10. Commutative Banach algebras - Aarhus Universitet

The Gelfand transform - the proof of Theorem 11.9
Proof.
a): Since ∆ is contained in the unit ball of A ∗ , and the latter is
compact in the weak*-topology by Banach-Alaoglu theorem
(Theorem 3.15), it suffices to show that ∆ is closed in the
weak*-topology.
Let therefore ω ∈ A ∗ be an element of the closure ∆ of ∆ in A ∗ .
We want to show that ω ∈ ∆.
There is a δ > 0 such that |st − ω(a)ω(b)| < ǫ when |s − ω(a)| < δ
and |t − ω(b)| < δ.
Let a,b ∈ A, and let ǫ > 0. Since U =
{ν ∈ A ∗ : |ν(a) − ω(a)| < δ, |ν(b) − ω(b)| < δ, |ν(ab) − ω(ab)| < ǫ}
is an open neighborhood of ω in the weak*-topology of A ∗ there is
a character ξ ∈ ∆ contained in U because ω ∈ ∆.
Klaus Thomsen 10. Commutative Banach algebras