Definition of a closed ideal in a $C^*$ algebra

Question

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.

Answer 1

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.