From observables and states to Hilbert space and back: a 2-categorical adjunction

TL;DR

This paper formalizes the GNS construction as a 2-categorical adjunction between observables and states in C*-algebras, revealing deep structural insights with physical interpretations.

Contribution

It introduces a novel 2-categorical framework for the GNS construction, showing it as a weak natural transformation and adjoint, connecting algebraic and physical perspectives.

Findings

GNS construction as a left adjoint in a 2-category

Weak naturality encodes functoriality and universal properties

Provides physical interpretations of the categorical structures

Abstract

Given a representation of a C*-algebra, thought of as an abstract collection of physical observables, together with a unit vector, one obtains a state on the algebra via restriction. We show that the Gelfand-Naimark-Segal (GNS) construction furnishes a left adjoint of this restriction. To properly formulate this adjoint, it must be viewed as a weak natural transformation, a 1-morphism in a suitable 2-category, rather than as a functor between categories. Weak naturality encodes the functoriality and the universal property of adjunctions encodes the characterizing features of the GNS construction. Mathematical definitions and results are accompanied by physical interpretations.