On Automatic Continuity of 3-Homomorphisms on Banach Algebras

Abstract

A linear map $ϕ:{\mathcal A}\to {\mathcal B} $ between (Banach) algebras is called 3-homomorphism if $ϕ(abc)=ϕ(a)ϕ(b)ϕ(c)$ for each $a, b, c \in {\mathcal A}$. We investigate 3-homomorphisms on Banach algebras with bounded approximate identities and establish in two ways that every involution preserving homomorphism between $C^*$-algebras is norm decreasing.