Extension of positive maps into B(\({\mathcal H})\)
\({\mathcal H},{\mathcal K}\) denote Hilbert spaces, A a norm closed self-adjoint linear subspace of B(\({\mathcal K})\) containing the identity and B(A,\({\mathcal H})_+\) the positive maps on the set B(A,\({\mathcal K})\) of bounded linear maps of A into B(\({\mathcal H})\). If \({\mathcal I}\) denotes the trace class operators on \({\mathcal H}\), then B(A,\({\mathcal H})\) is isometric to (A\({\hat \otimes}{\mathcal I})^*\) and B(A,\({\mathcal H})_+\) corresponds to the functionals which are positive on \(A+\otimes {\mathcal I}_+\). For a cone K in B(B(\({\mathcal H}),{\mathcal H})_+\) satisfying some topological and invariance conditions is associated a cone P(A,K) in A\({\hat \otimes}{\mathcal I}\) and \(\phi\in B(A,{\mathcal H})\) is called K-positive if its image \({\tilde \phi}\in (A{\hat \otimes}{\mathcal I})^*\) is positive on P(A,K). The author proves that if \(\phi\) is K-positive than it has a K- positive extension in B(B(\({\mathcal K}),{\mathcal H})_+\) and shows that \(\phi\) is completely positive if and only if \({\tilde \phi}\) is positive on the cone (A\({\hat \otimes}{\mathcal I})_+\) generated by positive operators in the algebraic tensor product. Arveson's extension theorem for completely positive maps and the result of Neumark and Stinespring on positive maps from an abelian \(C^*\)-algebra into B(\({\mathcal H})\) and Choi's characterization of completely positive maps from the complex matrices \(M_ m\) into \(M_ n\) are obtained as easy consequences of this general study. Also is given a concrete description of K-positive maps and is deduced that for a nuclear \(C^*\)-algebra A every map in B(A,\({\mathcal H})_+\) has an extension in B(B(\({\mathcal K})),{\mathcal H})_+\).
Arveson's extension theorem for completely positive maps
scientific article; zbMATH DE number 4038537
scientific article