Why is $e \in C$ (a commutative subalgebra of $A$)?

Question

I am reading the following proposition:

Here $e$ represents the identity element of $A$ and $\sigma_C(x)$ and $\sigma_A(x)$ denote the spectrum of an element $x$ in $C$ and $A$ respectively.

I am not sure why $e \in C$ is true though? Can anyone please explain to me why this is the case?

Answer 1

If $e\not\in C$, let $C'$ be the subalgebra generated by $C$ and $e$. Then $C'$ is still commutative (since $e$ commutes with everything), and is strictly larger than $C$. This contradicts maximality of $C$.