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

M. Brešar; M. Cabrera; M. Fošner; A. R. Villena

Studia Mathematica (2005)

  • Volume: 169, Issue: 3, page 207-228
  • ISSN: 0039-3223

Abstract

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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 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 Ĵ, 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 → Ĵ 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 epimorphism between Jordan-Banach algebras H and J lies in the center modulo the radical of J.

How to cite

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M. Brešar, et al. "Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan-Banach algebras." Studia Mathematica 169.3 (2005): 207-228. <http://eudml.org/doc/286608>.

@article{M2005,
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 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 Ĵ, 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 → Ĵ 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 epimorphism between Jordan-Banach algebras H and J lies in the center modulo the radical of J. },
author = {M. Brešar, M. Cabrera, M. Fošner, A. R. Villena},
journal = {Studia Mathematica},
language = {eng},
number = {3},
pages = {207-228},
title = {Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan-Banach algebras},
url = {http://eudml.org/doc/286608},
volume = {169},
year = {2005},
}

TY - JOUR
AU - M. Brešar
AU - M. Cabrera
AU - M. Fošner
AU - A. R. Villena
TI - Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan-Banach algebras
JO - Studia Mathematica
PY - 2005
VL - 169
IS - 3
SP - 207
EP - 228
AB - 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 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 Ĵ, 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 → Ĵ 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 epimorphism between Jordan-Banach algebras H and J lies in the center modulo the radical of J.
LA - eng
UR - http://eudml.org/doc/286608
ER -

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