PreprintCC BY 4.0arXiv
Orthogonality of Defect Ideals for Jordan Homomorphisms
Abstract
Let φ: A→ B be a Jordan homomorphism of associative algebras over a field 𝔽 of characteristic different from $2$, with defect ideals Vφ and Wφ. If A=Alg([[A,A],[A,A]]), then VφWφ=WφVφ=0 and Vφ∩ Wφ⊆Ann(Alg(φ(A))). Moreover, the induced map modulo Vφ∩ Wφ is standard. The annihilator bound is sharp, even in finite dimension. As an application, when A is unital and A=K(A), every Jordan homomorphism defined on A is splittable and standard. In this setting, the corresponding result of Brešar and Zelmanov holds without the surjectivity hypothesis.
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BibTeX
@misc{nam2026orthogonality,
title = {{Orthogonality of Defect Ideals for Jordan Homomorphisms}},
author = {Cao Minh Nam},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17518},
eprint = {2610.17518},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17518},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Orthogonality of Defect Ideals for Jordan Homomorphisms
AU - Cao Minh Nam
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17518
UR - https://arxiv.org/abs/2610.17518
AB - Let φ: A→ B be a Jordan homomorphism of associative algebras over a field 𝔽 of characteristic different from $2$, with defect ideals V_φ and W_φ. If A=Alg([[A,A],[A,A]]), then V_φW_φ=W_φV_φ=0 and V_φ∩ W_φ⊆Ann(Alg(φ(A))). Moreover, the induced map modulo V_φ∩ W_φ is standard. The annihilator bound is sharp, even in finite dimension. As an application, when A is unital and A=K(A), every Jordan homomorphism defined on A is splittable and standard. In this setting, the corresponding result of Brešar and Zelmanov holds without the surjectivity hypothesis.
ER -