10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
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The Gelfand transform - the proof of Theorem 11.9<br />
Proof.<br />
Since ξ(ab) = ξ(a)ξ(b) and ξ ∈ U it follows that<br />
|ω(a)ω(b) − ω(ab)| < 2ǫ.<br />
Since a,b ∈ A and ǫ > 0 were arbitrary we conclude that<br />
ω(a)ω(b) = ω(ab) for all a,b ∈ A.<br />
In order to conclude that ω ∈ ∆ it remains only to show that<br />
ω = 0. But {ν ∈ A ∗ : |ν(1) − ω(1)| < 1} is an open neighborhood<br />
of ω in the weak*-topology of A ∗ and hence it contains an element<br />
ξ from ∆.<br />
Since ξ(1) = 1 we conclude that ω(1) = 0, and the proof of a) is<br />
complete.<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>