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: (1) If $\mu$ is $\sigma$-finite, then for every finitely generated, nonzero, proper ideal $\mathfrak{n}$ of $L^{\infty }(X,\mu )$, there holds $\mathfrak{n}{L}^2(X,\mu )⊊L^2(X,\mu )$. (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).

17 pages, 0 figures. The new version corrects typos detected in the previous version. To appear in the Nagoya Mathematical Journal