Completely contractive factorizations of \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Completely contractive factorizations of \(C^*\)-algebras
zbMATH Open document ID: 0585.46050
DOI: 10.1016/0022-1236(85)90063-1
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: A map \(\phi\) :A\(\to B\) between \(C^*\)-algebras is said to be completely bounded if \(\sup_{n}\) \(\| \phi_ n\|\) is finite, where \(\phi_ n:A\otimes M_ n\to B\otimes M_ n\) is defined by \(\phi_ n(a_{ij})=(\phi (a_{ij}))\). If \(\sup_{n}\) \(\| \phi_ n\| \leq 1\) then \(\phi\) is said to be a complete contraction. If each \(\phi_ n\) is positive then \(\phi\) is called completely positive. Nuclearity for \(C^*\)-algebras can be characterized by the following property (among several): there exists a net of completely positive finite rank maps \(\phi\) \({}_{\alpha}:A\to A\) such that \(\lim_{\alpha}\| \phi_{\alpha}(x)-x\| =0\) for all \(x\in A.\) If the maps are only complete contractions, then the characterization no longer holds. There are examples of \(C^*\)-algebras for which the identity map is the point norm limit of finite rank complete contractions but which are not nuclear. The main result of this paper is that a \(C^*\)-algebra A is nuclear if and only if there exists a net \(\phi_{\alpha}:A\to A\) of complete contractions which converge in the point norm topology to the identity, and which factor through \(M_ n\).
Mathematics Subject Classification ID: 46L30
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 3937832
zbMATH Keywords: completely bounded
zbMATH Keywords: complete contraction
zbMATH Keywords: completely positive
zbMATH Keywords: Nuclearity for \(C^*\)-algebras
zbMATH Keywords: point norm topology
author: Roger R. Smith
MaRDI profile type: Publication
full work available at URL: 0022-1236(85)90063-1
OpenAlex ID: W1997368183
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description: scientific article; zbMATH DE number 3937832