Characterization of mixed n-Jordan homomorphisms and pseudo n-Jordan homomorphisms

Abstract

In this article, a new notion of n-Jordan homomorphism namely the mixed n-Jordan homomorphism is introduced. It is proved that how a mixed (n+1)-Jordan homomorphism can be a mixed n-Jordan homomorphism and vice versa. By means of some examples, it is shown that the mixed n-Jordan homomorphisma are different from the n-Jordan homomorphisms and the pseudo n-Jordan homomorphisms. As a consequence, it shown that every mixed Jordan homomorphism from Banach algebra A into commutative semisimple Banach algebra B is automatically continuous. Under some mild conditions, every unital pseudo 3-Jordan homomorphism can be a homomorphism.