10. Commutative Banach algebras - Aarhus Universitet

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The Gelfand transform - the proof of Theorem 11.9 Proof. a): Since ∆ is contained in the unit ball of A ∗ , and the latter is compact in the weak*-topology by Banach-Alaoglu theorem (Theorem 3.15), it suffices to show that ∆ is closed in the weak*-topology. Let therefore ω ∈ A ∗ be an element of the closure ∆ of ∆ in A ∗ . We want to show that ω ∈ ∆. There is a δ > 0 such that |st − ω(a)ω(b)| < ǫ when |s − ω(a)| < δ and |t − ω(b)| < δ. Let a,b ∈ A, and let ǫ > 0. Since U = {ν ∈ A ∗ : |ν(a) − ω(a)| < δ, |ν(b) − ω(b)| < δ, |ν(ab) − ω(ab)| < ǫ} is an open neighborhood of ω in the weak*-topology of A ∗ there is a character ξ ∈ ∆ contained in U because ω ∈ ∆. Klaus Thomsen 10. Commutative Banach algebras
The Gelfand transform - the proof of Theorem 11.9 Proof. Since ξ(ab) = ξ(a)ξ(b) and ξ ∈ U it follows that |ω(a)ω(b) − ω(ab)| < 2ǫ. Since a,b ∈ A and ǫ > 0 were arbitrary we conclude that ω(a)ω(b) = ω(ab) for all a,b ∈ A. In order to conclude that ω ∈ ∆ it remains only to show that ω = 0. But {ν ∈ A ∗ : |ν(1) − ω(1)| < 1} is an open neighborhood of ω in the weak*-topology of A ∗ and hence it contains an element ξ from ∆. Since ξ(1) = 1 we conclude that ω(1) = 0, and the proof of a) is complete. Klaus Thomsen 10. Commutative Banach algebras
Page 1 and 2: 10. Commutative Banach algebras Kla
Page 3 and 4: The Gelfand-Mazur theorem Let A and
Page 5 and 6: Commutative Banach algebras - ideal
Page 7 and 8: The proof of Theorem 11.3 Proof. (b
Page 9 and 10: Quotients Lemma Let X be a Banach s
Page 11 and 12: Quotients Proof. First we observe t
Page 13 and 14: Theorem 11.5 Let A be a Banach alge
Page 15 and 16: Theorem 11.5 Proof. It follows that
Page 17 and 18: Theorem 11.5 Proof. (e) : If λ ∈
Page 19: The Gelfand transform - Theorem 11.
Page 23: The Gelfand transform - the proof o

10. Commutative Banach algebras - Aarhus Universitet

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