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 $C^{\ast }$-algebra, and if $\ddot{I}•:A\mapsto \mathbb{C}$ is a continuous function satisfying $\ddot{I}•(1)=1$ and $\ddot{I}•(x)\ddot{I}•(y)\in \ddot{I}ƒ(xy)$ for all $x,y\in A$ (where $\ddot{I}ƒ$ denotes the spectrum), then $\ddot{I}•$ generates a corresponding character $\ddot{I}{ˆ}_{\ddot{I}}•$ on $A$ which coincides with $\ddot{I}•$ 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, $\ddot{I}•$ is always a character.