Is the Carrier Space of a Commutative Banach Algebra Always Nontrivial? Let $ A $ be a commutative Banach algebra and let $ \Phi_{A} $ be the carrier space of $ A $, that is, the space consisting of all multiplicative linear functionals from $ A $ to $ \mathbb{C} $.  
Certainly the zero functional in always in $ \Phi_A $, but is it ever the case that $ 0 $ is the only member of $ \Phi_A $ when $ A $ is commutative?  If so, what are some well known criteria for when $ \Phi_A \neq \{ 0 \} $?

A partial answer, assuming that $A$ is unital and not simply the zero-algebra.
If we assume Zorn's lemma, then there must exist at least one non-zero character.  In particular, it suffices to note that $A$ must have a maximal ideal $I$.  The quotient map $A \mapsto A + I$ is multiplicative, and $A/I \cong \Bbb C$ by the maximality of $I$, the closedness of maximal ideals, and the Gelfand-Mazur theorem.
