/compile/output.dvi (-1)-Weak Amenability of Unitized Banach Algebras S. Alireza Hosseinioun1, Arezou Valadkhani 2,∗ 1 University of Arkansas, Department of Mathematical sciences, Fayetteville, AR 72703, USA 2 Department of Mathematics, Shahid Beheshti University, Tehran, Iran Abstract. For a Banach algebra A, its second dual A′′ is (-1)-weakly amenable if A′ is a Banach A′′- bimodule and the first cohomology group of A′′ with coefficients in A′ is zero i.e. H1(A′′,A′) = {0}. We first show that under certain conditions A′ is a Banach A′′-bimodule. We then consider the relationships between (-1)-weak amenability of A and A#, where A# is the unitization of A. 2010 Mathematics Subject Classifications: 46H25 Key Words and Phrases: Banach algebra, (-1)-Weak amenability, Arens products Unitization. 1. Introduction Let A be a Banach algebra and E be a Banach A-bimodule, then a bounded derivation from A into E is a bounded linear mapping D : A −→ E such that D(a · b) = Da · b + a · Db, for each a, b ∈ A. For example let x ∈ X and define δx : A −→ E by δx a = a · x − x · a, then δx is a bounded derivation which is called an inner derivation. Let Z1(A, E) be the space of all bounded derivations from A into E, N1(A, E) be the space of all inner derivations from A into E and the first cohomology group of A with coefficients in E be the quotient space H1(A, X ) = Z1(A, X )/N1(A, X ). A Banach algebra A is amenable if H1(A, E′) = {0} for each Banach A-bimodule E, this concept was introduced by B. E. Johnson in [8]. The notion of weak amenability for commutative Banach algebras was introduced by W. G. Bade, P. C. Curtis and H. G. Dales in [2]. Later Johnson defined weak amenability for arbitrary Banach algebras in [9], in fact a Banach algebra A is weakly amenable if H1(A,A′) = {0}. In [10], A. Medghalchi and T. Yazdanpanah introduced the notion of (-1)-weak amenability. A Banach algebra A is (-1)-weakly amenable if A′ is a Banach A′′-bimodule and H1(A′′,A′) = {0}. There are some examples of non (-1)-weakly amenable Banach algebras. For instance, in [7] we proved that (LipαK)′′ for α ∈ (0,1) and infinite compact metric space K is not ∗Corresponding author. Email addresses: ahosseinioun@yahoo.com (S. Hosseinioun), arezou.valadkhani@yahoo.com (A. Valadkhani) http://www.ejpam.com 231 c© 2016 EJPAM All rights reserved. EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 2, 2016, 231-239 ISSN 1307-5543 – www.ejpam.com S. Hosseinioun, A. Valadkhani / Eur. J. Pure Appl. Math, 9 (2016), 231-239 232 (-1)-weakly amenable. The space lp for 1 < p <∞ is reflexive and weakly amenable, so is (-1)-weakly amenable which is not amenable since it doesn’t factor. Furthermore, the second dual of a C∗-algebra is (-1)-weakly amenable and in the case A′′ is a non-nuclear C∗-algebra, we can conclude that A′′ is (-1)-weakly amenable which is not amenable. Therefore, the notion of (-1)-weak amenability is different from amenability. For more examples see [7] and [9]. Although there are some main theorems and examples which may suggest that the notion of (-1)-weak amenability is close to the notion of weak amenability, there are some examples which prove that these two notions are different, see [8]. Let A be a Banach algebra and A′′ be its second dual, for each a, b ∈ A, f ∈ A′ and F, G ∈ A′′ we define f · a, a · f and F · f , f · F ∈ A′ by f · a(b) = f (a · b), a · f (b) = f (b · a) F · f (a) =F( f · a), f · F(a) = F(a · f ). Now we define F · G, F × G ∈ A′′ as follows F · G( f ) = F(G · f ), F × G( f ) = G( f · F). Then A′′ is a Banach algebra with respect to either of the products · and ×. These products are called the first and the second Arens products on A′′, respectively. A is called Arens regular if F · G = F × G, for all F, G ∈ A′′. Let E be a Banach A-bimodule, then the iterated conjugates of E, denoted by E′, E′′, E′′′, . . . are Banach A-bimodules, and the map ρ : E′′′ −→ E′ with ρ(Γ) = Γ | is an A-bimodule homomorphism which is called natural projection. All concepts and definitions which are not defined in this paper may be found in [4]. 2. When A′ is a Banach A′′-bimodule? In the notion of (-1)-weak amenability, a necessary condition is that "A′ is a Banach A′′- bimodule". Throughout this paper, we shall consider the second dual A′′ with the first Arens product. For the relations between (-1)-weak amenability of (A′′, ·) and (A′′,×), see [9]. Theorem 1. Let A be a Banach algebra. Then in each of the following cases, A′ is a Banach A′′-bimodule: (1) A is Arens regular; (2)  is a left ideal in A′′; (3)  is a right ideal in A′′ and A′′ = A′′ · A. Proof. (1) and (2) are proved in [6]. (3) Let  be a right ideal in A′′ and A′′ · A= A′′. Let a ∈ A, F, G ∈ A′′ and f ∈ A′, then there exist F1 ∈ A′′ and b, c ∈ A such that F = F1 · b and b · G = ĉ, so we have ( f · F) · G(a) = � f · (F1 · b) � · G(a) = ( f · F1) · (b · G)(a) = ( f · F1) · ĉ(a) S. Hosseinioun, A. Valadkhani / Eur. J. Pure Appl. Math, 9 (2016), 231-239 233 =ĉ(a · ( f · F1)) = f · F1(c · a) = F1 · c(a · f ) = � F1 · (b · G) � (a · f ) = F · G(a · f ) = f · (F · G)(a) So A′ is a right A′′-module. On the other hand, there exists d ∈ A such that a · F = d̂ and we have � F · f ) · G � a) =G(a · (F · f )) = G((a · F) · f ) = G(d · f ) =d̂( f · G) = (a · F)( f · G) = F · ( f · G)(a). Therefore A′ is a Banach A′′-bimodule. Remark 1. Dales, Rodrigues-palacios and Velasco in [5] proved that for a Banach algebra A, A′ is an A′′-submodule of A′′′ if and only if A is Arens regular. So under the condition ”A′ is a Banach A′′-bimodule” we can consider a larger class of Banach algebras. Example 1. In each of the following cases by using Theorem 1, A′ is a Banach A′′-bimodule. (1) Let A be a C∗-algebra, then A is Arens regular and A′ is a Banach A′′-bimodule [3]. (2) Let A= l1(N) with product f · g = f (1)g . Then A is a Banach algebra with l1-norm and A is a left ideal in A′′. So A′ is a Banach A′′-bimodule [6]. (3) Let S be an infinite set with product s · t = t for all s, t ∈ S. Then l1(S) is a left ideal in (l1(S))′′ and so (l1(S))′ is a Banach (l1(S))′′ -bimodule. But l1(S) is not a right ideal in (l1(S))′′ [6] (so the third condition in Theorem 1 is not a necessary condition). (4) We know that for each semisimple annihilator Banach algebra A, A is an ideal in A′′ [13]. So A′ is a Banach A′′-bimodule and we have the following assertion: • Let G be an infinite compact group, then L1(G) is not Arens regular but L1(G) is an ideal in � L1(G) �′′ . So � L1(G) �′ is a Banach � L1(G) �′′ -bimodule, whereas L1(G) is not Arens regular (so the first condition in Theorem 1 is not a necessary condition). • Let G be an finite group then M(G) is an ideal in M(G)′′. So M(G)′ is a Banach M(G)′′-bimodule [11] and [12]. (5) Let X be a reflexive Banach space and K L(X ) be the algebra of compact operators on X . Then K L(X ) is an ideal in K L(X )′′ and so (K L(X ))′ is a Banach (K L(X ))′′-bimodule. Note that in the case X has not approximation property K L(X ) is not an annihilator algebra [1]. Now we give an example of a Banach algebra A for which A′ is not a Banach A′′-bimodule. Example 2. Consider A = (l1,∗) for n, m ∈ N. Set an = δ22n , bm = δ22m+1−1 and x = δ1 that (an)n, (bm)m are bounded sequences in l1. There are F, G ∈ A′′ for which F = w∗ − limn ân, G = w∗ − limm b̂m. Now, let S = � 22n + 22m+1 : n, m ∈ N, n< m S. Hosseinioun, A. Valadkhani / Eur. J. Pure Appl. Math, 9 (2016), 231-239 234 and set λ = χS, where χS is characteristic function on S. So (bm ∗ x) ∗ an = δ22n+22m+1 and we have lim n→∞ λ(bm ∗ x ∗ an) = 0, lim m→∞ λ(bm ∗ x ∗ an) = 1. So, (F ·λ) · G(x) =G(x · (F ·λ)) = lim m F(λ · (bm ∗ x)) = lim m lim n λ(bm ∗ x ∗ an) = 0. On the other hand, F · (λ · G)(x) =F((λ · G) · x) = lim n λ · G(x ∗ an) = lim n lim m λ(bm ∗ x ∗ an) = 1. Therefore A′ is not a Banach A′′-bimodule and so A′′ is not (-1)-weakly amenable. Question. Is there any Banach algebra A such that A is amenable but A′ is not A′′-bimodule? 3. Unitization Let A has not unit element and A# = A⊕Ce be the unitization of A. For e ∈ A#, by Hahn-Banach Theorem there exists e′ ∈ A#′ such that e′(e) = 1 and e′(a) = 0 for each a ∈ A, and we can extend λ ∈ A′ to an element of A#′ with λ(e) = 1. So A#′ = Ce′⊕∞A′ and ‖αe′+λ‖=max{|α|,‖λ‖} for α ∈ C and λ ∈ A′. Moreover, A#′ is a Banach space and is a Banach A#-bimodule by module multiplications (αe+ a) · (γe′ +λ) =(αγ+λ(a))e′ +αλ+ a ·λ (γe′ +λ) · (αe+ a) =(αγ+λ(a))e′ +αλ+λ · a where α,γ ∈ C, a ∈ A and λ ∈ A′. Let ê ∈ A′′ with ê(λ) = λ(e), then (A#)′′ = A′′ ⊕Cê. For more details see [4]. Lemma 1. Let A be an Arens regular Banach algebra. Then A′ and A#′ are Banach A#′′-bimodule. Proof. The proof is straightforward. Theorem 2. Let A be an Arens regular Banach algebra and A′′2 = A′′. If A′′ is (-1)-weakly amenable, Then A′′# is (-1)-weakly amenable. Proof. Suppose that A has not unit element and A# = A⊕ Ce be its unitization. By the previous Lemma, A′ is a Banach A′′-bimodule and A#′ is a Banach A#′′-bimodule. Since A′′# is a unital Banach algebra and A′# is a unital A′′#-bimodule and A′′ is a maximal ideal of codimension one in A′′#, by 2.8.23 (iii) in [4] we can conclude that H1(A#′′,A#′) = H1(A′′,A#′). Let D : A′′ −→ A#′ be a bounded derivation. We define S. Hosseinioun, A. Valadkhani / Eur. J. Pure Appl. Math, 9 (2016), 231-239 235 D : A′′ −→ A′ by D(F) = DF |A×{0}, for each F ∈ A′′. Then D is a bounded derivation (Note that DF(a) = DF(a+ 0e). So DF ∈ A′). By (-1)-weakly amenability of A′′, there exists f0 ∈ A′ such that for each F ∈ A′′, DF = δ f0 F . Let D1 = D − D, then D1 is a bounded derivation. Now we show that D1 = 0 (Consider DF as an element in A′ with its extension). For F, G ∈ A′′, there is (b j) j in A with b̂ j w∗ −→ G, then e′ · G(a+αe) = G((a+αe)(0+ e′)) = G(αe′) = lim j b̂ j(αe′) = lim j αe′(b j) = 0. On the other hand, since D : A′′ −→ A#′ and A#′ = A′ ⊕ Ce′, for each F ∈ A′′ there are unique elements, λF ∈ A′ and αF ∈ C such that DF = λF + αF e′. Since DF = DF |A×{0}, and DF = λF then D1F = αF e′. So we have D1(F · G) = D1F · G + F · D1G = αF (e ′ · G) +αG(F · e ′) = 0. Since D1 is bounded then D1|A′′2 = 0. So by the essentiality of A′′, D1 = 0, so D = δ f0 where f0 = f0 + 0e′ ∈ A′ ⊕Ce′ = A#′. Therefore H1(A#′′,A#′) = H1(A′′,A#′) = {0}. Theorem 3. Let A be an Arens regular Banach algebra, A#′′ be (-1)-weakly amenable and H2(A′′,C0) = (0). Then A′′ is (-1)-weakly amenable. Proof. We may suppose that A has not unit element and A# = A⊕C0e. Then Σ : 0 −→ A−→ A# −→ C0 −→ 0 is an admissible short exact sequence and hence so is its dual, Σ′ : 0 −→ C0 −→ A#′ −→ A′ −→ 0. Using 2.8.25 in [4] we have exact sequence S : . . . −→ H1(A′′,C0) −→ H1(A′′,A#′) −→ H1(A′′,A′) −→ H2(A′′,C0) −→ . . . , from 2.8.23 (iii) in [4], H1(A′′,A#′) = H1(A′′#,A#′) = (0), since A′′# is (-1)-weakly amenable. Moreover H2(A′′,C0) = (0), so in the exact sequence S, H1(A′′,A#′) = H2(A′′,C0) = (0) then H1(A′′,A′) = (0). Remark 2. The condition H2(A′′,C0) = (0) in Theorem 3 is not trivial. To this end, let B = � f ∈ A(D) : f (0) = f ′(0) = 0 then B is a closed subalgebra of the disc algebra A(D). Con- sider C0 as the annihilator B-module i.e. B acts trivially on the left and right on C0. Now we define µ : B × B −→ C0, by ( f , g) 7→ f ′′′(0)g ′′′(0). Then µ is a continuous functional for which µ( f , g) = µ(g, f ). If H2(B,C0) = {0}, then for some λ ∈ B′ we have µ= δ1(λ) where δ1(λ)( f , g) = f ·λg −λ( f · g) +λ f · g. (1) S. Hosseinioun, A. Valadkhani / Eur. J. Pure Appl. Math, 9 (2016), 231-239 236 If for z ∈ D we define f , g,h ∈ B by f (z) = z2, g(z) = z4 and h(z) = z3 then f ′′′(z) = 0, g ′′′(z) = 24z, h′′′(z) = 6 and we have µ( f , g) = f ′′′(0)g ′′′(0) = 0 and µ(h,h) = 36. Since C is an annihilator B-module then f ·λg = λ f · g = 0. On the other hand by (1) we have µ( f , g) =δ1(λ)( f , g) = −λ( f · g), µ(h,h) =δ1(λ)(h,h) = −λ(h · h). So λ( f · g) = 0 and λ(h ·h) = −36. But f · g(z) = z2 ·z4 = z6 = h ·h(z), which is a contradiction. So H2(B,C0) 6= {0}. Now consider µ′′ : B′′×B′′ −→ C0, and suppose that for some Λ ∈ B′′′ we have µ′′ = δ1(Λ) and soΛ(Õf · g) = 0 andΛ( ˆh · h) = −36, but f ·g = h·h. So there is noΛ ∈ B′′′ with µ′′ = δ1(Λ). Therefore H2(B′′,C0) 6= {0}. The ext example shows that the converse of Theorem 1 is not true. Example 3. By 4.1.42 in [4], H2(lp,C0) 6= {0} for p > 1 and lp is weakly amenable and reflexive. So lp is (-1)-weakly amenable. Since lp has an approximate identity, then lp = (lp)2 and by Theorem 2, (lp)# is (-1)-weakly amenable (note that (lp)# ′′ = (lp)′′# ≃ lp#). A normed algebra A has π-property if there is a constant c > 0 with 9a9π ≤ c‖a‖, for a ∈ A2, where 9a9π = inf ¦∑∞ j=1 ‖a j‖‖b j‖ : a = ∑∞ j=1 a j b j © , for more details see [4]. By 2.8.21 in [4], a Banach algebra with H2(A,C0) = {0} hasπ-property. Now, by using Theorem 3 we have the following corollary. Corollary 1. Let A be a Banach algebra for which A′′ has π-property. If A#′′ is (-1)-weakly amenable, then A′′ is (-1)-weakly amenable. Theorem 4. Let A be an Arens regular Banach algebra and A′′# is (-1)-weakly amenable. If G(DF) = −F(DG), for each D ∈ Z1(A′′,A′) and each F, G ∈ A′′. Then A′′ is (-1)-weakly amenable. Proof. Let D ∈ Z1(A′′,A′). We define D# : A′′ −→A#′ D#(F)(αe+ a) :=D(F)(a). We prove D# is a derivation. Let F, G ∈ A′′, then there are nets (ai)i and (b j) j in A such that ai w∗ −→ F and b j w∗ −→ G and for each α ∈ C, a ∈ A we have D#F · G(αe+ a) =G((αe+ a) · D#F) = lim j � (αe+ a) · D#F � (b j) = lim j DF(αb j + b ja) = lim j α · b̂ j(DF) + lim j b̂ j(a · DF). So � D#F · G) � αe+ a) = αG(DF) + G(a · DF) and similarly � F · D#G � (αe+ a) = αF(DG) + F(DG · a). S. Hosseinioun, A. Valadkhani / Eur. J. Pure Appl. Math, 9 (2016), 231-239 237 Then we have � D#F · G + F · D#G � (αe+ a) =αG(DF) + G(a · DF) +αF(DG) + F(DG · a) =α (G(DF) + F(DG)) + F · DG(a) + DF · G(a) =(F · DG + DF · G)(a) = D#(F · G)(a+αe). Therefore D# is a bounded derivation and there exists λ1 = λ0 + α0e′ ∈ A#′ such that D#(F) = δλ1 (F), for F ∈ A′′ (Note that since A#′ = A′ ⊕Ce′, λ0 ∈ A′ and α0 ∈ C are unique and H1(A′′,A#′) = H1(A′′#,A#′) = (0)). We show that D = δλ0 . Toward this end, let F ∈ A′′ and a ∈ A, we have (DF)(a) =D#F(a+ 0e) = δλ1 (F)(a+ 0e) = (F ·λ1 −λ1 · F)(a+ 0e) =F(λ1 · (a+ 0e)− (a+ 0e) ·λ1) =F � (λ0 +α0e′)(a+ 0e)− (a+ 0e)(λ0 +α0e′) � =F(λ0 · a− a ·λ0) = (F ·λ0 −λ0 · F)(a) = δλ0 (F)(a). So D = δλ0 . Therefore A′′ is (-1)-weakly amenable. The following example shows that the condition in Theorem 4, is not trivial. Example 4. Let T be the unit circle and A= l ipαT. Let (F̂(n))n∈Z and ( ĝ(n))n∈Z are the Fourier coefficients of F ∈ LipαT and g ∈ l ipαT. We define D as follows D :A′′→ A′ DF(g) = +∞∑ n=−∞ nĝ(n)F̂(n). So D is a derivation which is not inner. Since (l ipαT) ′′ = LipαT, then for F, G ∈ LipαT there are ( fα)α and (gβ)β in l ipαT such that F = w∗ − limα f̂α and G = w∗ − limβ ĝβ . Then we have DFα(gβ) = +∞∑ n=−∞ nĝβ (n) f̂α(−n) = +∞∑ n=−∞ (−n) ĝβ(−n) f̂α(n) =− +∞∑ n=−∞ nĝβ (−n) · f̂α(n) = −(Dgβ )( fα). On the other hand lim β lim α D fα(gβ) = lim β DF(gβ ) = lim β ĝβDF = G(DF), lim β lim α Dgβ ( fα) = lim α lim β Dgβ ( fα) = lim α DG( fα) = F(DG). So F(DG) = −G(DF) where D is a non-inner derivation. S. Hosseinioun, A. Valadkhani / Eur. J. Pure Appl. Math, 9 (2016), 231-239 238 Theorem 5. Let A be a unital Banach algebra and A′′ is commutative and (-1)-weakly amenable. Then Z1(A′′, E) = (0), for each Banach A′′-module E. Proof. Let E be a Banach left A′′-module and define x · F =: F · x for each F ∈ A′′ and x ∈ E. Then E is a Banach right A′′-module and commutativity of A′′ implies that E is a Banach A′′-bimodule (of course E is an A-bimodule and E′ is an A′′-bimodule). Let e be the unit element in A, and let D be a non-zero derivation in Z1(A′′, E). Then for some F0 ∈ A′′, we have DF0 6= 0, so there exists λ ∈ E′ such that λ(DF0) = 1. We define R : E −→A′ R(x)(a) =λ(â · x), (a ∈ A, x ∈ X ). R is a bounded linear map. Now R ◦ D : A′′ −→ A′ is a bounded derivation since R ◦ D(F · G)(a) =R(DF · G + F · DG)(a) = λ(â · (DF · G) + â · (F · DG)) =G ·λ(â · DF) + F ·λ(â · DG). On the other hand for G = w∗− limα b̂α and x ∈ E, the net ( b̂α · x)α is a bounded net in E′′, so ×bα · x w∗ −→ G · x , especially λ(G · x) = limλ( b̂α · x) and we have (R(DF) · G) (a) = lim α (R(DF) · a)(bα) = lim α R(DF)(a · bα) = lim α λ(Ôa · bα · DF) =λ · G(â · DF) = G ·λ(â · DF). Similarly (F · R(DG)) (a) = F ·λ(a · DG). Therefore R ◦ D is a derivation in Z1(A′′,A′). Now, since A′′ is (-1)-weakly amenable and commutative then R ◦ D = 0. But R ◦ D(F0)(e) = R(DF0)(e) = λ(e · DF0) = 1, which is a contradiction. So D = 0 and we have Z1(A′′, E) = 0 Now we recall some Theorems which are used in the following corollaries. Theorem 6. For a commutative Banach algebra A, if A is weakly amenable, then Z1(A, E) = (0) for each Banach A-module E. Theorem 7. Let A be a commutative Banach algebra. 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