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

TL;DR

This paper proves that *-preserving n-homomorphisms between C*-algebras are necessarily trivial or decomposable into simpler *-homomorphisms, resolving a question about their continuity and structure.

Contribution

It establishes that all *-preserving n-homomorphisms of C*-algebras are either ordinary *-homomorphisms or differences of two orthogonal *-homomorphisms, showing nonexistence of nontrivial cases.

Findings

All *-preserving n-homomorphisms are continuous.

For even n > 2, such maps are *-homomorphisms.

For odd n >= 3, such maps are differences of two orthogonal *-homomorphisms.

Abstract

An n-homomorphism between algebras is a linear map $\phi : A \to B$ such that $\phi(a_1 ... a_n) = \phi(a_1)... \phi(a_n)$ 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 $\phi$ 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*-algebras.