Generalized Derivations and Bilocal Jordan Derivations of Nest Algebras

Abstract

Let 𝐻 be a complex Hilbert space and 𝐵(𝐻) the collection of all linear bounded operators, 𝔄 is the closed subspace lattice including 0 an 𝐻, then 𝔄 is a nest, accordingly alg 𝔄={𝑇∈𝐵(𝐻)∶𝑇𝑁⊆𝑁,∀𝑁∈𝔄} is a nest algebra. It will be shown that of nest algebra, generalized derivations are generalized inner derivations, and bilocal Jordan derivations are inner derivations.

1. Introduction

The concept of local derivations was introduced by Kadison [1] who showed that on a Von-nemann algebra all norm-continuous local derivations are derivations. Larson and Sourour [2] proved that on the algebra 𝐵(𝑋) local derivations are derivations. M. Brešar and P. Šemrl [3, 4] generalized the results of the three authors above under a weaker condition. Shulman [5] showed that all local derivations on 𝐶^∗-algebra are derivations.

Based on a great deal of research works of many mathematicians, some scholars paid more interests in similar kind of problems under more generalized conditions, such as considering local derivations on nest algebras and generalized derivation. Zhu and Xiong [6, 7] proved that local derivations of nest algebra and standard operator algebra are derivations, Zhang [8] considered the Jordan derivations of nest algebras, Lee [9] discussed generalized derivations of left faithful rings. Recently, some scholars discussed some new types of derivations, as Li and Zhou [10] and Majieed and Zhou [11] investigated some new types of generalized derivations associated with Hochschild 2 cocycles, other examples are in [12–15]. In fact, under appropriate conditions, local derivations are derivations.

In this paper, we will show that of nest algebra, a generalized derivation is a generalized inner derivation, and bilocal Jordan derivations are inner derivations.

2. Some Notations and Definitions

In what follows, some notations and basic definitions are introduced.

Let 𝐻 be a complex Hilbert space and 𝐵(𝐻) the collection of all linear bounded operators on 𝐻, 𝔄 is the closed subspace lattice including 0 an 𝐻, then 𝔄 is a nest, correspondingly the Nest algebra is alg𝔄={𝑇∈𝐵(𝐻)∶𝑇𝑁⊆𝑁,∀𝑁∈𝔄}.

If 𝑁≠0, we denote 𝑁₋=∨{𝑀∈𝔄∶𝑀⊊𝑁} and if 𝑁≠𝐻, denote 𝑁₊=∧{𝑀∈𝔄∶𝑀⊋𝑁}, where ⊊ is real inclusion, and we define 0₋=0, 𝐻₊=𝐻.

For all 𝑁∈𝔄, 𝑃(𝑁) represent the project operator from 𝐻 to 𝑁, and 𝑁^⟂={𝑓∈𝐻∶⟨𝑥,𝑓⟩=0,∀𝑥∈𝑁}.

Let 𝔄 be a Banach algebra and 𝔄₁ a subalgebra of 𝔄, we call the linear map 𝜑∶𝔄₁→𝔄 a generalized inner derivation if and only if for all 𝑇∈𝔄₁, there exist operators 𝐴 and 𝐵 in 𝔄 such that 𝜑(𝑇)=𝐴𝑇+𝑇𝐵; if for all 𝑇∈𝔄₁, we have 𝜑(𝑇²)=𝜑(𝑇)𝑇+𝑇𝜑(𝑇), then 𝜑 is called a Jordan derivation; if for all 𝑇∈𝔄₁, there is a Jordan derivation 𝜑_𝑇∶𝔄₁→𝔄, such that 𝜑(𝑇)=𝜑_𝑇(𝑇), then 𝜑 is said to be a local Jordan derivation.

Definition 2.1. Let 𝜑∶alg𝔄→alg𝔄 be an additive mapping, if there exists a derivation 𝛿∶alg𝔄→alg𝔄 that 𝜑(𝑆𝑇)=𝜑(𝑆)𝑇+𝑆𝛿(𝑇), for all 𝑆,𝑇∈alg𝔄, then 𝜑 is called a generalized derivation.

Definition 2.2. We call the linear mapping 𝜑∶alg𝔄→alg𝔄 a bilocal Jordan derivation, if for every 𝑢∈𝐻, there is a Jordan derivation 𝛿_𝑇,𝑢∶alg𝔄→alg𝔄, such that 𝜑(𝑇)𝑢=𝛿_𝑇,𝑢(𝑇)𝑢.

3. Main Results

Next to give out the main conclusions.

Theorem 3.1. If 𝜑∶alg𝔄→alg𝔄 is a generalized derivation, then there are operators 𝐴 and 𝐵 in alg𝔄, such that 𝜑(𝑇)=𝐴𝑇+𝑇𝐵, for all 𝑇∈alg𝔄.

Proof. From the definition of generalized derivation, we can find a derivation: 𝛿∶alg𝔄→alg𝔄, such that 𝜑(𝑆𝑇)=𝜑(𝑆)𝑇+𝑆𝛿(𝑇), for all 𝑆,𝑇∈alg𝔄, so when 𝑆=𝐼, we have 𝜑(𝑇)=𝜑(𝐼)𝑇+𝛿(𝑇), for all 𝑇∈alg𝔄, denote 𝜑(𝐼)=𝐶, apparently 𝐶∈alg𝔄 and 𝜑(𝑇)=𝐶𝑇+𝛿(𝑇), for all 𝑇∈alg𝔄.
Since 𝛿∶alg𝔄→alg𝔄 is a derivation, by [6], it is an inner derivation, namely, there exists 𝐷∈alg𝔄, such that 𝛿(𝑇)=𝐷𝑇−𝑇𝐷, consequently 𝜑(𝑇)=𝐶𝑇+𝐷𝑇−𝑇𝐷=(𝐶+𝐷)𝑇−𝑇𝐷. (3.1) Denote 𝐴=𝐶+𝐷, 𝐵=−𝐷, then 𝜑(𝑇)=𝐴𝑇+𝑇𝐵, for all 𝑇∈alg𝔄.

Theorem 3.2. If 𝜑∶alg𝔄→alg𝔄 is a local Jordan derivation, then 𝜑 is an inner derivation.

Proof. Since 𝜑 is a local Jordan derivation, there exists a Jordan derivation 𝜑_𝑇∶alg𝔄→alg𝔄, such that 𝜑(𝑇)=𝜑_𝑇(𝑇), from Theorem  2.12 in [8], we know that the Jordan derivation of nest algebra alg𝔄 is an inner derivation, so there exists 𝐴_𝑇∈alg𝔄, such that 𝜑_𝑇(𝑇)=𝑇𝐴_𝑇−𝐴_𝑇𝑇, by imitating the proof in [6], we can conclude that 𝜑_𝑇(𝑇)=𝑇𝐴−𝐴𝑇, so 𝜑(𝑇)=𝑇𝐴−𝐴𝑇, namely, 𝜑 is an inner derivation.

The following is the main result.

Theorem 3.3. If 𝜑∶alg𝔄→alg𝔄 is a bilocal Jordan derivation, then it is an inner derivation.

Proof. We will prove this proposition by the following three steps.
(1) 𝜑(𝑇)(ker𝑇)⊆ran𝑇, where ker𝑇 and ran𝑇 are the kernal of 𝑇 and range of 𝑇, respectively. In fact, since 𝜑(𝑇)𝑢=𝛿_𝑇,𝑢(𝑇)𝑢 for all 𝑇∈alg𝔄, for all 𝑢∈𝐻, and 𝛿_𝑇,𝑢 is a Jordan derivation, by Theorem  2.12 in [8], there is an 𝐴_𝑇,𝑢∈alg𝔄, such that 𝜑(𝑇)𝑢=(𝑇𝐴_𝑇,𝑢−𝐴_𝑇,𝑢𝑇)𝑢, so if 𝑢∈ker𝑇, we have 𝜑(𝑇)𝑢=𝑇𝐴_𝑇,𝑢𝑢∈ran𝑇.
(2) For all 𝑁∈𝔄, {𝑜}⊂𝑁⊂𝐻, there exists 𝐶_𝑁∈𝐵(𝐻) and 𝐵_𝑁∈𝐵(𝐻), such that 𝜑(𝑥⊗𝑓)=𝑥⊗𝐶_𝑁𝑓+𝐵_𝑁𝑥⊗𝑓, for all 𝑥∈𝑁, 𝑓∈𝑁^⟂₋.
For arbitrary fixed 𝑓∈𝑁^⟂₋, 𝑓≠0, and for all 𝑥∈𝑁, we know that 𝑥⊗𝑦∈alg𝔄, from step (1), we have 𝜑(𝑥⊗𝑓){𝑓}^⟂⊆span{𝑥}, so there exists a linear function 𝜆_𝑥,𝑓over{𝑓}^⟂, such that 𝜑(𝑥⊗𝑓)(𝑢)=⟨𝑢,𝜆_𝑥,𝑓⟩𝑥, for all 𝑢∈{𝑓}^⟂, in succession we will prove that 𝜆_𝑥,𝑓 is independent of 𝑥. Take a 𝑧∈𝑁 which is linear independent of 𝑥, we have 𝑧⊗𝑓∈alg𝔄, then 𝜑(𝑥+𝑧)⊗𝑓(𝑢)=𝜑𝑥⊗𝑓(𝑢)+𝜑𝑧⊗𝑓(𝑢)=𝑢,𝜆_𝑥,𝑓𝑥+𝑢,𝜆_𝑧,𝑓𝑧. (3.2) On the other hand, 𝜑((𝑥+𝑧)⊗𝑓)(𝑢)=⟨𝑢,𝜆_{𝑥+𝑧,𝑓}⟩(𝑥+𝑧), so 𝑢,𝜆_{𝑥+𝑧,𝑓}−𝜆_𝑥,𝑓𝑥=𝑢,𝜆_𝑧,𝑓−𝜆_{𝑥+𝑧,𝑓}𝑧. (3.3) Since 𝑥 is linear independent of 𝑧, we know that 𝜆_𝑥,𝑓=𝜆_{𝑥+𝑧,𝑓}, that is, 𝜆_𝑥,𝑓 is independent of 𝑥, so 𝜆_𝑥,𝑓 can be denoted by 𝜆_𝑓, and 𝜑𝑥⊗𝑓(𝑢)=𝑢,𝜆_𝑓𝑥,∀𝑢∈𝑓^⟂. (3.4) Let 𝑔_𝑓 be the linear continuous span on 𝐻 of 𝜆_𝑓, we define 𝐵_𝑢,𝑓∶𝑁→𝑁 as follows: 𝐵_𝑢,𝑓(𝑥)=1𝜑(𝑢)−𝑢,𝑓𝑥⊗𝑓𝑢,𝑔_𝑓𝑥,where𝑢∈𝑓^⟂. (3.5) Obviously, 𝐵_𝑢,𝑓 is linear and 𝜑(𝑥⊗𝑓)(𝑢)=⟨𝑢,𝑔_𝑓⟩𝑥+⟨𝑢,𝑓⟩𝐵_𝑢,𝑓𝑥, 𝑥∈𝑁, 𝑢∈𝐻.
Next 𝐵_𝑢,𝑓 is independent of 𝑢, which reduce to show (i) 𝐵_{𝑎𝑢,𝑓}=𝐵_𝑢,𝑓, 𝑎∈𝐶; (ii) 𝐵_𝑢,𝑓=𝐵_𝑣,𝑓, where 𝑣≠𝑢, 𝑣∈𝐻.
In fact, (i) is evident. For (ii), since for all 𝑥∈𝑁, 𝜑(𝑥⊗𝑓)(𝑣)=⟨𝑣,𝑔_𝑓⟩𝑥+⟨𝑣,𝑓⟩𝐵_𝑣,𝑓𝑥 and 𝜑(𝑥⊗𝑓)(𝑢+𝑣)=⟨𝑢+𝑣,𝑔_𝑓⟩𝑥+⟨𝑢+𝑣,𝑓⟩𝐵_{𝑢+𝑣,𝑓}𝑥=⟨𝑢+𝑣,𝑔_𝑓⟩𝑥+⟨𝑣,𝑓⟩𝐵_𝑣,𝑓𝑥+⟨𝑢,𝑓⟩𝐵_𝑢,𝑓𝑥, we have ⟨𝑢,𝑓⟩𝐵_{𝑢+𝑣,𝑓}+⟨𝑣,𝑓⟩𝐵_{𝑢+𝑣,𝑓}=⟨𝑢,𝑓⟩𝐵_𝑢,𝑓+⟨𝑣,𝑓⟩𝐵_𝑣,𝑓, namely, ⟨𝑢,𝑓⟩(𝐵_{𝑢+𝑣,𝑓}−𝐵_𝑢,𝑓)=⟨𝑣,𝑓⟩(𝐵_𝑣,𝑓−𝐵_{𝑢+𝑣,𝑓}), on account of 𝑢≠𝑣, so 𝐵_{𝑢+𝑣,𝑓}=𝐵_𝑢,𝑓, that is, 𝐵_𝑢,𝑓 is independent of 𝑢, so we can mark 𝐵_𝑢,𝑓 by 𝐵_𝑓, as a result, we have 𝜑𝑥⊗𝑓(𝑢)=𝑢,𝑔_𝑓𝑥+𝑢,𝑓𝐵_𝑓𝑥=𝑥⊗𝑔_𝑓(𝑢)+𝐵_𝑓𝑥⊗𝑓(𝑢),∀𝑢∈𝐻. (3.6) Consequently 𝜑𝑥⊗𝑓=𝑥⊗𝑔_𝑓+𝐵_𝑓𝑥⊗𝑓. (3.7)
Define 𝐶_𝑁∶𝑁^⟂₋→𝑁^⟂₋∶𝐶_𝑁𝑓→𝑔_𝑓, now we will show that 𝐶_𝑁 is a linear bounded operator. Because 𝑔_𝑓 is a continuous linear function, so 𝑔_𝑓 is bounded, consequently 𝐶_𝑁 is bounded, according to (3.4), we know 𝜑𝑥⊗𝑎𝑓(𝑢)=𝑢,𝜆_𝑎𝑓𝑥=𝑎𝜑𝑥⊗𝑓(𝑢)=𝑎𝑢,𝜆_𝑓𝑥=𝑢,𝑎𝜆_𝑓𝑥, (3.8) so 𝜆_𝑎,𝑓=𝑎𝜆_𝑓; on the other hand, 𝜑𝑥⊗𝑓₁+𝑓₂(𝑢)=𝑢,𝜆_𝑓1+𝑓2𝑥=𝜑𝑥⊗𝑓₁(𝑢)+𝜑𝑥⊗𝑓₂(𝑢)=𝑢,𝜆_𝑓1𝑥+𝑢,𝜆_𝑓2𝑥=𝑢,𝜆_𝑓1+𝜆_𝑓2𝑥, (3.9) so 𝜆_𝑓1+𝑓2=𝜆_𝑓1+𝜆_𝑓2, this is enough to show that 𝑔_𝑎𝑓=𝑎𝑔_𝑓, 𝑔_𝑓1+𝑓2=𝑔_𝑓1+𝑔_𝑓2, so when 𝑎𝑓₁+𝑓₂∈𝑁^⟂₋, 𝐶_𝑁𝑎𝑓₁+𝑓₂=𝑔_{𝑎𝑓1+𝑓2}=𝑔_𝑎𝑓1+𝑔_𝑓2=𝑎𝐶_𝑁𝑓₁+𝐶_𝑁𝑓₂. (3.10) That is to say 𝐶_𝑁 is linear, so (3.7) has the form of 𝜑𝑥⊗𝑓=𝑥⊗𝐶_𝑁𝑓+𝐵_𝑓𝑥⊗𝑓. (3.11) In succession we will prove that 𝐵_𝑓 is independent of 𝑓, arbitrarily choose 𝑦∈𝑁^⟂₋, where 𝑦 is linear independent of 𝑓, then 𝑦+𝑓∈𝑁^⟂₋ and 𝜑𝑥⊗𝑓+𝑦=𝑥⊗𝐶_𝑁𝑓+𝑦+𝐵_𝑓+𝑦𝑥⊗𝑓+𝑦=𝑥⊗𝐶_𝑁𝑓+𝑥⊗𝐶_𝑁𝑦+𝐵_𝑓+𝑦𝑥⊗𝑓+𝐵_𝑓+𝑦𝑥⊗𝑦. (3.12) On the other hand, 𝜑𝑥⊗𝑓+𝑦=𝜑𝑥×⊗𝑓+𝜑𝑥⊗𝑦=𝑥⊗𝐶_𝑁𝑓+𝐵_𝑓𝑥⊗𝑓+𝑥⊗𝐶_𝑁𝑦+𝐵_𝑦𝑥⊗𝑦. (3.13) So (𝐵_𝑓+𝑦−𝐵_𝑓)𝑥⊗𝑓=(𝐵_𝑦−𝐵_𝑦+𝑓)𝑥⊗𝑓, then 𝐵_𝑓=𝐵_𝑓+𝑦=𝐵_𝑦, that is, 𝐵_𝑓 is independent of 𝑓, which can be marked by 𝐵_𝑁, so (3.11) has the form of 𝜑𝑥⊗𝑓=𝑥⊗𝐶_𝑁𝑓+𝐵_𝑁𝑥⊗𝑓. (3.14)We proved that𝐵_𝑁 is bounded, because ‖𝐵_𝑁𝑥⊗𝑓‖≤‖𝜑(𝑥⊗𝑓)‖+‖𝑥⊗𝐶_𝑁𝑓‖≤𝑀+‖𝐶_𝑁‖‖𝑥‖‖𝑓‖.
On account of the boundary of 𝐶_𝑁 and 𝜑(𝑥⊗𝑓)∈alg𝑁⊆𝐵(𝐻), we know that 𝐵_𝑁 is bounded, namely, 𝐵_𝑁∈𝐵(𝑁), 𝐶_𝑁∈𝐵(𝑁^⟂₋).
(3) For arbitrary 𝑥⊗𝑦∈alg𝔄, there is 𝜑(𝑥⊗𝑓)=𝑥⊗𝐶𝑓+𝐵𝑥⊗𝑓.
For all 𝑀,𝑁∈𝔄, {0}⊂𝑁⊂𝑀⊂𝐻, select 𝑥∈𝑁, 𝑓∈𝑀^⟂₋, then 𝑥∈𝑀, 𝑓∈𝑁^⟂₋ and 𝑥⊗𝑓∈alg𝔄, from the result of step (2), it is easy to know that 𝜑𝑥⊗𝑓=𝑥⊗𝐶_𝑁𝑓+𝐵_𝑁𝑥⊗𝑓,𝜑𝑥⊗𝑓=𝑥⊗𝐶_𝑀𝑓+𝐵_𝑀𝑥⊗𝑓. (3.15) Consequently 𝑥⊗(𝐶_𝑁−𝐶_𝑀)𝑓=(𝐵_𝑀−𝐵_𝑁)𝑥⊗𝑓, so there exists a scalar 𝜆(𝑁,𝑀), such that 𝐶_𝑁−𝐶_{𝑀𝑀⟂−}=𝜆𝑃𝑀^⟂₋,𝐵_𝑁−𝐵_𝑀𝑁=−𝜆𝑃(𝑁). (3.16) By imitating Lemma  2 mentioned in [6], we can prove that 𝜑𝑥⊗𝑓=𝑥⊗𝐶𝑓+𝐵𝑥⊗𝑓,𝑥⊗𝑦∈alg𝑁. (3.17)
Since the collection of all rank one operators is dense in alg𝔄, so for every 𝑇∈alg𝔄, we have 𝜑(𝑇)=𝑇𝐶^∗+𝐵𝑇, let 𝑇=𝐼, then 𝜑(𝐼)=𝐵+𝐶^∗, considering 𝜑 to be a bilocal Jordan derivation, namely, 𝜑(𝐼)(𝑢)=𝛿_𝐼,𝑢(𝐼²)𝑢=(𝐼𝛿_𝐼,𝑢(𝐼)+𝛿_𝐼,𝑢(𝐼)𝐼)𝑢=2𝛿_𝐼,𝑢(𝐼)𝑢, we can conclude that 𝛿_𝐼,𝑢(𝐼)=0, so 𝜑(𝐼)=0, thereby 𝐵+𝐶^∗=0 and 𝜑(𝑇)=𝐵𝑇−𝑇𝐵, which shows that 𝜑 is an inner derivation.

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable suggestions. This work was supported by the Natural Science Foundation Project of CQ CSTC under Contract no. 2010BB2240.