Weak Korovkin approximation by completely positive linear maps on \(\beta\) (H) (Q801500)

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)].