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

Volume 169 / 2005

M. Brešar, M. Cabrera, M. Fošner, A. R. Villena Studia Mathematica 169 (2005), 207-228 MSC: 17C10, 17C65, 46H40, 46H70. DOI: 10.4064/sm169-3-1

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 $J$.

Authors

  • M. BrešarDepartment of Mathematics
    University of Maribor
    PEF, Koroška 160
    2000 Maribor, Slovenia
    e-mail
  • M. CabreraDepartamento de Análisis Matemático
    Facultad de Ciencias
    Universidad de Granada
    18071 Granada, Spain
    e-mail
  • M. FošnerInstitute of Mathematics, Physics, and Mechanics
    Jadranska 19
    1000 Ljubljana, Slovenia
    e-mail
  • A. R. VillenaDepartamento de Análisis Matemático
    Facultad de Ciencias
    Universidad de Granada
    18071 Granada, Spain
    e-mail

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