Abstract
In this chapter, we will be primarily interested in the question of which associative algebras, viewed either as Lie algebras under the Lie product or as Jordan algebras under the Jordan product, are zero product determined. We will see that the results in the Jordan case are similar to those in the associative case, while the Lie case is quite different. For example, a unital associative algebra (over a field of characteristic not 2) that is generated by idempotents is zpd when considered as a Jordan algebra, but not necessarily when considered as a Lie algebra. On the other hand, it is easy to find examples of associative algebras that are zpd as Lie algebras, but not as associative algebras.
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References
J. Alaminos, M. Brešar, M. Černe, J. Extremera, A.R. Villena, Zero product preserving maps on C 1[0, 1]. J. Math. Anal. Appl. 347, 472–481 (2008)
A.R. Villena, Orthogonally additive polynomials on Banach function algebras. J. Math. Anal. Appl. 448, 447–472 (2017)
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Brešar, M. (2021). Zero Product Determined Nonassociative Banach Algebras. In: Zero Product Determined Algebras. Frontiers in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-80242-4_4
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DOI: https://doi.org/10.1007/978-3-030-80242-4_4
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