Cones of completely bounded maps

Abstract

By analogy with the Choi matrix we associate an operator $$C_{\varphi }\in \mathrm{B}({\mathscr {H}})$$ to each weak* continuous $${\mathscr {A}}$$ -bimodule map $$\varphi :\mathrm{B}({\mathscr {K}})\rightarrow \mathrm{B}({\mathscr {H}})$$ , where $${\mathscr {K}}$$ and $${\mathscr {H}}$$ are normal Hilbert modules over a von Neumann algebra $${\mathscr {A}}$$ and $${\mathscr {K}}$$ contains a cyclic vector for $${\mathscr {A}}$$ . If $${\mathscr {A}}\subseteq \mathrm{B}({\mathscr {K}})$$ has no central summands of type I ( $${\mathscr {K}}$$ cyclic), every normal $${\mathscr {A}}$$ -bimodule map on $$\mathrm{B}({\mathscr {K}})$$ , which is positive on $${\mathscr {A}}^{\prime }$$ , is shown to be completely positive on $${\mathscr {Z}}^{\prime }$$ , where $${\mathscr {A}}^{\prime }$$ and $${\mathscr {Z}}^{\prime }$$ are the commutant of $${\mathscr {A}}$$ and the center $${\mathscr {Z}}$$ of $${\mathscr {A}}$$ . We investigate cones of bimodule maps, introduce the corresponding dual cones of operators and show that in an appropriate context these notions reduce to those studied earlier by Stormer. We also consider positive maps relative to a mapping cone and positivity in operator projective tensor product of suitable operator bimodules.