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

Abstract

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.

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

“…Let S = Der(A) be the set of all derivations on a C * -algebra A. Theorem 3.4 in [13] (see also [14]) proves that every weak-local Der(A) map on A lies in Der(A). This phenomenon also holds for other sets S, for example when S is the set of all triple derivations on a JB * -triple (see [8]). To provide an example in the setting of general Banach spaces, let X and Y be Banach spaces with Y infinite dimensional, and let F be a closed proper subspace of Y .…”

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

“…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)). Local and weak-local maps have been intensively studied by a long list of authors (see, for example, [8,9,12,15,16,17,26,30,32,36] and [37]).…”