n-Jordan homomorphisms on C*-algebras

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