Korovkin-type theorems for Schwarz maps and operator monotone functions in \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Korovkin-type theorems for Schwarz maps and operator monotone functions in \(C^*\)-algebras
zbMATH Open document ID: 0931.47015
DOI: 10.1007/PL00004717
author: Mitsuru Uchiyama
published in: Mathematische Zeitschrift
publication date: 2000-02-23T00:00:00Z
review text: Let \({\mathcal A}\) be a \(C^*\)-algebra with unit. After giving simple proofs on estimates of the norms related to Schwarz maps, the author proves extensions of known Korovkin-type theorems by using operator monotone functions. Given a certain operator monotone function \(f\) it is proved that the *-subalgebra \({\mathcal J}^*(S)\) generated by some subset \(S\) of \({\mathcal A}\) is contained in the Korovkin closure of some set associated with \(S\) and \(f\). These results lead to a generalization of Robertson's theorem on \(C^*\)-subalgebras generated by a sequence of Schwarz maps. The obtained results are finally applied to \(C(X)\), the \(C^*\)-algebra of all continuous functions on a compact Hausdorff space \(X\), and to \({\mathcal B}(H)\), the \(C^*\)-algebra of all bounded linear operators on a Hilbert space \(H\).
reviewed by: André Noll
Mathematics Subject Classification ID: 47A58
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 1336859
zbMATH Keywords: \(C^*\)-algebra with unit
zbMATH Keywords: Korovkin-type theorems
zbMATH Keywords: operator monotone functions
zbMATH Keywords: Korovkin closure
zbMATH Keywords: Robertson's theorem on \(C^*\)-subalgebras generated by a sequence of Schwarz maps
zbMATH Keywords: compact Hausdorff space
zbMATH Keywords: bounded linear operators
MaRDI profile type: Publication
full work available at URL: pl00004717
OpenAlex ID: W2086705791
description: scientific article; zbMATH DE number 1336859