Weak-2-local symmetric maps on C⁎-algebras

Abstract

We introduce and study weak-2-local symmetric maps between C * -algebras A and B as non necessarily linear nor continuous maps ∆ : A → B such that for each a, b ∈ A and φ ∈ B * , there exists a symmetric linear map T a,b,φ : A → B, depending on a, b and φ, satisfying φ∆(a) = φT a,b,φ (a) and φ∆(b) = φT a,b,φ (b). We prove that every weak-2-local symmetric map between C * -algebras is a linear map. Among the consequences we show that every weak-2-local * -derivation on a general C * -algebra is a (linear) * -derivation. We also establish a 2-local version of the Kowalski-S lodkowski theorem for general C *algebras by proving that every 2-local * -homomorphism between C * -algebras is a (linear) * -homomorphism.

2-Local standard isometries on vector-valued Lipschitz function spaces

“…Let S be a subset of the space L(E, F ) of all continuous linear maps from E into F . We shall follow the notation in [19,20,8,9] and [21]. Accordingly to those references, a (non-necessarily linear nor continuous) mapping ∆ : E → F is a 2-local S-map if for any x, y ∈ E, there exists T x,y ∈ S, depending on x and y, such that ∆(x) = T x,y (x) and ∆(y) = T x,y (y).…”

“…The same conclusion holds when K is the closure of a strictly pseudoconvex domain in C 2 with boundary of class C 2 (Hatori, Miura, Oka and Takagi [11]); ( ) If K is a σ-compact metric space and E is a smooth reflexive Banach space, then C 0 (K, E) is 2-iso-reflexive if and only if E is 2-iso-reflexive (Al-Halees and Fleming [1]); ( ) Every weak-2-local isometry between uniform algebras is linear (Li, Peralta, Wang and Wang [16]). 2-local derivations on C * -algebras have been studied in [21,2,3,19,20,8,9,14] and [15].…”

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

“…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)).…”

“…We shall write Iso(X) instead of Iso(X, X). Accordingly to the notation in [39,40,13,14] and [43], we shall say that a (non-necessarily linear nor continuous) mapping ∆ : X → Y is a weak-2-local Iso(X, Y )-map or a weak-2-local isometry (respectively, a 2-local Iso(X, Y )-map or a 2-local isometry) if for each x, y ∈ X and φ ∈ Y * , there exists T x,y,φ in Iso(X, Y ), depending on x, y, and φ (respectively, for each x, y ∈ X, there exists T x,y in Iso(X, Y ), depending on x and y), satisfying φ∆(x) = φT x,y,φ (x), and φ∆(y) = φT x,y,φ (y) Date: May 11, 2017May 11, . 2000 Mathematics Subject Classification.…”

“…For 1 ≤ p < ∞ and p = 2, Al-Halees and Fleming [1] showed that ℓ p is 2-iso-reflexive. In the setting of B(H), C * -algebras and JB * -triples there exists a extensive literature on different classes of (weak-)2-local of maps (see, for example, [2,3,4,10,11,13,14,19,20,23,31,35,39,40] and [43]).…”