Why is $f(e)=1$?

Question

Let $A$ be a commutative complex Banach Algebra with unit element $e$. Now, let $f \in A^*$ be a non-zero multiplicative linear functional.

Why does it follow, directly from the above, that $f(e)=1$?

Answer 1

Since $e^2=e$, $f(e)^2=f(e)$. This means $f(e)$ must be either $0$ or $1$. But if $f(e)=0$, then $f(x)=f(ex)=f(e)f(x)=0$ for all $x\in A$.