The relative projectivity of modules $L_p$

We give relative projectivity criteria for two types of modules that are $L_p$-spaces. The first type consists of the spaces $L_p(X,\mu)$ for $1\leq p\leq \infty$ considered as modules over the algebra of bounded measurable functions on a measure space $(X,\Sigma,\mu)$. The second type consists of the spaces $L_p(S,\mu)$ for $1 \leq p < +\infty$ considered as modules over the algebra $C_0(S)$, where $S$ is a locally compact space equipped with a regular Borel measure $\mu$. Both criteria can be roughly stated as an isomorphism (in the category of modules) with $\ell_p(\Lambda)$ for some index set $\Lambda$. Remarkably, for $p=+\infty$ we have only a necessary, but a very restrictive condition for the second type of modules, namely the pesudocompactness of the support of the measure and the inner regularity with respect to open sets.