On derivations of the Jordan triple product in classical Cartan factors

Abstract

Let U be a classical Cartan factor of any of the types In, IIn, IIIn or IVn with dim⁡(U)=∞, and let δ:U→U be a map (neither linearity nor continuity of δ is assumed) that satisfies the functional equation δ{aba}={δ(a)ba}+{aδ(b)a}+{abδ(a)}, (a,b∈U), where {abc} stands for the Jordan triple product in U. We prove that δ is additive and homogeneous over the field Q of rational numbers. Moreover δ is real-linear if and only if it is continuous. This gives a new description of the real Banach Lie algebra of derivations of U.