Local Triple Derivations on Real C $$^*$$ ∗ -Algebras and JB $$^*$$ ∗ -Triples

Abstract

Abstract. We study when a local triple derivation on a real JB * -triple is a triple derivation. We find an example of a (real linear) local triple derivation on a rank-one Cartan factor of type I which is not a triple derivation. On the other hand, we find sufficient conditions on a real JB * -triple E to guarantee that every local triple derivation on E is a triple derivation.

A survey on local and 2-local derivations on C*- and von Neuman algebras

“…. We shall revisit here the answer to the above problem provided by Fernández Polo, Molino and the third author of this note in [31]. The first goal is to study this problem in the case of generalized Cartan factors.…”

“…Local triple derivations on rank one generalized Cartan factor are the next objective. The next two results and the counterexample following them are taken from [31]. Proof.…”

“…25 [31,. Theorem 3.4] Let E be a real JB * -triple whose second dual contains no rank-one generalized real Cartan factors of types I C , I H and V O := M 1,2 (O).…”

Derivations and projections on Jordan triples: An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis

“…Algebraic reflexivity of the set of local triple derivations on a C * -algebra and on a JB * -triple have been studied in [173,52,55] and [92]. More precisely, Mackey proves in [173] that the set D = Der t (M ), of all triple derivations on a JBW * -triple M satisfies (7).…”

2-local triple derivations on von Neumann algebras

“…Algebraic reflexivity of the set of local triple derivations on a C * -algebra and on a JB * -triple have been studied in [24,9,12] and [14]. More precisely, Mackey proves in [24] that the set D = Der t (M ), of all triple derivations on a JBW * -triple M satisfies (1.1).…”