n-Homomorphisms

Abstract

Let $\mathcal A$ and $\mathcal B$ be two (complex) algebras. A linear map $ϕ:{\mathcal A}\to{\mathcal B}$ is called $n$-homomorphism if $ϕ(a_{1}... a_{n})=ϕ(a_{1})...ϕ(a_{n})$ for each $a_{1},...,a_{n}\in{\mathcal A}.$ In this paper, we investigate $n$-homomorphisms and their relation to homomorphisms. We characterize $n$-homomorphisms in terms of homomorphisms under certain conditions. Some results related to continuity and commutativity are given as well.