1 Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran

2 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran


Let $n\in \mathbb{N}$. An additive map $h:A\to B$ between algebras $A$ and $B$ is called $n$-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $a\in A$. We show that every $n$-Jordan homomorphism between commutative Banach algebras is a $n$-ring homomorphism when $n < 8$. For these cases, every involutive $n$-Jordan homomorphism between commutative C-algebras is norm continuous.


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