This is an extract from Douglas: "Banach Algebra Techniques in operator theory".
"For Banach algebras and, in particular, for $C(X)$ the importan idea is that of multiplicative linear functional. [...] Except for the zero functional, which is obviously both multiplicative and linear, every multiplicative linear functional $\varphi$ satisfies $\varphi(1) = 1$ since $\varphi \not= 0$ means there exists an $f$ in $C(X)$ with $\varphi(f) \not= 0$ and then the equation $\varphi(1)\varphi(f)=\varphi(f)$ implies $\varphi(1)=1$"
I don't understand the reason that he want to prove that $\varphi(1)=1$. Isn't that property in the definition itself of Banach Algebra?
And after that, I didn't understand the "proof" he provides:
$\varphi \not= 0$ means there exists an $f$ in $C(X)$ with $\varphi(f) \not= 0$ OK
and then $\varphi(1)\varphi(f)=\varphi(f)$ implies $\varphi(1)=1$" WHY?
Ok, that's because "multiplicative" means "homomorphic", but why can I say that I can take $\varphi(f 1)$? The identity element in C(X) should be something like 1(x) = x no?
Then $\varphi(f 1)=\varphi(f(x) 1(x))$ and I cannot conclude that $\varphi(1)\varphi(f)=\varphi(f)$.
Please, can you explain me the reason of this in detail? Thank you!