2-local triple derivations on von Neumann algebras

Abstract

We prove that every (not necessarily linear nor continuous) 2-local triple derivation on a von Neumann algebra M is a triple derivation, equivalently, the set Dert(M ), of all triple derivations on M, is algebraically 2-reflexive in the set M(M ) = M M of all mappings from M into M . 1991 Mathematics Subject Classification. Primary 46L05; 46L40.

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]).…”

Boundedness of completely additive measures with application to 2-local triple derivations

“…Applications can be found in quantum physics and quantum information (cf. [18], [43], [37], [36], [22,Chapter 7], and [16], among many others), and in functional analysis with studies on vector-valued measures on von Neumann algebras and 2-local maps on von Neumann algebras, JBW * -algebras and JBW * -triples (see [20], [3], [4], [13] [14] and [32]).…”

“…The Dorofeev-Shertsnev theorem was used in [32] in order to apply the Bunce-Wright-Mackey-Gleason theorem to prove the main result of that paper, namely, that a 2-local triple derivation on a von Neumann algebra is a triple derivation (see the next subsection). In section 3 of this paper, we shall establish the first main result of this paper, namely, a Jordan version of Dorofeev's boundedness theorem (Theorem 3.1).…”

“…A 2-local derivation (respectively, 2-local triple derivation) from an algebra (respectively, triple system) into itself is a mapping (not necessarily linear) whose values at each pair of points in the algebra (respectively, triple system) coincides with the values of some derivation (respectively, triple derivation) at those two points. 2-local derivations were introduced in 1997 by Semrl [38] and 2-local triple derivations were introduced in 2014 by Kudaybergenov, Oikhberg, Peralta, and Russo [32] although the concept was mentioned by Michael Mackey in a lecture in 2012 at a conference in Hong Kong celebrating the 65th birthday of Cho-Ho Chu. It is now known that, for von Neumann algebras, a 2-local derivation is in fact a derivation (Ayupov-Kudaybergenov [3]) and, as noted above, a 2-local triple derivation is a triple derivation (Kudaybergenov-Oikhberg-Peralta-Russo [32]).…”

“…2-local derivations were introduced in 1997 by Semrl [38] and 2-local triple derivations were introduced in 2014 by Kudaybergenov, Oikhberg, Peralta, and Russo [32] although the concept was mentioned by Michael Mackey in a lecture in 2012 at a conference in Hong Kong celebrating the 65th birthday of Cho-Ho Chu. It is now known that, for von Neumann algebras, a 2-local derivation is in fact a derivation (Ayupov-Kudaybergenov [3]) and, as noted above, a 2-local triple derivation is a triple derivation (Kudaybergenov-Oikhberg-Peralta-Russo [32]).…”