I am reading the following proposition:

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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?


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 $C$.


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