Von Neumann Categories

TL;DR

This paper introduces von Neumann categories, a categorical generalization of von Neumann algebras inspired by algebraic quantum field theory, providing a framework for incorporating relativistic effects into quantum mechanics.

Contribution

It defines von Neumann categories, explores their properties, and extends classical von Neumann algebra constructions, like crossed products, to the categorical setting.

Findings

Defined von Neumann categories as premonoidal dagger categories.

Extended von Neumann algebra constructions to categories.

Presented a crossed product construction for *-premonoidal categories.

Abstract

In this paper, we introduce the notion of a von Neumann category, as a generalization and categorification of von Neumann algebra. A von Neumann category is a premonoidal category with compatible dagger structure which embeds as a double commutant into a suitable premonoidal category of Hilbert spaces. The notion was inspired by algebraic quantum field theory. In AQFT, one assigns to open regions in Minkowski space a C*-algebra, called the local algebra. The local algebras are patched together to form a global algebra associated to the AQFT. The key relativistic assumption is Einstein Causality, which says that the algebras associated to spacelike separated regions commute in the global algebra. Premonoidal categories provide a natural framework for lifting such structure from algebras to categories. Thus von Neumann categories serve as a basis for extending the abstract quantum mechanics of Abramsky and Coecke to include relativistic effects. In this paper, we focus on the structure of von Neumann categories. After giving the basic definitions and examples, we consider constructions typically associated to von Neumann algebras, and examine their extensions to the category setting. In particular, we present a crossed product construction for *-premonoidal categories.