The Multiplicative Kowalski-Slodkowski Theorem for Hermitian Algebras

Abstract

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 $\varphi :A\mapsto \mathbb{C}$ is a continuous function satisfying $\varphi (x)\varphi (y)\in \sigma (xy)$ for all $x,y\in A$ (where $\sigma$ denotes the spectrum), then either $\varphi$ or $-\varphi$ 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.

7 pages