On the faithful flatness of some modules arising in analysis

The notion of faithful flatness of a module over a commutative ring is studied for two $R$-modules $M$ arising in functional analysis, where $R$ is a Banach algebra and $M$ is a Hilbert space. The following results are shown: If $X$ is a locally compact Hausdorff topological space, and $\mu$ is a positive Radon measure on $X$, then $L^2(X,\mu)$ is a flat $L^\infty(X,\mu)$-module. Moreover: (1) If $\mu$ is $\sigma$-finite, then for every finitely generated, nonzero, proper ideal $\mathfrak{n}$ of $L^\infty(X,\mu)$, there holds $\mathfrak{n}L^2(X,\mu)\subsetneq L^2(X,\mu)$. (2) If $X$ is the union of an increasing family of Borel sets $U_n$, $n\in \mathbb{N}$, such that for each $n\in \mathbb{N}$, $\overline{U_n}$ is compact and $\mu(U_{n+1}\setminus U_n)>0$, then $L^2(X,\mu)$ is not a faithfully flat $L^\infty(X,\mu)$-module. It is shown that the Hardy space $H^2$ is a flat, but not a faithfully flat $H^\infty$-module (answering a 2005 question of Alban Quadrat).