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Relations Between n-Jordan Homomorphisms and n-Homomorphisms

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Abstract

For \(n\ge 2\), an additive map f between two rings A and B is called an n-Jordan homomorphism, or an n-homomorphism if \(f(a^n)=f(a)^n\), for all \(a\in A\), or \(f(a_1a_2\cdots a_n)=f(a_1)f(a_2)\cdots f(a_n)\), for all \(a_1,a_2,\ldots ,a_n\in A\), respectively. In particular, if \(n=2\) then f is simply called a Jordan homomorphism or a homomorphism, respectively. The notion of n-Jordan homomorphism between rings was introduced in 1956 by Herstein and the concept of n-homomorphism between algebras was introduced in 2005 by Hejazian et al. Properties of n-Jordan homomorphisms as well as n-homomorphisms have been studied by many authors since then. One of the main questions is that, “under what conditions n- Jordan homomorphisms are n-homomorphism?”. Another natural question is that “under what conditions certain properties of homomorphisms may be extended to n-homomorphisms”. We provide conditions under which these questions have affirmative answers. We also study the continuity problem for n-Jordan homomorphisms on Banach algebras, while extending some known results in this field.

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The authors would like to thank the referee whose comments helped us to improve this article.

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Correspondence to Taher Ghasemi Honary.

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Communicated by Mohammad B. Asadi.

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Honary, T.G., Hosseinzadeh, H. & Mohammadi, S. Relations Between n-Jordan Homomorphisms and n-Homomorphisms. Bull. Iran. Math. Soc. 47, 689–700 (2021). https://doi.org/10.1007/s41980-020-00407-4

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