What is the result that implies $\ell^1$ is isomorphic to $C(K)$

Question

In this answer here t.b. writes that $\ell^1(\mathbb Z)$ would then have to be isomorphic to a space of the form $C(K)$ with $K$ compact (metrizable and infinite).

What is the result they are referring to?

Answer 1

Every commutative unital $C^*$ algebra is of the form $C(K)$. This follows from the Gelfand theory. For more references, see here.