Weak Korovkin approximation by completely positive linear maps on \(\beta\) (H)
type: Item
instance of: scholarly article
title: Weak Korovkin approximation by completely positive linear maps on \(\beta\) (H)
zbMATH Open document ID: 0552.41028
DOI: 10.1016/0021-9045(84)90039-X
author: M. N. N. Namboodiri
author: Balmohan V. Limaye
published in: Journal of Approximation Theory
publication date: 1984-01-01T00:00:00Z
review text: Let \(\beta\) (H) denote the set of all bounded operators on a complex Hilbert space H. A subset \({\mathcal S}\) of \(\beta\) (H) is called a weak Korovkin set if, for every net \((\phi_{\alpha})\) of completely positive maps on \(\beta\) (H) with \(\phi_{\alpha}(I)\leq I\), the relation \(\phi_{\alpha}(S)\to S\) weakly, \(S\in {\mathcal S}\), implies \(\phi_{\alpha}(T)\to T\) weakly, \(T\in \beta (H)\). Consider a subset \({\mathcal S}\) of \(\beta\) (H) which contains the identity operator I and is irreducible. Theorem: Let \(C^*({\mathcal S})\), the \(C^*\)-algebra generated by \({\mathcal S}\) in \(\beta\) (H), contain a non-zero compact operator. (i) \({\mathcal S}\) is a weak Korovkin set in \(\beta\) (H) if and only if \(id|_{{\mathcal S}}\) has a unique completely positive extension to \(C^*({\mathcal S})\). (ii) \({\mathcal S}\cup \{S^*S+SS^*: S\in {\mathcal S}\}\) is a weak Korovkin set in \(\beta\) (H). Corollary. If there are \(T\in span\{{\mathcal S}+{\mathcal S}^*\}\) and a compact operator K with \(\| T-K\| <\| T\|\), then \({\mathcal S}\) is a weak Korovkin set in \(\beta\) (H). Various examples are given to illustrate these results. The work is based on the boundary theorem of \textit{W. Arveson} [Acta Math. 128, 271-308 (1972; Zbl 0245.46098)].
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 3879477
zbMATH Keywords: complex Hilbert space
zbMATH Keywords: weak Korovkin set
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3879477