Pre-Hilbert $*$-categories: The Hilbert-space analogue of abelian categories

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.