Characterizations of Jordan derivations and Jordan homomorphisms

Li, Jiankui; Zhou, Jiren

doi:10.1080/03081080903304093

Cited by 26 publications

(9 citation statements)

References 19 publications

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“…So we have g 12 (Y Z) = Y h 22 (Z). Through a similar discussion as the proof of Claim 4, we obtain h 22 (Z) = Zh 22 (I 2 ) for all Z in B and g12 (Y ) = Y h 22 (I 2 ) for all Y in M.Thus we have thatφ X Y 0 Z = f 11 (I 1 )X f 11 (I 1 )Y 0 Zh 22 (I 2 ) = f 11 (I 1 )X Y h 22 (I 2 ) 0 Zh 22 (I 2 )for everyX Y 0 Z in J .So it is sufficient to show that f 11 (I 1 )X = Xf 11 (I 1 ) for all X in A, and h 22 (I 2 )Z = Zh 22 (I 2 ) for all Z in B. Since f 11 (I 1 )Y = Y h 22 (I 2 ) for all Y in M, we have f 11 (I 1 )XY = XY h 22 (I 2 ) = Xf 11 (I 1 )Y .…”

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confidence: 61%

Characterizations of Centralizers and Derivations on Some Algebras

He¹,

Li²,

Qian³

2017

Journal of the Korean Mathematical Society

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Abstract. A linear mapping φ on an algebra A is called a centralizable mapping at G ∈ A if φ(AB) = φ(A)B = Aφ(B) for each A and B in A with AB = G, and φ is called a derivable mapping at G ∈ A if φ(AB) = φ(A)B + Aφ(B) for each A and B in A with AB = G. A point G in A is called a full-centralizable point (resp. full-derivable point) if every centralizable (resp. derivable) mapping at G is a centralizer (resp. derivation). We prove that every point in a von Neumann algebra or a triangular algebra is a full-centralizable point. We also prove that a point in a von Neumann algebra is a full-derivable point if and only if its central carrier is the unit.

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confidence: 61%

Characterizations of Centralizers and Derivations on Some Algebras

He¹,

Li²,

Qian³

2017

Journal of the Korean Mathematical Society

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“…Thus, by Theorem 2.3 we have the following result which is a generalization of main result in [6]. Downloaded by [New York University] at 12:27 23 June 2015 Theorem 2.4 Let R be a prime ring with the unit 1 and a nontrivial idempotent p = 1 1 .…”

Section: Downloaded By [New York University] At 12:27 23 June 2015mentioning

confidence: 84%

“…It is surprising that there do exist full-derivable points for some algebras. In [5,6], the authors shown that 'every nontrivial idempotent and left or right separating points are full-derivable points of a prime Banach algebra'. In [2], we proved that 'every nonzero operator is a full-derivable point of the Hilbert space nest algebra AlgN '.…”

Section: Introductionmentioning

confidence: 99%

“…It is natural and interesting to ask the question whether or not an additive map is a derivation if it is only derivable at one given point. Following, [2,5,6] we say that an element β is a full-derivable point of a ring R if every additive map from R into its bimodule M which is derivable at β is in fact a derivation. It is surprising that there do exist full-derivable points for some algebras.…”

Section: Introductionmentioning

confidence: 99%

“…So z 1 x 2 + w 1 z 2 = 0 entails δ 22 (1 2 )z 1 x 2 + w 1 z 2 δ 11 (1 1 ) = z 1 x 2 δ 11 (1 1 ) + w 1 z 2 δ 11 (1 1 ) = 0. Therefore, condition (6) and Equation …”

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confidence: 96%

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Equivalent characterization of derivations on operator algebras

Xue

Hou

2013

Linear and Multilinear Algebra

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Let R be a unital ring with a nontrivial idempotent p, and let M be an R-bimodule. We say that an additive map δ : R → M is derivable at β ∈ R if δ(x y) = δ(x)y + xδ(y) for any x, y ∈ R with x y = β. In this paper, we give a necessary and sufficient condition for an additive map δ : R → M to be derivable at β with β = pβ = βp. Moreover, we show that if R is a prime Banach algebra with the unit 1, then an additive map δ : R → R is derivable at β with β = pβ = βp if and only if there is a derivation τ : R → R such that δ(x) = τ (x) + δ(1)x for all x ∈ R. As an application, we get a full characterization of derivable maps on some reflexive algebras and von Neumann algebras with no abelian summands. In particular, we show that an additive map δ : B(X ) → B(X ) is derivable at any nonzero finite rank operator if and only if it is a derivation. New equivalent characterization of derivations on these algebras are obtained.

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