1)-Weak Amenability of Second Dual of Real Banach Algebras

Sahand Communications in Mathematical Analysis (SCMA) Vol. 12 No. 1 (2018), 59-88 http://scma.maragheh.ac.ir DOI: 10.22130/scma.2018.88929.466 (−1)-Weak Amenability of Second Dual of Real Banach Algebras Hamidreza Alihoseini1 and Davood Alimohammadi2∗ Abstract. Let (A, ∥ · ∥) be a real Banach algebra, a complex algebra AC be a complexification of A and ∥| · ∥| be an algebra norm on AC satisfying a simple condition together with the norm ∥ · ∥ on A. In this paper we first show that A∗ is a real Banach A∗∗ -module if and only if (AC )∗ is a complex Banach (AC )∗∗ -module. Next we prove that A∗∗ is (−1)-weakly amenable if and only if (AC )∗∗ is (−1)-weakly amenable. Finally, we give some examples of real Banach algebras which their second duals of some them are and of others are not (−1)-weakly amenable. 1. Introduction And Preliminaries The symbol F denotes a field that can be either R or C. For a Banach space X over F we denote by X∗ and X∗∗ the dual space and the second dual space of X, respectively. Let B be an algebra over F and X be a B-module over F with the module operations (a, x) 7−→ a · x, (a, x) 7−→ x · a : B × X −→ X. A linear map D : B −→ X over F is called an X-derivation on B over F if D(ab) = D(a) · b + a · D(b) for all a, b ∈ B. For each x ∈ X, the map δx : B −→ X defined by δx (a) = a · x − x · a (a ∈ B), is an X-derivation on B over F. An X-derivation D on B is called inner if D = δx for some x ∈ X. Let (B, ∥ · ∥) be a Banach algebra over F. A B-module X over F is called a Banach B-module if X is a Banach space with a norm ∥ · ∥ and 2010 Mathematics Subject Classification. 46H25, 46H20. Key words and phrases. Banach algebra, Banach module, Complexification, Derivation, (−1)-Weak amenability. Received: 27 June 2018, Accepted: 11 October 2018. ∗ Corresponding author. 59 60 H. ALIHOSEINI AND D. ALIMOHAMMADI there exists a positive constant k such that ∥a · x∥ ≤ k∥a∥∥x∥, ∥x · a∥ ≤ k∥a∥∥x∥, for all a ∈ B and x ∈ X. Clearly, B is a Banach B-module over F with the module operations a · b = ab and b · a = ba for all a, b ∈ B. Let X be a Banach B-module over F with the module operations (a, x) 7→ a · x, (a, x) 7→ x · a : B × X −→ X. Then X∗ is a Banach B-module over F with the natural module operations (λ, a) 7−→ a · λ, (λ, a) 7−→ λ · a : B × X∗ −→ X∗ given by (a · λ)(x) = λ(x · a), (λ · a)(x) = λ(a · x), (a ∈ B, λ ∈ X∗ , x ∈ X), and with the operator norm ∥ · ∥op . In particular, B ∗ is a Banach Bmodule over F. We denote by ZF1 (B, X) the set of all continuous Xderivations on B over F. Clearly, ZF1 (B, X) is a linear space over F which contains all inner X-derivations on B over F. We denote by NF1 (B, X) the set of all inner X-derivations on B over F. Clearly, NF1 (B, X) is a linear subspace of ZF1 (B, X) over F. We denote by HF1 (B, X) the quotient space ZF1 (B, X)⧸NF1 (B, X) which it is called the first cohomology group of B over F with coefficients in X. A Banach algebra B over F is called amenable if HF1 (B, X∗ ) = {0} for all Banach B-module X over F. This concept was first introduced by Johnson in [12]. The notion of weak amenability was first introduced by Bade, Curtis and Dales for commutative Banach algebras in [4] and later defined for Banach algebras, not necessarily commutative, by Johnson in [13]. In fact, a Banach algebra B over F is called weakly amenable if HF1 (B, B ∗ ) = {0}. Let B be a Banach algebra over F. For each (λ, Λ) ∈ B ∗ × B ∗∗ the F-valued functions λ · Λ and Λ · λ on B are defined by (λ · Λ)(a) = Λ(a · λ), (a ∈ B), (Λ · λ)(a) = Λ(λ · a), (a ∈ B). Then λ · Λ ∈ B ∗ , ∥λ · Λ∥op ≤ ∥λ∥op ∥Λ∥op , Λ · λ ∈ B ∗ and ∥Λ · λ∥op ≤ ∥Λ∥op ∥λ∥op . For each Λ, Γ ∈ B ∗∗ , the F-valued functions Λ□Γ and Λ△Γ on B ∗ are defined by (Λ□Γ)(λ) = Λ(Γ · λ), (λ ∈ B ∗ ), (Λ△Γ)(λ) = Γ(λ · Λ), (λ ∈ B ∗ ). Then Λ□Γ ∈ B ∗∗ , ∥Λ□Γ∥op ≤ ∥Λ∥op ∥Γ∥op , Λ△Γ ∈ B ∗∗ and ∥Λ□Γ∥op ≤ ∥Λ∥op ∥Γ∥op . Moreover, B ∗∗ is a Banach algebra over F with respect to either of the products □ and △ and with the operator norm ∥ · ∥op . These products are called the first and second Arens products on B ∗∗ , respectively. The Banach algebra B over F is called Arens regular if two products □ and △ coincide on B ∗∗ . For the general theory of Arens (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 61 products, see [3, 7, 18], for example. For the product □ on B ∗∗ one can show that B ∗ is a Banach B ∗∗ -module over F if and only if the following statements hold: (i) (Λ · λ) · Γ = Λ · (λ · Γ) for all (Λ, λ, Γ) ∈ B ∗∗ × B ∗ × B ∗∗ , (ii) λ · (Λ□Γ) = (λ · Λ) · Γ for all (λ, Λ, Γ) ∈ B ∗ × B ∗∗ × B ∗∗ , (iii) (Λ□Γ) · λ = Λ · (Γ · λ) for all (Λ, Γ, λ) ∈ B ∗∗ × B ∗∗ × B ∗ . Definition 1.1. Let (B, ∥ · ∥) be a Banach algebra over F and × be one of the Arens products □ and △ on B ∗∗ . We say that B ∗∗ (with the product ×) is (−1)-weakly amenable if B ∗ is a Banach B ∗∗ -module over F and HF1 (B ∗∗ , B ∗ ) = {0}. Medghalchi and Yazdanpanah introduced the concept of (−1)-weak amenability for Banach algebras in [17] and obtained some results in this area. Eshaghi Gordji, Hosseinioun and Valadkhani in [8] gave some examples of complex Banach algebras that their second duals which are and some others which are not (−1)-weakly amenable. Hosseinioun and Valadkhani obtained interesting results in (−1)-weak amenability of complex Banach algebras in [10, 11]. Let E be a real linear space (real algebra, respectively). A complex linear space (complex algebra, respectively) EC is called a complexification of E if there exists an injective real linear map (real algebra homomorphism, respectively) J : E −→ EC such that EC = J(E) ⊕ iJ(E). If X is a real linear space, then X × X with the additive operation and scalar multiplication defined by (x1 , x2 , y1 , y2 ∈ X), (x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y2 ), (α + iβ)(x, y) = (αx − βy, αy + βx), (α, β ∈ R, x, y ∈ X), is a complexification of X with respect to the injective linear map J : X −→ X × X defined by J(x) = (x, 0), x ∈ X. If A is a real algebra, then A × A with the algebra operations (a1 , b1 ) + (a2 , b2 ) = (a1 + a2 , b1 + b2 ), (α + iβ)(a, b) = (αa − βb, αb + βa), (a1 , a2 , b1 , b2 ∈ A), (α, β ∈ R, a, b ∈ A), (a1 , b1 )(a2 , b2 ) = (a1 a2 − b1 b2 , a1 b2 + b1 a2 ), (a1 , b1 , a2 , b2 ∈ A), is a complexification of A with the injective real algebra homomorphism J : A −→ A × A defined by J(a) = (a, 0), a ∈ A. It is known [5, Proposition I.1.13] that if (E, ∥ · ∥) is a real normed algebra (real normed space, respectively), then there exists an algebra norm (a norm, respectively) ∥| · ∥| on E × E satisfying ∥|(a, 0)∥| = ∥a∥ for all a ∈ E and max{∥a∥, ∥b∥} ≤ ∥|(a, b)∥| ≤ 2 max{∥a∥, ∥b∥}, for all a, b ∈ E. 62 H. ALIHOSEINI AND D. ALIMOHAMMADI Definition 1.2. Let (E, ∥·∥) be a real normed linear space (real normed algebra, respectively), let a complex linear space (algebra, respectively) EC be a complexification of E with respect to an injective real linear map (real algebra homomorphism, respectively) J : E −→ EC and let ∥| · ∥| be a norm (an algebra norm, respectively) on EC . We say that ∥| · ∥| satisfies the (∗) condition if there exist positive constants k1 and k2 such that max{∥a∥, ∥b∥} ≤ k1 ∥|J(a) + iJ(b)∥| ≤ k2 max{∥a∥, ∥b∥}, for all a, b ∈ E. Note that the (∗) condition implies that (E, ∥ · ∥) is a Banach space (Banach algebra, respectively) if and only if (EC , ∥| · ∥|) is Banach space (Banach algebra, respectively). Moreover, the existence of a norm (an algebra norm, respectively) ∥| · ∥| on EC satisfying the (∗) condition guarantees by [5, Proposition I.1.13]. It is shown [2] that if (A, ∥ · ∥) is a real Banach algebra and if ∥| · ∥| is an algebra norm on complex algebra A × A satisfying max{∥a∥, ∥b∥} ≤ k1 ∥|(a, b)∥| ≤ k2 max{∥a∥, ∥b∥} for some positive constants k1 and k2 and for all a, b ∈ A, then (i) A is amenable if and only if A× A is amenable [2, Theorem 2.4]. (ii) A is weakly amenable if and only if A × A is weakly amenable [2, Theorem 2.5]. In Section 2 we assume that (A, ∥ · ∥) is a real Banach algebra, a complex algebra AC is the complexification of A with respect to an injective real algebra homomorphism J : A −→ AC , ∥| · ∥| is an algebra norm on AC satisfying the (∗) condition and (AC )∗ is the dual space of (AC , ∥| · ∥|). We first show that A is Arens regular if and only if AC is Arens regular. Next we prove that A∗ is a real Banach A∗∗ -module if and only if (AC )∗ is a complex Banach (AC )∗∗ -module. Moreover, we prove that if A is a real Banach algebra such that A∗ is a real Banach A∗∗ -module, then A∗∗ is (−1)-weakly amenable if and only if (AC )∗∗ is (−1)-weakly amenable. Finally, we give some examples of real Banach algebras which their second duals of some them are and of others are not (−1)-weakly amenable. 2. Main Results and Applications We first give some lemmas which they will use in the sequel to prove of the main results. Lemma 2.1. Let (X, ∥ · ∥) be a real Banach space, let XC be a complexification of X with respect to an injective real linear map J : X −→ XC , (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 63 let ∥|·∥| be a norm on XC satisfying the (∗) condition with respect to positive constants k1 and k2 and let (XC )∗ be the dual space of the complex Banach space (XC , ∥| · ∥|). (i) Let ϕ ∈ X∗ and define the map ϕC : XC −→ C by ϕC (J(x) + iJ(y)) = ϕ(x) + iϕ(y) (x, y ∈ X). Then ϕC (J(x)) = ϕ(x) for all x ∈ X, ϕC ∈ (XC )∗ , ∥ϕC ∥op ≤ 2k1 ∥ϕ∥op and ∥ϕ∥op ≤ kk12 ∥ϕC ∥op . (ii) Let λ ∈ (XC )∗ and define the map λR : X −→ R by λR (x) = Re λ(J(x)) (x ∈ X). k2 k1 ∥λ∥op . X∗ Then λR ∈ and ∥λR ∥op ≤ ∗ (iii) Let λ ∈ (XC ) and define the map λI : X −→ R by λI (x) = Im λ(J(x)) Then λI ∈ X∗ and ∥λI ∥op ≤ (x ∈ X). k2 k1 ∥λ∥op . Proof. Let x ∈ X. Then ϕC (J(x)) = ϕC (J(x) + iJ(0)) = ϕ(x) + iϕ(0) = ϕ(x) + i0 = ϕ(x). It is easy to see that ϕC is a complex linear functional on XC . Since |ϕC (J(x) + iJ(y))| = |ϕ(x) + iϕ(y)| ≤ |ϕ(x)| + |ϕ(y)| ≤ 2∥ϕ∥op max{∥x∥, ∥y∥} ≤ 2k1 ∥ϕ∥op ∥|J(x) + iJ(y)∥| for all x, y ∈ X, we deduce that ϕC ∈ (XC )∗ and ∥ϕC ∥op ≤ 2k1 ∥ϕ∥op . On the other hand, we have |ϕ(x)| = |ϕC (J(x))| ≤ ∥ϕC ∥op ∥|J(x)∥| k2 ≤ ∥ϕC ∥op ∥x∥, k1 for all x ∈ X. Hence, ∥ϕ∥op ≤ kk21 ∥ϕC ∥op . Therefore, (i) holds. Clearly, λR is a real linear functional on X. Since |λR (x)| = |Re λ(J(x))| ≤ |λ(J(x))| ≤ ∥λ∥op ∥|J(x)∥| 64 H. ALIHOSEINI AND D. ALIMOHAMMADI ≤ ∥λ∥op k2 ∥x∥, k1 for all x ∈ X, we deduce that λR ∈ X∗ and ∥λR ∥op ≤ kk21 ∥λ∥. Hence, (ii) holds. It is easy to see that λI is a real linear functional on X. Moreover, for each x ∈ X we have |λI (x)| = |Im λ(J(x))| ≤ |λ(J(x))| ≤ ∥λ∥op ∥|J(x)∥| k2 ≤ ∥λ∥op ∥x∥. k1 Hence, λI ∈ X∗ and ∥λI ∥op ≤ k2 k1 ∥λ∥op . Therefore, (iii) holds. □ Lemma 2.2. Let (X, ∥ · ∥) be a real Banach space, let XC be a complexification of X with respect to an injective real linear map J : X −→ XC , let ∥| · ∥| be a norm on XC satisfying (∗) condition with respect to positive constants k1 and k2 and let (XC )∗ be the dual space of the complex Banach space (XC , ∥| · ∥|). Define the map J1 : X∗ −→ (XC )∗ by (2.1) Then: (i) (ii) (iii) (iv) (v) J1 (ϕ) = ϕC , (ϕ ∈ X∗ ). J1 (ϕ)(J(x) + iJ(y)) = ϕ(x) + iϕ(y) for all ϕ ∈ X∗ and x, y ∈ X. J1 is a real linear map from X∗ into (XC )∗ . If λ ∈ (XC )∗ , then λ = J1 (λR ) + iJ1 (λI ). J1 is injective and (XC )∗ = J1 (X∗ ) ⊕ iJ1 (X∗ ). (XC )∗ is a complexification of X∗ with respect to the map J1 : X∗ −→ (XC )∗ defined by (2.1) and k2 ∥J1 (ϕ) + iJ1 (ψ)∥op k1 ≤ 4k2 max{∥ϕ∥op , ∥ψ∥op }, max{∥ϕ∥op , ∥ψ∥op } ≤ for all ϕ, ψ ∈ X∗ . Proof. By part (i) of Lemma 2.1, J1 is well-defined. Let ϕ ∈ X∗ and x, y ∈ X. Then, by part (i) of Lemma 2.1, we have J1 (ϕ)(J(x) + iJ(y)) = ϕC (J(x) + iJ(y)) = ϕC (J(x)) + iϕC (J(y)) = ϕ(x) + iϕ(y). Hence, (i) holds. (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 65 It is easy to see that (ϕ + ψ)C = ϕC + ψC for all ϕ, ψ ∈ X∗ and (αϕ)C = αϕC for all α ∈ R and ϕ ∈ X∗ . Hence, (ii) holds. Let λ ∈ (XC )∗ . By parts (ii) and (iii) of Lemma 2.1, λR , λI ∈ X∗ . Since λ(J(x) + iJ(y)) =λ(J(x)) + iλ(J(y))) = (Re λ(J(x)) + iIm λ(J(x))) + i (Re λ(J(y)) + iIm λ(J(y))) = (λR (x) + iλI (x)) + i (λR (y) + iλI (y)) = (λR (x) + iλR (y)) + i (λI (x) + iλI (y)) =(λR )C (J(x) + iJ(y)) + i(λI )C (J(x) + iJ(y)) = (((λR )C ) + i((λI )C )) (J(x) + iJ(y)) = (J1 (λR ) + iJ1 (λI )(J(x) + iJ(y)) , for all x, y ∈ X, we have λ = J1 (λR ) + iJ1 (λI ). Hence, (iii) holds. Let ϕ ∈ X∗ and J1 (ϕ) = 0. Then ϕC = 0 and so ϕC (J(x)) = 0 for all x ∈ X. This implies that ϕ(x) = 0 for all x ∈ X by part (ii) of Lemma 2.1. Hence, ϕ = 0 and so J1 is injective. By the definition of the map J1 : X∗ −→ (XC )∗ and (iii), we conclude that (2.2) (XC )∗ = J1 (X∗ ) + iJ1 (X∗ ). Let λ ∈ J1 (X∗ ) ∩ iJ1 (X∗ ). Then there exist ϕ, ψ ∈ X∗ such that λ = J1 (ϕ) = iJ1 (ψ). This implies that ϕ(x) = iψ(x) for all x ∈ X and so ϕ(x) = 0 for all x ∈ X since ϕ and ψ are real-valued functions on X. Hence, ϕ = 0 and so λ = J1 (ϕ) = 0. Thus (2.3) J1 (X∗ ) ∩ iJ1 (X∗ ) = {0}. From (2.2) and (2.3) we have (XC )∗ = J1 (X∗ ) ⊕ iJ1 (X∗ ). Therefore, (iv) holds. Applying (ii) and (iv), we deduce that (XC )∗ is a complexification of X∗ with respect to the injective real linear map J1 : X∗ −→ (XC )∗ which is defined by (2.1). Let ϕ, ψ ∈ X∗ . Since |ϕ(x)| ≤ |ϕ(x) + iψ(x)| = |J1 (ϕ)(J(x)) + iJ1 (ψ)(J(x))| = |(J1 (ϕ) + iJ1 (ψ))(J(x))| ≤ ∥J1 (ϕ) + iJ1 (ϕ)∥op ∥|J(x)∥| k2 ≤ ∥J1 (ϕ) + iJ1 (ψ)∥op ∥x∥, k1 66 H. ALIHOSEINI AND D. ALIMOHAMMADI for all x ∈ X, we deduce that ∥ϕ∥op ≤ kk21 ∥J1 (ϕ) + iJ1 (ψ)∥op . Similarly, we have ∥ψ∥op ≤ kk12 ∥J1 (ϕ) + iJ1 (ψ)∥op . Hence, (2.4) max{∥ϕ∥op , ∥ψ∥op } ≤ k2 ∥J1 (ϕ) + iJ1 (ψ)∥op . k1 Since |(J1 (ϕ) + iJ1 (ψ))(J(x) + iJ(y))| = |J1 (ϕ)(J(x) + iJ(y)) + iJ1 (ψ)(J(x) + iJ(y))| = |(ϕ(x) + iϕ(y)) + i(ψ(x) + iψ(y))| ≤ |ϕ(x)| + |ϕ(y)| + |ψ(x)| + |ψ(y)| ≤ ∥ϕ∥op ∥x∥ + ∥ϕ∥op ∥y∥ + ∥ψ∥op ∥x∥ + ∥ψ∥op ∥y∥ ≤ 2∥ϕ∥op max{∥x∥, ∥y∥} + 2∥ψ∥op max{∥x∥, ∥y∥} ≤ 4k1 ∥|J(x) + iJ(y)∥| max{∥ϕ∥op , ∥ψ∥op } for all x, y ∈ X, we deduce that (2.5) ∥J1 (ϕ) + iJ1 (ψ)∥op ≤ 4k1 max{∥ϕ∥op , ∥ψ∥op }. From (2.4) and (2.5) we have k2 ∥J1 (ϕ) + iJ1 (ψ)∥op k1 ≤ 4k2 max{∥ϕ∥op , ∥ψ∥op }. max{∥ϕ∥op , ∥ψ∥op } ≤ Hence, (v) holds. □ Lemma 2.3. Let (X, ∥ · ∥) be a real Banach space, let XC be a complexification of X with respect to an injective real linear map J : X −→ XC , let ∥|·∥| be a norm on XC satisfying (∗) condition with positive constants k1 and k2 and let (XC )∗ be the dual space of (XC , ∥| · ∥|). Define the map J2 : X∗∗ −→ (XC )∗∗ by (2.6) J2 (Φ) = ΦC (Φ ∈ X∗∗ ). Then: (i) J2 (Φ)(J1 (ϕ) + iJ1 (ψ)) = Φ(ϕ) + iΦ(ψ) for all Φ ∈ X∗∗ and ϕ, ψ ∈ X∗ . (ii) J2 is a real linear map from X∗∗ into (XC )∗∗ . (iii) If Λ ∈ (XC )∗∗ , then the maps ΛR , ΛI : X∗ −→ R defined by ΛR (ϕ) = Re Λ(J1 (ϕ)) (ϕ ∈ X∗ ), ΛI (ϕ) = Im Λ(J1 (ϕ)) (ϕ ∈ X∗ ), belong to X∗∗ and Λ = J2 (ΛR ) + iJ2 (ΛI ). (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 67 (iv) J2 is injective and (XC )∗∗ = J2 (X∗∗ ) ⊕ iJ2 (X∗∗ ). (v) (XC )∗∗ is a complexification of X∗∗ with respect to the map J2 : X∗∗ −→ (XC )∗∗ defined by (2.6) and max{∥Φ∥op , ∥Ψ∥op } ≤ 4k1 ∥J2 (Φ) + iJ2 (Ψ)∥op ≤ 16k2 max{∥Φ∥op , ∥Ψ∥op }, X∗∗ . for all Φ, Ψ ∈ (vi) J2 ◦ πX = πXC ◦ J, whenever πY : Y −→ Y ∗∗ is the natural embedding Y in Y ∗∗ defined by πY (y)(λ) = λ(y) (y ∈ Y, λ ∈ Y ∗ ). (vii) X is reflexive if and only if XC is reflexive. Proof. By Lemma 2.2, we deduce that the map J1 : X∗ −→ (XC )∗ defined by (2.1) is an injective real linear map, the complex linear space (XC )∗ is a complexification of X∗ with respect to J1 , λ = J1 (λR ) + iJ1 (λI ) (λ ∈ (XC )∗ ), k2 ∥J1 (ϕ) + iJ1 (ψ)∥op k1 ≤ 4k2 max{∥ϕ∥op , ∥ psi∥op }, max{∥ϕ∥op , ∥ψ∥op } ≤ for all ϕ, ψ ∈ X∗ , and J1 (ϕ)(J(x) + iJ(y)) = ϕ(x) + iϕ(y) for all ϕ ∈ X∗ and x, y ∈ X. Hence, by the definition of J2 , we deduce that (i), (ii), (iii), (iv) and (v) hold. To prove (vi), suppose that x ∈ X. Then for each λ ∈ (XC )∗ we have ((πXC ◦ J)(x))(λ) = (πXC (J(x)))(λ) = λ(J(x)) = (J1 (λR ) + iJ1 (λI ))(J(x)) = (J1 (λR ))(J(x)) + i(J1 (λI ))(J(x)) = λR (x) + iλI (x) = πX (x)(λR ) + iπX (x)(λI ) = J2 (πX )(x)(J1 (λR )) + iJ2 (πX )(x))(J1 (λI )) = J2 (πX )(x)(J1 (λR ) + iJ1 (λI )) = (J2 ◦ πX )(x)(λ). This implies that (2.7) (πXC ◦ J)(x) = (J2 ◦ πX )(x). 68 H. ALIHOSEINI AND D. ALIMOHAMMADI Since (2.7) holds for all x ∈ X, we deduce that πXC ◦ J = J2 ◦ πX . Hence (vi) holds. To prove (vii) we first assume that X is reflexive. Then πX (X) = X∗∗ . Let Λ ∈ (XC )∗∗ . By part (iii) we have Λ = J2 (ΛR ) + iJ2 (ΛI ). Since ΛR , ΛI ∈ X∗∗ , there exist x, y ∈ X such that πX (x) = ΛR and πX (y) = ΛI . Hence, by part (vi) we have Λ = J2 (πX (x)) + iJ2 (πX (y)) = (J2 ◦ πX )(x) + i(J2 ◦ πX )(y) = (πXC ◦ J)(x) + i(πXC ◦ J)(y) = πXC (J(x) + iJ(y)), and so Λ ∈ πXC (XC ). Therefore, πXC is surjective and so XC is reflexive. We now assume that XC is reflexive. Then πXC (XC ) = (XC )∗∗ . Let Φ ∈ X∗∗ . Then J2 (Φ) ∈ (XC )∗∗ and so there exist x, y ∈ X such that J2 (Φ) = πXC (J(x) + iJ(y)). Hence, by part (vi) we have J2 (Φ) + iJ2 (0) = J2 (Φ) = (πXC ◦ J)(x) + i(πXC ◦ J)(y) = (J2 ◦ πX )(x) + i(J2 ◦ πX )(y) = J2 (πX (x)) + iJ2 (πX (y)). This implies that J2 (Φ) = J2 (πX (x)) since (XC )∗∗ = J2 (X∗∗ ) ⊕ iJ2 (X∗∗ ). Therefore, Φ = πX (x) since J2 is injective. Hence, πX is surjective and so X is reflexive. Thus, (vii) holds. □ Lemma 2.4. Let (A, ∥ · ∥) be a real Banach algebra, let AC be a complexification of A with respect to an injective real algebra homomorphism J : A −→ AC , let ∥| · ∥| be an algebra norm on AC satisfying the (∗) condition and let (AC )∗ be the dual space of (AC , ∥| · ∥|). (i) If a ∈ A and ϕ ∈ A∗ , then J1 (a · ϕ) = J(a) · J1 (ϕ), J1 (ϕ · a) = J1 (ϕ) · J(a). (ii) If ϕ ∈ A∗ and Φ ∈ A∗∗ , then J1 (ϕ · Φ) = J1 (ϕ) · J2 (Φ), J1 (Φ · ϕ) = J2 (Φ) · J1 (ϕ). (iii) If Φ, Ψ ∈ A∗∗ , then J2 (Φ□Ψ) = J2 (Φ)□J2 (Ψ), J2 (Φ△Ψ) = J2 (Φ)△J2 (Ψ). (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 69 (iv) If Λ ∈ (AC )∗∗ and λ ∈ (AC )∗ , then Λ · λ = J1 (ΛR · λR − ΛI · λI ) + iJ1 (ΛR · λI + ΛI · λR ), λ · Λ = J1 (λR · ΛR − λI · ΛI ) + iJ1 (λR · ΛI + λI · ΛR ) Proof. Let a ∈ A and ϕ ∈ A∗ . Then, by Lemma 2.3, we have J1 (a · ϕ)(J(b)) = (a · ϕ)(b) = ϕ(ba) = J1 (ϕ)(J(ba)) = J1 (ϕ)(J(b)J(a)) = (J(a) · J1 (ϕ))(J(b)), for all b ∈ A. This implies that J1 (a · ϕ)(J(b) + iJ(c)) = J1 (a · ϕ)(J(b)) + iJ1 (a · ϕ)(J(c)) = (J(a) · J1 (ϕ))(J(b)) + i(J(a) · J1 (ϕ))(J(c)) = (J(a) · J1 (ϕ))(J(b) + iJ(c)), for all b, c ∈ A. Hence, J1 (a · ϕ) = J(a) · J1 (ϕ). Similarly, we can show that J1 (ϕ · a) = J1 (ϕ) · J(a). Therefore, (i) holds. Let ϕ ∈ A∗ and Φ ∈ A∗∗ . Then, by (i), we have J1 (ϕ · Φ)(J(a)) = J1 (ϕ · Φ)(a) = Φ(a · ϕ) = J2 (Φ)(J1 (a · ϕ)) = J2 (Φ)(J(a) · J1 (ϕ)) = (J1 (ϕ) · J2 (Φ))(J(a)) for all a ∈ A. This implies that J1 (ϕ · Φ)(J(a) + iJ(b)) =J1 (ϕ · Φ)(J(a)) + iJ1 (ϕ · Φ)(J(b)) =(J1 (ϕ) · J2 (Φ))(J(a)) + i(J1 (ϕ) · J2 (Φ))(J(b)) =(J1 (ϕ) · J2 (Φ))(J(a) + iJ(b)), for all a, b ∈ A. Hence, J1 (ϕ · Φ) = J1 (ϕ) · J2 (Φ). 70 H. ALIHOSEINI AND D. ALIMOHAMMADI Similarly, we can show that J1 (Φ · ϕ) = J2 (Φ) · J1 (ϕ). Therefore, (ii) holds. Let Φ, Ψ ∈ A∗∗ . Then, by (ii), we have J2 (Φ□Ψ)(J1 (ϕ)) = (Φ□Ψ)(ϕ) = Φ(Ψ · ϕ) = J2 (Φ)(J1 (Ψ · ϕ)) = J2 (Φ)(J2 (Ψ) · J1 (ϕ)) = (J2 (Φ)□J2 (Ψ))(J1 (ϕ)), for all ϕ ∈ A∗ . This implies that J2 (Φ□Ψ)(J1 (ϕ) + iJ1 (ψ)) =J2 (Φ□Ψ)(J1 (ϕ)) + iJ2 (Φ□Ψ)(J1 (ψ)) =(J2 (Φ)□J2 (Ψ)(J1 (ϕ)) + i(J2 (Φ)□J2 (Ψ))(J1 (ψ)) =(J2 (Φ)□J2 (Ψ))(J1 (ϕ) + iJ1 (ψ)), for all ϕ, ψ ∈ A∗ . Hence, J2 (Φ□Ψ) = J2 (Φ)□J2 (Ψ). Similarly, we can show that J2 (Φ△Ψ) = J2 (Φ)△J2 (Ψ). Therefore, (iii) holds. Let Λ ∈ (AC )∗∗ and λ ∈ (AC )∗ . Then by part (iii) of Lemma 2.3 and part (iii) of Lemma 2.2, we have (2.8) Λ = J2 (ΛR ) + iJ2 (ΛI ), λ = J1 (λR ) + iJ1 (λI ). Applying (2.8) and (ii), we get Λ · λ =(J2 (ΛR + iJ2 (ΛI )) · (J1 (λR ) + iJ1 (λI ) =(J2 (ΛR ) · J1 (λR ) − J2 (ΛI ) · J1 (λI )) + i(J2 (ΛR ) · J1 (λI ) + J2 (ΛI ) · J1 (λR )) =(J1 (ΛR · λR ) − J1 (ΛI · λI )) + i(J1 (ΛR · λI ) + J1 (ΛI · λR )) =J1 (ΛR · λR − ΛI λI ) + iJ1 (ΛR · λI + ΛI · λR ). Similarly, we can show that λ · Λ = J1 (λR · ΛR − λI · ΛI ) + iJ1 (λR · ΛI + λI · ΛR ). Hence, (iv) holds. □ (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 71 Theorem 2.5. Let (A, ∥ · ∥) be a real Banach algebra, let AC be a complexification of A with respect to an injective real algebra homomorphism J : A −→ AC , let ∥| · ∥| be an algebra norm on AC satisfying the (∗) condition and let (AC )∗ be the dual space of (AC , ∥| · ∥|). Then A is Arens regular if and only if AC is Arens regular. Proof. We first assume that A is Arens regular. Then (2.9) Φ□Ψ = Φ△Ψ, for all Φ, Ψ ∈ A∗∗ . Let Λ, Γ ∈ (AC )∗∗ . Then, by part (iii) of Lemma 2.3, we have ΛR , ΛI , ΓR , ΓI ∈ A∗∗ and (2.10) Λ = J2 (ΛR ) + iJ2 (λI ), Γ = J2 (ΓR ) + iJ2 (ΓI ). Since (2.9) holds for all Φ, Ψ ∈ A∗∗ , we have ΛR □ΓR = ΛR △ΓR , (2.11) ΛI □ΓR = ΛI △ΓR , ΛR □ΓI = ΛR △ΓI , ΛI □ΓI = ΛI △ΓI . By Lemma 2.4 and according to (2.10) and (2.11), we get Λ□Γ = (J2 (ΛR ) + iJ2 (ΛI )) □ (J2 (ΓR ) + iJ2 (ΓI )) = (J2 (ΛR )□J2 (ΓR ) − (J2 (ΛI )□J2 (ΓI ))) + i ((J2 (ΛR )□J2 (ΓI )) + (J2 (ΛI □J2 (ΓR )) = (J2 (ΛR □ΓR ) − J2 (ΛI □ΓI )) + i (J2 (ΛR □ΓI ) + J2 (ΛI □ΓR )) = (J2 (ΛR △ΓR ) − J2 (ΛI △ΓI )) + i (J2 (ΛR △ΓI ) + J2 (ΛI △ΓR )) = (J2 (ΛR )△J2 (ΓR ) − J2 (ΛI )△J2 (ΓI )) + i (J2 (ΛR )△J2 (ΓI ) + J2 (ΛI )□J2 (ΓR )) = (J2 (ΛR ) + iJ2 (ΛI )) △ (J2 (ΓR ) + iJ2 (ΓI )) =Λ△Γ. Therefore, (AC )∗∗ is Arens regular. We now assume that AC is Arens regular. Then (2.12) Λ□Γ = Λ△Γ, )∗∗ . for all Λ, Γ ∈ (AC Let Φ, Ψ ∈ A∗∗ . Then, by Lemma 2.3, we have ∗∗ J2 (Φ), J2 (Ψ) ∈ (AC ) and so by (2.12) we have (2.13) J2 (Φ)□J2 (Ψ) = J2 (Φ)△J2 (Ψ). Moreover, (2.14) J2 (Φ□Ψ) = J2 (Φ)□J2 (Ψ), J2 (Φ)△J2 (Ψ) = J2 (Φ△Ψ), 72 H. ALIHOSEINI AND D. ALIMOHAMMADI by part (iii) of Lemma 2.4. From (2.13) and (2.14) we get J2 (Φ□Ψ) = J2 (Φ△Ψ). This implies that Φ□Ψ = Φ△Ψ, since J2 is injective. Therefore, A is Arens regular. □ Theorem 2.6. Let (A, ∥ · ∥) be a real Banach algebra, let AC be a complexification of A with respect to an injective real algebra homomorphism J : A −→ AC , let ∥| · ∥| be an algebra norm on AC satisfying the (∗) condition and let (AC )∗ be the dual space of (AC , ∥| · ∥|). Then A∗ is a real Banach A∗∗ -module if and only if (AC )∗ is a complex Banach (AC )∗∗ module. Proof. We prove the result for the first Arens product □ on A∗∗ and (AC )∗∗ . Similarly, on can show that the result hold for the second Arens product △ on A∗∗ and (AC )∗∗ . We first assume that A∗ is a real Banach A∗∗ -module. Then (2.15) (Φ · ϕ) · Ψ = Φ · (ϕ · Ψ), (2.16) ϕ · (Φ□Ψ) = (ϕ · Φ) · Ψ, (Φ□Ψ) · ϕ = Φ · (Ψ · ϕ), (2.17) A∗ ×A∗∗ ×A∗∗ . for all (ϕ, Φ, Ψ) ∈ Let (Λ, λ, Γ) ∈ (AC )∗∗ ×(AC )∗ ×(AC )∗∗ . Then ΛR , ΛI ∈ A∗∗ , λR , λI ∈ A∗ and ΓR , ΓI ∈ A∗∗ . Applying part (iv) of Lemma 2.4 and (2.15), we get (Λ · λ) · Γ = (J1 (ΛR · λR − ΛI · λI ) + iJ1 (ΛR · λI + ΛI · λR )) · Γ =J1 ((ΛR · λR − ΛI · λI ) · ΓR − (ΛR · λI + ΛI · λR ) · ΓI ) + iJ1 ((ΛR · λR − ΛI · λI ) · ΓI + (ΛR · λI + ΛI · λR ) · ΓR ) =J1 ((ΛR · λR ) · ΓR − (ΛI · λI ) · ΓR − (ΛR · λI ) · ΓI − (ΛI · λR ) · ΓI ) + iJ1 ((ΛR · λR ) · ΓI − (ΛI · λI )) · ΓI + (ΛR · λI ) · ΓR + (ΛI · λR ) · ΓR ) =J1 (ΛR · (λR · ΓR ) − ΛI · (λI · ΓR ) − ΛR · (λI · ΓI ) − ΛI · (λR · ΓI )) + iJ1 (ΛR · (λR · ΓI ) − ΛI · (λI · ΓI ) + ΛR · (λI · ΓR ) + ΛI · (λR · ΓR )) =J1 (ΛR · (λR · ΓR − λI · ΓI ) − ΛI · (λI · ΓR + λR · ΓI ) + iJ1 (ΛI · (λR · ΓI − λI · ΓR ) + ΛI · (λR · ΓR − λI · ΓI )) =Λ · (J1 (λR · ΓR − λI · λI ) + iJ1 (λR · ΓI + λI · ΓR )) =Λ · (λ · Γ). Applying part (ii) of Lemma 2.4 and (2.16), we get λ · (Λ□Γ) =(J1 (λR ) + iJ1 (λI )) (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 73 · (J2 (ΛR □ΓR − ΛI □ΓI ) + iJ2 (ΛR □ΓI + ΛI □ΓR )) =J1 (λR · (ΛR □ΓR − ΛI □ΓI ) − λI · (ΛR □ΓI + ΛI □ΓR )) + iJ1 (λR · (ΛR □ΓI + ΛI □ΓR ) + λI · (ΛR □ΓR − ΛI □ΓI )) =J1 (λR · (ΛR □ΓR ) − ΛR · (ΛI □ΓI ) − λI · (ΛR □ΓI ) − λI · (ΛI □ΓR )) + iJ1 (λR · (ΛR □ΓI ) + λR · (ΛI □ΓR ) + λI · (ΛR □ΓR ) − λI · (ΛI □ΓI )) =J1 ((λR · ΛR ) · ΓR − (λR · ΛI ) · ΓI − (λI · ΛR ) · ΓI − (λI · ΛI ) · ΓR ) + iJ1 ((λR · ΛR ) · ΓI + (λR · ΛI ) · ΓR + (λI · ΛR ) · ΓR − (λI · ΛI ) · ΓI ) =J1 ((λR · ΛR − λI · ΛI ) · ΓR − (λR · ΛI + λI · ΛR ) · ΓI ) + iJ1 ((λR · ΛR − λI · ΛI ) · ΓI + (λR · ΛI + λI · ΛR ) · ΓR ) = (J1 (λR · ΛR − λI · ΛI ) + iJ1 (λR · ΛI + λI · ΛR )) · (J2 (ΓR ) + iJ2 (ΓI )) =(λ · Λ) · Γ. Similarly, applying part (ii) of Lemma 2.4 and (2.17) we get (Λ□Γ) · λ = Λ · (Γ · λ). )∗ Therefore, (AC is a complex Banach (AC )∗∗ -module. We now assume that (AC )∗ is a complex Banach (AC )∗∗ -module. Then (2.18) (Λ · λ) · Γ = Λ · (λ · Γ), (2.19) λ · (Λ□Γ) = (λ · Λ) · Γ, (2.20) (Λ□Γ) · λ = Λ · (Γ · λ), for all (Λ, λ, Γ) ∈ (AC )∗∗ ×(AC )∗ ×(AC )∗∗ . Let (Φ, ϕ, Ψ) ∈ A∗∗ ×A∗ ×A∗∗ . Then (J2 (Φ), J1 (ϕ), J2 (Ψ)) ∈ (AC )∗∗ × (AC )∗ × (AC )∗∗ . By (2.18), we have (2.21) (J2 (Φ) · J1 (ϕ)) · J2 (Ψ) = J2 (Φ) · (J1 (ϕ) · J2 (Ψ)). Applying part (ii) of Lemma 2.4 and (2.21), we get J1 ((Φ · ϕ) · Ψ) = J1 (Φ · ϕ) · J2 (Ψ) = (J2 (Φ) · J1 (ϕ)) · J2 (Ψ) = J2 (Φ) · (J1 (ϕ) · J2 (Ψ)) = J2 (Φ) · J1 (ϕ · Ψ) = J1 (Φ · (ϕ · Ψ)). This implies that (Φ · ϕ) · Ψ = Φ · (ϕ · Ψ), since J1 is injective. By (2.19), we have (2.22) J1 (ϕ) · (J2 (Φ) · J2 (Ψ)) = (J1 (ϕ) · J2 (Φ)) · J2 (Ψ). 74 H. ALIHOSEINI AND D. ALIMOHAMMADI Applying part (iii) of Lemma 2.4 and (2.22), we get J1 (ϕ · (Φ□Ψ)) = J1 (ϕ) · J2 (Φ□Ψ) = J1 (ϕ) · (J2 (Φ)□J2 (Ψ)) = (J1 (ϕ) · J2 (Φ)) · J2 (Ψ) = J1 (ϕ · Φ) · J2 (Ψ) = J1 ((ϕ · Φ) · Ψ). This implies that ϕ · (Φ□Ψ) = (ϕ · Φ) · Ψ, since J1 is injective. Similarly, we can show that (Φ□Ψ) · ϕ = Φ · (Ψ · ϕ). Therefore, A∗ is a real Banach A∗∗ -module. □ Applying Theorem 2.6 and [11, Example 2], we give an example of a real Banach algebra A for which A∗ is not a real Banach A∗∗ -module. Example 2.7. Let Z be the set of all integer numbers and l1 (Z) denote the complex Banach algebra consisting of all sequence {an }∞ n=−∞ in C ∑∞ for which n=−∞ |an | < ∞ with convolution product ∗ defined by ∞ 1 a ∗ b = {cn }∞ a = {an }∞ n=−∞ , n=−∞ , b = {bn }n=−∞ ∈ l (Z), ∑∞ 1 where cn = j=−∞ an−j bj for all n ∈ Z and with the l -norm ∥ · ∥1 defined by ∥a∥1 = ∞ ∑ |an |, 1 a = {an }∞ n=−∞ ∈ l (Z). n=−∞ It is shown [11, Example 2] that (l1 (Z))∗ is not a complex Banach (l1 (Z))∗∗ -module. Let τ : Z −→ Z be a bijection additive map. Define } { 1 l1 (Z, τ ) = {an }∞ n=−∞ ∈ l (Z) : aτ (n) = an (n ∈ Z) . It is easy to see that l1 (Z, τ ) is a real closed subalgebra of l1 (Z) and l1 (Z) = l1 (Z, τ ) ⊕ il1 (Z, τ ). Hence, l1 (Z, τ ) is a real Banach algebra with the algebra norm ∥ · ∥1 and l1 (Z) is the complexification of l1 (Z, τ ) with respect to the injective real algebra homomorphism J : l1 (Z, τ ) −→ l1 (Z) defined by J(a) = a, 1 a = {an }∞ n=−∞ ∈ l (Z, τ ). ∞ Since ∥a − ib∥1 = ∥a + ib∥1 for all a = {an }∞ n=−∞ , b = {bn }n=−∞ ∈ 1 l (Z, τ ), we deduce that max{∥a∥1 , ∥b∥1 } ≤ ∥a + ib∥1 ≤ 2 max{∥a∥1 , ∥b∥1 } (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 75 ∞ 1 1 ∗ for all a = {an }∞ n=−∞ , b = {bn }n=−∞ ∈ l (Z, τ ). Therefore, (l (Z, τ )) 1 ∗∗ is not a real Banach (l (Z, τ )) -module by Theorem 2.6. Note that the map τ : Z −→ Z is a bijection additive map if and only if either τ (n) = n for all n ∈ Z or τ (n) = −n for all n ∈ Z. We now discuss the relationship between the (−1)-weak amenability of A∗∗ and (−1)-weak amenability of (AC )∗∗ . For this purpose we need the following lemma. Lemma 2.8. Let (A, ∥ · ∥) be a real Banach algebra, let AC be a complexification of A with respect to an injective real algebra homomorphism J : A −→ AC , let ∥| · ∥| be an algebra norm on AC satisfying (∗) condition and let (AC )∗∗ be the second dual of (AC , ∥| · ∥|). Suppose that A∗ is a real Banach A∗∗ -module. Then: (i) If d ∈ ZR1 (A∗∗ , A∗ ) and Φ ∈ A∗∗ , then J1 (d(Φ)) ∈ (AC )∗ . (ii) If d ∈ ZR1 (A∗∗ , A∗ ) then ∆d ∈ ZC1 ((AC )∗∗ , (AC )∗ ), where the map ∆d : (AC )∗∗ −→ (AC )∗ is defined by (2.23) ∆d (J2 (Φ) + iJ2 (Ψ)) = J1 (d(Φ)) + iJ1 (d(Ψ)), Φ, Ψ ∈ A∗∗ . (iii) The map JZ : ZR1 (A∗∗ , A∗ ) −→ ZC1 ((AC )∗∗ , (AC )∗ ) defined by (2.24) JZ (d) = ∆d , d ∈ ZR1 (A∗∗ , A∗ ) is an injective real linear map. (iv) The complex linear space ZC1 ((AC )∗∗ , (AC )∗ ) is a complexification of the real linear space ZR1 (A∗∗ , A∗ ) with respect to the injective linear map JZ . (v) If ϕ ∈ A∗ , then JZ (δϕ ) = δJ1 (ϕ) . (vi) If λ ∈ (AC )∗ , then δλ = JZ (δλR ) + iJZ (δλI ). (vii) HR1 (A∗∗ , A∗ ) = {0} if and only if HC1 ((AC )∗∗ , (AC )∗ ) = {0}. Proof. Let d ∈ ZR1 (A∗∗ , A∗ ) and Φ ∈ A∗∗ . Then d(Φ) ∈ A∗ and so J1 (d(Φ)) ∈ (AC )∗ by Lemma 2.2. Hence, (i) holds. Let d ∈ ZR1 (A∗∗ , A∗ ) and define ∆d : (AC )∗∗ −→ (AC )∗ by (2.23). Then ∆d is well-defined by (i). It is easy to see that ∆d is a complex linear map from (AC )∗∗ to (AC )∗ . Since ∥| · ∥| be an algebra norm on AC satisfying (∗) condition, there exist positive constants k1 and k2 such that max{∥a∥, ∥b∥} ≤ k1 ∥|J(a) + iJ(b)∥| ≤ k2 max{∥a∥, ∥b∥} for all a, b ∈ A. Applying part (v) of Lemma 2.2 and part (v) of Lemma 2.3, we get ∥∆d (J2 (Φ) + iJ2 (Ψ))∥op = ∥J1 (d(Φ) + iJ1 (d(Ψ)))∥op ≤ 4k1 max{∥d(Φ)∥op , ∥d(Ψ)∥op } ≤ 4k1 ∥d∥op max{∥Φ∥op , ∥Ψ∥op } 76 H. ALIHOSEINI AND D. ALIMOHAMMADI ≤ 4k1 ∥d∥op ∥d∥op 4k1 ∥J2 (Φ) + iJ2 (Ψ)∥op = 16k12 ∥d∥op ∥J2 (Φ) + iJ2 (Ψ)∥op , for all Φ, Ψ ∈ A∗∗ . Therefore, ∆d is a bounded complex linear operator and ∥∆d ∥op ≤ 16k12 ∥d∥op . By Theorem 2.6, (AC )∗ is complex Banach (AC )∗∗ -module. Since d ∈ ZR1 (A∗∗ , A∗ ), by Lemma 2.4, for all Φ, Ψ ∈ A∗∗ we have ∆d (J2 (Φ)□J2 (Ψ)) = ∆d (J2 (Φ□Ψ)) = J1 (d(Φ□Ψ)) = J1 (d(Φ) · Ψ + Φ · d(Ψ)) = J1 (d(Φ) · Ψ) + J1 (Φ · d(Ψ)) = J1 (d(Φ)) · J2 (Ψ) + J2 (Φ) · J1 (d(Ψ)) = ∆d (J2 (Φ))J2 (Ψ)) + J2 (Φ) · ∆d (J2 (Ψ)). This implies that for all Φ, Ψ, Φ′ , Ψ′ ∈ A∗∗ we have ∆d (J2 (Φ)+iJ2 (Ψ))□(J2 (Φ′ + iJ2 (Ψ′ ))) =∆d ((J2 (Φ)□J2 (Φ′ )) − (J2 (Ψ)□J2 (Ψ′ )) + i((J2 (Φ)□J2 (Ψ′ )) + (J2 (Ψ)□J2 (Φ′ ))) =∆d (J2 (Φ)□J2 (Φ′ )) − ∆d (J2 (Ψ)□J2 (Ψ′ )) + i∆d (J2 (Φ)□J2 (Ψ′ )) + i∆d (J2 (Ψ)□J2 (Φ′ )) =(∆d (J2 (Φ)) · J2 (Φ′ ) + J2 (Φ) · ∆d (J2 (Φ′ )) − ∆d (J2 (Ψ)) · J2 (Ψ′ ) − J2 (Ψ) · ∆d (J2 (Ψ′ ))) + i(∆d (J2 (Φ)) · J2 (Ψ′ ) + J2 (Φ) · ∆d (J2 (Ψ′ ))) ( ) + i ∆d (J2 (Ψ)) · J2 (Φ′ ) + J2 (Ψ) · ∆d (J2 (Φ′ )) ) ( = (∆d (J2 (Φ)) + i∆d (J2 (Ψ))) · J2 (Φ′ ) + iJ2 (Ψ′ ) ( ) + (J2 (Φ) + iJ2 (Ψ)) · ∆d (J2 (Φ′ )) + i∆d (J2 (Ψ′ )) ) ( =∆d (J2 (Φ) + iJ2 (Ψ)) · J2 (Φ′ ) + iJ2 (Ψ′ ) + (J2 (Φ) + iJ2 (Ψ)) · ∆d (J2 (Φ′ ) + iJ2 (Ψ′ )). Therefore, ∆d ∈ ZC1 ((AC )∗∗ , (AC )∗ ). Hence, (ii) holds. It is clear that the map JZ : ZR1 (A∗∗ , A∗ ) −→ ZC1 ((AC )∗∗ , (AC )∗ ), defined by (2.24), is a real linear map. Let d ∈ ZR1 (A∗∗ , A∗ ) and JZ (d) = 0. Then ∆d = 0 and so for each Φ ∈ A∗∗ we have 0 = ∆d (J2 (Φ)) = J1 (d(Φ)). (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 77 This implies that d(Φ) = 0 for all Φ ∈ A∗∗ , since J1 is injective. Hence, d = 0 and so JZ is injective. Assume that D ∈ Z1C ((AC )∗∗ , (AC )∗ ). Define the maps DR , DI : A∗∗ −→ A∗ by (2.25) DR (Φ) = (D(J2 (Φ)))R , (2.26) DI (Φ) = (D(J2 (Φ)))I , (Φ ∈ A∗∗ ), (Φ ∈ A∗∗ ). By Lemma 2.1, DR is well-defined. It is easy to see that DR is a real linear map from A∗∗ to A∗ . Applying part (iii) of Lemma 2.1 and part (v) of Lemma 2.3, we have ∥DR (Φ)∥op = ∥(D(J2 (Φ)))R ∥op k2 ≤ ∥D(J2 (Φ))∥op k1 k2 ≤ ∥D∥op ∥J2 (Φ)∥op k1 k2 4k2 ≤ ∥D∥op ∥Φ∥op k1 k1 4k 2 = 22 ∥D∥op ∥Φ∥op k1 for all Φ ∈ A∗∗ . Hence, DR is a bounded real linear operator and ∥DR ∥op ≤ 4k22 ∥D∥op . k12 On the other hand, for all Φ, Ψ ∈ A∗∗ we have DR (Φ□Ψ) = (D(J2 (Φ□Ψ)))R = (D(J2 (Φ)□J2 (Ψ))R = (D(J2 (Φ)) · J2 (Ψ) + J2 (Φ) · D(J2 (Ψ)))R = (D(J2 (Φ)) · J2 (Ψ))R + (J2 (Φ) · D(J2 (Ψ)))R = (D(J2 (Φ)))R · Ψ + Φ · D(J2 (Ψ)))R = DR (Φ) · Ψ + Φ · DR (Ψ). Therefore, DR is a real A∗ -derivation on A∗∗ and so DR ∈ ZR1 (A∗∗ , A∗ ). Similarly, we can show that DI ∈ ZR1 (A∗∗ , A∗ ). Now we show that (2.27) D = JZ (DR ) + iJZ (DI ). Let Φ ∈ A∗∗ . For each a ∈ A we have D(J2 (Φ))(J(a)) =Re D(J2 (Φ))(J(a)) + iIm D(J2 (Φ))(J(a)) =DR (Φ)(a) + iDI (Φ)(a) 78 H. ALIHOSEINI AND D. ALIMOHAMMADI =J1 (DR (Φ))(J(a)) + iJ1 (DI (Φ))(J(a)) =(J1 (DR (Φ)) + iJ1 (DI (Φ))(J(a)) =(JZ (DR )(J2 (Φ)) + iJZ (DI )(J2 (Φ)))(J(a)) =((JZ (DR ) + iJZ (DI ))(J2 (Φ)))(J(a)). This implies that (2.28) D(J2 (Φ)) = (JZ (DR ) + iJZ (DI ))(J2 (Φ)), since D(J2 (Φ)) and (JZ (DR )+iJZ (DI ))(J2 (Φ)) are complex linear mappings from AC to C. Since D and JZ (DR ) + iJZ (DI ) are complex linear mappings from (AC )∗∗ to (AC )∗ and (2.28) holds for each Φ ∈ A∗∗ , we deduce that D(J2 (Φ) + iJ2 (Ψ)) =(JZ (DR ) + iJZ (DI ))(J2 (Φ) + iJ2 (Ψ)) for all Φ, Ψ ∈ A∗∗ . Hence, (2.27) holds. Since (2.27) holds for all D ∈ ZC1 ((AC )∗∗ , (AC )∗ ), we have (2.29) ZC1 ((AC )∗∗ , (AC )∗ ) = JZ (ZR1 (A∗∗ , A∗ )) + iJZ (ZR1 (A∗∗ , A∗ )). Let D ∈ JZ (ZR1 (A∗∗ , A∗ )) ∩ iJZ (ZR1 (A∗∗ , A∗ )). Then there exist two functions d1 , d2 ∈ ZR1 (A∗∗ , A∗ ) such that D = JZ (d1 ) = iJZ (d2 ). Hence, for each Φ ∈ A∗∗ we have J1 (d1 (Φ)) = (JZ (d1 ))(J2 (Φ)) = (iJZ (d2 ))(J2 (Φ)) = i(JZ (d2 ))(J2 (Φ)) = iJ1 (d2 (Φ)), and so J1 (d1 (Φ)) = 0, since J1 (A∗ ) ∩ iJ1 (A∗ ) = {0}. This implies that d1 (Φ) = 0 for all Φ ∈ A∗∗ , since J1 is injective. Hence, d1 = 0 and so D = JZ (d1 ) = 0. Therefore, (2.30) JZ (ZR1 (A∗∗ , A∗ )) ∩ iJZ (ZR1 (A∗∗ , A∗ )) = {0}. From (2.29) and (2.30) we obtain ZC1 ((AC )∗∗ , (AC )∗ ) = JZ (ZR1 (A∗∗ , A∗ )) ⊕ iJZ (ZR1 (A∗∗ , A∗ )). Therefore, (iv) holds. Let ϕ ∈ A∗ . Since JZ (δϕ )(J2 (Φ) + iJ2 (Ψ)) =J1 (δϕ (Φ)) + iJ1 (δϕ (Ψ)) =J1 (Φ · ϕ − ϕ · Φ) + iJ1 (Ψ · ϕ − ϕ · Ψ) = (J1 (Φ · ϕ) − J1 (ϕ · Φ)) + i (J1 (Ψ · ϕ) − J1 (ϕ · Ψ)) = (J2 (Φ) · J1 (ϕ) − J1 (ϕ) · J2 (Φ)) (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 79 + i (J2 (Ψ) · J1 (ϕ) − J1 (ϕ) · J2 (Ψ)) = (J2 (Φ) + iJ2 (Ψ)) · J1 (ϕ) − J1 (ϕ) · (J2 (Φ) + iJ2 (Ψ)) =δJ1 (ϕ) (J2 (Φ) + iJ2 (Ψ)) A∗∗ , for all Φ, Ψ ∈ we deduce that JZ (δϕ ) = δJ1 (ϕ) . Hence (v) holds. ∗ Let λ ∈ (AC ) . By parts (ii) and (iii) of Lemma 2.1 and part (iii) of Lemma 2.2, we have λR , λI ∈ A∗ and (2.31) λ = J1 (λR ) + iJ1 (λI ). Since JZ (δλR ), δJ1 (λR ) ∈ ZC1 ((AC )∗∗ , (AC )∗ ) and JZ (δλR )(J2 (Φ)) = J1 (δλR (Φ)) = J1 (Φ · λR − λR · Φ) = J1 (Φ · λR ) − J1 (λR · Φ) = J2 (Φ) · J1 (λR ) − J1 (λR ) · J2 (Φ) = δJ1 (λR ) (J2 (Φ)) for all Φ ∈ A∗∗ , we conclude that JZ (δλR )(J2 (Φ) + iJ2 (Ψ)) = δJ1 (δR ) (J2 (Φ) + iJ2 (Ψ)) for all Φ, Ψ ∈ A∗∗ . Hence, (2.32) JZ (δλR ) = δJ1 (λR ) . Similar to the argument above we can obtian (2.33) JZ (δλI ) = δJ1 (λI ) . Applying (2.32), (2.33) and (2.31), we get JZ (δλR ) + iJZ (δλI ) = δJ1 (λR ) + iδJ1 (λI ) = δJ1 (λR )+iJ1 (λI ) = δλ . Hence, (vi) holds. To prove (vii), we first assume that (2.34) HR1 (A∗∗ , A∗ ) = {0}. Let D ∈ ZC1 ((AC )∗∗ , (AC )∗ ). By (iv), there exist unique elements d, d′ ∈ ZR1 (A∗∗ , A∗ ) such that (2.35) D = JZ (d) + iJZ (d′ ). By (2.34), there exist ϕ, ϕ′ ∈ A∗ such that (2.36) d = δϕ , d′ = δϕ′ . 80 H. ALIHOSEINI AND D. ALIMOHAMMADI Set λ = J1 (ϕ) + iJ1 (ϕ′ ). Then λ ∈ (AC )∗ and (2.37) ϕ = λR , ϕ′ = λ I . From (2.35), (2.36) and (2.37) we obtain (2.38) D = JZ (δλR ) + iJZ (δλI ). Since λ ∈ (AC )∗ , we deduce that (2.39) δλ = JZ (δλR ) + iJZ (δλI ), by (vi). From (2.38) and (2.39), we have D = δλ and so HC1 ((AC )∗∗ , (AC )∗ ) = {0}. We now assume that (2.40) HC1 ((AC )∗∗ , (AC )∗ ) = {0}. Let d ∈ ZR1 (A∗∗ , A∗ ). Then JZ (d) ∈ ZC1 ((AC )∗∗ , (AC )∗ ). By (2.40), there exists λ ∈ (AC )∗ such that JZ (d) = δλ , and so by (vi) we have (2.41) JZ (d) + iJZ (0) = JZ (d) = JZ (δλR ) + iJZ (δλI ). Applying (2.41) and (iv), we deduce that JZ (d) = JZ (δλR ) and so d = δλR , since JZ is injective. Therefore, HR1 (A∗∗ , A∗ ) = {0} and so (vii) holds. □ Theorem 2.9. Let (A, ∥ · ∥) be a real Banach algebra, let AC be a complexifiction of A with respect to an injective real algebra homomorphism J : A −→ AC , let |∥ · ∥| be an algebra norm on AC satisfying the (∗) condition, and let (AC )∗ be the dual space of (AC , ∥| · ∥|). Then A∗∗ is (−1)-weakly amenable if and only if (AC )∗∗ is (−1)-weakly amenable. Proof. We first assume that A∗∗ is (−1)-weakly amenable. Then A∗ is a real Banach A∗∗ -module and HR1 (A∗∗ , A∗ ) = {0}. Hence, (AC )∗ is a complex Banach (AC )∗∗ -module by Theorem 2.6 and HC1 ((AC )∗∗ , (AC )∗ ) = {0} by part (vii) of Lemma 2.8. Therefore, (AC )∗∗ is (−1)-weakly amenable. We now assume that (AC )∗∗ is (−1)-weakly amenable. Then (AC )∗ is a complex Banach (AC )∗∗ -module and HC1 ((AC )∗∗ , (AC )∗ ) = {0}. Hence, A∗ is a real Banach A∗∗ -module by Theorem 2.6 and so we conclude that HR1 (A∗∗ , A∗ ) = {0} by part (vii) of Lemma 2.8. Therefore, A∗∗ is (−1)weakly amenable. □ Here, as applications of Theorem 2.9, we give some examples of real Banach algebras which their second duals of some them are and of others are not (−1)-weakly amenable. (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 81 Example 2.10. Let A = R with the zero multiplication.Then A is a real Banach algebra with the Euclidean norm | · |. Set AC = C with the zero multiplication. Clearly, AC is a complex Banach algebra with Euclidean norm | · | and AC = A ⊕ iA. Hence, AC is a complexification of A with respect to the injective real algebra homomorphism J : A −→ AC defined by J(a) = a (a ∈ R). Moreover, max{|a|, |b|} ≤ |a + ib| ≤ 2 max{|a|, |b|}, for all a, b ∈ A. It is known [11, Example 2.2] that (AC )∗∗ is not (−1)-weakly amenable. Therefore, A∗∗ is not (−1)-weakly amenable by Theorem 2.9. 1 Example 2.11. Let S be a discrete semigroup. We denote ∑ by l (S) the set of all complex-valued functions f on S for which s∈S |f (s)| < ∞. Then l1 (S) is a self-adjoint complex Banach algebra with the convolution product ∗ defined by ∑ (f ∗ g)(r) = f (s)g(t), f, g ∈ l1 (S), s,t∈S,st=r and with the algebra norm ∥ · ∥1 defined by ∑ |f (s)|, f ∈ l1 (S). ∥f ∥1 = s∈S Let τ : S −→ S be a self-map of S satisfying τ (st) = τ (s)τ (t) for all s, t ∈ S and τ (τ (s)) = s for all s ∈ S. It is easy to see f¯ ◦ τ ∈ l1 (S) for all f ∈ l1 (S). Define l1 (S, τ ) = {f ∈ l1 (S) : f¯ ◦ τ = f }. Then l1 (S, τ ) is a real closed subalgebra of l1 (S) and l1 (S) = l1 (S, τ ) ⊕ il1 (S, τ ). Hence, l1 (S) is the complexification of l1 (S, τ ) with respect to the injective real algebra homomorphism J : l1 (S, τ ) −→ l1 (S) defined by J(f ) = f (f ∈ l1 (S, τ )). Since ∥f − ig∥1 = ∥f + ig∥1 for all f, g ∈ l1 (S, τ ), we deduce that max{∥f ∥1 , ∥g∥1 } ≤ ∥f + ig∥1 ≤ 2 max{∥f ∥1 , ∥g∥1 }, for all f, g ∈ l1 (S, τ ). It is known [11, Example 2.3] that if S 2 ̸= S then (l1 (S))∗∗ is not (−1)-weakly amenable. Therefore, if S 2 ̸= S then (l1 (S, τ ))∗∗ is not (−1)-weakly amenable by Theorem 2.9. Example 2.12. Let N<ω = ∪k∈N Nk and let P be the set of all elements p = (p1 , . . . , pk ) ∈ N<ω such that k ≥ 2 and pj < pj+1 for all j ∈ 82 H. ALIHOSEINI AND D. ALIMOHAMMADI {1, . . . , k−1}. For a sequence α = {αn }∞ n=1 in F and for p = (p1 , . . . pk ) ∈ P , define N (α, p) by   k−1 ∑ |αpj+1 − αpj |2  + |αpn − αp1 |2 . 2(N (α, p))2 =  j=1 For each sequence α = {αn }∞ n=1 in F, we set N (α) = sup{N (α, p) : p ∈ P }. Then N (α) ∈ [0, ∞] for all sequence α = {αn }∞ n=1 in F. Define JF = {α = {αn }∞ n=1 : α ∈ F, N (α) < ∞}. Then JF is a closed subalgebra of Banach algebra (lF∞ (N), ∥ · ∥∞ ) over F, where lF∞ (N) is the set of all sequence α = {αn }∞ n=1 in F for which sup{|αn | : n ∈ N} < ∞ and ∥ · ∥∞ is the algebra norm on lF∞ (N) over F defined by ∥α∥∞ = sup{|αn | : n ∈ N}, ∞ (α = {αn }∞ n=1 ∈ lF (N)). JF is called the James algebra over F. It is clear that JR is a real subalgebra of JC and JC = JR ⊕ iJR . Hence, JC is a complexification of JR with the injective real algebra homomorphism J : JR −→ JC defined by J(α) = α (α ∈ JR ). It is easy to see that max{∥α∥∞ , ∥β∥∞ } ≤ ∥α + iβ∥∞ ≤ 2 max{∥α∥∞ , ∥β∥∞ }, ∞ for all α = {αn }∞ n=1 , β = {β}n=1 ∈ JR . By [6, Theorem 4.1.45], we have some properties of JC as: (i) JC is Arens regular, (ii) JC is weakly amenable, (iii) JC is not amenable. It is shown [8, Example 2.2] that (JC )∗∗ is (−1)-weakly amenable. Therefore, we deduce that JR is weakly amenable by [2, Theorem 2.5], JR is not amenable by [2, Theorem 2.4], JR is Arens regular by Theorem 2.5 and (JR )∗∗ is (−1)-weakly amenable by Theorem 2.9. Example 2.13. Let 1 < p < ∞ and let ∑ lp (Z) denote the set of all ∞ p sequences α = {αn }∞ n=−∞ in C for which n=−∞ |αn | < ∞. Then p l (Z) with the pointwise addition and scalar multiplication is a complex Banach space with the norm ∥ · ∥p defined by ( ∞ )1 p ∑ p , (α = {αn }∞ ∥α∥p = |αn |p n=−∞ ∈ l (Z)). n=−∞ lp (Z) Moreover, with the pointwise multiplication becomes a complex algebra and ∥ · ∥p is a complete algebra norm on lp (Z). Hence, (lp (Z), ∥ · (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 83 ∥p ) is a complex Banach algebra. For each m ∈ Z we have em ∈ lp (Z) and em em = em whenever em = {em,n }∞ n=−∞ and { 1 n=m em,n = (n ∈ Z). 0 n ̸= m Moreover, lp (Z) generates by {em : m ∈ Z}. Hence, lp (Z) is weakly amenable by [6, Proposition 2.8.72(i)]. Therefore, (lp (Z))∗∗ is (−1)weakly amenable since lp (Z) is reflexive. Let τ : Z −→ Z be a bijection additive map. Define } { p (n ∈ Z) . lp (Z, τ ) = α = {αn }∞ n=−∞ ∈ l (Z) : ατ (n) = αn , It is easy to see that lp (Z, τ ) is closed real subalgebra of lp (Z) and lp (Z) = lp (Z, τ ) ⊕ ilp (Z, τ ). Hence, (lp (Z, τ ), ∥ · ∥p ) is a real Banach algebra and lp (Z) is a complexification of lp (Z, τ ) with respect to the injective real algebra homomorphism J : lp (Z, τ ) −→ lp (Z) defined by J(α) = α (α ∈ lp (Z, τ )). Since ∥α − iβ∥p = ∥α + iβ∥p for all α = ∞ p {αn }∞ n=−∞ , β = {βn }n=−∞ ∈ l (Z, τ ), we deduce that max {∥α∥p , ∥β∥p } ≤ ∥α + iβ∥p ≤ 2 max {∥α∥p , ∥β∥p } ∞ p p for all α = {αn }∞ n=−∞ , β = {βn }n=−∞ ∈ l (Z, τ ). Therefore, l (Z, τ ) is reflexive by the reflexivity of lp (Z) and part (vii) of Lemma 2.3, lp (Z, τ ) is weakly amenable by [2, Theorem 2.5] and (lp (Z), τ )∗∗ is (−1)-weakly amenable by Theorem 2.9. Example 2.14. Let X be a compact Hausdorff space. We denote by CF (X) the algebra of all F-valued continuous functions on X over F. Then CF (X) is a Banach algebra over F with the uniform norm ∥ · ∥X defined by ∥f ∥X = sup{|f (x)| : x ∈ X}, (f ∈ C(X)). We write C(X) instead of CC (X). A self-map τ : X −→ X is called a topological involution on X if τ is continuous and τ (τ (x)) = x for all x ∈ X. Clearly, f¯ ◦ τ ∈ C(X) for all f ∈ C(X). Define C(X, τ ) = {f ∈ C(X) : f¯ ◦ τ = f }. Then C(X, τ ) is a real closed subalgebra of C(X), 1X ∈ C(X, τ ) and i1X ∈ / C(X, τ ), where 1X is the constant function on X with value 1. Moreover, C(X) = C(X, τ ) ⊕ iC(X, τ ). Hence, C(X) is a complexification of C(X, τ ) with respect to the injective real algebra homomorphism J : C(X, τ ) −→ C(X) defined by J(f ) = f, (f ∈ C(X, τ )). Since ∥f − ig∥X = ∥f + ig∥X for all f, g ∈ C(X, τ ), we deduce that max{∥f ∥X , ∥g∥X } ≤ ∥f + ig∥X ≤ 2 max{∥f ∥X , ∥g∥X }, 84 H. ALIHOSEINI AND D. ALIMOHAMMADI for all f, g ∈ C(X, τ ). Real Banach algebra C(X, τ ) was first defined by Kulkarni and Limaye in [14]. For further general facts about C(X, τ ) and certain real subalgebras we refer to [15]. Clearly, C(X) is a complex C ∗ -algebra with the natural algebra involution f 7→ f¯ : C(X) −→ C(X). Hence, C(X) is Arens regular and, by [11, Corollary 3.7], (C(X))∗∗ is (−1)-weakly amenable. Therefore, if τ is a topological involution on X then C(X, τ ) is Arens regular by Theorem 2.5 and (C(X, τ ))∗∗ is (−1)-weakly amenable by theorem 2.9. Example 2.15. Let (X, d) be an infinite compact metric space and let α ∈ (0, 1]. We denote by LipF (X, dα ) the set of all F−valued functions f on X for which } { |f (x) − f (y)| : x, y ∈ X, x ̸= y < ∞. p(X,dα ) (f ) = sup dα (x, y) Clearly, LipF (X, dα ) is a subalgebra of CF (X) and 1X ∈ LipF (X, dα ). Moreover, LipF (X, dα ) is a Banach algebra over F with the α-Lipschitz norm ∥ · ∥Lip(X,dα ) defined by ∥f ∥Lip(X,dα ) = ∥f ∥X + p(X,dα ) (f ), (f ∈ LipF (X, dα )). LipF (X, dα ) is called the Lipschits algebra of order α on (X, d) over F. This algebra was first introduced by Sherbert in [19]. We write Lip(X, dα ) instead of LipC (X, dα ). Let (X, d) be a metric space. A Lipschitz mapping on (X, d) is a self-map τ : X −→ X for which there exist a positive constant M such that d(τ (x), τ (y)) ≤ M d(x, y) for all x, y ∈ X. For a Lipschitz mapping τ : X −→ X on (X, d), the constant Lipschitz of τ is denoted by p(τ ) and defined by { } d(τ (x), τ (y)) p(τ ) = sup : x, y ∈ X, x ̸= y . d(x, y) A self-map τ : X −→ X is called a Lipschitz involution on (X, d) if τ is a Lipschitz mapping and τ (τ (x)) = x for all x ∈ X. Let (X, d) be a compact metric space, let α ∈ (0, 1] and let τ : X −→ X be a Lipschitz involution on (X, d). It is easy to see that f¯ ◦ τ ∈ Lip(X, dα ) for all f ∈ Lip(X, dα ). Define { } Lip(X, dα , τ ) = f ∈ Lip(X, dα ) : f¯ ◦ τ = f . Then Lip(X, dα , τ ) is a real closed subalgebra of Lip(X, dα ), containing 1X , i1X ∈ / Lip(X, dα , τ ) and Lip(X, dα ) = Lip(X, dα , τ ) ⊕ iLip(X, dα , τ ). ) Hence, Lip(X, dα , τ ), ∥ · ∥Lip(X,dα ) is a real Banach algebra and the complex algebra Lip(X, dα ) is a complexification of Lip(X, dα , τ ) with ( (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 85 respect to the injective real algebra homomorphism J : Lip(X, dα , τ ) −→ Lip(X, dα ) by J(f ) = f (f ∈ Lip(X, dα , τ )). Moreover, max{∥f ∥Lip(X,dα ) , ∥g∥Lip(X,dα ) } ≤ C∥f + ig∥Lip(X,dα ) } { ≤ 2C max ∥f ∥Lip(X,dα ) , ∥g∥Lip(X,dα ) for all f, g ∈ Lip(X, dα , τ ), where C = (p(τ ))α (see [1]). By [20, Theorem 9.2], Lip(X, dα ) has a nonzero continuous point derivation. Hence, (Lip(X, dα ))∗∗ is not (−1)-weakly amenable by [11, Theorem 2.6]. Therefore, if τ : X −→ X is a Lipschitz involution on (X, d) then (Lip(X, dα , τ ))∗∗ is not (−1)-weakly amenable by Theorem 2.9. Example 2.16. Let (X, d) be a compact metric space, let K be a nonempty compact subset of X and let α ∈ (0, 1]. We denote by Lip(X, K, dα ) the set of all f ∈ C(X) for which f |K ∈ Lip(K, dα ). Then Lip(X, K, dα ) is a complex subalgebra of C(X) and Lip(X, dα ) is a complex subalgebra of Lip(X, K, dα ). Moreover, Lip(X, K, dα ) = C(X) if K is finite and Lip(X, K, dα ) = Lip(X, dα ) if X \ K is finite. Furthermore, Lip(X, K, dα ) is a complex Banach algebra with the algebra norm ∥ · ∥Lip(X,K,dα ) defined by ∥f ∥Lip(X,K,dα ) = ∥f ∥X + p(K,dα ) (f ), f ∈ Lip(X, K, dα ). Lip(X, K, dα ) is called extended Lipschitz algebra of order α on (X, d) with respect to K. This algebra was first studied in [9]. By [16, Theorem 3.3], Lip(X, K, dα ) has a nonzero continuous point derivation if int(K) ∩ K ′ ̸= ∅ where int(K) is the set of all interior points of K and K ′ is the set of all limit points of K in (X, d). Therefore, if int(K) ∩ K ′ ̸= ∅ then (Lip(X, K, dα ))∗∗ is not (−1)-weakly amenable by [12, Theorem 2.6]. Let (X, d) be a compact metric space, let K be compact subset of X, let α ∈ (0, 1] and let τ be a Lipschitz involution on (X, d) such that τ (K) = K. Clearly, f¯ ◦ τ ∈ Lip(X, K, dα ) for all f ∈ Lip(X, K, dα ). Define Lip(X, K, dα , τ ) = {f ∈ Lip(X, K, dα ) : f¯ ◦ τ = f }. It is easy to see that Lip(X, K, dα , τ ) is a real closed subalgebra of Lip(X, K, dα ), 1X ∈ Lip(X, K, dα , τ ) and Lip(X, K, dα ) = Lip(X, K, dα , τ ) ⊕ iLip(X, K, dα , τ ). Hence, (Lip(X, K, dα , τ ), ∥ · ∥Lip(X,K,dα ) ) is a real Banach algebra and Lip(X, K, dα ) is a complexification of Lip(X, K, dα , τ ) with the injective real algebra homomorphism J : Lip(X, K, dα , τ ) −→ Lip(X, K, dα ) 86 H. ALIHOSEINI AND D. ALIMOHAMMADI defined by J(f ) = f (f ∈ Lip(X, K, dα , τ ). Moreover, max{∥f ∥B , ∥g∥B } ≤ C∥f + ig∥B ≤ 2C max{∥f ∥B , ∥g∥B }, for all f, g ∈ Lip(X, K, dα , τ ) where B = Lip(X, K, dα ) and C = (p(τ ))α . Therefore, if int(K)∩K ′ ̸= ∅ and τ : X −→ X is a Lipschitz involution on (X, d) with τ (K) = K, then Lip(X, K, dα , τ ) is not weakly amenable by [2, Theorem 2.5] and (Lip(X, K, dα , τ ))∗∗ is not (−1)-weakly amenable by Theorem 2.9. Example 2.17. Let (X, d) be an infinite compact metric space and α ∈ (0, 1). We denote by lipF (X, dα ) the set of all f ∈ LipF (X, dα ) |f (x) − f (y)| for which limd(x,y)→0 = 0, i.e., for each ε > 0 there exists dα (x, y) |f (x) − f (y)| < ε whenever x, y ∈ X with 0 < d(x, y) < δ > 0 such that dα (x, y) δ. Then lipF (X, dα ) is a closed subalgebra of LipF (X, dα ) over F, and 1X ∈ lipF (X, dα ). Hence, (lipF (X, dα ), ∥ · ∥Lip(X,dα ) ) is a Banach algebra over F. This algebra is called the little Lipschitz algebra of order α on (X, d) over F and was first introduced by Sherbert in [20]. We write lip(X, dα ) instead of lipC (X, dα ). Let (X, d) be an infinite compact metric space, let α ∈ (0, 1) and let B = lip(X, dα ). For each x ∈ X the map eB,x : B −→ C defined by eB,x (f ) = f (x), f ∈ B, belongs to B ∗ . Moreover, ∥eB,x − eB,y ∥op ≤ dα (x, y) for all x, y ∈ X and so the map EB,X : X −→ B ∗ defined by EB,X (x) = eB,x , x ∈ X, is a continuous function from (X, d) to (B ∗ , ∥·∥op ). We know [4, Theorem 3.5] that the map η : B ∗∗ −→ Lip(X, dα ) defined by η(Λ) = Λ ◦ EB,X , Λ ∈ B ∗∗ , is a complex linear isometry from (B ∗∗ , ∥ · ∥op ) onto (Lip(X, dα ), ∥ · ∥Lip(X,dα ) ). It is shown [4, Theorem 3.8] that B is Arens regular and η is an algebra homomorphism. This implies that B ∗ is a complex Banach B ∗∗ -module. Let τ : X −→ X be a Lipschitz involution on (X, d). It is easy to see that f¯ ◦ τ ∈ B for all f ∈ B. Define lip(X, dα , τ ) = {f ∈ B = lip(X, dα ) : f¯ ◦ τ = f }. Then lip(X, dα , τ ) is a real closed subalgebra of B and lip(X, dα ) = lip(X, dα , τ ) ⊕ ilip(X, dα , τ ). (−1)-WEAK AMENABILITY OF REAL BANACH ALGEBRAS 87 Therefore, (lip(X, dα , τ ), ∥ · ∥lip(X,dα ) ) is a real Banach algebra and the complex algebra lip(X, dα ) is a complexification of lip(X, dα , τ ) with respect to the injective real algebra homomorphism J : lip(X, dα , τ ) −→ lip(X, dα ) defined by J(f ) = f (f ∈ lip(X, dα , τ )). Moreover, max{∥f ∥Lip(X,dα ) , ∥g∥Lip(X,dα ) } ≤ C∥f + ig∥Lip(X,dα ) ≤ 2C max{∥f ∥Lip(X,dα ) , ∥g∥Lip(X,dα ) }, for all f, g ∈ lip(X, dα , τ ) where C = (p(τ ))α (see [1]). By Theorem 2.5, we deduce that lip(X, dα , τ ) is Arens regular. Let T = {Z ∈ C : |z| = 1}, let d be the Euclidean metric on T and let α ∈ ( 21 , 1). By [8, Theorem 2.2], (lip(T, dα ))∗∗ is not (−1)-weakly amenable. Therefore, if τ : T −→ T be a Lipschitz involution on T then (lip(T, dα , τ ))∗∗ is not (−1)-weakly amenable by Theorem 2.9. 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Math., (13) (1963), pp. 1387-1399. 20. D.R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc, 111 (1964), pp. 240-272. 1 Department of Mathematics, Faculty of Science, University of Arak, 38156-8-8349, Arak, Iran. E-mail address: hr [email protected] 2 Department of Mathematics, Faculty of Science, University of Arak, 38156-8-8349, Arak, Iran. E-mail address: [email protected]