10. Commutative Banach algebras - Aarhus Universitet

The proof of Theorem 11.3
Proof.
(b) Let J be a maximal ideal in A. Then J is an ideal in A by (b) of
Lemma 2.
If 1 ∈ J there is an element x ∈ J such that 1 − x < 1. But then
x = 1 − (1 − x) is invertible, which is not possible since J is a
proper ideal, cf. (a) of Lemma 2. Thus 1 /∈ J, proving that J is a
proper ideal.
Since J ⊆ J and J is maximal, we conclude that J = J, i.e. J is
closed.
Klaus Thomsen 10. Commutative Banach algebras