Generalized derivable mappings at zero point on some reflexive operator algebras

Zhu, Jiang; Xiong, Changping

doi:10.1016/j.laa.2004.11.012

Cited by 41 publications

(22 citation statements)

References 11 publications

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“…They are closely related to the notion of a generalized derivation that first appeared in [7]. We point out that there are several papers which are concerned with the characterization of (generalized) derivations by their action on the zero product elements [2,9,18,23,33,37].…”

Section: (Ab)t (C) = T (A)t (Bc) (A B C ∈ A)mentioning

confidence: 96%

“…The condition (D2) has been considered in [9,23,37]. Furthermore, ξ can be chosen in Z A (X) in each of the following cases:…”

Section: 3mentioning

confidence: 99%

“…Over the last years a considerable attention has been paid to characterizations of such kind [2,8,9,23,37]. In the recent papers [2,8], it was observed that characterizing derivations through the zero product analysis can be a powerful tool in studying local derivations, which is another popular research area.…”

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confidence: 99%

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Maps preserving zero products

et al. 2009

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Abstract. A linear map T from a Banach algebra A into another B preserves zero products if T (a)T (b) = 0 whenever a, b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T : A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras.Our method involves continuous bilinear maps φ : A × A → X (for some Banach space X) with the property that φ(a, b) = 0 whenever a, b ∈ A are such that ab = 0. We prove that such a map necessarily satisfies φ(aµ, b) = φ(a, µb) for all a, b ∈ A and for all µ from the closure with respect to the strong operator topology of the subalgebra of M(A) (the multiplier algebra of A) generated by doubly power-bounded elements of M(A). This method is also shown to be useful for characterizing derivations through the zero products.Introduction. S. Banach [4] was the first to describe isometries on L p ([0, 1]) with p = 2. Although Banach did not give the full proof for this case (this was provided by J. Lamperti [26]), he made the key observation that isometries on L p ([0, 1]) must take functions with disjoint support into functions with disjoint support. This property arises in a variety of situations and has been considered by several authors. For example, in the theory of Banach lattices there is an extensive literature about linear maps T : X → Y , where X and Y are Banach lattices, with the property that |T (x)| ∧ |T (y)| = 0 whenever x, y ∈ X are such that |x| ∧ |y| = 0. Such maps are usually called disjointness preserving operators or d-homomorphisms. The reader interested in this setting is referred to the monograph [1]. The concept of a disjointness preserving operator was exported to function algebras by E. Beckenstein and L. Narici (see [5] for general information).

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Section: (Ab)t (C) = T (A)t (Bc) (A B C ∈ A)mentioning

confidence: 96%

“…The condition (D2) has been considered in [9,23,37]. Furthermore, ξ can be chosen in Z A (X) in each of the following cases:…”

Section: 3mentioning

confidence: 99%

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Maps preserving zero products

et al. 2009

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“…Over the last years a considerable attention has been paid to the question of determining derivations through the action on the zero product elements [1,2,4,5,8,9,17,19]. There are several natural conditions for a linear operator T : A → A on a given Banach algebra A, which relate zero products with derivations, namely:…”

Section: Approximate Derivations On the Zero Productsmentioning

confidence: 99%

Hyperreflexivity of the derivation space of some group algebras

2009

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Let T be a continuous linear operator on a Banach algebra A. We address the question of whether the constant sup{ aT (b)c : a, b, c ∈ A, ab = bc = 0, a = b = c = 1} being small implies that the distance from T to the space Der(A) of all continuous derivations on A is small. We show that this is the case for amenable group algebras. As a consequence, we deduce that Der(L 1 (G)) is hyperreflexive for each amenable group in [SIN].

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“…For more studies concerning Jordan derivations we refer the reader to [5,8,10,12,16,17,18,19] and the references therein. Also, there have been a number of papers concerning the study of conditions under which (generalized or Jordan) derivations of rings can be completely determined by the action on some sets of points [1,2,3,7,9,13,14,15,21,22].…”

Section: Introductionmentioning

confidence: 99%

Characterizing Jordan Derivations of Matrix Rings Through Zero Products

Ghahramani¹

2015

Mathematica Slovaca

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Abstract. Let Mn(R) be the ring of all n × n matrices over a unital ring R, let M be a 2-torsion free unital Mn(R)-bimodule and let D : Mn(R) → M be an additive map. We prove that if (1)a = 0 whenever ab = ba = 0. We apply this results to provide that any (generalized) Jordan derivation from Mn(R) into a 2-torsion free Mn(R)-bimodule (not necessarily unital) is a (generalized) derivation. Also, we show that if ϕ : Mn(R) → Mn(R) is an additive map satisfying ϕ(ab + ba) = aϕ(b) + ϕ(b)a (a, b ∈ Mn(R)), then ϕ(a) = aϕ(1) for all a ∈ Mn(R), where ϕ(1) lies in the centre of Mn(R). By applying this result we obtain that every Jordan derivation of the trivial extension of Mn(R) by Mn(R) is a derivation.

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Generalized derivable mappings at zero point on some reflexive operator algebras

Cited by 41 publications

References 11 publications

Maps preserving zero products

Maps preserving zero products

Hyperreflexivity of the derivation space of some group algebras

Characterizing Jordan Derivations of Matrix Rings Through Zero Products

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