Positive operators on tensor products which are the tensor product of their restrictions
type: Item
instance of: scholarly article
title: Positive operators on tensor products which are the tensor product of their restrictions
zbMATH Open document ID: 0672.47027
DOI: 10.1007/BF01199117
author: Michael Pannenberg
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: Let E, F, \(E_ 1\), \(F_ 1\) be Banach lattices possessing quasi-interior positive elements e, u, \(e_ 1\), \(u_ 1\). Further on, let \(\psi\) : \(E\to {\mathbb{R}}\) be a positive linear functional, \(\delta\) : \(F\to {\mathbb{R}}^ a \)lattice homomorphism, \(S_ 0: E\to E_ 1\) a positive operator, \(T_ 0: F\to F_ 1\) a lattice homomorphism, such that \(\psi (e)=\delta (u)=1\), \(S_ 0(e)=e_ 1\) and \(T_ 0(u)=u_ 1\). We show that \(\psi\) \(\otimes \delta\) is the only positive linear functional on E\({\tilde \otimes}_{\ell}F\) coinciding with \(\psi\) on E and with \(\delta\) on F, resp. that \(S_ 0\otimes T_ 0\) is the only positive operator on E\({\tilde \otimes}_{\ell}F\) which coincides with \(S_ 0\) on E and with \(T_ 0\) on F. The proof uses the Kakutani-Krein theorem and an extreme point argument. A variant of this result for a Banach lattice F with the principal projection property is proved by elementary calculations with components. Some possible applications of the results are indicated. The paper finishes with analogous results for tensor products of states and pure states resp. tensor products of positive operators and pure completely positive operators on \(C^*\)-algebras, as well as with variants of this result for particular involutive Banach algebras.
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 47A80
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 4099988
zbMATH Keywords: Banach lattices possessing quasi-interior positive elements
zbMATH Keywords: positive linear functional
zbMATH Keywords: lattice homomorphism
zbMATH Keywords: Kakutani-Krein theorem
zbMATH Keywords: extreme point
zbMATH Keywords: principal projection property
zbMATH Keywords: tensor products of states and pure states
zbMATH Keywords: tensor products of positive operators and pure completely positive operators on \(C^ *\)-algebras
zbMATH Keywords: involutive Banach algebras
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01199117
OpenAlex ID: W2078581536
description: scientific article; zbMATH DE number 4099988