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

Matthew Di Meglio

TL;DR

The paper develops pre-Hilbert $*$-categories as the Hilbert-space analogue of abelian categories, unifying algebraic aspects of Hilbert theory under a categorical framework. It introduces a robust structure—zero objects, orthonormal biproducts, isometric kernels, and diagonal kernels—and proves additivity and quasi-abelianness, making these categories homological. A central contribution is the Gram–Schmidt process generalised to categorical biproducts, enabling orthogonal decompositions and a rich theory of orthogonal complements. The work further introduces positivity, order, contractions, and codilators, including a categorical analogue of Sz.-Nagy’s dilation via codilators and a Douglas-type factorisation theory, culminating in a Douglian framework for contractions and a universal codilation construction. Together, these developments yield a cohesive, algebraic, and homological approach to Hilbert-space structures with broad examples such as unitary representations, inner-product modules over ordered rings, and self-dual Hilbert modules over W*-algebras, highlighting the deep links between operator-algebraic intuitions and categorical formalism.

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.

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

TL;DR

The paper develops pre-Hilbert -categories as the Hilbert-space analogue of abelian categories, unifying algebraic aspects of Hilbert theory under a categorical framework. It introduces a robust structure—zero objects, orthonormal biproducts, isometric kernels, and diagonal kernels—and proves additivity and quasi-abelianness, making these categories homological. A central contribution is the Gram–Schmidt process generalised to categorical biproducts, enabling orthogonal decompositions and a rich theory of orthogonal complements. The work further introduces positivity, order, contractions, and codilators, including a categorical analogue of Sz.-Nagy’s dilation via codilators and a Douglas-type factorisation theory, culminating in a Douglian framework for contractions and a universal codilation construction. Together, these developments yield a cohesive, algebraic, and homological approach to Hilbert-space structures with broad examples such as unitary representations, inner-product modules over ordered rings, and self-dual Hilbert modules over W*-algebras, highlighting the deep links between operator-algebraic intuitions and categorical formalism.

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.
Paper Structure (28 sections, 45 theorems, 129 equations)

This paper contains 28 sections, 45 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Understanding the definition
  3. Hilbert spaces
  4. Involution
  5. Closed subobjects
  6. Zero objects
  7. Isometric kernels
  8. Orthonormal biproducts
  9. Pre-Hilbert *-categories
  10. Similarities with abelian categories
  11. Enrichment in commutative monoids
  12. Additivity
  13. Quasi-abelianness
  14. Range factorisation
  15. Orthogonal biproducts and the Gram–Schmidt process
  16. ...and 13 more sections

Key Result

Theorem 1

An object $X$ in a category with finite biproducts is abelian if and only if the diagonal $\Delta \colon X \to X\oplus X$ is a kernel of a split epimorphism.

Theorems & Definitions (104)

  • Definition
  • Theorem
  • Definition
  • Theorem
  • Theorem
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • ...and 94 more