Relating Operator Spaces via Adjunctions

TL;DR

This paper employs categorical methods to relate different operator classes on Hilbert spaces, revealing connections with quantum structures through dual adjunctions and module categories.

Contribution

It introduces a systematic categorical framework to describe relations between operator sets and their links to quantum structures using adjunctions and monads.

Findings

Categorical descriptions of operator relations

Dual adjunctions connecting convex sets and effect modules

Use of Eilenberg-Moore algebras for module categories

Abstract

This chapter uses categorical techniques to describe relations between various sets of operators on a Hilbert space, such as self-adjoint, positive, density, effect and projection operators. These relations, including various Hilbert-Schmidt isomorphisms of the form tr(A-), are expressed in terms of dual adjunctions, and maps between them. Of particular interest is the connection with quantum structures, via a dual adjunction between convex sets and effect modules. The approach systematically uses categories of modules, via their description as Eilenberg-Moore algebras of a monad.