Categories of abstract and noncommutative measurable spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915810223587328 |
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| author | Fritz, Tobias Lorenzin, Antonio |
| author_facet | Fritz, Tobias Lorenzin, Antonio |
| contents | Gelfand duality is a fundamental result that justifies thinking of general unital $C^*$-algebras as noncommutative versions of compact Hausdorff spaces. Inspired by this perspective, we investigate what noncommutative measurable spaces should be. This leads us to consider categories of monotone $σ$-complete $C^*$-algebras as well as categories of Boolean $σ$-algebras, which can be thought of as abstract measurable spaces. Motivated by the search for a good notion of noncommutative measurable space, we provide a unified overview of these categories, alongside those of measurable spaces, and formalize their relationships through functors, adjunctions and equivalences. This includes an equivalence between Boolean $σ$-algebras and commutative monotone $σ$-complete $C^*$-algebras, as well as a Gelfand-type duality adjunction between the latter category and the category of measurable spaces. This duality restricts to two equivalences: one involving standard Borel spaces, which are widely used in probability theory, and another involving the more general Baire measurable spaces. Moreover, this result admits a probabilistic version, where the morphisms are $σ$-normal cpu maps and Markov kernels, respectively. We hope that these developments can also contribute to the ongoing search for a well-behaved Markov category for measure-theoretic probability beyond the standard Borel setting - an open problem in the current state of the art. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Categories of abstract and noncommutative measurable spaces Fritz, Tobias Lorenzin, Antonio Operator Algebras Category Theory Probability Quantum Physics Gelfand duality is a fundamental result that justifies thinking of general unital $C^*$-algebras as noncommutative versions of compact Hausdorff spaces. Inspired by this perspective, we investigate what noncommutative measurable spaces should be. This leads us to consider categories of monotone $σ$-complete $C^*$-algebras as well as categories of Boolean $σ$-algebras, which can be thought of as abstract measurable spaces. Motivated by the search for a good notion of noncommutative measurable space, we provide a unified overview of these categories, alongside those of measurable spaces, and formalize their relationships through functors, adjunctions and equivalences. This includes an equivalence between Boolean $σ$-algebras and commutative monotone $σ$-complete $C^*$-algebras, as well as a Gelfand-type duality adjunction between the latter category and the category of measurable spaces. This duality restricts to two equivalences: one involving standard Borel spaces, which are widely used in probability theory, and another involving the more general Baire measurable spaces. Moreover, this result admits a probabilistic version, where the morphisms are $σ$-normal cpu maps and Markov kernels, respectively. We hope that these developments can also contribute to the ongoing search for a well-behaved Markov category for measure-theoretic probability beyond the standard Borel setting - an open problem in the current state of the art. |
| title | Categories of abstract and noncommutative measurable spaces |
| topic | Operator Algebras Category Theory Probability Quantum Physics |
| url | https://arxiv.org/abs/2504.13708 |