10. Commutative Banach algebras - Aarhus Universitet

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Commutative Banach algebras - the definition Let A be a Banach algebra. A is commutative or abelian when ab = ba for all a,b ∈ A. Let A be a commutative Banach algebra. An ideal in A is a subspace J ⊆ A such that aj ∈ J when a ∈ A and j ∈ J. Lemma (Proposition 11.2) Let A be a commutative and unital Banach algebra. (a) The only ideal in A which contains an invertible element is A itself. (b) If J ⊆ A is an ideal then the closure J of J is also an ideal in A. Klaus Thomsen 10. Commutative Banach algebras
Commutative Banach algebras - ideals An ideal J ⊆ A is proper when J = A. A proper ideal J ⊆ A is maximal when there is no proper ideal I ⊆ A such that J ⊆ I and J = I. Theorem (Theorem 11.3) Let A be a commutative unital Banach algebra. Then (a) every proper ideal is contained in a maximal ideal in A, and (b) every maximal ideal is closed. Klaus Thomsen 10. Commutative Banach algebras
Page 1 and 2: 10. Commutative Banach algebras Kla
Page 3: The Gelfand-Mazur theorem Let A and
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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