Representations of product systems over semigroups and dilations of commuting CP maps

We prove that every pair of commuting CP maps on a von Neumann algebra M can be dilated to a commuting pair of endomorphisms (on a larger von Neumann algebra). To achieve this, we first prove that every completely contractive representation of a product system of C∗-correspondences over the semigroup N2 can be dilated to an isometric (or Toeplitz) representation. © 2005 Elsevier Inc. All rights reserved.