Corollary 2

Let A be a Banach algebra and I a regular maximal left ideal (resp. right ideal) of A. Then I is closed.

Let $(\widetilde{A}, e)$ be the unital Banach algebra deduced from A by adjunction of a unit element. There exists a maximal left ideal (resp. right) J of $\widetilde{A}$ such that $J\cap A = I$ (A, VIII, p. 428, prop. 4). Then J is disjoint from the open ball of center $e$ and of radius 1 (cor. 1), and therefore $\overline{J}\not=\widetilde{A}$. Since J is a maximal ideal, this implies that $\overline{J}= J$, and consequently that $I = J\cap A =\overline{J}\cap A$ is closed in A.