Let Alg N be a nest algebra associated with the nest N on a (real or complex) Banach space X. Suppose that there exists a non-trivial idempotent P∈Alg N with range P(X)∈?N, and δ:Alg N→Alg N is a continuous linear mapping (generalized) left derivable at P, i.e. δ(ab) = aδ(b) + bδ(a) (δ(ab) = aδ(b) + bδ(a) - baδ(I)) for any a,b∈Alg N with ab = P, where I is the identity element of AlgN. We show that δ is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps on some nest algebras Alg???? with the property that δ(P) = 2Pδ(P) or δ(P) = 2Pδ(P) - Pδ(I) for every idempotent P in Alg N

Ghahramani, Hoger

Sattari, Saman

English

Platforma czasopism naukowych Uniwersytetu Śląskiego w Katowicach

Annales Mathematicae Silesianae 33 (2019), 97–105DOI: 10.2478/amsil-2019-0001LEFT DERIVABLE MAPS AT NON-TRIVIALIDEMPOTENTS ON NEST ALGEBRASHoger Ghahramani, Saman SattariAbstract. Let AlgN be a nest algebra associated with the nest N on a (realor complex) Banach space X. Suppose that there exists a non-trivial idempo-tent P ∈ AlgN with range P (X) ∈ N , and δ : AlgN → AlgN is a continuouslinear mapping (generalized) left derivable at P , i.e. δ(ab) = aδ(b) + bδ(a)(δ(ab) = aδ(b) + bδ(a) − baδ(I)) for any a, b ∈ AlgN with ab = P , whereI is the identity element of AlgN . We show that δ is a (generalized) Jordanleft derivation. Moreover, in a strongly operator topology we characterize con-tinuous linear maps δ on some nest algebras AlgN with the property thatδ(P ) = 2Pδ(P ) or δ(P ) = 2Pδ(P )− Pδ(I) for every idempotent P in AlgN .1. IntroductionThroughout this paper, all algebras and vector spaces will be over F, whereF is either the real field R or the complex field C. Let A be an algebra withunity 1, M be a left A-module and δ : A → M be a linear mapping. Themapping δ is said to be a left derivation (or a generalized left derivation) ifδ(ab) = aδ(b) + bδ(a) (or δ(ab) = aδ(b) + bδ(a) − baδ(1)) for all a, b ∈ A. Itis called a Jordan left derivation (or a generalized Jordan left derivation) ifδ(a2) = 2aδ(a) (or δ(a2) = 2aδ(a) − a2δ(1)) for any a ∈ A. Obviously, any(generalized) left derivation is a (generalized) Jordan left derivation, but ingeneral the converse is not true (see [15, Example 1.1]). The concepts of leftderivation and Jordan left derivation were introduced by Brešar and VukmanReceived: 14.01.2018. Accepted: 10.02.2019. Published online: 05.04.2019.(2010) Mathematics Subject Classification: 47B47, 47L35.Key words and phrases: nest algebra, left derivable, left derivation.98 Hoger Ghahramani, Saman Sattariin [4]. For results concerning left derivations and Jordan left derivations werefer the readers to [10] and the references therein.In recent years, several authors studied the linear (additive) maps thatbehave like homomorphisms, derivations or left derivations when acting onspecial products (for instance, see [3, 7, 9, 6, 11, 12, 16] and the referencestherein). In this article we study the linear maps on nest algebras behavinglike left derivations at idempotent-product elements.Let A be an algebra with unity 1,M be a left A-module and δ : A→M be alinear mapping. We say that δ is left derivable (or generalized left derivable) ata given point z ∈ A if δ(ab) = aδ(b)+ bδ(a) (or δ(ab) = aδ(b)+ bδ(a)− baδ(1))for any a, b ∈ A with ab = z. In this paper, we characterize the continuouslinear maps on nest algebras which are (generalized) left derivable at a non-trivial idempotent operator P . Moreover, in a strongly operator topology wedescribe continuous linear maps δ on some nest algebra AlgN with the pro-perty that δ(P ) = 2Pδ(P ) or δ(P ) = 2Pδ(P ) − Pδ(I) for every idempotentP in AlgN , where I is the identity element of AlgN .The following are the notations and terminologies which are used through-out this article.Let X be a Banach space. We denote by B(X) the algebra of all boundedlinear operators on X, and F(X) denotes the algebra of all finite rank operatorsin B(X). A subspace lattice L on a Banach space X is a collection of closed(under norm topology) subspaces of X which is closed under the formationof arbitrary intersection and closed linear span (denoted by ∨), and whichincludes {0} and X. For a subspace lattice L, we define AlgL byAlgL = {T ∈ B(X) |T (N) ⊆ N for allN ∈ L}.A totally ordered subspace lattice N on X is called a nest and AlgN is calleda nest algebra. When N 6= {{0},X}, we say that N is non-trivial. It is clearthat if N is trivial, then AlgN = B(X). Denote AlgF N := AlgN ∩ F(X),the set of all finite rank operators in AlgN and for N ∈ N , let N− = ∨{M ∈N |M ⊂ N}. The identity element of a nest algebra will be denoted by I.An element P in a nest algebra is called a non-trivial idempotent if P 6= 0, Iand P 2 = P .Let N be a non-trivial nest on a Banach space X. If there exists a non-trivial idempotent P ∈ AlgN with range P (X) ∈ N , then we have (I −P )(AlgN )P = {0} and henceAlgN = P (AlgN )P +̇P (AlgN )(I − P )+̇(I − P )(AlgN )(I − P )as sum of linear spaces. This is so-called Peirce decomposition of AlgN . Thesets P (AlgN )P , P (AlgN )(I − P ) and (I − P )(AlgN )(I − P ) are closedLeft derivable maps at non-trivial idempotents on nest algebras 99in AlgN . In fact, P (AlgN )P and (I − P )(AlgN )(I − P ) are Banach sub-algebras of AlgN whose unit elements are P and I − P , respectively andP (AlgN )(I−P ) is a Banach (P (AlgN )P, (I−P )(AlgN )(I−P ))-bimodule.Also P (AlgN )(I − P ) is faithful as a left P (AlgN )P -module as well as aright (I − P )(AlgN )(I − P )-module. For more information on nest algebras,we refer to [5].A subspace lattice L on a Hilbert space H is called a commutative subspacelattice, or a CSL, if the projections of H onto the subspaces of L commutewith each other. If L is a CSL, then AlgL is called a CSL algebra. Each nestalgebra on a Hilbert space is a CSL-algebra.2. Main resultsIn order to prove our results we need the following result.Theorem 2.1 ([8]). Let A be a Banach algebra with unity 1, X be a Banachspace and let φ : A × A → X be a continuous bilinear map with the propertythata, b ∈ A, ab = 1⇒ φ(a, b) = φ(1, 1).Thenφ(a, a) = φ(a2, 1)for all a ∈ A.Proposition 2.2. Let A be a Banach algebra with unity 1, and M be aunital Banach left A-module. Let δ : A→ M be a continuous linear map. If δis left derivable at 1, then δ is a Jordan left derivation.Proof. Since 1 ·1 = 1, it follows that δ(1) = 2δ(1) and therefore δ(1) = 0.Define a continuous bilinear map φ : A × A → M by φ(a, b) = aδ(b) + bδ(a).Then φ(a, b) = φ(1, 1) for all a, b ∈ A with ab = 1, since δ is left derivable at1. By applying Theorem 2.1, we obtain φ(a, a) = φ(a2, 1) for all a ∈ A. So,δ(a2) = 2aδ(a) (a ∈ A). 100 Hoger Ghahramani, Saman SattariCorollary 2.3. Let A be a Banach algebra with unity 1, and M be aunital Banach left A-module. Let x, y ∈ A with x+y = 1 and let δ : A→M bea continuous linear map. If δ is left derivable at x and y, then δ is a Jordanleft derivation.Proof. For a, b ∈ A with ab = 1, we have abx = x and aby = y. Since δis left derivable at x and y, it follows thatδ(x) = δ(abx) = aδ(bx) + bxδ(a)andδ(y) = δ(aby) = aδ(by) + byδ(a).Combining the two above equations, we get thatδ(1) = δ(x+ y) = aδ(bx) + bxδ(a) + aδ(by) + byδ(a) = aδ(b) + bδ(a),i.e. δ is left derivable at 1. It follows from Proposition 2.2 that δ is a Jordanleft derivation. Remark 2.4. If A is a CSL-algebra or a unital semisimple Banach algebra,then by [12] and [14] every continuous Jordan left derivation on A is zero.Hence it follows from Proposition 2.2 that every continuous linear map δ : A→A which is left derivable at 1 is zero.The following is our main result.Theorem 2.5. Let N be a nest on a Banach space X such that there ex-ists non-trivial idempotent P ∈ AlgN with range P (X) ∈ N . If δ : AlgN →AlgN is a continuous left derivable map at P , then δ is a Jordan left deriva-tion.Proof. For a notational convenience, we denote A = AlgN , A11 = PAP ,A12 = PA(I−P ) and A22 = (I−P )A(I−P ). As mentioned in the introductionA = A11+̇A12+̇A22. Throughout the proof, aij and bij will denote arbitraryelements in Aij for 1 ≤ i, j ≤ 2.First we show that δ(P ) = 0. Since P 2 = P , we have 2Pδ(P ) = δ(P ).It follows from equation 2Pδ(P ) = δ(P ) that Pδ(P ) = 0 and it implies thatδ(P ) = 0.We complete the proof by verifying the following steps.Left derivable maps at non-trivial idempotents on nest algebras 101Step 1. Pδ(a211)P = 2a11Pδ(a11)P and Pδ(a211)(I−P ) = 2a11Pδ(a11)(I−P ).For any a11, b11 with a11b11 = P , we have(2.1) a11δ(b11) + b11δ(a11) = δ(P ).Multiplying this identity by P both from the left and from the right, we finda11Pδ(b11)P + b11Pδ(a11)P = Pδ(P )P (a11b11 = P ).Define a continuous linear map d : A11 → A11 by d(a11) = Pδ(a11)P . Byabove identity d is left derivable at P . Hence by Proposition 2.2, d is a Jordanleft derivation, which impliesPδ(a211)P = 2a11Pδ(a11)P (a11 ∈ A11).By multiplying (2.1) by P from the left and by (I − P ) from the right, wearrive ata11Pδ(b11)(I − P ) + b11Pδ(a11)(I − P ) = Pδ(P )(I − P ) (a11b11 = P ).Define a continuous linear map D : A11 → A12 by D(a11) = Pδ(a11)(I − P ).It is easy to see that D is a left derivable at P . It follows from Proposition 2.2that D is a Jordan left derivation. Thus,Pδ(a211)(I − P ) = 2a11Pδ(a11)(I − P ) (a11 ∈ A11).Step 2. Pδ(a22) = 0.Since (P + a22)P = P , we have(P + a22)δ(P ) + Pδ(P + a22) = δ(P ).From δ(P ) = 0 we getPδ(a22) = 0.Step 3. Pδ(a12) = 0.Applying δ to (P + a12)P = P , we get(P + a12)δ(P ) + Pδ(P + a12) = δ(P ).Since δ(P ) = 0, it follows thatPδ(a12) = 0.102 Hoger Ghahramani, Saman SattariStep 4. (I − P )δ(a11)(I − P ) = 0.For any a11, b11 with b11a11 = P , we have (I −P + b11)a11 = P and hence(I − P + b11)δ(a11) + a11δ(I − P + b11) = δ(P ).Multiplying this identity by I − P both from the left and from the right wearrive at(I − P )δ(a11)(I − P ) = 0.Since any element in a Banach algebra with unit element is a sum of itsinvertible elements ([1]), by the linearity of δ and above identity we have(I − P )δ(a11)(I − P ) = 0for all a11 ∈ A11.Step 5. (I − P )δ(a12)(I − P ) = 0.Since (P − a12)(I + a12) = P , it follows that(P − a12)δ(I + a12) + (I + a12)δ(P − a12) = δ(P ).Multiplying this identity by I − P both from the left and from the right andusing the fact that δ(P ) = 0, we find(I − P )δ(a12)(I − P ) = 0.Step 6. (I − P )δ(a22)(I − P ) = 0.Applying δ to (P + a12)(P − a12a22 + a22) = P , we see that(P + a12)δ(P − a12a22 + a22) + (P − a12a22 + a22)δ(P + a12) = δ(P ).Now, multiplying this identity from the left by P , from the right by I−P andby Steps 2,3 and 5 and the fact that δ(P ) = 0, we get a12(I−P )δ(a22)(I−P ) =0. Since a12 ∈ A12 is arbitrary, we have A12((I−P )δ(a22)(I−P )) = {0}. Fromthe fact that A12 is faithful as right A22-module, we arrive at(I − P )δ(a22)(I − P ) = 0.Since ab = PaPbP + PaPb(I − P ) + Pa(I − P )b(I − P ) + (I − P )a(I −P )b(I − P ), for any a, b ∈ A, by Steps 1–6, it follows that δ is a Jordan leftderivation. Left derivable maps at non-trivial idempotents on nest algebras 103Our next result characterizes the linear mappings on AlgN which aregeneralized left derivable at P .Theorem 2.6. Let N be a nest on a Banach space X such that there existsa non-trivial idempotent P ∈ AlgN with range P (X) ∈ N . If δ : AlgN →AlgN is a continuous generalized left derivable map at P , then δ is a gener-alized Jordan left derivation.Proof. Define ∆: AlgN → AlgN by ∆(a) = δ(a) − aδ(1). It is easy tosee that ∆ is a continuous left derivable map at P . By Theorem 2.5, ∆ is aJordan left derivation. Thereforeδ(a2) = ∆(a2) + a2δ(1)= 2a∆(a) + a2δ(1)= 2a(δ(a)− aδ(1)) + a2δ(1)= 2aδ(a)− a2δ(1)for all a ∈ AlgN . So δ is a generalized Jordan left derivation. Since every continuous Jordan left derivation on a CSL algebra is zero ([12]),we have the following result.Corollary 2.7. Let N be a non-trivial nest on a Hilbert space H. Let Pbe a non-trivial idempotent in AlgN with range P (H) ∈ N and δ : AlgN →AlgN be a continuous linear map.(i) If δ is left derivable at P , then δ is zero.(ii) If δ is generalized left derivable at P , then δ(a) = aδ(1) for all a ∈ AlgN .Proof. (i) Since every continuous Jordan left derivation on a CSL algebrais zero ([12]), by Theorem 2.5, δ is zero.(ii) By Theorem 2.6, δ is a generalized Jordan left derivation, so the mapping∆: AlgN → AlgN defined by ∆(a) = δ(a) − aδ(1) is a continuous Jordanleft derivation. Therefore ∆ = 0 and hence δ(a) = aδ(1) for all a ∈ AlgN . Now, we characterize (generalized) left Jordan derivations which are con-tinuous in the strongly operator topology, but in order to prove our resultwe must assume an additional (mild) condition concerning the nest N . Atpresent we have no counter-example, so it remains an open problem if thisadditional condition can be omitted.The idea of the proof of Proposition 2.8 (i) comes from [2].104 Hoger Ghahramani, Saman SattariProposition 2.8. Let N be a nest on a Banach space X, with each N ∈ Ncomplemented in X whenever N− = N . Let δ : AlgN → AlgN be a continuouslinear map in a strong operator topology.(i) If δ(P ) = 2Pδ(P ) for every idempotent P in AlgN , then δ = 0.(ii) If δ(P ) = 2Pδ(P )−Pδ(I) for every idempotent P in AlgN , then δ(a) =aδ(I) for all a ∈ AlgN .Proof. (i) For arbitrary idempotent operator P ∈ AlgN , by hypothe-sis we have δ(P ) = 2Pδ(P ). It follows from equation 2Pδ(P ) = δ(P ) thatPδ(P ) = 0 and it implies that δ(P ) = 0.Notice that AlgF N is contained in the linear span of the idempotentsin AlgN (see [11]), which implies that δ(F ) = 0 for all finite rank operatorF in AlgN . Since δ is continuous under the strong operator topology andAlgF NSOT= AlgN (see [13]), we find that δ(a) = 0 for all a ∈ AlgN .(ii) Define ∆: AlgN → AlgN by ∆(a) = δ(a) − aδ(I). It is easy tosee that ∆ is a continuous left map satisfying ∆(P ) = 2P∆(P ) for everyidempotent P in AlgN . So by (i) we have ∆ = 0 and hence δ(a) = aδ(I) forall a ∈ AlgN . It is obvious that the nests on Hilbert spaces, finite nests and the nestshaving order-type ω + 1 or 1 + ω∗, where ω is the order-type of the naturalnumbers, satisfy the condition in Proposition 2.8 automatically.References[1] F.F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, Berlin, 1973.[2] M. Brešar, Characterizations of derivations on some normed algebras with involution,J. Algebra 152 (1992), 454–462.[3] M. Brešar, Characterizing homomorphisms, derivations and multipliers in rings withidempotents, Proc. Roy. Soc. Edinburgh. Sect. A 137 (2007), 9–21.[4] M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer.Math. Soc. 110 (1990), 7–16.[5] K.R. Davidson, Nest Algebras, Pitman Res. Notes in Math., vol. 191, Longman, Lon-don, 1988.[6] B. Fadaee and H. Ghahramani, Jordan left derivations at the idempotent elements onreflexive algebras, Publ. Math. Debrecen 92 (2018), 261–275.[7] H. Ghahramani, Additive mappings derivable at non-trivial idempotents on Banachalgebras, Linear Multilinear Algebra 60 (2012), 725–742.[8] H. Ghahramani, On centralizers of Banach algebras, Bull. Malays. Math. Sci. Soc. 38(2015), 155–164.[9] H. Ghahramani, Characterizing Jordan maps on triangular rings through commutativezero products, Mediterr. J. Math. 15 (2018), Art. 38, 10 pp., DOI: 10.1007/s00009-018-1082-3.Left derivable maps at non-trivial idempotents on nest algebras 105[10] N.M. Ghosseiri, On Jordan left derivations and generalized Jordan left derivations ofmatrix rings, Bull. Iranian Math. Soc. 38 (2012), 689–698.[11] J.C. Hou and X.L. Zhang, Ring isomorphisms and linear or additive maps preservingzero products on nest algebras, Linear Algebra Appl. 387 (2004), 343–360.[12] J.K. Li and J. Zhou, Jordan left derivations and some left derivable maps, Oper.Matrices 4 (2010), 127–138.[13] N.K. Spanoudakis, Generalizations of certain nest algebra results, Proc. Amer. Math.Soc. 115 (1992), 711–723.[14] J. Vukman, On left Jordan derivations of rings and Banach algebras, AequationesMath. 75 (2008), 260–266.[15] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin. 32(1991), 609–614.[16] J. Zhu and C.P. Xiong, Derivable mappings at unit operator on nest algebras, LinearAlgebra Appl. 422 (2007), 721–735.Department of MathematicsUniversity of KurdistanP. O. Box 416SanandajIrane-mail: h.ghahramani@uok.ac.ir; hoger.ghahramani@yahoo.come-mail: saman.satari84@gmail.com

Left derivable maps at non-trivial idempotents on nest algebras

https://journals.us.edu.pl/index.php/AMSIL/article/download/13653/10873

Left derivable maps at non-trivial idempotents on nest algebras

Abstract

Similar works

Full text

Available Versions

Platforma czasopism naukowych Uniwersytetu Śląskiego w Katowicach