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Classification of $E_0$–semigroups by product systems

About this Title

Michael Skeide

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 240, Number 1137
ISBNs: 978-1-4704-1738-3 (print); 978-1-4704-2826-6 (online)
Published electronically: October 9, 2015
Keywords:Quantum dynamics, Hilbert modules, product systems, endomorphism semigroups, Markov semigroups, dilations, representations noises

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Morita equivalence and representations
  • Chapter 3. Stable Morita equivalence for Hilbert modules
  • Chapter 4. Ternary isomorphisms
  • Chapter 5. Cocycle conjugacy of $E_0$–semigroups
  • Chapter 6. $E_0$–Semigroups, product systems, and unitary cocycles
  • Chapter 7. Conjugate $E_0$–Semigroups and Morita equivalent product systems
  • Chapter 8. Stable unitary cocycle (inner) conjugacy of $E_0$–semigroups
  • Chapter 9. About continuity
  • Chapter 10. Hudson-Parthasarathy dilations of spatial Markov semigroups
  • Chapter 11. Von Neumann case: Algebraic classification
  • Chapter 12. Von Neumann case: Topological classification
  • Chapter 13. Von Neumann case: Spatial Markov semigroups
  • Appendix A: Strong type I product systems
  • Appendix B: $E_0$–Semigroups and representations for strongly continuous product systems


In his Memoir from 1989, Arveson started the modern theory of product systems. More precisely, with each $E_{0}$-semigroup (that is, a unital endomorphism semigroup) on $B(H)$ he associated a product system of Hilbert spaces (Arveson system, henceforth). He also showed that the Arveson system determines the $E_{0}$-semigroup up to cocycle conjugacy. In three successor papers, Arveson showed that every Arveson system comes from an $E_{0}$-semigroup. There is, therefore, a one-to-one correspondence between $E_{0}$-semigroups on $B(H)$ (up to cocycle conjugacy) and Arveson systems (up to isomorphism). In the meantime, product systems of correspondences (or Hilbert bimodules) have been constructed from Markov semigroups on general unital $C^{*}$-algebras or on von Neumann algebras. These product systems showed to be an efficient tool in the construction of dilations of Markov semigroups to $E_{0}$-semigroups and to automorphism groups. In particular, product systems over correspondences over commutative algebras (as they arise from classical Markov processes) or other algebras with nontrivial center, show surprising features that can never happen with Arveson systems. A dilation of a Markov semigroup constructed with the help of a product system always acts on $B^{a}(E)$, the algebra of adjointable operators on a Hilbert module $E$. (If the Markov semigroup is on $B(H)$ then $E$ is a Hilbert space.) Only very recently, we showed that every product system can occur as the product system of a dilation of a nontrivial Markov semigroup. This makes it necessary to extend the theory to the relation between $E_{0}$-semigroups on $B^{a}(E)$ and product systems of correspondences. In these notes we present a complete theory of classification of $E_{0}$-semigroups by product systems of correspondences. As an application of our theory, we answer the fundamental question if a Markov semigroup admits a dilation by a cocycle perturbations of noise: It does if and only if it is spatial.

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