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

M. Brešar ¹   ; M. Cabrera ²   ; M. Fošner ³   ; A. R. Villena ²

¹ Department of Mathematics University of Maribor PEF, Koroška 160 2000 Maribor, Slovenia
² Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain
³ Institute of Mathematics, Physics, and Mechanics Jadranska 19 1000 Ljubljana, Slovenia

Studia Mathematica, Tome 169 (2005) no. 3, pp. 207-228

Résumé

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 $J$.

Détail

DOI : 10.4064/sm169-3-1

Keywords: linear subspace jordan algebra said lie triple ideal nbsp subseteq where cdot cdot cdot denotes associator every lie triple ideal nondegenerate jordan algebra nbsp either contained center contains nonzero lie triple ideal where ideal generated jordan algebra prime nondegenerate jordan algebra extended centroid unital central closure widehat mit phi rightarrow lie triple epimorphism linear surjection preserving associators assume hbox deg geq there exist homomorphism mit psi rightarrow widehat linear map tau rightarrow satisfying tau either mit phi mit psi tau mit phi mit psi tau using preceding results separating space lie triple epimorphism between jordan banach algebras lies center modulo radical nbsp