On the faithful flatness of some modules arising in analysis
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910031743549440 |
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| author | Sasane, Amol |
| author_facet | Sasane, Amol |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14452 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the faithful flatness of some modules arising in analysis Sasane, Amol Functional Analysis Commutative Algebra Complex Variables Rings and Algebras Primary 16D40, Secondary 46E25, 46J15, 28C05 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). |
| title | On the faithful flatness of some modules arising in analysis |
| topic | Functional Analysis Commutative Algebra Complex Variables Rings and Algebras Primary 16D40, Secondary 46E25, 46J15, 28C05 |
| url | https://arxiv.org/abs/2409.14452 |