10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
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The Gelfand transform - Theorem 11.9<br />
The map<br />
is called the Gelfand transform.<br />
Theorem<br />
A ∋ a ↦→ â ∈ C(∆)<br />
a) The character space ∆ is compact in the weak*-topology.<br />
b) The Gelfand transform is a homomorphism from A onto a<br />
subalgebra  of C(∆) whose kernel is the radical of A, i.e. the<br />
intersection of the maximal ideals in A.<br />
c) For each a ∈ A,<br />
max {|â(t)| : t ∈ ∆} = ρ(a) ≤ a.<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>