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

TL;DR

This paper introduces pre-Hilbert $*$-categories, abstracting Hilbert-space properties into category theory, showing they are quasi-abelian and homological, with applications to unitary representations and modules.

Contribution

It formalizes pre-Hilbert $*$-categories, extending abelian category concepts to Hilbert-space analogues, and demonstrates their quasi-abelian and homological nature.

Findings

Finite biproducts can be orthogonalised via Gram-Schmidt.

Supports a variant of Sz.-Nagy's unitary dilation theorem.

Examples include categories of unitary representations and Hilbert modules.

Abstract

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.