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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.13063 |
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| _version_ | 1866916427220385792 |
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| author | Schötz, Matthias |
| author_facet | Schötz, Matthias |
| contents | The following representation theorem is proven: A partially ordered commutative ring $R$ is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space $X$ if and only if $R$ is archimedean and localizable. Here we assume that the positive cone of $R$ is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on $X$ is one that is defined on a dense open subset of $X$. A partially ordered commutative ring $R$ is archimedean if the underlying additive partially ordered abelian group is archimedean, and $R$ is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the $σ$-bounded case, lattice-ordered commutative rings ($f$-rings), partially ordered fields, and commutative operator algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rings of almost everywhere defined functions Schötz, Matthias Rings and Algebras Operator Algebras 06F25, 13J25, 16G30, 47L60 The following representation theorem is proven: A partially ordered commutative ring $R$ is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space $X$ if and only if $R$ is archimedean and localizable. Here we assume that the positive cone of $R$ is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on $X$ is one that is defined on a dense open subset of $X$. A partially ordered commutative ring $R$ is archimedean if the underlying additive partially ordered abelian group is archimedean, and $R$ is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the $σ$-bounded case, lattice-ordered commutative rings ($f$-rings), partially ordered fields, and commutative operator algebras. |
| title | Rings of almost everywhere defined functions |
| topic | Rings and Algebras Operator Algebras 06F25, 13J25, 16G30, 47L60 |
| url | https://arxiv.org/abs/2406.13063 |