Let $A$ be a unital Banach algebra and $J$ maximal ideal of $A$. Why is $\overline{J}$ an ideal of $A$?

Question

Let $A$ be a unital Banach algebra with maximal ideal $J$.

Why is it true that $\overline{J}$ is also an ideal of $A$, i.e. the closure of a maximal ideal of $A$ is an ideal of $A$?

Answer 1

The fact that $\overline{J}$ is an ideal follows from the continuity of the algebraic operations. For example, if $x \in \overline{J}$ and $y \in A$ then we can choose $x_n \in J$ such that $x_n \rightarrow x$ and then $yx_n \rightarrow yx$ and since $J$ is an ideal, $yx_n \in J$ and so $yx \in \overline{J}$. Similarly for right multiplication and addition. The interesting thing is that if $J$ is a proper ideal, then $\overline{J}$ must also be a proper ideal. This implies for example that a maximal ideal of $A$ must be closed.