It is well known that a compact Hausdorff topological space can be fully reconstructed from its $C^\ast$-algebra of complex valued continuous functions with the sup norm.
Are there similar (partial) results for reconstructing topological/geometrical information from the (Hilbert space structure of) $L^2(X)$ for (say) a compact manifold $X$, or more general spaces with a Borel measure? To what extent does it depend on the particular measure?