On the faithful flatness of some modules arising in analysis
Amol Sasane
TL;DR
The paper characterizes faithful flatness for two analytic modules: $L^2(X, μ)$ as an $L∞(X, μ)$-module and $H^2$ as an $H∞$-module. It proves $L^2(X, μ)$ is flat and identifies conditions under which it fails to be faithfully flat, notably when $μ$ is σ-finite and under certain decompositions of $X$; it also shows $H^2$ is flat but not faithfully flat, resolving Quadrat's 2005 question. The methods combine a pointwise projection argument to prove flatness, Bézout and inner-function theory to analyze ideals, and ultrafilter-based maximal ideals to demonstrate nonfaithfulness, with a half-plane analogue via a conformal map. These results clarify the algebraic structure of central function spaces used in analysis and have implications for stability questions in related contexts.
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 $μ$ is a positive Radon measure on $X$, then $L^2(X,μ)$ is a flat $L^\infty(X,μ)$-module. Moreover: (1) If $μ$ is $σ$-finite, then for every finitely generated, nonzero, proper ideal $\mathfrak{n}$ of $L^\infty(X,μ)$, there holds $\mathfrak{n}L^2(X,μ)\subsetneq L^2(X,μ)$. (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 $μ(U_{n+1}\setminus U_n)>0$, then $L^2(X,μ)$ is not a faithfully flat $L^\infty(X,μ)$-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).