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 />
(e) : If λ ∈ σ(x), λ1 − x is not invertible, and hence<br />
µ (λ1 − x) = 0 for some µ ∈ ∆. It follows that<br />
λ ∈ {ω(x) : ω ∈ ∆} since µ(1) = 1 by Proposition <strong>10.</strong>6.<br />
If λ = ω(x) for some ω ∈ ∆, we conclude from (c) that λ1 − x is<br />
not invertible (since ω(λ1 − x) = 0), and it follows that λ ∈ σ(x).<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>