Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan–Banach algebras
Volume 169 / 2005
Abstract
A linear subspace $M$ of a Jordan algebra $J$ is said to be a Lie triple ideal of $J$ if $[M,J,J] \subseteq M$, where $[\cdot ,\cdot ,\cdot ]$ 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 $\widehat{J}$, and let ${\mit\Phi}: H\rightarrow J$ be a Lie triple epimorphism (i.e. a linear surjection preserving associators). Assume that $\hbox{deg}(J) \geq 12$. Then we show that there exist a homomorphism ${\mit\Psi} : H \rightarrow \widehat{J}$ and a linear map $\tau : H \rightarrow C$ satisfying $\tau([H,H,H])=0$ such that either ${\mit\Phi} = {\mit\Psi} + \tau$ or ${\mit\Phi} = -{\mit\Psi} + \tau$.
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 $J$.
