An outline of proving Gelfand-Neimark Theorem for Commutative $C^{*}$ algebras

Question

I would like to look at a sketch or an outline of the proof of the Gelfand-Neimark Theorem for commutative $C^{*}$ algebra's. I am doing a final year essay and I just want the major points or ideas to proving the theorem. Can you help me with this? Thank you very much!

Answer 1

In the unital case, there are three basic steps:

1. For a unital C*-algebra $A$, take $\Omega(A)$ to be the set of non-zero multiplicative linear functionals on $A$. Banach-Aluoglu tells you that this is compact in the weak-$\ast$ topology.

2. For each $a\in A$, define $\hat{a} : \Omega(A)\to \mathbb{C}$ by evaluation. $\hat{a}$ is naturally continuous in the weak-$\ast$ topology, so we get a map $\Gamma : A \to C(\Omega(A))$, the space of continuous complex valued functions on $A$. This map is shown to be a homomorphism.

3. That $\Gamma$ is injective follows from the description of the spectrum of an element. That $\Gamma$ is surjective follows from the Stone-Weierstrass theorem. This completes the proof.