Korovkin-type theorems for Schwarz maps and operator monotone functions in \(C^*\)-algebras
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 \(H\).
Robertson's theorem on \(C^*\)-subalgebras generated by a sequence of Schwarz maps
scientific article; zbMATH DE number 1336859