2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101087We study completely contractive representations of product systems of $C^*$-correspondences over semigroups. For a product system of $C^*$-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 JFAOperator Algebras46L08, 46L10, 46L55, 46L57, 47L30text