Some Character Generating Functions on Banach Algebras

Abstract

We consider a multiplicative variation on the classical Kowalski-Słodkowski Theorem which identifies the characters among the collection of all functionals on a Banach algebra $A$. In particular we show that, if $A$ is a $C^*$-algebra, and if $ϕ:A\mapsto\mathbb C$ is a continuous function satisfying $ϕ(\mathbf 1)=1$ and $ϕ(x)ϕ(y) \in σ(xy)$ for all $x,y\in A$ (where $σ$ denotes the spectrum), then $ϕ$ generates a corresponding character $ψ_ϕ$ on $A$ which coincides with $ϕ$ on the principal component of the invertible group of $A$. We also show that, if $A$ is any Banach algebra whose elements have totally disconnected spectra, then, under the aforementioned conditions, $ϕ$ is always a character.