Local triple derivations on C∗-algebras and JB∗-triples

Abstract

In a first result, we prove that every continuous local triple derivation on a JB * -triple is a triple derivation. We also give an automatic continuity result, that is, we show that local triple derivations on a JB * -triple are continuous even if not assumed a priori to be so. These results provide positive answers to the conjectures posed by Mackey (Bull. London Math. Soc. 45 (2013) 811-824). In particular, every local triple derivation on a C * -algebra is a triple derivation. We also explore the connections between (bounded local) triple derivations and generalized (Jordan) derivations on a C * -algebra.

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

“…In this case, the mapping δ a,b is a derivation on E and T (a + ib) = T (a) + iT (b) = δ a,b (a) + iδ a,b (b) = δ a,b (a + ib). Therefore, T is a local triple derivation on the complex JB * -triple E, and hence Theorems 2.4 and 2.8 in [8] assure that T (and hence T ) is a (continuous) triple derivation. …”

“…Corollary 2.5 in [8] assures that T {a, a, a} = 2 {T (a), a, a}+ {a, T (a), a}, for every a ∈ E. If we consider the symmetrized triple product < a, b, c >:= 1 3 ({a, b, c} + {c, a, b} + {b, c, a}) , which is trilinear and symmetric, a real polarisation formula gives that T is a triple derivation of the symmetrized Jordan triple product < ., ., . > .…”

“…> . In other words, [8,Corollary 2.5] shows that every local triple derivation on a real JB * -triple is a derivation of the symmetrized triple product. We shall see in this section that the reciprocal statement is true whenever E is a generalized real Cartan factor.…”

“…Let T : E → E be a local triple derivation on E. We have already commented that Corollary 2.5 in [8] assures that T is a triple derivation of the symmetrized triple product. Lemma 3.2 implies that T * * : E * * → E * * is a triple derivation of the symmetrized triple product.…”

“…A similar result for bounded local triple derivations on unital C * -algebras was established by M. Burgos, J.J. Garcés and the first and third authors of this note (see [7]). The problems and questions about triple derivations have been completely solved in the recent note [8], where M. Burgos and the first and third authors of this note establish that every continuous local triple derivation on a JB * -triple is a triple derivation; moreover, local triple derivations on a JB * -triple are continuous even if not assumed a priori to be so. In the same paper, the authors open the scope to the study of real-linear local triple derivations on JB * -triples and to the study of local triple derivations in the wider class of real JB * -triples (i.e.…”