Pre-Hilbert -categories: The Hilbert-space analogue of abelian categories
arXiv:2312.02883
Abstract
This article introduces pre-Hilbert -categories: an abstraction of categories exhibiting "algebraic" aspects of the theory of Hilbert spaces. Notably, finite biproducts in pre-Hilbert -categories can be orthogonalised using the Gram-Schmidt process, and generalised notions of positivity and contraction support variants of Douglas' lemma and 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 axioms for pre-Hilbert -categories are otherwise inspired by those 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 groupoid, 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.
Rewrote parts of the sections on Douglas' lemma and contractions in light of the new Theorem 7.3