Stability of derivations under weak-2-local continuous perturbations

Abstract

Let Ω be a compact Hausdorff space and let A be a C * -algebra. We prove that if every weak-2-local derivation on A is a linear derivation and every derivation on C(Ω, A) is inner, then every weak-2-local derivation ∆ : C(Ω, A) → C(Ω, A) is a (linear) derivation. As a consequence we derive that, for every complex Hilbert space H, every weak-2-local derivation ∆ : C(Ω, B(H)) → C(Ω, B(H)) is a (linear) derivation. We actually show that the same conclusion remains true when B(H) is replaced with an atomic von Neumann algebra. With a modified technique we prove that, if B denotes a compact C * -algebra (in particular, when B = K(H)), then every weak-2-local derivation on C(Ω, B) is a (linear) derivation. Among the consequences, we show that for each von Neumann algebra M and every compact Hausdorff space Ω, every 2-local derivation on C(Ω, M ) is a (linear) derivation.

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

“…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

“…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]).…”

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

“…Inspired by the Kowalski-S lodkowski theorem (see [22]), P.Šemrl introduced in [32] the notions of 2-local derivations and automorphisms. This notion and subsequent generalizations have been intensively explored in recent papers (see, for example, [25,26,9,10] and [19]). More recent contributions deal with the following general notion: Let S be a subset of the space L(X, Y ), of all linear maps between Banach spaces X and Y , a (non-necessarily linear nor continuous) mapping ∆ : X → Y is said to be a weak-2-local S map (respectively, a 2-local S-map) if for every x, y ∈ X and φ ∈ Y * (respectively, for every x, y ∈ X), there exists T x,y,φ ∈ S, depending on x, y and φ (respectively, T x,y ∈ S, depending on x and y), satisfying φ∆(x) = φT x,y,φ (x), and φ∆(y) = φT x,y,φ (y) (respectively, ∆(x) = T x,y (x), and ∆(y) = T x,y (y)).…”

“…The linearity of a 2-local S-map is not always guaranteed. For example, as noted in [19], for S = K(X, Y ) the space of compact linear mappings from X to Y , every 1-homogeneous map ∆ : X → Y , i.e. ∆(αx) = α∆(x) for each α ∈ C, is a 2-local S-map.…”

“…In certain cases, there exist derivations which are not inner however they are very similar to inner derivations. For example, by Proposition 2.10 in [19], for each compact Hausdorff space Ω with a topology τ , each complex Hilbert space H, and each derivation D : C(Ω, K(H)) → C(Ω, K(H)), there exists a τ -weak *continuous, bounded mapping Z 0 : Ω → B(H) satisfying D(X) = [Z 0 , X], for every X ∈ C(Ω, K(H)). Remark 2.11 in [19] shows that the mapping Z 0 need not be, in general, τ -norm continuous.…”