On the nonexistence of nontrivial involutive $n$-homomorphisms of $C^{\star }$-algebras

An $n$-homomorphism between algebras is a linear map $\phi : A \to B$ such that $\phi (a_1 \cdots a_n) = \phi (a_1)\cdots \phi (a_n)$ for all elements $a_1, \dots , a_n \in A.$ Every homomorphism is an $n$-homomorphism for all $n \geq 2$, but the converse is false, in general. Hejazian et al. (2005) ask: Is every $*$-preserving $n$-homomorphism between $C^{\star }$-algebras continuous? We answer their question in the affirmative, but the even and odd $n$ arguments are surprisingly disjoint. We then use these results to prove stronger ones: If $n >2$ is even, then $\phi$ is just an ordinary $*$-homomorphism. If $n \geq 3$ is odd, then $\phi$ is a difference of two orthogonal $*$-homomorphisms. Thus, there are no nontrivial $*$-linear $n$-homomorphisms between $C^{\star }$-algebras.