Arveson's extension theorem for conditionally unital completely positive maps
type: Item
instance of: scholarly article
title: Arveson's extension theorem for conditionally unital completely positive maps
author: Q6339611
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2024-07-16T00:00:00Z
review text: Motivated by those properties that are satisfied by generators of uniformly continuous semigroups of unital completely positive maps, the author defines and studies the notion of \(\phi\)-conditionally unital completely positive maps. More precisely, given two unital operator systems \(V\) and \(W\), we denote by \(\mathrm{ucp}(V,W)\) the set of all unital completely positive maps from \(V\) to \(W\). For \(\phi\in \mathrm{ucp}(V, W)\), a linear map \(A:V\to W\) is said to be \textit{\(\phi\)-conditionally completely positive} if for each \(n\in \mathbb{N}\), \(v\in M_n(V)\) and \(\rho\in \mathrm{ucp}(M_n(W), \mathbb{C})\) with \(\rho\big(\phi^{(n)}(v)\big) = 0\), one has \(\rho\big(A^{(n)}(v)\big) = 0\). Moreover, a \(\phi\)-conditionally completely positive map \(A\) is called a \textit{\(\phi\)-conditionally unital completely positive map} if \(A(e) = 0\), where \(e\) is the order unit of \(V\).\N\NThe main results of this paper are a form of Choi-Jamiołkowski duality for \(\phi\)-completely positive maps and a form of Arveson's extension theorem for \(\phi\)-conditionally unital completely positive maps.
reviewed by: Chi-Keung Ng
Mathematics Subject Classification ID: 46L07
Mathematics Subject Classification ID: 47L25
Mathematics Subject Classification ID: 46L57
zbMATH DE Number: 7881440
zbMATH Keywords: operator systems
zbMATH Keywords: completely positive maps
zbMATH Keywords: one-parameter semigroups
zbMATH Keywords: quantum channels
zbMATH Keywords: noncommutative probability
MaRDI profile type: Publication
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DOI: 10.1134/S0081543824010218
description: scientific article; zbMATH DE number 7881440