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Commuting U-operators and nondegenerate imbeddings of Jordan systems

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Abstract

Over an arbitrary ring of scalars, we build a Jordan algebra J having two elements x,y∈J such that x◦y=0, but whose U-operators do not commute. This shows that nondegeneracy is a necessary condition in the main theorem of Commuting U-Operators in Jordan Algebras by J. A Anquela, T. Cortés and H. P. Petersson (Trans. Amer. Math. Soc. 366 (2014), 5877–5902). As a consequence, we obtain examples of Jordan systems over arbitrary rings of scalars that cannot be imbedded in nondegenerate systems.

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Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Oviedo, C/Federico Garca Lorca 18, 33007, Oviedo, Spain

    José A. Anquela & Teresa Cortés

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, CEP 05508-090, São Paulo, SP, Brazil

    Ivan Shestakov

  3. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    Ivan Shestakov

Authors
  1. José A. Anquela
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  2. Teresa Cortés
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  3. Ivan Shestakov
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Corresponding author

Correspondence to José A. Anquela.

Additional information

Partially supported by the Spanish Ministerio de Econom´ıa y Competitividad and Fondos FEDER, MTM2014-52470-P and MTM2017-84194-P (AEI/FEDER, UE).

Partially supported by a FAPES grant 2014/09310-5 and CNPq grant 303916/2014-1.

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Anquela, J.A., Cortés, T. & Shestakov, I. Commuting U-operators and nondegenerate imbeddings of Jordan systems. Isr. J. Math. 225, 871–887 (2018). https://doi.org/10.1007/s11856-018-1681-5

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  • DOI: https://doi.org/10.1007/s11856-018-1681-5

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