Arthur J. Parzygnat - Stinespring's construction as an adjunction

compositionality:13505 - Compositionality, December 20, 2019, Volume 1 (2019) -
Stinespring's construction as an adjunctionArticle

Authors: Arthur J. Parzygnat ORCID1

Given a representation of a unital $C^*$-algebra $\mathcal{A}$ on a Hilbert space $\mathcal{H}$, together with a bounded linear map $V:\mathcal{K}\to\mathcal{H}$ from some other Hilbert space, one obtains a completely positive map on $\mathcal{A}$ via restriction using the adjoint action associated to $V$. We show this restriction forms a natural transformation from a functor of $C^*$-algebra representations to a functor of completely positive maps. We exhibit Stinespring's construction as a left adjoint of this restriction. Our Stinespring adjunction provides a universal property associated to minimal Stinespring dilations and morphisms of Stinespring dilations. We use these results to prove the purification postulate for all finite-dimensional $C^*$-algebras.

Volume: Volume 1 (2019)
Published on: December 20, 2019
Imported on: May 2, 2024
Keywords: Mathematics - Operator Algebras,Mathematical Physics,Mathematics - Category Theory,47A20 (Primary), 18D05, 46L05, 47A67, 81R15 (Secondary)

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