Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan-Banach algebras

A linear subspace M of a Jordan algebra J is said to be a Lie triple ideal of J if [M, J, J] subset of M, where [(.), (.), (.)] denotes the associator. We show that every Lie triple ideal M of a nondegenerate Jordan algebra J is either contained in the center of J or contains the nonzero Lie triple ideal [U, J, J], where U is the ideal of J generated by [M, M, M]. Let H be a Jordan algebra, let J be a prime nondegenerate Jordan algebra with extended centroid C and unital central closure (J) over cap, and let Phi : H -> J be a Lie triple epimorphism (i.e. a linear surjection preserving associators). Assume that deg(J) >= 12. Then we show that there exist a homomorphism Psi : H -> (J) over cap and a linear map tau : H -> C satisfying tau([H, H, H]) = 0 such that either Phi = Psi + tau or Phi = -Psi + tau. Using the preceding results we show that the separating space of a Lie triple epimorphism between Jordan-Banach algebras H and J lies in the center modulo the radical of J.