10. Commutative Banach algebras - Aarhus Universitet

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The Gelfand transform - the proof of Theorem 11.9 Proof. b): It should be obvious that the Gelfand transform is a homomorphism: a + λb(t) = t(a+λb) = a(t)+λt(b) = â(t)+λˆ b(t) = â + λˆ b (t) and ab(t) = t(ab) = t(a)t(b) = a(t) b(t) The kernel of the Gelfand transform is {a ∈ A : t(a) = 0 ∀t ∈ ∆}. By Theorem 11.5 this is the same as the set of elements of A that lie in every maximal ideal of A. Klaus Thomsen 10. Commutative Banach algebras
The Gelfand transform - the proof of Theorem 11.9 Proof. c): By e) of Theorem 11.5 σ(a) = {â(t) : t ∈ ∆} so we see that sup {|â(t)| : t ∈ ∆} = sup {|λ| : λ ∈ σ(a)} . Thus a∞ ≤ ρ(a). By Theorem 10.13 ρ(a) = infn an 1 n. But a n 1 n ≤ (a n ) 1 n = a. 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 and 20: The Gelfand transform - Theorem 11.
Page 21: The Gelfand transform - the proof o

10. Commutative Banach algebras - Aarhus Universitet

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