Pre-Hilbert $\ast$-categories: The Hilbert-space analogue of abelian categories

This article introduces pre-Hilbert $\ast$-categories: an abstraction of categories exhibiting “algebraic” aspects of Hilbert-space theory. Notably, finite biproducts in pre-Hilbert $\ast$-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 $\ast$-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 $\ast$-categories are quasi-abelian and thus also homological. Examples include the $\ast$-category of unitary representations of a group, the $\ast$-category of finite-dimensional inner product modules over an ordered division $\ast$-ring, and the $\ast$-category of self-dual Hilbert modules over a W*-algebra.