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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.02883 |
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Table of 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.