Compactness of Gelfand topology

Question

I have a trouble to understand the proof of Theorem 16 of Chapter 18 in Lax's functional analysis. I post the proof and relevant definitions below.

Here's my question. In the proof, he defines $p(N)=t_N$ and proves that it is homomorphism. But how can we know this homomorphism comes from the Gelfand representation? Or does Gelfand representation contains all homomorphisms?

Answer 1

The function $p:J \times \mathcal{L} \to \mathbb{C}$ does "contain" all complex homomorfisms:

For each $h:\mathcal{L} \to \mathbb{C}$ nonzero homomorfism, $\mathcal{M}:=\ker(h)$ has codimension 1 and is a proper ideal ($I\not\in \mathcal{M}$), thus $\mathcal{M} \in J$. Therefore $h = p(\mathcal{M},\cdot).$

So every $t$ in the closure of (17) is the image of some $\mathcal{M} \in J$ under (17).

Note: I think he means that the Gelfand representation is the function $N \in \mathcal{L} \mapsto p(\cdot, N)$, and that the function $p(\cdot, N): J \to \mathbb{C}$ is the representation of $N$. So it is not correct to say "the Gelfand representation contains all homomorphisms", because the homomorfisms are the functions of the form $p(\mathcal{M}, \cdot)$.