Ideal amenability of second duals of Banach algebras

International Mathematical Forum, 2, 2007, no. 16, 765 - 770 Ideal Amenability of Second Duals of Banach Algebras M. Eshaghi Gordji(1) , F. Habibian(2) and B. Hayati(3) (1) Department of Mathematics, Faculty of Sciences, Semnan University, Semnan, Iran (2) Department of Mathematics, Isfahan University, Isfahan, Iran (3) Department of Mathematics, Shahid Beheshti University, Tehran, Iran eimail:[email protected] & [email protected] & [email protected] Abstract In this paper we study the ideal amenability of second duals of Banach algebras. We investigate relations between ideal amenability of the second dual of a Banach algebra with the first and the second Arens products. Mathematics Subject Classification: 46H25 Keywords: Ideally amenable, Banach algebra, Derivation 1 Introduction Let A be a Banach algebra and let A∗∗ be the second dual algebra of A endowed with the first or the second Arens product. Throughout this paper, the first and the second Arens product are respectively denoted by  and ♦. These products can be defined by F G = w ∗ lim lim f̂i gˆj i j and F ♦G = w ∗ lim lim f̂i gˆj j i where (fi ) and (gj ) are nets of elements of A such that fi −→ F and gj −→ G in w ∗ -topology (see [A] and [D-H]). If X is a Banach A-bimodule, then a derivation from A into X is a linear map D such that for every a, b ∈ A, D(ab) = D(a).b + a.D(b). If x ∈ X and we define δx : A −→ X by δx (a) = 766 M. Eshaghi Gordji , F. Habibian and B. Hayati a.x − x.a (a ∈ A), then δx is a derivation. Derivations of this form are called inner derivations. Let A be a Banach algebra and X be a Banach A-bimodule, then X ∗ is a Banach A-bimodule if for each a ∈ A and x ∈ X and x∗ ∈ X ∗ we define ax∗ , x = x∗ , xa, x∗ a, x = x∗ , ax. A Banach algebra A is amenable if every derivation from A into every dual A-bimodule is inner, equivalently if H 1 (A, X ∗ ) = {0} for every Banach Abimodule X, where H 1 (A, X ∗ ) is the first cohomology group of A with coefficients in X ∗ [J1], [R], [H]. A Banach algebra A is weakly amenable if H 1 (A, A∗) = {0} (the special case of X = A) [B-C-D], [J2]. A Banach algebra A is ideally amenable if H 1 (A, I ∗ ) = {0} for every closed two-sided ideal I in A. Eshaghi-Gordji and Yazdanpanah have introduced the concept of ideal amenability of Banach algebras in [G-Y], (see [E-H]). 2 Some examples In this section, let A be a Banach algebra, X a Banach A−bimodule and let Y be a Banach A−submodule of X. We give some examples to show that H 1(A, X ∗ ) = {0} ( H 1(A, X) = {0}) is not sufficient condition for H 1 (A, Y ∗ ) = {0} ( H 1 (A, Y ) = {0}). So, there are some examples to show that weak amenability dose not imply ideal amenability. Example 1. Let A be a non weakly amenable Banach algebra with bounded left(or right) approximate identity, such that H 2 (A, A∗ ) = {0}. We know that i ∆ ˆ {0} −→ ker ∆ −−−→ A⊗A −−−→ A −→ {0} is an exact sequence of Banach A−bimodules, and ˆ ∗ −→ (ker ∆)∗ −→ {0} {0} −→ A∗ −→ (A⊗A) ˆ ∗ and is an admissible exact sequence. A∗ is a Banach A−submodule of (A⊗A) 1 ∗ 2 ∗ ˆ by Theorem 2.4.6 of [R] we have H (A, (A⊗A) ) ≃ H (A, A ) = {0}. Now, ∗ ∗ 1 ˆ let X = (A⊗A) and Y = A , then H (A, X) = {0} but H 1 (A, Y ) = {0}. Example 2. Let G = P S(2, R), A = X = L1 (G) and let Y be the augmentation ideal of A. Then we have H 1 (A, Y ∗ ) = {0} [J-W]. On the other hand A is weakly amenable and so H 1 (A, X ∗ ) = {0}. A is also an example of a weakly amenable Banach algebra which is not ideally amenable. Ideal amenability of second duals of Banach algebras 767 Example 3. Let G = P S(2, R) and let I be the augmentation ideal of L1 (G). Suppose that X = A = I # and Y = I. Then H 1 (A, X ∗) = {0} =  H 1(A, Y ∗ ) [G-Y]. Example 4. Let A be a Banach algebra. Then A × A is a Banach algebra by the following product and norm: (a, x)(b, y) = (ab, ay + xb) (a, b, x, y ∈ A), (a, b) = a + b (a, b ∈ A). Let A be a weakly amenable Banach algebra with a bounded approximate identity such that span {ab−ba : a, b ∈ A} is dense in A. Then by Theorem 5.4 of [Zh], A × A is weakly amenable. Now let 0 = ϕ ∈ ΩA , then D : A × A −→ {0} × A∗ defined by D(a, x) = (0, ϕ(a)ϕ) is a derivation. Since D((a, x)(b, y)) = = = = = = D(ab, ay + xt) (0, ϕ(ay + xb)ϕ) (0, (ϕ(a)ϕ(y) + ϕ(x)ϕ(t))ϕ) (0, (ϕ(x)ϕ)b) + (0, (aϕ(y))ϕ) (0, ϕ(x)ϕ)(b, y) + (a, x)(0, ϕ(y)ϕ) D(a, x)(b, y) + (a, x)D(b, y). If D = δ(0,f ) for some f ∈ A∗ , then we have (0, ϕ(x)ϕ) = = = = D(a, x) (a, x)(0, f ) − (0, f )(a, x) (xf, af ) − (xf, f a) (xf − f x, af − f a). It shows that (0, ϕ(x)ϕ) = D(a, x) = D(0, x) = (0, 0), which is a contradiction with ϕ = 0. Now let B = X = A×A and Y = {0}×A. We have H 1 (B, X ∗ ) = {0} and H 1 (B, Y ∗ ) = {0}. 3 Ideal Amenability of Second dual of a Banach algebra In [Gh-L-W] Ghahramani, Loy and Willis studied some implications of amenability and weak amenability of A∗∗ . In this paper we study the ideal amenability of the second dual of a Banach algebra. 768 M. Eshaghi Gordji , F. Habibian and B. Hayati Theorem 3.1. Let A be a commutative Banach algebra. Suppose that A∗∗ is ideally amenable. If one of the following assertions holds: (i) A is a left ideal in A∗∗ . (ii) A is a dual algebra. (iii) A is Arens regular and every derivation from A into its dual is weakly compact. Then A is ideally amenable. Proof. Let A∗∗ be ideally amenable, then it is weakly amenable. If one of the (i) , (ii) or (iii) holds, then by Theorem 3.2 of [Gh-L-W], Theorem 2.2 of [Gh-L] and Corollary 7.5 of [D-G-V], A is weakly amenable. Since A is commutative, then A is ideally amenable [G-Y]. Theorem 3.2. Let A be a Banach algebra. Consider A∗∗ endowed with the first Arens product. If A is an ideal of A∗∗ with a bounded approximate identity, then ideal amenability of A∗∗ implies ideal amenability of A. Proof. Let I be a closed two sided ideal of A. Then I is an ideal in A∗∗ . Now let D : A −→ I ∗ be a derivation, then there is a bounded derivation D̄ : A∗∗ −→ I ∗ which is an extension of D [R]. Since A∗∗ is ideally amenable, then there exists f ∈ I ∗ such that D̄ = δf . Thus D = δf and A is ideally amenable. For a Banach algebra A, let Aop be the Banach algebra with underlying Banach space same as A and with product ◦ given by a ◦ b = ba. Lemma 3.3. Let A be a Banach algebra. Then A is ideally amenable if and only if Aop is ideally amenable. Proof. It is clear that I is an ideal of A if and only if I is an ideal of Aop . Now let A be ideally amenable and let D : Aop −→ I ∗ be a continuous derivation, where I is a closed ideal of Aop . Since for every a, b ∈ A, D(ab) = D(b ◦ a) and I is a closed ideal of A, then D : A −→ I ∗ is a continuous derivation. But A is ideally amenable, so there is ξ ∈ I ∗ such that D(a) = a.ξ −ξ.a = −(a◦ξ −ξ ◦a). This means that Aop is ideally amenable. The converse is similar. Lemma 3.4. Let A, B be Banach algebras and let θ : A −→ B be a continuous isomorphism. If A is ideally amenable, then B is ideally amenable. 769 Ideal amenability of second duals of Banach algebras Proof. It is evident. Theorem 3.5. Let A be a Banach algebra with a continuous anti-isomorphism ϕ : A −→ A. Then (A∗∗ , ) is ideally amenable if and only if (A∗∗ , ♦) is ideally amenable. Proof. Let (A∗∗ , ) be ideally amenable and let ϕ : A −→ A be a continuous anti-isomorphism. Take F, G ∈ A and let (fi ), (gj ) be nets in A such that weak ∗ − lim f̂i = F and i w ∗ − lim gˆj = G. j Let ϕ∗∗ : (A∗∗ , ) −→ (A∗∗ , ♦) be the second adjoint of ϕ. We have ϕ∗∗ (F G) = w ∗ − lim w ∗ − lim ϕ∗∗ (f̂i gˆj ) i j = w − lim w − lim (ϕ(f i )ϕ(gj )) ∗ ∗ i j = w ∗ − lim w ∗ − lim ϕ∗∗ (gˆj )ϕ∗∗ (f̂i ) i ∗∗ j ∗∗ = ϕ (G)♦ϕ (F ). Then ϕ∗∗ is a continuous isomorphism from (A∗∗ , ) onto (A∗∗ , ♦)op and so by lemma 3.4, (A∗∗ , ♦)op is ideally amenable. Now, by lemma 3.3, (A∗∗ , ♦) is ideally amenable. The converse is similar. Corollary 3.6. Let A be a commutative Banach algebra. Then (A∗∗ , ) is ideally amenable if and only if (A∗∗ , ♦) is ideally amenable. Proof. If A is a commutative Banach algebra then the identity map is a continuous anti-isomorphism on A. Now apply theorem 3.5. Acknowledgement. The second author would like to thank the University of Isfahan for its financial support. References [A] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2(1951), 839-848. [B-C-D] W. G. Bade, P. G. Curtis and H. G. Dales, Amenability and weak amenability for Beurling and Lipschits algebra , Proc. London Math. Soc. 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