The correspondence associated to an inner completely positive map
type: Item
instance of: scholarly article
title: The correspondence associated to an inner completely positive map
zbMATH Open document ID: 0646.46056
DOI: 10.1007/BF01443509
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164544
review text: Let M be a \(\sigma\)-finite von Neumann algebra. A completely positive map \(\Phi\) : \(M\to M\) is inner if there is a sequence \(\{a_ i\}\subseteq M\) with \(\Phi (x)=\sum _{i}a\) \(*_ ixa_ i\) for \(x\in M\). If \(\Phi\) is any completely positive map on M then associated to \(\Phi\) is a correspondence \(H_{\Phi}\) from M to M. Inner completely positive maps are characterized as those maps for which \(H_{\Phi}\subseteq \sum ^{\oplus}L\) 2(M), where L 2(M) is the identity correspondence. For an inner completely positive map \(\Phi\), we give a complete invariant of the unitary equivalence class of the representation of \(M\otimes M^{op}\) associated to \(H_{\Phi}\); it is a decreasing sequence of central projections of M, \(\{d_ i\}\), such that \(H_{\Phi}\cong \Sigma ^{\oplus}d_ iL\) 2(M). The relation of the \(d_ i's\) to the \(a_ i's\) is as follows; \(d_ 1\) is the supremum of the central supports of the \(a_ i's\) and \(d_ n\) is the central projection over which at least n of the \(a_ i's\) are independent over the centre of M.
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 4054341
zbMATH Keywords: \(\sigma \) -finite von Neumann algebra
zbMATH Keywords: inner completely positive maps
zbMATH Keywords: decreasing sequence of central projections
author: James A. Mingo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4054341