10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
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Theorem 11.5<br />
Let A be a <strong>Banach</strong> algebra. Recall that a complex homomorphism<br />
or character on A is a non-zero homomorphism A → C.<br />
Theorem<br />
Let A be a unital commutative <strong>Banach</strong> algebra and ∆ the set of<br />
characters on A.<br />
a) Every maximal ideal in A is the kernel of an element of ∆.<br />
b) Let ω ∈ ∆. The the kernel of ω is a maximal ideal in A.<br />
c) An element a ∈ A is invertible if and only if ω(a) = 0 for all<br />
ω ∈ ∆.<br />
d) An element of A is invertible if and only it is not contained in<br />
a proper ideal of A.<br />
e) Let x ∈ A. Then<br />
σ(x) = {ω(x) : ω ∈ ∆}.<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>