  We consider a multiplicative variation on the classical
Kowalski-S\l{}odkowski 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 $\phi:A\mapsto\mathbb C$ is a
continuous function satisfying $\phi(\mathbf 1)=1$ and $\phi(x)\phi(y) \in
\sigma(xy)$ for all $x,y\in A$ (where $\sigma$ denotes the spectrum), then
$\phi$ generates a corresponding character $\psi_\phi$ on $A$ which coincides
with $\phi$ 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, $\phi$ is
always a character.
