Matrix regular operator spaces
type: Item
instance of: scholarly article
title: Matrix regular operator spaces
zbMATH Open document ID: 0898.46017
author: Walter J. Schreiner
published in: Journal of Functional Analysis
publication date: 1998-11-02T00:00:00Z
review text: A norm on an ordered real Banach space \(E\) is called regular (or Riesz norm) if \(-x\leq y\leq x\) implies \(\| y\|\leq\| x\|\), and \(\| y\|< 1\) implies the existence of \(x\in E\) with \(\| x\|< 1\) and \(-x\leq y\leq x\). This concept is generalized to matrix ordered complex operator spaces [as introduced by \textit{M.-D. Choi} and \textit{E. G. Effros}, J. Funct. Anal. 24, 156-209 (1977; Zbl 0341.46049)] in a way which respects the matricial structure of the operator space. Such a space is called matrix regular if, at each matrix level, the restriction of the norm to the selfadjoint elements is regular in the above sense. The author studies hereditary properties of this notion. In particular, he shows that, if \(X\), \(V\) are operator spaces and \(V\) is matrix regular, then so is \(X^*\otimes^h V\otimes^h X\), where \(\otimes^h\) denotes the Haagerup tensor product. As an application, the author presents new proofs of both the multilinear representation theorem and the multilinear decomposition theorem of \textit{E. Christensen} and \textit{A. M. Sinclair} [J. Funct. Anal. 72, 151-181 (1987; Zbl 0622.46040)].
Mathematics Subject Classification ID: 46B40
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 1121257
zbMATH Keywords: ordered real Banach space
zbMATH Keywords: Riesz norm
zbMATH Keywords: matrix ordered complex operator spaces
zbMATH Keywords: matricial structure of the operator space
zbMATH Keywords: matrix regular
zbMATH Keywords: Haagerup tensor product
zbMATH Keywords: multilinear representation theorem
zbMATH Keywords: multilinear decomposition theorem
reviewed by: Frank Mantlik
MaRDI profile type: Publication
full work available at URL: jfan.1997.3160
OpenAlex ID: W2076807832
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DOI: 10.1006/JFAN.1997.3160
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description: scientific article; zbMATH DE number 1121257