10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
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Theorem 11.5<br />
Proof.<br />
It follows that A/J = C1 by the Gelfand-Mazur theorem. Define<br />
ω : A → C such that ω(a)1 = q(a), where q : A → A/J is the<br />
quotient map. Then ω ∈ ∆ and J = ker ω.<br />
b): Let µ ∈ ∆. Then ker µ is clearly (!) an ideal in A. Let I ⊆ A be<br />
a proper ideal such that ker µ ⊆ I and ker µ = I.<br />
Let x ∈ I be an element such that µ(x) = 0. Then λµ(x) = 1 for<br />
some λ ∈ C, and it follows that µ (x − µ(x)1) = 0. Hence<br />
x − µ(x)1 ∈ ker µ ⊆ I, and we deduce that µ(x)1 ∈ I.<br />
This contradicts the properness of I, and we conclude that ker µ is<br />
a maximal ideal.<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>