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 />
Let A be a commutative <strong>Banach</strong> algebra with unit. For any<br />
character ω : A → C of A,we know from (c) of Theorem <strong>10.</strong>7 that<br />
x < 1 ⇒ |ω(a)| < 1.<br />
This shows that every character is continuous and has norm ≤ 1.<br />
(Since ω(1) = 1 by Proposition <strong>10.</strong>6 we have in fact that ω = 1).<br />
Thus<br />
∆ ⊆ {τ ∈ A ∗ : τ ≤ 1} .<br />
In particular, ∆ comes equipped with a natural topology - namely<br />
the weak*-topology inherited from A ∗ .<br />
By Theorem 11.5 there is a bijective correpsondance between the<br />
character space ∆ and the set of maximal ideals of A. That’s why<br />
Rudin calls it the maximal ideal space of A. We will here call it the<br />
character space.<br />
Now, every a ∈ A defines a continuous function â : ∆ → C such<br />
that<br />
â(ω) = ω(a). (1)<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>