is a Banach A-bimodule with module homomorphism ∆ : E → A, and X is an essential Banach I-module, then X is (canonically) a Banach A-module and each ∆|I-derivation D : I → X ∗ uniquely extends to a ∆-derivation D̃ : A → X∗ which is continuous with respect to the strict topology of A (induced by I) and w∗-topology of X∗. Proof Each x ∈ X decomposes (not uniquely) as x = a.y, for some a ∈ I and y ∈ X. It is easy to see that X is a left Banach A-module under the action b.x = ba.y (a ∈ I,b ∈ A,x,y ∈ X,x = a.y). This is well defined, as I has a bounded approximate identity. Define D̃ : A → X∗ by D̃(b) = w∗- ĺım α (D(beα) − b.D(eα)), where {eα} is a bounded approximate identity of I. Now beα → b strictly, for each b ∈ A. Hence, given b ∈ A and e ∈ E, we have D̃(∆(b.e)) = D̃(b∆(e)) = w∗- ĺım α w∗- ĺım β D(beα∆(e)eβ) = w∗- ĺım α w∗- ĺım β [beαD(∆(e)eβ) + D(beα).∆(e)eβ] = bD̃(∆(e)) + D̃(b).∆(e). Hence D̃ is a ∆-derivation. The rest of the proof is similar to [Ru, 2.1.6]. Proposition 2.8. If ∆ : E → A has a dense range, then ∆-amenability of E is equivalent to amenabil- ity of A. Proof If E is ∆-amenable, then each derivation D : A → X∗, where X is a Banach A-module, is a module derivation and so inner on ∆(E). By continuity, D is inner on A. Conversely each module derivation D : A → X∗ is a derivation. Indeed, given b ∈ A, there is a sequence {xn} ⊆ E such that ∆(xn) → b, and so D(ab) = ĺım n D(a∆(xn)) = ĺım n (D(a).∆(xn) + a.D(∆(xn))) = D(a).b + a.D(b), for each a ∈ A. Hence, if A is amenable, then E is ∆-amenable. 132 D. Ebrahimi Bagha, M. Amini CUBO 13, 2 (2011) Definition 2.9. Let π : A⊗̂A → A be the continuous lift of the multiplication map of A to the projective tensor product A⊗̂A. A module approximate diagonal of E is a bounded net {eα} in A⊗̂A such that ‖eα.∆(x) − ∆(x).eα‖ → 0, ‖π(eα).∆(x) − ∆(x)‖ → 0 (x ∈ E), as α → ∞. A module virtual diagonal of E is an element M in (A⊗̂A)∗∗ such that M.∆(x) − ∆(x).M = 0, π∗∗(M).∆(x) − ∆(x) = 0 (x ∈ E). It is clear that if E has a module virtual diagonal, then A contains a bounded approximate identity. Theorem 2.10. Consider the following assertions. (i) E is module amenable, (ii) E has a module virtual diagonal, (iii) E has a module approximate diagonal. We have (i) → (ii) ↔ (iii). If moreover ∆ has a dense range, all the assertions are equivalent. Proof (i) → (ii). By Proposition 2.3, we may choose a bounded approximate identity {eα} for E. We may assume that {eα ⊗ eα} is w ∗-convergent to a point P ∈ (A⊗̂A)∗∗. Then for each x ∈ E and f ∈ A∗, 〈π∗∗((DP ◦ ∆)(x),f〉 = w ∗- ĺım α 〈π∗(f).∆(x) − ∆(x).π∗(f),eα ⊗ eα〉 = w∗- ĺım α f(∆(x.eα)eα − eα∆(eα.x)) = 0. Hence Im(DP ◦ ∆) ⊆ Ker(π ∗∗). Now Ker(π∗∗) is isometrically isomorphic to X∗, where X = (∆(E)⊗̂A)∗∗/Im(π∗)⊥, so by assumption, there is Q ∈ Ker(π∗∗) with DP ◦ ∆ = DQ ◦ ∆. It is easy to see that M := P − Q is a module virtual diagonal for E. (ii) → (iii). Let M be a module virtual diagonal and let {eα} be a net in A⊗̂A which w ∗- clusters to M. Then clearly eα.∆(x) − ∆(x).eα → 0, ∆◦π(eα).∆(x) − ∆(x) → 0 as α → ∞ for each x ∈ E in the w∗-topology of (A⊗̂A)∗∗. A standard argument based on Mazur’s theorem shows that the same holds in the norm topology for a net consisting of appropriate convex combinations of elements of {eα}. (iii) → (ii). Just take any w∗-cluster point. (iii) → (i). Now assume that ∆ has a dense range. Let {mα} be a module approximate diagonal for E with w∗-cluster point M, then {π(mα)} is a bounded approximate identity for E. By Lemma 2.5(i), it is enough to show that for each essential A-module Y, all module derivation D from A to Y∗ are inner. Each y ∈ Y could be regarded as a bounded linear functional ŷ on A⊗̂A via 〈ŷ,b ⊗ a〉 := 〈b.D(a),y〉 (a,b ∈ A). CUBO 13, 2 (2011) Module amenability for Banach modules 133 Then for each x,x ′ ∈ E, a ∈ A, and y ∈ Y 〈(y.∆(x) − ∆(x).y)̂,∆(x ′ ) ⊗ a〉 = 〈ŷ.∆(x) − ∆(x).ŷ,∆(x ′ ) ⊗ a〉 + 〈∆(x ′ )a.D ◦ ∆(x),y〉. It follows that 〈(y.∆(x) − ∆(x).y)̂,m〉 = 〈ŷ.∆(x) − ∆(x).ŷ,m〉 + 〈π(m).D ◦ ∆(x),y〉, for each m ∈ A⊗̂A. If we identify M with an element of Y∗ with M(y) = 〈ŷ,M〉, for y ∈ Y, then 〈DM ◦ ∆(x),y〉 = w ∗- ĺım α 〈y.∆(x) − ∆(x).y,mα〉 = 〈M,ŷ.∆(x) − ∆(x).ŷ〉 + w∗- ĺım α 〈π(mα).D ◦ ∆(x),y〉. Now in the last equation, the first term is zero, as M is a module virtual diagonal, and the second term is easily seen to be equal to 〈D◦∆(x),y〉, using the fact that y = z.∆(x ′ ), for some z ∈ Y and x ′ ∈ E. Therefore D = DM on ∆(E), as required. 3. Weak Module Amenability In this section we study weak module amenability of Banach modules. All over this section E is a commutative Banach A-module (that is a.x = x.a, for each a ∈ A,x ∈ E) and ∆ : E → A is a bounded Banach A-module homomorphism. A Banach A-module X is called ∆-commutative (or more specifically ∆(E)-commutative) if a.x = x.a (a ∈ ∆(E),x ∈ X). Definition 3.1. E is called weak module amenable (or more specifically weak ∆-amenable as an A-module) if each ∆-derivation from A to ∆(E)∗ is inner on ∆(E). Clearly A is weak A-module amenable (with ∆ = id) if and only if it is weakly amenable as a Banach algebra. The following result could be proved as in the classical case. Proposition 3.2. (i) If E ′ is a commutative A-module and Φ : E ′ → E is a module homomorphism with dense range, and E is weak ∆-amenable then E ′ is weak ∆ ◦ Φ-amenable. (ii) If I is a closed ideal of A with IE = EI = {0} and q : A → A/I is the quotient map, then E is weak q ◦ ∆-amenable as an A/I-module if it is ∆-amenable as an A-bimodule. Proposition 3.3. If E is weak ∆-amenable, then the closed linear span F of A∆(E) is dense in ∆(E). Proof If not, there is a nonzero bounded linear functional λ in ∆(E)∗ which vanishes on F. By Hahn-Banach Theorem λ extends to an element of A∗, which we still denote by λ. Define D : A → ∆(E)∗ by D(a) = λ(a)λ. This is a module derivation which is not inner, a contradiction. Now if A is a (commutative) Banach algebra with maximal ideal space MA and φ ∈ MA, then C is a Banach A-module with respect to the module action a.z = z.a = φ(a)z (a ∈ A,z ∈ C), 134 D. Ebrahimi Bagha, M. Amini CUBO 13, 2 (2011) which is denoted by Cφ. Each module derivation D : A → Cφ is called a module point derivation (at φ). Clearly when a commutative Banach A-module E is ∆-weak amenable, all module point derivations vanish on ∆(E). This holds in general. Proposition 3.4. If E is weak ∆-amenable, there is no nonzero point derivation on A. Proof Let d : A → Cφ be a nonzero module point derivation. Let ψ be the restriction of φ to ∆(E) and define D : A → ∆(E)∗ by D(a) = d(a)ψ. Then D is a ∆-derivation and so D = Dλ on ∆(E), for some λ ∈ ∆(E) ∗. Choose e,f ∈ E so that ψ(∆(e)) = 1, ψ(∆(f)) = 0, and d(∆(f)) = 1. Then for a = ∆(e) + (1−d(∆(e)))∆(f), we have ψ(a) = d(a) = 1, hence D(a)(a) = 1, a contradiction. Theorem 3.5. If E is a commutative A-module and there is a ∆-commutative A-module X and a module derivation D0 : A → X which is not identically zero on ∆(E), then there is a nonzero module derivation D : A → ∆(E)∗. Proof We consider two cases. First assume that A∆(E) is not dense in ∆(E). By Hahn-Banach Theorem, there is a nonzero functional λ ∈ (∆(E))∗ whose kernel contains A∆(E). Extend λ to an element of A∗ (still denoted by λ) and define D : A → ∆(E)∗ by D(a) = λ(a)λ (a ∈ A), Next consider the case where A∆(E) is dense in ∆(E). We know that there is a ∆-commutative A-module X and a module derivation D0 : A → X which is not identically zero on ∆(E). Choose a ∈ A, e ∈ E, and λ ∈ X∗ such that D0(a∆(e)) 6= 0 and λ(D0(a∆(e))) 6= 0. Define D : A → ∆(E) ∗ by 〈D(a),∆(e)〉 = λ(∆(e).D0(a)) (e ∈ E,a ∈ A). In both cases D is a nonzero module derivation. 4. Examples In this sections we give three examples in which strong and weak module amenability of some Banach modules are demonstrated. Example 4.1. Let S be an inverse semigroup and ES be the commutative sub-semigroup of idem- potents in S. Then A = ℓ1(ES) is a commutative Banach algebra and E = ℓ 1(S) is a commutative Banach A-bimodule with the module actions δe.δx = δx.δe = δex (e ∈ ES,x ∈ S). Also there is a surjective module homomorphism ∆ : ℓ1(S) → ℓ1(ES) defined by ∆(δx) = δxx∗ (x ∈ S). CUBO 13, 2 (2011) Module amenability for Banach modules 135 We show that ℓ1(S) is always module weakly amenable. If D : ℓ1(ES) → ℓ ∞ (ES) is a ∆-derivation, then for each e ∈ ES, D(δe) = D(δee∗ ) = D(∆(δe)) = D(∆(δe.δe)) = ∆(δe).D(δe) + δe.D(∆(δe)) = 2δeD(δe). Applying the same formula to the right hand side, δe.D(δe) = 2δe.(δe.D(δe)) = 2(δe ∗ δe).D(δe)) = 2δe.D(δe), hence D(δe) = δe.D(δe) = 0. Example 4.2. In the above example, if ℓ1(S) is ∆-amenable, then ℓ1(ES) is amenable (Proposition 2.8). Hence ES is finite, ℓ 1(S) is weakly amenable, and it has a bounded approximate identity [DN]. Example 4.3. It is well known that the disk algebra A(D) is non amenable [BD]. A(D) is a C- module with respect to the scalar product. Now evaluation at zero defines a module epimorphism ∆ : A(D) → C and A(D) is ∆-amenable. Example 4.4. If A is an amenable Banach algebra, the canonical map π : A⊗̂A → A is an A- module epimorphism (it is surjective, since A has a bounded approximate identity) and A⊗̂A is π-amenable. Example 4.5. For a locally compact group G, L1(G) is a closed two sided ideal in M(G), so we can consider it as a Banach M(G) module. Now if G is a non discrete amenable group, M(G) is not amenable [DGH] but L1(G) is i-amenable, where i : L1(G) → M(G) is the canonical injection. Example 4.6. Let 1 < p < ∞ and 1 p + 1 q = 1. Then ℓ1 is a Banach algebra and ℓp is a Banach ℓ1-bimodule, both with respect to pointwise multiplication. Also each f ∈ ℓq defines a module homomorphism ∆f : ℓ p → ℓ1 by ∆f(g) = g ∗ f. If f = ∑∞ k=−∞ 1 k δk, then ∆f has dense range(as its range contains all finitely supported element) and ℓp is ∆f-amenable, by Proposition 2.8. This example could also be stated for any discrete group G, where ℓp(G) is considered as a Banach ℓ1(G)-bimodule. Same is true for Lp(G) with convolution, when G is a compact group [D, 3.3.19]. In this case we have the module homomorphism ∆f : L p(G) → L1(G) defined by ∆f(g) = g ∗ f, where f ∈ Lq(G). If ∆f has a dense range, then L p(G) is ∆f -amenable. This is always the case when G is an abelian compact group. We illustrate this for G = T. The same proof basically works for arbitrary abelian compact groups as well. Take f = ∑∞ k=−∞ 1 k e2πikt ∈ Lq(T) (which is basically the Fourier transform of the above function f used in the discrete case). Then, for each g ∈ Lp(G), ∆f(g) = ∞∑ k=−∞ 1 k ĝ(k)e2πikt, where ĝ ∈ c0 is the Fourier transform of g. In particular, range of ∆f includes all trigonometric functions which are dense in L1(G). Example 4.7. If G is a discrete group (with identity e) which acts on a C∗-algebra A, then A× {e} could be identified with A and A × {g} is a Banach A-module under a.(b,g) = ((g.a)b,g) , (b,g).a = (ba,g) (a,b ∈ A,g ∈ G), 136 D. Ebrahimi Bagha, M. Amini CUBO 13, 2 (2011) and there is a natural surjective module homomorphism ∆g : A × {g} → A which sends (a,g) to a. The crossed product C∗-algebra A ⋊ G is nuclear iff A is nuclear and G is amenable [Ro] iff G is amenable and modules A × {g} are ∆g-amenable, for each g ∈ G. Example 4.8. If A is a Banach algebra such that A∗ ⊆ A and A∗ is a dense subspace of A, then A∗ is a Banach A-bimodule (with canonical Arens actions) and ∆ = id : A∗ → A has dense range. Therefore A∗ is ∆-amenable as a Banach A-bimodule iff A is amenable as a Banach algebra. There are many examples of this type. If G is a compact group, then the Fourier algebra A(G) is dense in the group C∗-algebra C∗(G). Indeed A(G) ⊆ C(G) ⊆ L1(G) ⊆ C∗(G), and each space in this chain is dense in the subsequent space (with respect to the norm of the bigger space). But the norms of the last three spaces satisfy ‖.‖C∗(G) ≤ ‖.‖1 ≤ ‖.‖∞ [Ey]. Hence A(G) is dense in C∗(G). Also, since G is compact, A(G) = B(G) ≃ C∗(G)∗, where B(G) is the Fourier-Stieltjes algebra [Ey]. But C∗(G) is amenable when G is compact [Ru]. Hence A(G) is id- amenable in this case. This becomes more interesting when we recall that there are compact groups for which the Fourier algebra A(G) is not amenable [J2]. Another example is ℓ1 which is dense in c0. It follows that ℓ 1 ≃ c∗0 is id-amenable as a c0-bimodule. Finally, for a compact group G, L1(G) is an amenable Banach algebra with convolution, and so L1(G) is id-amenable as a Banach L1(G)-bimodule. Received: December 2009. Revised: April 2010. Referencias [A] M. Amini, Module amenability for semigroup algebras, Semigroup Forum 69 (2004) 243-254. [AE] M. Amini and D. Ebrahimi Bagha, Weak module amenability for semigroup algebras, Semigroup Forum 71 (2005) 18-26. [BD] F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, New York, 1973. [D] H.G. Dales, Banach algebras and automatic continuity, London Math. Soc. 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Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, 1972. [J2] B.E. Johnson,Non-amenability of the Fourier algebra of a compact group, J. London Math. Soc. (2), 50 (1994), 361-374. [Ro] J. Rosenberg, Amenability of crossed products of C∗-algebras, Comm. Math. Phys. 57 (1977), no. 2, 187-191. [Ru] V. Runde, Lectures on amenability, Lecture Notes in Mathematics 1774, Springer-Verlag, Berlin, 2002. Introduction Module Amenability Weak Module Amenability Examples