Abstract
Recently Anquela et al. (Trans. AMS 366(11):5877–5902, 2014) proved that for elements x, y in a non-degenerate Jordan algebra J, the relation x ∘ y = 0 implies that the U-operators of x and y commute: U x U y = U y U x . We show that the result may be not true without the assumption on non-degeneracity of J. We give also a more simple proof of the mentioned result in the case of linear Jordan algebras, that is, when char F ≠ 2.
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Acknowledgements
The author acknowledges the support by FAPESP, Proc. 2014/09310-5 and CNPq, Proc. 303916/ 2014-1. He is grateful to professor Holger Petersson for useful comments and suggestions, and to professors José Ángel Anquela and Teresa Cortés for correction the proof of Theorem 6.2.1 in the case of characteristic 2. He thanks all of them for pointing out some misprints.
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Dedicated to Professor Amin Kaidi on the occasion of his 65-th anniversary
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Shestakov, I. (2016). On Commuting U-Operators in Jordan Algebras. In: Gueye, C., Molina, M. (eds) Non-Associative and Non-Commutative Algebra and Operator Theory. Springer Proceedings in Mathematics & Statistics, vol 160. Springer, Cham. https://doi.org/10.1007/978-3-319-32902-4_6
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DOI: https://doi.org/10.1007/978-3-319-32902-4_6
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