0
$\begingroup$

I am reading the following proposition:

enter image description here

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?

$\endgroup$
1
$\begingroup$

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 $C$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.