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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?

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

voldemort
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    Additionally, in the case that the $C^*$ algebra is not unital, it is of the form $C_0(X)$ for some locally compact Hausdorff space $X$. – Cameron Williams Oct 31 '14 at 23:16