Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan-Banach algebras
Brešar, Matej ...
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 ▫$\Phi : H \to J$▫ be a Lie triple epimorphism (i.e. a linear surjection preserving associators). Assume that ▫${\rm deg}(J) \geq 12$▫. Then we show that there exist a homomorphism ▫$\Psi : H \to \widehat{J}$▫ and a linear map ▫$\tau : H \to C$▫ satisfying ▫$\tau([H,H,H]) = 0$▫ such that either ▫$\Phi = \Psi + \tau$▫ or ▫$\Phi = -\Psi + \tau$▫. 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 ▫$J$▫.
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