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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2312.02883 |
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| _version_ | 1866914159218655232 |
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| author | Di Meglio, Matthew |
| author_facet | Di Meglio, Matthew |
| contents | This article introduces pre-Hilbert $*$-categories: an abstraction of categories exhibiting "algebraic" aspects of Hilbert-space theory. Notably, finite biproducts in pre-Hilbert $*$-categories can be orthogonalised using the Gram-Schmidt process, and generalised notions of positivity and contraction support a variant of Sz.-Nagy's unitary dilation theorem. Underpinning these generalisations is the structure of an involutive identity-on-objects contravariant endofunctor, which encodes adjoints of morphisms. The pre-Hilbert $*$-category axioms are otherwise inspired by the ones for abelian categories, comprising a few simple properties of products and kernels. Additivity is not assumed, but nevertheless follows. In fact, the similarity with abelian categories runs deeper: pre-Hilbert $*$-categories are quasi-abelian and thus also homological. Examples include the $*$-category of unitary representations of a group, the $*$-category of finite-dimensional inner product modules over an ordered division $*$-ring, and the $*$-category of self-dual Hilbert modules over a W*-algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02883 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pre-Hilbert $*$-categories: The Hilbert-space analogue of abelian categories Di Meglio, Matthew Category Theory 18M40, 18E99, 46M15, 06F25 This article introduces pre-Hilbert $*$-categories: an abstraction of categories exhibiting "algebraic" aspects of Hilbert-space theory. Notably, finite biproducts in pre-Hilbert $*$-categories can be orthogonalised using the Gram-Schmidt process, and generalised notions of positivity and contraction support a variant of Sz.-Nagy's unitary dilation theorem. Underpinning these generalisations is the structure of an involutive identity-on-objects contravariant endofunctor, which encodes adjoints of morphisms. The pre-Hilbert $*$-category axioms are otherwise inspired by the ones for abelian categories, comprising a few simple properties of products and kernels. Additivity is not assumed, but nevertheless follows. In fact, the similarity with abelian categories runs deeper: pre-Hilbert $*$-categories are quasi-abelian and thus also homological. Examples include the $*$-category of unitary representations of a group, the $*$-category of finite-dimensional inner product modules over an ordered division $*$-ring, and the $*$-category of self-dual Hilbert modules over a W*-algebra. |
| title | Pre-Hilbert $*$-categories: The Hilbert-space analogue of abelian categories |
| topic | Category Theory 18M40, 18E99, 46M15, 06F25 |
| url | https://arxiv.org/abs/2312.02883 |