On the faithful flatness of some modules arising in analysis

TL;DR

This paper investigates the conditions under which certain modules in functional analysis, specifically $L^2$ spaces over measure spaces and Hardy spaces, are flat or faithfully flat over their associated Banach algebras, revealing new insights into their algebraic properties.

Contribution

It establishes new criteria for flatness and faithful flatness of $L^2$ modules over $L^ Infty$ algebras and answers a longstanding question about the Hardy space $H^2$.

Findings

$L^2(X,)$ is flat over $L^(X,)$ for any Radon measure.

If  is , then certain ideals do not act faithfully on $L^2(X,)$.

$H^2$ is flat but not faithfully flat over $H^$, answering a 2005 question.

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 $\mathfrak{n}L^2(X,\mu)\subsetneq L^2(X,\mu)$. (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).