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Relating Operator Spaces via Adjunctions

03 Feb 2016 [ invited chapter · arXiv ]

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.