On the Nonexistence of Nontrivial Involutive n-Homomorphisms of C*-algebras

An n-homomorphism between algebras is a linear map $\varphi :A\to B$ such that $\varphi (a_1\mathrm{.}\mathrm{.}\mathrm{.}{a}_n)=\varphi (a_1)\mathrm{.}\mathrm{.}\mathrm{.}\varphi (a_n)$ for all elements $a_1, >…, a_n \in A.$ Every homomorphism is an n-homomorphism, for all n >= 2, but the converse is false, in general. Hejazian et al. [7] ask: Is every -preserving n-homomorphism between C-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 $\varphi$ is just an ordinary *-homomorphism. If n >= 3 is odd, then $\varphi$ is a difference of two orthogonal *-homomorphisms. Thus, there are no nontrivial -linear n-homomorphisms between C-algebras.