ON THE NONEXISTENCE OF NONTRIVIAL INVOLUTIVE n-HOMOMORPHISMS OF C☆-ALGEBRAS

An n-homomorphism between algebras is a linear map phi : A -> B such that phi(a1...a(n)) = phi(a(1))...phi(a(n)) for all elements a(1),...,a(n) is an element of A. Every homomorphism is an n-homomorphism for all n >= 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 >= 3 is odd, then phi is a difference of two orthogonal *-homomorphisms. Thus, there are no nontrivial *-linear n-homomorphisms between C-star-algebras.