10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
The Gelfand transform - the proof of Theorem 11.9<br />
Proof.<br />
b): It should be obvious that the Gelfand transform is a<br />
homomorphism:<br />
a +<br />
λb(t) = t(a+λb) = a(t)+λt(b) = â(t)+λˆ <br />
b(t) = â + λˆ <br />
b (t)<br />
and<br />
ab(t) = t(ab) = t(a)t(b) = a(t) b(t)<br />
The kernel of the Gelfand transform is {a ∈ A : t(a) = 0 ∀t ∈ ∆}.<br />
By Theorem 11.5 this is the same as the set of elements of A that<br />
lie in every maximal ideal of A.<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>