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

Abstract

We study completely contractive representations of product systems of $C^{\ast }$-correspondences over semigroups. For a product system of $C^{\ast }$-correspondences over the semigroup ${\mathbb{N}}^2$, we prove that every such representation can be dilated to an isometric (or Toeplitz) representation. We use it to 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).

Revised version. References added. Changes in exposition. To appear in JFA

References in corpus (3)

• Commutants of von Neumann Modules, Representations of B^a(E) and Other Topics Related to Product Systems of Hilbert Modules
• Crossed products by abelian semigroups via transfer operators
• The $H^{\infty }$ algebras of higher rank graphs

Cited by in corpus (1)

• Regular dilations of representations of product systems