R*-categories: The Hilbert-space analogue of abelian categories

Abstract: This article introduces R*-categories: an abstraction of categories exhibiting the "algebraic" aspects of the theory of Hilbert spaces. Notably, finite biproducts in R*-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 R*-category axioms 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: R*-categories are quasi-abelian and thus homological. Examples include the category of unitary representations of a group, the category of finite-dimensional inner product modules over a partially ordered division ring, and the category of self-dual Hilbert modules over a W*-algebra.