On the faithful flatness of some modules arising in analysis

Abstract 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: • 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 $ $L^2(X,\mu )$ . • 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. In addition, it is shown that the classical Hardy space $H^2$ is a flat, but not a faithfully flat $H^\infty $ -module, which answers a 2005 question of Alban Quadrat.