Complete positivity of mapping valued linear maps (Q1089876)

We consider the matrix order structure of ordered Banach space. This notion is an extended version of the order structure of a \(C^ *\)- algebra or a predual of von Neumann algebra induced by the cone of its positive elements. Corresponding to the case that the associated algebra is abelian, we introduce the notion, a matrix ordered Banach space of order 1. For matrix ordered Banach space E, F and G, we can consider a positive element of L(E,F), that is, a positive element of L(E,F) is a completely positive map of E to F. Using the canonical embedding of \(M_ n(L(F,G))\) into \(L(M_ n(F),M_ n(G))\), we can define completely positive map of E to L(F,G). Then we can get the following result. Any positive map of E to L(F,G) is completely positive if and only if two of E, F and G are of order 1.