A multiplicative Gleason-Kahane-Żelazko theorem for C⋆-algebras

Abstract

If A is a unital complex Banach algebra, and if σ(a) denotes the spectrum of an element a∈A, then the famous Gleason-Kahane-Żelazko Theorem says that any linear functional ϕ:A→C satisfying ϕ(a)∈σ(a) for each a∈A, is multiplicative and continuous. In this paper we establish a multiplicative Gleason-Kahane-Żelazko theorem for the case where A is a C⋆-algebra. Specifically, if A is a C⋆-algebra, then any continuous multiplicative functional ϕ:A→C satisfying ϕ(a)∈σ(a) for each a∈A, is linear and hence a character of A.