10. Commutative Banach algebras - Aarhus Universitet

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We read in W. Rudin: Functional Analysis Chapter 10, 10.14, followed by the beginning of Chapter 11. Klaus Thomsen 10. Commutative Banach algebras
The Gelfand-Mazur theorem Let A and B be Banach algebras. An isometric isomorphism or just an isomorphism between A and B is a linear bijection α : A → B such that α(ab) = α(a)α(b) and α(a) = a for all a,b ∈ A. We write A ≃ C in this case. Theorem (Theorem 10.14) Let A be a unital Banach algebra with the property that every non-zero element is invertible. Then A ≃ C. Proof. Define α : C → A by α(λ) = λ1. Note that α is an algebra homomorphism and α(λ) = λ1 = |λ|1 = |λ|. It remains only to show that α is surjective. To this end consider an element a ∈ A. By Theorem 10.13 there is an element λ in the spectrum of a. Then λ1 − a is not invertible. Under the present assumption this implies that a = λ1. Klaus Thomsen 10. Commutative Banach algebras
Page 1: 10. Commutative Banach algebras Kla
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 and 22: The Gelfand transform - the proof o
Page 23: The Gelfand transform - the proof o

10. Commutative Banach algebras - Aarhus Universitet

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