STUDIA MATHEMATICA 169 (3) (2005) Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan–Banach algebras by M. Breˇ sar (Maribor), M. Cabrera (Granada), M. Foˇ sner (Ljubljana) and A. R. Villena (Granada) Abstract. A linear subspace M of a Jordan algebra J is said to be a Lie triple ideal of J if [M, J, J ] ⊆ 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 con- tains 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 , and let Φ : 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 Ψ : H →  J and a linear map τ : H → C satisfying τ ([H, H, H]) = 0 such that either Φ = Ψ + τ or Φ = -Ψ + τ . Using the preceding results we show that the separating space of a Lie triple epimor- phism between Jordan–Banach algebras H and J lies in the center modulo the radical of J . 1. Introduction. An associative algebra A becomes a Lie algebra A − under the commutator [x, y]= xy − yx, and a linear Jordan algebra A + under the Jordan product x ◦ y = 1 2 (xy + yx). In the case A has a linear involution ∗, the set of skew elements K(A, ∗)= {x ∈ A : x ∗ = −x} is a subalgebra of A − , and the set of symmetric elements H (A, ∗)= {x ∈ A : x ∗ = x} is a subalgebra of A + . I. N. Herstein initiated the study of the ideals of A − , A + , K(A, ∗), and H (A, ∗) in the context of simple algebras. His work has 2000 Mathematics Subject Classification : 17C10, 17C65, 46H40, 46H70. The first named author is supported by M ˇ SZ ˇ S grant P0-0501-0101. The second named author is supported by Junta de Andalucia grant FQM290. The fourth named author is supported by MCYT grant BFM2003-01681. A preliminary version of this paper was received by the Editors on April 18, 2003. [207]