Stinespring's construction as an adjunction
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arXiv
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866911977706618880 |
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| author | Parzygnat, Arthur J. |
| author_facet | Parzygnat, Arthur J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1807_02533 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Stinespring's construction as an adjunction Parzygnat, Arthur J. Operator Algebras Mathematical Physics Category Theory 47A20 (Primary), 18D05, 46L05, 47A67, 81R15 (Secondary) 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. |
| title | Stinespring's construction as an adjunction |
| topic | Operator Algebras Mathematical Physics Category Theory 47A20 (Primary), 18D05, 46L05, 47A67, 81R15 (Secondary) |
| url | https://arxiv.org/abs/1807.02533 |