Let $A$ and $B$ be unital Banach algebras‎, ‎$X$ be an unital $A-B-$module and $T$ be the triangular Banach algebra associated to $A‎, ‎B$ and $X$‎. The structure of some derivations on triangular Banach algebras was studied by some authors. ‏‎Note that despite the apparent similarity between derivations and biderivations and also inner derivations and inner biderivations‎, ‎there are fundamental differences between them‎. Although there are some studying of biderivations on triangular Banach algebras, any of them do not completely determine the structure of biderivations on triangular Banach algebras. In this paper, we ‎completely characterize biderivations and inner biderivations from $T\times T$ to $T^*$‎ and we show that the first bicohomology group $BH^1(T, T^*)$ is equal to $BH^1(A, A^*)\oplus BH^1(B, B^*)$‎‏

Barootkoob, Sedigheh

English

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NIŠ)Ser. Math. Inform. Vol. 35, No 4 (2020), 929–937https://doi.org/10.22190/FUMI2004929BCHARACTERIZATION OF SOME BIDERIVATIONS ONTRIANGULAR BANACH ALGEBRASSedigheh Barootkoobc© 2020 by University of Nǐs, Serbia | Creative Commons Licence: CC BY-NC-NDAbstract. Let A and B be unital Banach algebras, X be a unital A-B-module andT be the triangular Banach algebra associated to A,B and X. The structure of somederivations on triangular Banach algebras was studied by some authors. Note thatdespite the apparent similarity between derivations and biderivations and also innerderivations and inner biderivations, there are fundamental differences between them.Although there are some studies of biderivations on triangular Banach algebras, any ofthem do not completely determine the structure of biderivations on triangular Banachalgebras. In this paper, we completely characterize biderivations and inner biderivationsfrom T × T to T ∗ and we show that the first bicohomology group BH1(T, T ∗) is equalto BH1(A,A∗)⊕BH1(B,B∗).Keywords: unital Banach algebras; triangular Banach algebra; bicohomology group;biderivations.1. IntroductionA derivation from a Banach algebra A to a Banach A-module X is a boundedlinear mapping d : A → X such that for each a, b ∈ A, d(ab) = d(a)b + ad(b). Foreach x ∈ X the mapping δx : a → ax− xa, (a ∈ A) is a bounded derivation, calledan inner derivation.Let A be a Banach algebra and X be an A-module. A bounded bilinear mappingD : A × A → X is called a biderivation if D is a derivation with respect to botharguments. That is, the mappings aD : A → X and Db : A → X whereaD(b) = D(a, b) = Db(a) (a, b ∈ A),are derivations. We denote the space of such biderivations by BZ1(A,X).Received October 17, 2019; accepted November 09, 20202020 Mathematics Subject Classification. Primary 15A78; Secondary 46H25929930 S. BarootkoobLet x ∈ Z(A,X) = {x ∈ X ; ax = xa ∀a ∈ A}. The map Dx : A×A → X thatDx(a, b) = x[a, b] = xab − xba (a, b ∈ A),is a basic example of a biderivation which is called an inner biderivation. Wedenote the space of such inner biderivations by BN1(A,X). Also we define the firstbicohomology group BH1(A,X) as follows,BH1(A,X) =BZ1(A,X)BN1(A,X).For more applications and details about biderivations see [6, Section 3]. Also see[5, 8], in which the structures of some biderivations on triangular algebras andgeneralized matrix algebras and when these biderivations on these algebras areinner, were studied.Let A and B be Banach algebras and X be an A-B-module. Then the algebraT ={(a x0 b); a ∈ A, x ∈ X, b ∈ B}equipped with the usual addition and multiplication of matrix and with the norm∥∥∥∥(a x0 b)∥∥∥∥= ‖a‖+ ‖x‖+ ‖b‖is a Banach algebra which is called triangular Banach algebra associated to X .Then the dual of triangular Banach algebra T isT ∗ ={(f h0 g); f ∈ A∗, h ∈ X∗, g ∈ B∗};where(f h0 g)((a x0 b))= f(a) + h(x) + g(b).Recall that for every Banach A-module X the dual space X∗ is a Banach A-module with module structures a · f and f · a that a · f(x) = f(xa) and f · a(x) =f(ax). So T ∗ is a T -module with the module actions(a x0 b)·(f h0 g)=(a · f + x · h b · h0 b · g)and(f h0 g)·(a x0 b)=(f · a h · a0 h · x+ g · b),for every(a x0 b)∈ T and(f h0 g)∈ T ∗.A Banach algebra A is called weakly amenable if every derivation from A to A∗is an inner derivation. The concept of weak amenability of Banach algebras wasCharacterization of Some Biderivations on Triangular Banach Algebras 931introduced by Bade, Curtis and Dales [1] for commutative Banach algebras andthen by Johnson [10] for a general Banach algebra.In this paper, we consider A and B as unital Banach algebras and X as aunital A-B-module, that is, 1Ax = x1B = x, for every x ∈ X . We characterizethe biderivations from T × T to T ∗. In particular, we show that BH1(T, T ∗) =BH1(A,A∗)⊕BH1(B,B∗).2. Biderivations and biamenability of triangular Banach algebrasSimilar to the definitions of amenability or weak amenability of Banach algebraswe may define the notions of biamenability [2] or weak biamenability of Banachalgebras [3].Definition 2.1. We say that a Banach algebra A is weakly biamenable ifBH1(A,A∗) = {0}.Example 2.1. (i) Let A be a Banach space and θ ∈ A∗. Then A with the productab = θ(a)b (a, b ∈ A),is a Banach algebra and θ becomes a homomorphism. Also for each h ∈ A∗ anda, b, c ∈ A we have h · a = θ(a)h and a · h = h(a)θ and since θ(ab) = θ(ba), we have[a, b]c = [b, a]c. Now consider f ∈ A∗ such that for some a0, b0 ∈ A, θ(a0)f(b0) 6=f(a0)θ(b0). Define the biderivation D : A× A → A∗ by D(a, b) = δδf (a)(b), for eacha, b ∈ A. Then since D is non zero and the only inner biderivation from A× A intoA∗ is zero, we conclude that A with this product is not weakly biamenable.(ii) Let B(H) be the Banach algebra of operators on Hilbert space H and D : B(H)×B(H) → B(H)∗ be a biderivation. Then similar to Lemma 1 of [5] D(T, S)[U, V ] =[T, S]D(U, V ) for each T, S, U, V ∈ B(H). Also by Lemma 5.8 of [12] B(H) =span{UV − V U ;V, U ∈ B(H)}. Therefore there exist {Ui}, {Vi} in B(H) such thatI =∑i[Ui, V i]. Now we haveD(T, S) = D(T, S)I= D(T, S)∑i[Ui, V i]=∑i[T, S]D(Ui, Vi)= [T, S]∑iD(Ui, Vi)and similarly D(T, S) =∑iD(Ui, Vi)[T, S]. So if we put x =∑iD(Ui, Vi), thenx ∈ Z(B(H),B(H)∗) and D(T, S) = x[T, S]. That is D is an inner biderivation andso B(H) is weakly biamenable.Note that despite the apparent similarity between derivations and biderivations andalso inner derivations and inner biderivations, there are fundamental differencesbetween them. Especially when a biderivation wants to be an inner bidetivationthese differences become more apparent. A part of these differences comes from thenature of biderivations which depend on two components. Another essential part932 S. Barootkoobof these differences goes back to the definition of inner biderivations which dependson the implemented elements that should be in Z(A,X). According to this, theconcept of amenability and also weak amenability have a different nature frombiamenability and weak biamenability, respectively. Indeed, there are examples ofBanach algebras that are biamenable while they are not amenable and there areBanach algebras that are amenable while they are not biamenable [2]. Also, ifwe consider the definition of a biamenable group G such that G × G is amenable,then we see that the Johnson’s theorem [11] is not valid for biamenability. Indeed,each abelian group G is biamenable while the commutative group algebra L1(G) isnot biamenable [2]. Of course, the situation of weak biamenability is better thanbiamenability, and many similar results of [4] and [7] are valid for weak biamenabilityof Banach algebras. Also, for each locally compact abelian groupG, L1(G) is weaklybiamenable. For more detales, see [3].The next theorem characterizes all biderivations from T × T to T ∗.Theorem 2.1. A bilinear mapping D : T × T → T ∗ is a biderivation if and onlyif there exist biderivations dA : A×A → A∗ and dB : B ×B → B∗ such thatD((a x0 b),(a′ x′0 b′))=(dA(a, a′) 00 dB(b, b′)).Proof. It is easy to verify that ifD((a x0 b),(a′ x′0 b′))=(dA(a, a′) 00 dB(b, b′)),for some biderivations dA : A × A → A∗ and dB : B × B → B∗, then D is abiderivation.Conversely, let D : T × T → T ∗ be a biderivation. Since for every a, a′ ∈A, b, b′ ∈ B and x, x′ ∈ X we havea x0 bD((a′ x′0 b′))= D((a x0 b),(a′ x′0 b′))=Da′ x′0 b′((a x0 b))and a x0 bD and Da′ x′0 b′are derivations, according to [9] there existderivations d(a′,x′,b′), d′(a,x,b) : A → A∗ and δ(a′,x′,b′), δ′(a,x,b) : B → B∗ and alsoCharacterization of Some Biderivations on Triangular Banach Algebras 933k(a′,x′,b′), k′(a,x,b) ∈ X∗ such that(d(a′,x′,b′)(a)− xk(a′,x′,b′) k(a′,x′,b′)a− bk(a′,x′,b′)0 k(a′,x′,b′)x+ δ(a′,x′,b′)(b))= Da′ x′0 b′((a x0 b))= D((a x0 b),(a′ x′0 b′))= a x0 bD((a′ x′0 b′))=(d′(a,x,b)(a′)− x′k′(a,x,b) k′(a,x,b)a′ − b′k′(a,x,b)0 k′(a,x,b)x′ + δ′(a,x,b)(b′)).In particular(d(a′,0,0)(a) k(a′,0,0)a0 0)= D((a 00 0),(a′ 00 0))=(d′(a,0,0)(a′) k′(a,0,0)a′0 0).Define dA : A × A → A∗ by dA(a, a′) = d(a′,0,0)(a) = d′(a,0,0)(a′). Then obviouslydA is a bounded biderivation.Similarly we can define the biderivation dB : B ×B → B∗ such that dB(b, b′) =δ(0,0,b′)(b) = δ′(0,0,b)(b′). Also we have(d(0,0,b′)(a) k(0,0,b′)a0 0)= D((a 00 0),(0 00 b′))=(0 −b′k′(a,0,0)0 δ′(a,0,0)(b′)).Sod(0,0,b′)(a) = 0, δ′(a,0,0)(b′) = 0 and k(0,0,b′)a = −b′k′(a,0,0).(2.1)On the other hand(d(0,x′,0)(a) k(0,x′,0)a0 0)= D((a 00 0),(0 x′0 0))=(−x′k′(a,0,0) 00 k′(a,0,0)x′).Therefore d(0,x′,0)(a) = −x′k′(a,0,0), k(a′,0,0)a = 0 and k′(a,0,0)x′ = 0. In particu-lar for each x′ ∈ X , −x′k′(1A,0,0) = d(0,x′,0)(1A) = 0 and since X is a unital A-module, k′(1A,0,0) = 0. On the other hand by (2.1) we have k(0,0,1B) = k(0,0,1B)1A =934 S. Barootkoob−1Bk′(1A,0,0)= 0 and henced(0,x′,0)(a) = −x′k′(a,0,0) = −x′1Bk′(a,0,0) = x′k(0,0,1B)a = 0.SoD((a 00 0),(a′ x′0 b′))= D((a 00 0),(a′ 00 0))+ D((a 00 0),(0 x′0 0))+ D((a 00 0),(0 00 b′))=(dA(a, a′) k(a′,0,0)a+ k(0,x′,0)a+ k(0,0,b′)a0 0).Now since k(a+a′,0,0) = k(a,0,0)+k(a′,0,0), k(0,x+x′,0) = k(0,x,0)+k(0,x′,0) and k(0,0,b+b′) =k(0,0,b) + k(0,0,b′), we can define the linear mappingh : A⊕X ⊕B → X∗(a, x, b) 7→ k(a,0,0) + k(0,x,0) + k(0,0,b)and we have D((a 00 0),(a′ x′0 b′))=(dA(a, a′) h(a′, x′, b′)a0 0).Similarly we haveD((0 00 b),(a′ x′0 b′))=(0 −bh(a′, x′, b′)0 dB(b, b′))andD((0 x0 0),(a′ x′0 b′))=(−xh(a′, x′, b′) 00 h(a′, x′, b′)x).So we haveD((a x0 b),(a′ x′0 b′))=(dA(a, a′)− xh(a′, x′, b′) h(a′, x′, b′)a− bh(a′, x′, b′)0 h(a′, x′, b′)x+ dB(b, b′)).Also we can show similarly there is a bounded linear mappings h′ : A⊕X⊕B → X∗such thatD((a x0 b),(a′ x′0 b′))=(dA(a, a′)− x′h′(a, x, b) h′(a, x, b)a′ − b′h′(a, x, b)0 h′(a, x, b)x′ + dB(b, b′)).Characterization of Some Biderivations on Triangular Banach Algebras 935Therefore h(a′, x′, b′)a− bh(a′, b′, x′) = h′(a, x, b)a′ − b′h′(a, x, b). Soh(a, x, b) = h(a, x, b)1A − 0Bh(a, x, b)= h′(1A.0, 0)a− bh′(1A, 0, 0)= k′(1A, 0, 0)a− bk′(1A, 0, 0)= 0.That is,D((a x0 b),(a′ x′0 b′))=(dA(a, a′) 00 dB(b, b′)).Proposition 2.1. The biderivation D : T × T → T ∗ which is defined for eacha, a′ ∈ A, b, b′ ∈ B and x, x′ ∈ X, byD((a x0 b),(a′ x′0 b′))=(dA(a, a′) 00 dB(b, b′))is an inner biderivation if and only if dA and dB are inner biderivations.Proof. If dA and dB are inner biderivations, then there are f ∈ Z(A,A∗) andg ∈ Z(B,B∗) such that for each a, a′ ∈ A, dA(a, a′) = f [a, a′] = faa′ − fa′a andfor each b, b′ ∈ B, dB(b, b′) = g[b, b′] = gbb′ − gb′b. Now we haveD((a x0 b),(a′ x′0 b′))=(dA(a, a′) 00 dB(b, b′))=(f [a, a′] 00 g[b, b′])=(f 00 g)[(a x0 b),(a′ x′0 b′)].Also it is easy to see that(f 00 g)∈ Z(T, T ∗) if and only if f ∈ Z(A,A∗) andg ∈ Z(B,B∗). Hence D is an inner biderivation.Conversely, if D is an inner biderivation, then there exists(f h0 g)∈ Z(T, T ∗)such thatD((a x0 b),(a′ x′0 b′))=(f h0 g)[(a x0 b),(a′ x′0 b′)].In particular(dA(a, a′) 00 0)= D((a 00 0),(a′ 00 0))=(f h0 g)[(a 00 0),(a′ 00 0)]=(f [a, a′] h[a, a′]0 0).936 S. BarootkoobHence dA(a, a′) = f [a, a′] and h[a, a′] = 0. On the other hand for each a ∈ A wehave(fa ha0 0)=(f h0 g)(a 00 0)=(a 00 0)(f h0 g)=(af 00 0).That is f ∈ Z(A,A∗) and ha = 0, that is h = 0. So dA is an inner biderivation.Similarly we can show that dB is an inner biderivation.Note that in the latter proposition it is also proved thatZ(T, T ∗) ={(f 00 g); f ∈ Z(A,A∗), g ∈ Z(B,B∗)}.Theorem 2.2. BH1(T × T, T ∗) = BH1(A×A,A∗)⊕BH1(B ×B,B∗)Proof. Defineϕ : BZ1(A×A,A∗)⊕BZ1(B ×B,B∗) → BH1(T × T, T ∗),(dA, dB) 7→[(dA 00 dB)]where[(dA 00 dB)]is the equivalent class of(dA 00 dB)in BH1(T × T, T ∗).Then by Theorem 2.1, ϕ is onto and by Proposition 2.1 we havekerϕ ={(dA, dB);(dA 00 dB)is inner}= {(dA, dB); dA and dB are inner}= BN1(A×A,A∗)⊕BN1(B ×B,B∗).ThereforeBH1(T × T, T ∗) =BZ1(A×A,A∗)⊕ BZ1(B ×B,B∗)BN1(A×A,A∗)⊕ BN1(B ×B,B∗)= BH1(A×A,A∗)⊕BH1(B ×B,B∗).Corollary 2.1. T is weakly biamenable if and only if A and B are weakly bia-menable.Characterization of Some Biderivations on Triangular Banach Algebras 937For example if A is a commutative Banach algebra and there is a non zero bideriva-tion from A×A into A∗, then since the only inner biderivation from A×A into A∗is zero, A and therefore T are not weakly biamenable.Acknowledgment. The useful comments of the anonymous referees are gratefullyacknowledged.REFERENCES1. W. G. Bade, P. C. Curtis, and H. G. Dales, Amenability and weak amenabilityfor Beurling and Lipschitz algebras. Proc. London Math. Soc. 55 (1987), 359–377.2. S. Barootkoob, On biamenability of Banach algebras. preprint.3. S. Barootkoob, On n-weak biamenability of Banach algebras. preprint.4. S. Barootkoob and H.R. Ebrahimi Vishki, Lifting derivations and n-weak amenabilityof the second dual of a Banach algebra, Bull. Austral. Math. Soc. 83 (2011), 222-129.5. D. Benkovič, Biderivations of triangular algebras. Linear Algebra Appl. 431 (2009),1587-1602.6. M. Brešar, Commuting maps: A survey. Taiwanese J. Math. 8 (2004), 361–397.7. H. G. Dales, F. Ghahramani and N. Grønbæk, Derivations into iterated dualsof Banach algebras. Studia Math., 128 (1) (1998), 19–54.8. Y. Du and Y. Wang, Biderivations of generalized matrix algebras. Linear Algebra.Appl. 438 (2013), 4483–4499.9. B. E. Forrest and L. W. Marcoux, Derivations on triangular Banach algebras.Indiana Univ. Math. J., 45 (1996), 441-462.10. B. E. Johnson, Weak amenability of group algebras. Bull. London Math. Soc. 23(1991), 281–284.11. V. Runde, Lectures On Amenability. Lecture Notes In Mathematics, 1774, Springer(2002).12. Y. Zhang, Weak amenability of module extension of Banach algebras. Trans. Amer.Math. Soc. 354 (2002), 4131-4151.Sedigheh BarootkoobFaculty of Basic SciencesDepartment of MathematicsP. O. Box 1339University of Bojnord, Irans.barutkub@ub.ac.ir

CHARACTERIZATION OF SOME BIDERIVATIONS ON TRIANGULAR BANACH ALGEBRAS

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CHARACTERIZATION OF SOME BIDERIVATIONS ON TRIANGULAR BANACH ALGEBRAS

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