Weak-2-local derivations on Mn

Abstract

We introduce the notion of weak-2-local derivation (respectively, * -derivation) on a C * -algebra A as a (non-necessarily linear) map ∆ : A → A satisfying that for every a, b ∈ A and φ ∈ A * there exists a derivation (respectively, a * -derivation) D a,b,φ : A → A, depending on a, b and φ, such that φ∆(a) = φD a,b,φ (a) and φ∆(b) = φD a,b,φ (b). We prove that every weak-2-local * -derivation on Mn is a linear derivation. We also show that the same conclusion remains true for weak-2-local * -derivations on finite dimensional C * -algebras.2000 Mathematics Subject Classification. Primary 47B49, 15A60, 16W25, 47B48 Secondary 15A86; 47L10.

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].…”

2-Local derivations on associative and Jordan matrix rings over commutative rings

“…Proof. Similar to proof of [22,Proposition 2.7] it can be proved that ∆ i,j is a 2-local derivation. Let x be an arbitrary element in eH n (ℜ)e and…”

On a generalized Šemrl’s theorem for weak 2-local derivations on $B(H)$

“…We recall some basic properties on weak-2-local maps which have been borrowed from [4] and [6]. Henceforth, H will denote an arbitrary complex Hilbert space.…”