Zhidong Pan DERIVABLE MAPS AND GENERALIZED DERIVATIONS ON NEST AND STANDARD ALGEBRAS

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1 DEMONSTRATIO MATHEMATICA Vol. 49 No Zhidong Pan DERIVABLE MAPS AND GENERALIZED DERIVATIONS ON NEST AND STANDARD ALGEBRAS Communicated by E. Weber Abstract. For an algebra A, an A-bimodule M, and m P M, define a relation on A by R Apm, 0q tpa, bq P A ˆ A : amb 0u. We show that generalized derivations on unital standard algebras on Banach spaces can be characterized precisely as derivable maps on these relations. More precisely, if A is a unital standard algebra on a Banach space X then P LpA, BpXqq is a generalized derivation if and only if is derivable on R ApM, 0q, for some M P BpXq. We give an example to show this is not the case in general for nest algebras. On the other hand, for an idempotent P in a nest algebra A algn on a Hilbert space H such that P is either left-faithful to N or right-faithful to N K, if δ P LpA, BpHqq is derivable on R ApP, 0q then δ is a generalized derivation. 1. Introduction For vector spaces U and V, we use LpU, Vq to denote the set of all linear maps from U to V. For a unital algebra A and an A-bimodule M, δ P LpA, Mq is called a derivation if for all a, b P A, δpabq δpaqb`aδpbq and δ is called a generalized derivation if for all a, b P A, δpabq δpaqb`aδpbq aδp1qb. Fix any u, v P M, then δ uv paq ua a P A is a generalized derivation; the study of such maps dates back at least to [13]. By a relation on A, we mean a nonempty subset R A Ď A ˆ A. We say δ P LpA, Mq is derivable on R A if δpabq δpaqb`aδpbq for all pa, bq P R A. Derivable maps have garnered interests of many researchers, for example, authors of [2], [4], [8], and [10] have studied maps that are derivable on R A tpa, bq P A ˆ A : a bu, such maps are called Jordan derivations. In [3], [6 7], [9], [11], and [14 16], the authors have studied derivable maps on relations R A pcq tpa, bq P A ˆ A : ab cu, for some c P A. Not all derivable maps are derivations. If a derivable map is not a derivation, it is natural to ask whether it is close to being a derivation, e.g. whether it is a generalized derivation. Every generalized derivation is 2010 Mathematics Subject Classification: Primary 47B47, 47L35. Key words and phrases: derivable map, derivation, nest algebra. DOI: /dema c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology

2 332 Z. Pan inherently derivable on a relation R A pm, 0q tpa, bq P A ˆ A : amb 0u, for some m P M, which leads to the study of maps preserving the derivation structure on these relations, i.e. derivable maps on R A pm, 0q, see [12] for more details. In this regard, it is more natural to consider maps that are derivable on relations R A pm, 0q. It should be noted that many of the techniques used to study derivable maps on R A pcq cannot be adapted to R A pm, 0q. For example, consider the following technique for derivable maps on R A pcq: If pa, bq P R A pcq, then both pt 1, tcq, pat 1, tbq P R A pcq, for any invertible t P A. From this, one can try to construct t based on the structure of A to achieve desired properties. This would not work on R A pm, 0q, because m blocks a and b from directly acting with each other with the multiplication operation of A, e.g. pa, bq P R A pm, 0q does not imply pat 1, tbq P R A pm, 0q. If A is an algebra with unit 1 and M is a left A-module, M is called a unital left A-module if 1m m P M. In this case a map ψ P LpA, Mq is called right-annihilator-preserving if aψpbq a, b P A with ab 0 and ψ is called a right multiplier if ψpaq a P A. Similarly, if M is a right A-module, M is called a unital right A-module if m1 m P M. In this case a map ψ P LpA, Mq is called left-annihilator-preserving if ψpaqb a, b P A with ab 0 and ψ is called a left multiplier if ψpaq a P A. Clearly, left multipliers are left-annihilator-preserving and right multipliers are right-annihilator-preserving. Let X be a separable complex Banach space and let BpXq be the set of all bounded linear operators on X. By a subspace lattice on X, we mean a collection L of subspaces of X with 0 and X in L such that for every family tm t u of elements of L, both XM t and _M t belong to L. For a subspace lattice L of X, we use algl to denote the algebra of all operators in BpXq that leave members of L invariant; and for a subalgebra A of BpXq, we use lata to denote the lattice of all subspaces of X that are invariant under all operators in A. A lattice L is called reflexive if L latpalglq. A totally ordered subspace lattice N is called a nest and the corresponding algebra algn is called a nest algebra. It is well known that nests are reflexive, see [1] for more on nest algebras. For any x P X and f P X, the rank-one operator x b f is defined by x b fy y P X. When X is a Hilbert space we change it to H. For a Hilbert space orthogonal projection, we use the same letter to denote the projection and its range space. While we use lower case letters for elements in algebras and modules in abstract settings, we will use capital letters for operators in operator algebras on Banach and Hilbert spaces. For a nest algebra A algn on a Hilbert space H and T P BpHq, we say T is left-faithful to N N P N, T N 0 iff N 0 and T is right-faithful to N K N P N, N K T 0 iff N K 0. In Section 2, we show that if P P A is an idempotent, then every left-

3 Derivable maps and generalized derivations on nest and standard algebras 333 annihilator-preserving map from P AP to BpHqP is a left multiplier and every right-annihilator-preserving map from P AP to P BpHq is a right multiplier. Using this we show that if P is either left-faithful to N or right-faithful to N K then every derivable map on R A pp, 0q is a generalized derivation. In Section 3, we show that if A is a unital standard algebra on a Banach space X, then P LpA, BpXqq is a generalized derivation iff is derivable on R A pm, 0q, for some M P BpXq. In particular, the above conclusion holds when A BpXq, which generalizes the main result of [12]. 2. Derivable maps and generalized derivations on nest algebras The following is the main result of this section. Theorem 2.1. Let A algn be a nest algebra on a Hilbert space H and P P A be an idempotent. If P is either left-faithful to N or right-faithful to N K, and δ P LpA, BpHqq is derivable on R A pp, 0q then δ is a generalized derivation; in this case δpiq P CP. We will proceed first with some lemmas, starting with the ones that hold in abstract settings of modules over algebras, which should be of more general interest. Note that these can be easily adapted to the context of modules over rings and additive maps, with essentially the same proofs. Let A be a unital algebra. If M is a left A-module and ψ P LpA, Mq, define R ψ ta P A : ψpaq aψp1qu. Clearly ψ is a right multiplier iff R ψ A. Similarly, if M is a right A-module and ψ P LpA, Mq, define L ψ ta P A : ψpaq ψp1qau. Then ψ is a left multiplier iff L ψ A. Lemma 2.2. Let A be a unital algebra. (i) Let M be a unital left A-module and ψ P LpA, Mq. If ψ is rightannihilator-preserving then R ψ contains the subalgebra of A generated by the idempotents of A. (ii) Let M be a unital right A-module and ψ P LpA, Mq. If ψ is leftannihilator-preserving then L ψ contains the subalgebra of A generated by the idempotents of A. Proof. To prove (i), for any idempotent u P A, p1 uqu up1 uq 0. Since ψ is right-annihilator-preserving, we have p1 uqψpuq 0 and uψp1 uq 0. It follows that ψpuq uψp1q, i.e. u P R ψ. For any b P A, define ψ b P LpA, Mq by ψ b paq a P A. If a, c P A satisfy ac 0 then acb 0. Since ψ is right-annihilator-preserving, aψpcbq 0. Thus aψ b pcq 0, i.e. ψ b is right-annihilator-preserving also. For any idempotents a, b P A, applying the previous paragraph to ψ and ψ b

4 334 Z. Pan yields ψpabq ψ b paq aψ b p1q aψpbq abψp1q. Thus ab P R ψ. Now (i) follows from induction and linearity of ψ. The proof of (ii) is similar. Suppose A is an algebra and A 1 is a subset of A. If M is a left A-module and M 1 is a subset of M, we say A 1 is left-faithful to M 1 if for any m P M 1, the condition A 1 m t0u implies m 0. If M is a right A-module and M 1 is a subset of M, we say A 1 is right-faithful to M 1 if the condition ma 1 t0u implies m 0. Lemma 2.3. Let A be a unital algebra. (i) If M is a unital left A-module and A has a right ideal I generated as an algebra by idempotents of A such that I is left-faithful to M, ψ P LpA, Mq, ψ is a right multiplier iff ψ is right-annihilatorpreserving. (ii) If M is a unital right A-module and A has a left ideal I generated as an algebra by idempotents of A such that I is right-faithful to M, ψ P LpA, Mq, ψ is a left multiplier iff ψ is left-annihilator-preserving. Proof. For (i), suppose ψ is right-annihilator-preserving. For any e P I and b P A, eb P I. Define ψ b paq a P A, then ψ b is also right-annihilatorpreserving. Since I is generated by idempotents of A, by Lemma 2.2 we have ebψp1q ψpebq ψ b peq eψ b p1q eψpbq. Thus erbψp1q ψpbqs 0, and bψp1q ψpbq 0, since I is left-faithful to M. The other direction of (i) is clear and the proof of (ii) is similar. For a unital algebra A, a unital A-bimodule M, and an idempotent p P A, we define A ij and M ij, the Peirce decompositions of A and M with respect to p as follows: Let p 1 p, p 2 1 p 1, A ij p i Ap j, and M ij p i Mp j, i, j 1, 2. Lemma 2.4. Let A be a unital algebra, M be a unital A-bimodule, p P A be an idempotent, A ij and M ij be the Peirce decompositions of A and M with respect to p. If d P LpA, Mq is derivable on R A pm, 0q for some m P M 11, then there exists a δ P LpA, Mq such that d δ is an inner derivation, δp1 pq 0, and δpa ij q Ď M ij. Proof. Let q 1 p, u pdpqqq qdpqqp and δ u paq ua P A. Let δ d δ u. It follows that δ is derivable on R A pm, 0q and one can verify pδpqqq qδpqqp 0. Therefore, δpqq pδpqqp ` pδpqqq ` qδpqqp ` qδpqqq pδpqqp ` qδpqqq. Since m P M 11, we see aqmb 0 and amqb 0, for all a, b P A. Since δ is

5 Derivable maps and generalized derivations on nest and standard algebras 335 derivable on R A pm, 0q, p2.1q and p2.2q δpaqbq δpaqqb ` aqδpbq δpaqbq δpaqqb ` aδpqbq. Setting a q and b p in Eq. 2.1, we get 0 δpqqp ` qδppq. It follows pδpqqp 0. Setting a b q in Eq. 2.2, we get δpqq δpqqq ` qδpqq. It follows qδpqqq 0. Thus δpqq 0. For any a 11 P A 11, setting a q and b a 11 in Eq. 2.1, we get 0 qδpa 11 q; setting a a 11 and b q in Eq. 2.2, we get 0 δpa 11 qq. Thus δpa 11 q P M 11. For any a 12 P A 12, setting a q and b a 12 in Eq. 2.1, we get 0 qδpa 12 q; setting a a 12 and b p in Eq. 2.1, we get 0 δpa 12 qp. Thus δpa 12 q P M 12. For any a 21 P A 21, setting a a 21 and b q in Eq. 2.2, we get 0 δpa 21 qq; setting a p and b a 21 in Eq. 2.2, we get 0 pδpa 21 q. Thus δpa 21 q P M 21. For any a 22 P A 22, setting a p and b a 22 in Eq. 2.2, we get 0 pδpa 22 q; setting a a 22 and b p in Eq. 2.1, we get 0 δpa 22 qp. Thus δpa 22 q P M 22. If A is a subalgebra of a unital algebra M such that 1 P A then M is a unital A-bimodule with the inherited algebraic operations. Moreover, if p P A is an idempotent, then pap is a unital algebra with unit p, pm is a unital left pap-module, and Mp is a unital right pap-module. If M is an algebra and S Ď M, we define the commutant of S by S 1 tt P M : st s P Su. Lemma 2.5. Let A be a subalgebra of a unital algebra M such that 1 P A and p P A be an idempotent. Suppose that the only right-annihilator-preserving maps from pap to pm are the right multipliers and the only left-annihilatorpreserving maps from pap to Mp are the left multipliers. If δ P LpA, Mq is derivable on R A pp, 0q then for any x P pap and y P A (i) pδpxyq pδpxqy ` xδppyq xδppqy. (ii) δpyxqp yδpxqp ` δpypqx yδppqx. Moreover, pδppqp P tpapu 1. Proof. For any a, b P pap such that ab 0 then apb ab 0 and apby P A. Since δ is derivable on R A pp, 0q, we have and δpaqb ` aδpbq δpabq 0 δpaqby ` aδpbyq δpabyq 0.

6 336 Z. Pan The above two equations yield p2.3q aδpbyq aδpbqy 0. For any x P pap, define ψpxq pδpxyq pδpxqy. By Eq. 2.3, ψ is a rightannihilator-preserving map from pap to pm, thus ψ is a right multiplier, so ψpxq xψppq, i.e. pδpxyq pδpxqy xδppyq xδppqy and Part (i) follows. Similarly, for any a, b P pap such that ab 0 then yapb yab P A. Since δ is derivable on R A pp, 0q, we have δpyaqb ` yaδpbq δpyabq 0. Combining with δpaqb ` aδpbq 0 we get p2.4q δpyaqb yδpaqb 0. For any x P pap, define φpxq δpyxqp yδpxqp. By Eq. 2.4, φ is a leftannihilator-preserving map from pap to Mp, thus φ is a left multiplier, so φpxq φppqx, i.e. δpyxqp yδpxqp δpypqx yδppqx and Part (ii) follows. For any x, y P pap such that xy 0 xpy, since δ is derivable on R A pp, 0q, On the other hand, by Part (i), 0 δpxyq δpxqy ` xδpyq. pδpxyq pδpxqy ` xδppyq xδppqy pδpxqy ` pxδpyq xδppqy. Thus xδppqy 0, which implies Φptq t P pap is a rightannihilator-preserving map from pap to pm, so Φ is a right multiplier and pδppqpt pδppqt Φptq tφppq tpδppqp, i.e. pδppqp P tpapu 1. Lemma 2.6. Let A be a subalgebra of BpHq containing rank-one operators x b f, y b f, x b g, and y b g. If gpyq 0 then x b f can be written as a linear combination of idempotents in A. Proof. Note that if a rank-one operator z b h P A satisfies hpzq 0 then z b h is a scalar multiple of an idempotent, so we only need to show x b f can be written as a linear combination of such operators. Without loss of generality, suppose fpxq 0. If fpyq 0, write x b f 1 2rpx ` yq b f ` px yq b fs. If gpxq 0, write x b f 1 2rx b pf ` gq ` x b pf gqs. If fpyq gpxq 0, write xbf 1 rpx`yqbpf `gq`px`yqbpf gq`px yqbpf `gq`px yqbpf gqs. 4

7 Derivable maps and generalized derivations on nest and standard algebras 337 Lemma 2.7. If A algn is a nest algebra on a Hilbert space H and P P A is an idempotent, then P AP has ideals I and J, both generated as algebras by idempotents in P AP, such that I is right-faithful to BpHqP and J is left-faithful to P BpHq. Proof. For any N P N, clearly P NP and P pn NT N K qp are idempotents in P AP, thus P NT N K P is a linear combination of idempotents in P AP. Let E be the orthogonal projection of H on to _tn H : N P N, NH Ğ P Hu. Then E P lata N, in particular, E P A. We will construct I in two separate cases: EH Ě P H and EH Ğ P H If EH Ě P H, take I spantp NT N K P : N P N, NH Ğ P H, T P BpHqu. The previous paragraph shows I is generated by idempotents in P AP. Clearly I is an ideal of P AP. Take any S P BpHqP such that SI t 0u. If N P N and NH Ğ P H, then N K P 0. Thus if SP NT N K P P BpHq then SP N 0, so SP E 0. Since EH Ě P H, SP 0, i.e. I is right-faithful to BpHqP. If EH Ğ P H, take I spantx b fe K P P P H, f P H u. For any x P P H, f P H, one can easily check that x b fe K P P P AP, I is an ideal of P AP, and I is right-faithful to BpHqP. Since EH Ğ P H, E K P 0. Thus there exist y P P H and g P H such that gpe K P yq 1. Applying Lemma 2.6 to x b fe K P, y b fe K P, x b ge K P, and y b ge K P, we see x b fe K P can be written as a linear combination of idempotents in P AP. To construct our J, note that A is also a nest algebra and P is an idempotent in A. From the previous paragraphs, P A P has an ideal J generated by idempotents of P A P and J is right-faithful to BpHqP. Let J be the adjoint of J, it follows that J is an ideal of P AP generated by idempotents of P AP, and J is left-faithful to P BpHq. The following theorem follows immediately from Lemmas 2.3 and 2.7. Theorem 2.8. If A is a nest algebra on a Hilbert space H and P P A is an idempotent, then any ψ P LpP AP, P BpHqq is a right multiplier iff it is right-annihilator-preserving and any ψ P LpP AP, BpHqP q is a left multiplier iff it is left-annihilator-preserving. For any N P N, define N _tf H : F P N, F Ĺ Nu. It is well known that for a nest algebra algn, a rank-one operator xbf P algn iff there exists an N P N such that x P N and f P pn q K. For vector spaces U and V, take any S P LpU, Vq, the one-dimensional subspace CS Ď LpU, Vq is algebraically reflexive in the sense that if T P LpU, Vq such that T x P P U then T P CS; a more general case for n-dimensional subspaces of LpU, Vq can be found in [5]. This will be used in the proofs of Lemmas 2.9 and 3.2.

8 338 Z. Pan Lemma 2.9. Let A algn be a nest algebra on a Hilbert space H and M BpHq. If P P A is an idempotent with A ij and M ij as the Peirce decompositions of A and M with respect to P, respectively, then ta 11 u 1 CP ` M 22. Proof. Let E be the orthogonal projection from H onto XtNH : N P N, NH Ě P Hu. Then clearly EH Ě P H, and E P lata N. Taking any T P ta 11 u 1, since T P P T, we see pi P qt P 0 P T pi P q, thus T P M 11 ` M 22. If E E, then E H Ğ P H. For any x P E, we can choose f P pe q K such that x b fp 0. Since T P x b fp P x b fp T, T P x P CP x. Since EH Ě P H, it follows that for any y P H, T P y T P P y P CP P y CP y, thus T P P CP. If E E, take any F P N such that F Ĺ E, then F H Ğ P H. For any x P F, take f P pf q K such that x b fp 0. Since T P x b fp P x b fp T, T P x P CP x. It follows T P F P CP F. Note that E _tf H : F P N, F Ĺ Eu and E E, it follows T P E P CP E, thus T P P CP. Lemma Let L be a reflexive lattice on a Hilbert space H, A algl, P P A be an idempotent, Q I P, M BpHq, and A ij be the corresponding Peirce decomposition of A with respect to P. Then 1) A 12 is left-faithful to QM iff P is left-faithful to L. 2) A 21 is right-faithful to MQ iff P is right-faithful to L K. Proof. To see 1q: For any T P QM such that P AQT P A, let N _taqt H : A P Au, then P N 0. Since L is reflexive, N P lata L. If P is left-faithful to L, then N 0, T 0. For any N P L such that P N 0, then P AQN P NAQN A P A. If A 12 is left-faithful to QM, then QN 0. Thus N QN 0. To see 2q: Take any T P MQ. Note that _ta Q T H : A P Au is an invariant subspace of A, it follows that _ta Q T H : A P Au N K for some N P L. If T QAP 0, for all A P A, then P A Q T 0, thus P N K 0 and N K P 0. If P is right-faithful to L K, then N K 0, thus Q T 0. Therefore T T Q 0. For any N P L such that N K P 0, then N K QAP N K QAN K P 0 for all A P A. If A 21 is right-faithful to MQ, then N K Q 0, thus N K 0. Corollary Let A algn be a nest algebra on a Hilbert space H, P P A be an idempotent, Q I P, M BpHq, and A ij be the Peirce decomposition of A with respect to P. Then 1) If P H Ě N 0 H for some 0 N 0 P N then A 12 is left-faithful to QM. 2) If P H Ě N K 0 H for some I N 0 P N then A 21 is right-faithful to MQ.

9 Derivable maps and generalized derivations on nest and standard algebras 339 Proof. To see 1q, for any N P N, if P N 0 then N 0 N P N 0 N P NN 0 0, thus N 0. The proof of 2q is similar. Proof of Theorem 2.1. Let M BpHq. If P 0, R A p0, 0q A ˆ A, the conclusion is clear. The case for P I follows from Lemma 2.5 and Theorem 2.8. Suppose P 0, I, let A ij and M ij, i, j P t1, 2u be the Peirce decompositions of A and M with respect to P. Set Q I P, subtracting an inner derivation from δ if necessary, by Lemma 2.4, we can assume δpqq 0 and δpa ij q Ď M ij. We will show that for any A ij P A ij and B kl P A kl, i, j, k, l P t1, 2u, p q δpa ij B kl q δpa ij qb kl ` A ij δpb kl q A ij δpp qb kl. By Eqs. p2.1q and p2.2q, p q holds if j 2 or k 2, so we assume j k 1. By Lemma 2.5, p q holds if i j 1 or k l 1. It remains to show δpa 21 B 12 q δpa 21 qb 12 ` A 21 δpb 12 q A 21 δpp qb 12. For any T 12 P A 12 and A 21 P A 21, δpt 12 A 21 Bq δpt 12 A 21 qb ` T 12 A 21 δpbq T 12 A 21 δpp qb On the other hand, δpt 12 qa 21 B ` T 12 δpa 21 qb ` T 12 A 21 δpbq T 12 A 21 δpp qb. δpt 12 A 21 Bq δpt 12 qa 21 B ` T 12 δpa 21 Bq. Combining the above two equations, we get T 12 δpa 21 Bq T 12 δpa 21 qb ` T 12 A 21 δpbq T 12 A 21 δpp qb. If P is left-faithful to N, then A 12 is left-faithful to QM by Lemma 2.10, thus δpa 21 Bq δpa 21 qb ` A 21 δpbq A 21 δpp qb, which implies p q. For any T 21 P A 21 and A 12 P A 12, δpba 12 T 21 q δpbqa 12 T 21 ` BδpA 12 T 21 q BδpP qa 12 T 21 δpbqa 12 T 21 ` BδpA 12 qt 21 ` BA 12 δpt 21 q BδpP qa 12 T 21. On the other hand, δpba 12 T 21 q δpba 12 qt 21 ` BA 12 δpt 21 q. Combining the above two equations, we get δpba 12 qt 21 δpbqa 12 T 21 ` BδpA 12 qt 21 BδpP qa 12 T 21. If P is right-faithful to N K, then A 21 is right-faithful to MQ by Lemma 2.10, thus δpba 12 q δpbqa 12 ` BδpA 12 q BδpP qa 12, which again implies p q.

10 340 Z. Pan Since δpiq δpp q, p q implies δ is a generalized derivation. By Theorem 2.8, Lemma 2.5, and Lemma 2.9, δpp q P δpp qp P ta 11 u 1 CP ` M 22. Thus δpiq δpp q P CP. Our next example shows that the assumption P is either left-faithful to N or right-faithful to N K " cannot be dropped in Theorem 2.1. Example. Let H C 3 with orthonormal basis e 1 p1, 0, 0q, e 2 p0, 1, 0q, and e 3 p0, 0, 1q. Take N 1 Ce 1 and N 2 Ce 1 ` Ce 2, then N t0, N 1, N 2, Hu forms a nest of H. Let A algn, then A can be identified with T 3, the set of all 3 ˆ 3 upper triangular matrices with respect to the orthonormal basis. Let E ij be the matrix units and P E 22, then P P A is an idempotent. In this case P is neither left-faithful to N nor right-faithful to N K. For any A pa ij q P algn T 3, define δ P LpA, Aq by δpaq E 11 AE 22 a 12 E 12, then δpiq 0. We will see that δ is drivable on R A pp, 0q, but it is not a generalized derivation. First note ą i, E jj AE ii P A. Let A pa ij q P A and B pb ij q P A. If AP B AE 22 B 0, then δpabq δpaqb ` AδpBq; indeed δpabq E 11 ABE 22 E 11 AIBE 22 E 11 ApE 11 ` E 22 ` E 33 qbe 22 and E 11 AE 11 BE 22 a 11 b 12 E 12 δpaqb ` AδpBq E 11 AE 22 B ` AE 11 BE 22 0 ` a 11 b 12 E 12. Thus, δ is drivable on R A pp, 0q. However, setting A E 12 and B E 23 we have δpabq δpe 12 E 23 q δpe 13 q 0, but δpaqb ` AδpBq AδpIqB δpe 12 qe 23 ` E 12 δpe 23 q 0 E 13, so δ is not a generalized derivation. Remarks. The following question is raised in [12]: For what Banach algebra A does it hold that δ P LpA, Aq is a generalized derivation iff δ is derivable on R A pm, 0q, for some m P A? In particular, does it hold for nest algebras in general and BpHq? The previous example gives a negative answer for general nest algebras. Our main result in the following section gives a positive answer for A BpHq, in fact for any unital standard algebra on a Banach space X. 3. Derivable maps and generalized derivations on standard algebras Theorem 3.1. If A is a unital standard algebra on a Banach space X then P LpA, BpXqq is a generalized derivation iff is derivable on R A pm, 0q for some M P BpXq; in this case piq P CM. In particular, the above holds if A BpXq.

11 Derivable maps and generalized derivations on nest and standard algebras 341 Remarks. The main result of [12] is a special case of Theorem 3.1 with X being finite-dimensional, the proofs there make extensive use of Jordan canonical forms of matrices, thus cannot be adapted to the infinite-dimensional settings. Lemma 3.2. Let A be a unital standard algebra on a Banach space X and M P BpXq. If P LpA, BpXqq is derivable on R A pm, 0q then piq P CM. Proof. By [5], we only need to show x P X, piqx P CMx. Let y Mx. If y 0 then for any 0 f P X, IMx b f 0. Since is derivable on R A pm, 0q, pix b fq piqx b f ` I px b fq. Thus piqx 0 P CMx. If y 0, take f P X such that fpyq 1 and fpxq 0. It follows pi y bfqmxbf 0. Since is derivable on R A pm, 0q, rpi y bfqxbfs pi y b fqx b f ` pi y b fq px b fq. Thus fpxq py b fq piqx b f py b fqx b f y b f px b fq. Applying both sides to x, we get fpxq py b fqx fpxq piqx fpxq py b fqx y b f px b fqx. It follows piqx P Cy CMx. Lemma 3.3. Let A be a unital standard algebra on a Banach space X and M P BpXq. If δ P LpA, BpXqq is derivable on R A pm, 0q and δpiq 0 x P X, y Mx, and f, g P X, we have δpy b g x b fq δpy b gqx b f ` y b gδpx b fq. Proof. If gpyq 0 then y b g Mx b f 0. Since δ is derivable on R A pm, 0q, the conclusion follows. If gpyq 0, rescaling if necessary, we can assume gpyq 1. Thus pi ybgqmxbf 0. Since δ is derivable on R A pm, 0q, δrpi ybgqxbfs δpi y b gqx b f ` pi y b gqδpx b fq. Since δpiq 0, again the conclusion follows. Lemma 3.4. Let A be a unital standard algebra on a Banach space X and M P BpXq. If δ P LpA, BpXqq is derivable on R A pm, 0q and δpiq 0 x b f, x b h P A, δpx b h x b fq δpx b hqx b f ` x b hδpx b fq. Proof. If Mx 0 then x b h Mx b f 0. Since δ is derivable on R A pm, 0q, the conclusion follows. Let y Mx 0. For any g P X, by Lemma 3.3, δpy b g x b h x b fq δpy b gpx b h x b fqq δpy b gqx b h x b f ` y b gδpx b h x b fq.

12 342 Z. Pan On the other hand, δpy b g x b h x b fq δrpy b g x b hqx b fs δpy b g x b hqx b f ` y b g x b hδpx b fq rδpy b gqx b h ` y b gδpx b hqsx b f ` y b g x b hδpx b fq δpy b gqx b h x b f ` y b gδpx b hqx b f ` y b g x b hδpx b fq. It follows y b gδpx b h x b fq y b gδpx b hqx b f ` y b g x b hδpx b fq. Thus δpx b h x b fq δpx b hqx b f ` x b hδpx b fq. Lemma 3.5. Let A be a unital standard algebra on a Banach space X and M P BpXq. If δ P LpA, BpXqq is derivable on R A pm, 0q and δpiq 0 then there exists a B P LpX, Xq such that for any f P X, there is a b f P X satisfying i) δpx b fq x b b f ` Bx b P X. ii) b f pxq ` fpbxq P X. Proof. For any 0 x b f P A, take h P X such that hpxq 1. By Lemma 3.4, δpx b fq δpx b h x b fq δpx b hqx b f ` x b hδpx b fq. It follows u P kerpfq, δpx b fqu P Cx, i.e. δpx b fq kerpfq Ď Cx, which yields Part i) using an argument similar to [11, Lemma 3.5(iii)]. To see Part ii), first by Part i), δpx b fqx b f rx b b f ` Bx b fsx b f b f pxqx b f ` fpxqbx b f. On the other hand, by Lemma 3.4 and Part i), δpx b fqx b f δpx b f x b fq x b fδpx b fq fpxqδpx b fq x b fδpx b fq fpxqrx b b f ` Bx b fs x b frx b b f ` Bx b fs fpxqbx b f fpbxqx b f. Thus b f pxqx b f ` fpxqbx b f fpxqbx b f fpbxqx b f, which gives Part ii). Lemma 3.6. Let A be a unital standard algebra on a Banach space X and M P BpXq. If δ P LpA, BpXqq is derivable on R A pm, 0q and δpiq 0 x, z P X and f, h P X, we have δpzbh xbfq δpzbhqxbf `zbhδpxbfq. Proof. By Lemma 3.5, there exists a B P LpX, Xq, b f, b h P X such that δpz b hqx b f ` z b hδpx b fq rz b b h ` Bz b hsx b f ` z b hrx b b f ` Bx b fs b h pxqz b f ` hpxqbz b f ` hpxqz b b f ` hpbxqz b f rb h pxq ` hpbxqsz b f ` hpxqrz b b f ` Bz b fs r0sz b f ` hpxqrδpz b fqs δpz b h x b fq.

13 Derivable maps and generalized derivations on nest and standard algebras 343 Lemma 3.7. Let A be a unital standard algebra on a Banach space X and M P BpXq. If δ P LpA, BpXqq is derivable on R A pm, 0q and δpiq 0 x P X, f P X and T P A, we have δpt x b fq δpt qx b f ` T δpx b fq. Proof. For any x P X and T P A, let y Mx and z T y. If y 0 then T Mx b f 0, the conclusion follows. If y 0, take h P X such that hpyq 1. Then pt z b hqmx b f 0. Since δ is derivable on R A pm, 0q, δrpt z b hqx b fs δpt z b hqx b f ` pt z b hqδpx b fq. By Lemma 3.6, δpz b h x b fq δpz b hqx b f ` z b hδpx b fq. By the above two equations we get δpt x b fq δpt qx b f ` T δpx b fq. Proof of Theorem 3.1. For any M P BpXq and P LpA, BpXqq, if is derivable on R A pm, 0q, by Lemma 3.2, there exists a k P C such that piq km. Define a left multiplier L P LpA, BpXqq by L paq P A. Let δ L. Then δ is derivable on R A pm, 0q and δpiq 0. For any A, B P A and 0 x b f P A, by Lemma 3.7, δpabx b fq δpaqbx b f ` AδpBx b fq On the other hand, also by Lemma 3.7, δpaqbx b f ` ArδpBqx b f ` Bδpx b fqs δpaqbx b f ` AδpBqx b f ` ABδpx b fq. δpabx b fq δpabqx b f ` ABδpx b fq. It follows δpabqx b f δpaqbx b f ` AδpBqx b f. Since x b f is arbitrary, δpabq δpaqb ` AδpBq, or equivalently pabq paqb ` A pbq A piqb. For the other direction, if P LpA, BpXqq is a generalized derivation then clearly it is derivable on R A pm 1, 0q, with M 1 piq. Acknowledgment. The author wishes to thank the referee for careful reading of the paper and valuable suggestions. References [1] K. Davidson, Nest Algebras, Pitman Research Notes in Math Series 191, [2] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), [3] W. Jing, S. Lu, P. Li, Characterizations of derivations on some operator algebras, Bull. Austral. Math. Soc. 66 (2002),

14 344 Z. Pan [4] B. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Cambridge Philos. Soc. 120 (1996), [5] J. Li, Z. Pan, Algebraic reflexivity of linear transformations, Proc. Amer. Math. Soc. 135 (2007), [6] J. Li, Z. Pan, Annihilator-preserving maps, multipliers, and derivations, Linear Algebra Appl. 432 (2010), [7] J. Li, Z. Pan, On derivable mappings, J. Math. Anal. Appl. 374 (2011), [8] J. Li, Z. Pan, Q. Shen, Jordan and Jordan higher all-derivable points of some algebras, Linear and Multilinear Algebra 61 (2013), [9] J. Li, Z. Pan, H. Xu, Characterizations of isomorphisms and derivations of some algebras, J. Math. Anal. Appl. 332 (2007), [10] F. Lu, Characterizations of derivations and Jordan derivations on Banach algebras, Linear Algebra Appl. 430 (2009), [11] Z. Pan, Derivable maps and derivational points, Linear Algebra Appl. 436 (2012), [12] Z. Pan, Derivable maps and generalized derivations, Oper. Matrices 8 (2014), [13] M. Rosenblum, On the operator equation BX XA Q, Duke Math. J. 23 (1956), [14] X. Qi, J. Hou, Characterizations of derivations of Banach space nest algebras: allderivable points, Linear Algebra Appl. 432 (2010), [15] J. Zhou, Linear mappings derivable at some nontrivial elements, Linear Algebra Appl. 435 (2011), [16] J. Zhu, C. Xiong, L. Zhang, All-derivable points in matrix algebras, Linear Algebra Appl. 430 (2009), Z. Pan DEPARTMENT OF MATHEMATICS SAGINAW VALLEY STATE UNIVERSITY UNIVERSITY CENTER MI 48710, USA pan@svsu.edu Received January 13, 2015; revised version April 20, 2015.