The Multiplicative Kowalski-Slodkowski Theorem for Hermitian Algebras

We prove, for Hermitian algebras, the multiplicative version of the Kowalski-Słodkowski Theorem which identifies the characters among the collection of all complex valued functions on a Banach algebra $A$ in terms of a spectral condition. Specifically, we show that, if $A$ is a Hermitian algebra, and if $ϕ:A\mapsto\mathbb C$ is a continuous function satisfying $ϕ(x)ϕ(y) \in σ(xy)$ for all $x,y\in A$ (where $σ$ denotes the spectrum), then either $ϕ$ or $-ϕ$ is a character of $A$; of course the converse holds as well. Our proof depends fundamentally on the existence of positive elements and square roots in these algebras.