Let $A$ be a commutative and unital $C^*$-algebra. What is the definition of a closed ideal of $A?$
My understanding:
A subset $I$ of $A$ is an ideal if it is a vector subspace of $A$ and for any $a$ in $A$ and any $i \in I,$ we have that $a \cdot i \in I$.
A subset $I$ of $A$ is a closed ideal if it is an ideal and it is closed in the topology induced by the norm on $A.$
Question: Is my definition correct or does a closed ideal need to also be closed under the star operation?
Bonus question: Is the quotient space $A/I$ also a commutative and unital $C^*$ algebra?
Thank you for all your help.
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$\begingroup$I think I found answers to my questions here.
Namely, the definition of an ideal does not require an ideal to be closed under the star operation. However, whenever $I$ is a (topologically) closed (two-sided) ideal, one can show that $I$ is also closed under the star operation. Also, one can show that $A/I$ is also a $C^*$ algebra.
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