Constructing Extensions of CP-Maps via Tensor Dilations with the Help of Von Neumann Modules

Abstract

We apply Hilbert module methods to show that normal completely positive maps admit weak tensor dilations. Appealing to a duality between weak tensor dilations and extensions of CP-maps, we get an existence proof for certain extensions. We point out that this duality is part of a far reaching duality between a von Neumann bimodule and its commutant in which also other dualities, known and new, find their natural common place.