Weak-local derivations and homomorphisms on C*-algebras

Essaleh, Ahlem Ben Ali; Peralta, Antonio M.; Ramírez, María Isabel

doi:10.1080/03081087.2016.1142171

Cited by 6 publications

(15 citation statements)

References 18 publications

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“…We can now obtain our main result. The following consequence is interesting by itself and it compliments the results in [10]. Corollary 2.19.…”

Section: Weak Local Triple Derivations Are Triple Derivationssupporting

confidence: 78%

“…Essaleh, M.I. Ramírez, and the third author of this note explore the notions of weak-local derivation and weak * -local derivation on C * -algebras and von Neumann algebras, respectively (see [10,11]). Going back in history, we remind that, according to Kadison's definition, a local derivation from a C * -algebra A into a Banach A-bimodule, X, is a linear map T : A → X such that for each a ∈ A there exits a derivation D a : A → X, depending on a, satisfying T (a) = D a (a).…”

Section: Introductionmentioning

confidence: 99%

“…Johnson proved in [17] that every local derivation from a C * -algebra A into a Banach A-bimodule is continuous and a derivation. Following [10], a linear mapping T : A → X is called a weak-local derivation if for each φ ∈ X * and each a ∈ A, there exists a derivation D a,φ : A → X, depending on φ and a, such that φT (a) = φD a,φ (a). It is shown in [10, Theorems 2.1 and 3.4] (see also [11]) that every weak-local derivation on a C * -algebra is continuous and a derivation.…”

Section: Introductionmentioning

confidence: 99%

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Weak-local triple derivations on C⁎-algebras and JB⁎-triples

Burgos¹,

Cabello

Peralta

2016

Linear Algebra and its Applications

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We prove that every weak-local triple derivation on a JB * -triple E (i.e. a linear map T : E → E such that for each φ ∈ E * and each a ∈ E, there exists a triple derivation δ a,φ : E → E, depending on φ and a, such that φT (a) = φδ a,φ (a)) is a (continuous) triple derivation. We also prove that conditions (h1) {a, T (b), c} = 0 for every a, b, c in E with a, c ⊥ b; (h2) P 2 (e)T (a) = −Q(e)T (a) for every norm-one element a in E, and every tripotent e in E * * such that e ≤ s(a) in E * * 2 (e), where s(a) is the support tripotent of a in E * * , are necessary and sufficient to show that a linear map T on a JB * -triple E is a triple derivation.

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“…We can now obtain our main result. The following consequence is interesting by itself and it compliments the results in [10]. Corollary 2.19.…”

Section: Weak Local Triple Derivations Are Triple Derivationssupporting

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Weak-local triple derivations on C⁎-algebras and JB⁎-triples

Burgos¹,

Cabello

Peralta

2016

Linear Algebra and its Applications

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“…To understand the whole picture it is worth to recall the notions of local and weak-local maps. Following [16,13], let S be a subset of the space L(X, Y ) of all linear maps between Banach spaces X and Y . A linear mapping ∆ : X → Y is said to be a local S map (respectively, a weak-local S-map) if for each x ∈ X (respectively, if for each x ∈ X and φ ∈ Y * ), there exists T x ∈ S, depending on x (respectively, there exists T x,φ ∈ S, depending on x and φ), satisfying ∆(x) = T x (x) (respectively, φ∆(x) = φT x,φ (x)).…”

Section: A Spherical Variant Of the Gleason-kahane-zelazko Theoremmentioning

confidence: 99%

Weak-2-local isometries on uniform algebras and Lipschitz algebras

Li¹,

Peralta²,

Wang³

et al. 2019

Publicacions Matemàtiques

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“…Let S be a subset of L(X, Y ), where X and Y are Banach spaces. Following [13,14], we say that a linear mapping T : X → Y is a weak-local S map (respectively, a local S map) if for each x ∈ X and φ ∈ Y * (respectively, for each x ∈ X) there exists S x,φ ∈ S, depending on x and φ (respectively, S x ∈ S, depending on x), such that φT (x) = φS x,φ (x) (respectively, T (x) = S x (x)).…”

Section: Weak-2-local Derivations On C * -Algebras Of Continuous Funcmentioning

confidence: 99%

Inner Derivations and Weak-2-Local Derivations on the $$\hbox {C}^*$$ C ∗ -Algebra $$\varvec{C_0(L,A)}$$

Jordá

Peralta

2017

Integr. Equ. Oper. Theory

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Let L be a locally compact Hausdorff space. Suppose A is a C *-algebra with the property that every weak-2-local derivation on A is a (linear) derivation. We prove that every weak-2-local derivation on C 0 (L, A) is a (linear) derivation. Among the consequences we establish that if B is an atomic von Neumann algebra or on a compact C *-algebra, then every weak-2-local derivation on C 0 (L, B) is a linear derivation. We further show that, for a general von Neumann algebra M , every 2-local derivation on C 0 (L, M) is a linear derivation. We also prove several results representing derivations on C 0 (L, B(H)) and on C 0 (L, K(H)) as inner derivations determined by multipliers.

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Weak-local derivations and homomorphisms on C*-algebras

Cited by 6 publications

References 18 publications

Weak-local triple derivations on C⁎-algebras and JB⁎-triples

Weak-local triple derivations on C⁎-algebras and JB⁎-triples

Weak-2-local isometries on uniform algebras and Lipschitz algebras

Inner Derivations and Weak-2-Local Derivations on the $$\hbox {C}^*$$ C ∗ -Algebra $$\varvec{C_0(L,A)}$$

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