If $A$ is a Banach algebra then let $\Omega (A)$ denote the character space of $A$. Apparently there exist abelian Banach algebras such that $\Omega (A) = \varnothing$. Also apparently, if $A$ is a (commutative) $C^\ast$ algebra then $\Omega (A) \neq \varnothing$. I am trying to prove this but got stuck.
Let $A \neq 0$ be a commutative $C^\ast$ algebra. Then there exists $a \in A$ with $\|a\|\neq 0$. By the Hahn Banach theorem there exists a continuous linear functional $\tau : A \to \mathbb C$ such that $|\tau (a)|=\|a\|$ and $\|\tau\|\le 1$. That's how it's stated on Wikipedia but of course $|\tau (a)|=\|a\|$ and $\|\tau\|\le 1$ implies $\|\tau\|=1$. Hence $\Omega(A) \neq \varnothing$.
To me this seems to be a correct proof. But in Murphy's book there is something quite different. Could someone help me and enlighten me about what is wrong with my proof above? I am including the text given in the book: