10. Commutative Banach algebras - Aarhus Universitet

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Quotients Let A be a Banach algebra - not necessarily abelian. A closed two-sided ideal in A is a closed subspace J ⊆ A such that aj ∈ J, ja ∈ J when j ∈ J and a ∈ A. Lemma Let A be a Banach algebra and J ⊆ A a closed two-sided ideal. Then A/J is a Banach algebra in the quotient norm, and the quotient map q : A → A/J is an algebra homomorphism. Proof. Except for some algebraic trivialities - which we omit here - it remains only to prove the submultiplicativity of the norm in A/J, i.e. we must show that q(x)q(y) ≤ q(x) q(y) for all x,y ∈ A. Let j,j ′ ∈ J and note that q(x)q(y) = xy + J ≤ xy − xj ′ − jy + jj ′ = (x − j)(y − j ′ ) ≤ x − j y − j ′ . The desired inequality follows by taking the infimum over j and j ′ . Klaus Thomsen 10. Commutative Banach algebras
Theorem 11.5 Let A be a Banach algebra. Recall that a complex homomorphism or character on A is a non-zero homomorphism A → C. Theorem Let A be a unital commutative Banach algebra and ∆ the set of characters on A. a) Every maximal ideal in A is the kernel of an element of ∆. b) Let ω ∈ ∆. The the kernel of ω is a maximal ideal in A. c) An element a ∈ A is invertible if and only if ω(a) = 0 for all ω ∈ ∆. d) An element of A is invertible if and only it is not contained in a proper ideal of A. e) Let x ∈ A. Then σ(x) = {ω(x) : ω ∈ ∆}. Klaus Thomsen 10. Commutative Banach algebras
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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: Quotients Proof. First we observe t
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
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10. Commutative Banach algebras - Aarhus Universitet

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