A characterisation of von Neumann regular Jordan triple systems

Abstract

In this note we give a characterisation of the Jacobson radical of a Jordan triple system in terms of principal inner ideals. If ( A , P ) (\mathfrak {A},P) is a Jordan triple system and Rad ⁡ A \operatorname {Rad} \mathfrak {A} the Jacobson radical of A \mathfrak {A} then x ∈ Rad ⁡ A x \in \operatorname {Rad} \mathfrak {A} iff P ( x ) A = P ( x + P ( x ) y ) A P(x)\mathfrak {A} = P(x + P(x)y)\mathfrak {A} for all y ∈ A y \in \mathfrak {A} . We use this to give a new characterisation of von Neumann regular Jordan triple systems. In particular, this gives another most elementary and short proof that semi-simple Jordan triple systems with dcc on principal inner ideals are von Neumann regular.