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.

To appear in Infinite Dimensional Analysis and Quantum Probability 2005. This version replaces the first one submitted under the titel "Normal CP-Maps admit Weak Tensor Dilations". The old titel was missleading, as it puts accent onto the wrong things. We rewrote the abstract and largely the second half of the introduction. A couple of typos has been eliminated, but for the rest the main body of the paper remained unchanged