For any $\sigma$-finite measure $\mu$, the space $L^\infty(\mu)$ is isometric (even as a Banach algebra) to the space $C(K)$ of continuous functions on a compact Hausdorff space $K$. For example, $\ell^\infty=L^\infty({\bf N})=C(\beta {\bf N})$.
Is there some canonical (or at least fairly explicit) $K$ for which $L^\infty(0,1)$ (with Lebesgue measure) is isometric to $C(K)$?
This question have been already answered on MO. For a more detailed discussion of the spectrum of arbitrary $L_\infty$-space see page 28 in Second duals of measure algebras. H. G. Dales, A. T.-M. Lau