2023/12/05 by Di Meglio, Matthew · 1 citation
#06F25 #18E99 #18M40 #46M15 #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2312.02883
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.