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