Smallest closed ideal containing an element in a $C^*$-algebra

Question

Let $A$ be a $C^*$-algebra and $a \in A$. I want to describe the smallest closed ideal containing $a$. If the algebra is unital, I think this ideal will be $\overline{AaA}$. But can we describe this ideal when $A$ is non-unital? Maybe something like

$$\overline{AaA + \Bbb{Z}a} $$

can work?

Answer 1

If there is no unit then the ideal is $$\overline{ AaA +Aa+aA+span(a)}.$$

Answer 2

For any $C^*$-algebra $A$ and any $a\in A$, the $C^*$subalgebra $\overline{AaA}$ would be the smallest ideal containing $a$. This contains $a$ because $A$ has an approximate unit, and is an ideal, as it is the closure of the (algebraic) ideal $AaA$. For the other direction, any ideal of $A$ containing $a$ certainly contains $AaA$, and thus contains its closure.