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

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

2005-02-20

arxiv.org/abs/math/0502423v2

Mathematics

Operator Algebras

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

Scientific paper

We 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).

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