10. Commutative Banach algebras - Aarhus Universitet

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The Gelfand transform - Theorem 11.9 Let A be a commutative Banach algebra with unit. For any character ω : A → C of A,we know from (c) of Theorem 10.7 that x < 1 ⇒ |ω(a)| < 1. This shows that every character is continuous and has norm ≤ 1. (Since ω(1) = 1 by Proposition 10.6 we have in fact that ω = 1). Thus ∆ ⊆ {τ ∈ A ∗ : τ ≤ 1} . In particular, ∆ comes equipped with a natural topology - namely the weak*-topology inherited from A ∗ . By Theorem 11.5 there is a bijective correpsondance between the character space ∆ and the set of maximal ideals of A. That’s why Rudin calls it the maximal ideal space of A. We will here call it the character space. Now, every a ∈ A defines a continuous function â : ∆ → C such that â(ω) = ω(a). (1) Klaus Thomsen 10. Commutative Banach algebras
The Gelfand transform - Theorem 11.9 The map is called the Gelfand transform. Theorem A ∋ a ↦→ â ∈ C(∆) a) The character space ∆ is compact in the weak*-topology. b) The Gelfand transform is a homomorphism from A onto a subalgebra Â of C(∆) whose kernel is the radical of A, i.e. the intersection of the maximal ideals in A. c) For each a ∈ A, max {|â(t)| : t ∈ ∆} = ρ(a) ≤ 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: Theorem 11.5 Proof. (e) : If λ ∈
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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