On derivable mappings

Li, Jiankui; Pan, Zhidong

doi:10.1016/j.jmaa.2010.08.020

Cited by 15 publications

(7 citation statements)

References 22 publications

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“…As usual, we say that an element Z ∈ A is an all-derivable point of A if every linear map on A derivable at Z is in fact a derivation. So far, we have known that there exist many all-derivable points (or full-drivable points) for certain (operator) algebras (see [1,4,7,8,13,14,16] and the references therein). However, zero point 0 is not an all-derivable point for any algebra because the generalized derivations are derivable at 0 [8].…”

Section: A Linear Map δ : a → A Where A Is An Algebra Is A Derivatimentioning

confidence: 99%

All-derivable subsets for nest algebras on Banach spaces

Zhang¹,

Hou²,

Qi³

2014

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Let N be a nest on a complex Banach space X and let AlgN be the associated nest algebra. We say that a subset S ⊂ AlgN is an all-derivable subset of AlgN if every linear map δ from AlgN into itself derivable on S (i.e. δ satisfies that, for each Z ∈ S, δ(A)B + Aδ(B) = δ(Z) for any A, B ∈ AlgN with AB = Z) is a derivation. In this paper, we show that S is an all-derivable subset of AlgN if span{ran(Z) : Z ∈ S} is dense in X or ∩{ker Z : Z ∈ S} = {0}. Mathematics Subject Classification: 47B47, 47L35

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Section: A Linear Map δ : a → A Where A Is An Algebra Is A Derivatimentioning

confidence: 99%

All-derivable subsets for nest algebras on Banach spaces

Zhang¹,

Hou²,

Qi³

2014

imf

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“…Remark 3.4. Several authors (see for example [1,11,12,18,19,24,30,36,37,38,39]) investigate derivable mappings at 0, invertible element, left (right) separating point, non-trivial idempotent, and the unit I on certain algebras. By Theorem 3.3, we can generalize these results to the higher derivation case.…”

Section: By Induction We Havementioning

confidence: 99%

Jordan and Jordan higher all-derivable points of some algebras

Pan

Shen

2013

Linear and Multilinear Algebra

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In this paper, we characterize Jordan derivable mappings in terms of Peirce decomposition and determine Jordan all-derivable points for some general bimodules. Then we generalize the results to the case of Jordan higher derivable mappings. An immediate application of our main results shows that for a nest N on a Banach X with the associated nest algebra algN , if there exists a non-trivial element in N which is complemented in X, then every C ∈ algN is a Jordan all-derivable point of L(algN , B(X)) and a Jordan higher all-derivable point of L(algN ). the relations among derivations, Jordan derivations as well as inner derivations (see for example [3,4,10,14,23,28,32], and the references therein).In general there are two directions in the study of the local actions of derivations of operator algebras. One is the well known local derivation problem (see for example [7,9,15,35,40]).The other is to study conditions under which derivations of operator algebras can be completely determined by the action on some subsets of operators (see for example [3,5,17,30,34,39]).obvious that a linear mapping is a Jordan derivation if and only if it is Jordan derivable at all points. It is natural and interesting to ask the question whether or not a linear mapping is a Jordan derivation if it is Jordan derivable only at one given point. If such a point exists, we call this point a Jordan all-derivable point. To be more precise, an element C ∈ A is called a Jordan all-derivable point of L(A, M) if every Jordan derivable mapping at C is a Jordan derivation. It is quite surprising that there do exist Jordan all-derivable points for some algebras. An and Hou [2] show that under some mild conditions on unital prime ring or triangular ring A, I is a Jordan all-derivable point of L(A). Jiao and Hou [13] study Jordan derivable mappings at zero point on nest algebras. Zhao and Zhu [33] prove that 0 and I are Jordan all-derivable points of the triangular algebra. In [16], the authors study some derivable mappings in the generalized matrix algebra A, and show that 0, P and I are Jordan all-derivable points, where P is the standard non-trivial idempotent. In [33], Zhao and Zhu prove that every element in the algebra of all n × n upper triangular matrices over the complex field C is a Jordan all-derivable point. In Section 2, we give some general characterizations of Jordan derivable mappings, which will be used to determine Jordan all-derivable points for some general bimodules. Let A be a unital algebra and N be the set of non-negative integers. A sequence of mappings {d i } i∈N ∈ L(A) with d 0 = I A is called a higher derivation if d n (AB) = i+j=n d i (A)d j (B) for all A, B ∈ A; it is called a Jordan higher derivation if d n (AB + BA) = i+j=n (d i (A)d j (B) + d i (B)d j (A)) for all A, B ∈ A. With the development of derivations, the study of higher and Jordan higher derivations has attracted much attention as an active subject of research in operator algebras, and the local action problem ranks among in the list. A sequence of mappingsJor...

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“…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 AˆA : a " bu, such maps are called Jordan derivations . In [3], [6][7], [9], [11], and [14][15][16], the authors have studied derivable maps on relations R A pcq " tpa, bq P AˆA : ab " cu, for some c P A. Not all derivable maps are derivations.…”

Section: Introductionmentioning

confidence: 99%

Derivable Maps and Generalized Derivations on Nest and Standard Algebras

Pan

2016

Demonstratio Mathematica

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Abstract. For an algebra A, an A-bimodule M, and m P M, define a relation on A by RApm, 0q " tpa, bq P AˆA : 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 RApM, 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 RApP, 0q then δ is a generalized derivation. IntroductionFor 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´av, @ 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 AˆA. 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 AˆA : 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 AˆA : 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

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On derivable mappings

Cited by 15 publications

References 22 publications

All-derivable subsets for nest algebras on Banach spaces

All-derivable subsets for nest algebras on Banach spaces

Jordan and Jordan higher all-derivable points of some algebras

Derivable Maps and Generalized Derivations on Nest and Standard Algebras

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