What type of dynamics arise in E0-dilations of commuting quantum Markov semigroups?

Orr Moshe Shalit

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

Let H be a separable Hilbert space, and let φ and θ be two strongly commuting CP0-semigroups on B(H). In a previous paper we constructed a Hilbert space K ⊇ H and two (strongly) commuting E 0-semigroups α and β such that φs ̂ θt (PH APH) =PH αŝ βt(A)PH for all s, t < 0 and all A ∈ B(K). In this note we prove that if φ is not an automorphism semigroup, then the semigroup α (given by the above mentioned construction) is cocycle conjugate to the minimal *-endomorphic dilation of φ, and that if φ is an automorphism semigroup, then α is also an automorphism semigroup. In particular, we conclude that if φ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional), then α is a type I E0-semigroup.

Original languageEnglish
Pages (from-to)393-403
Number of pages11
JournalInfinite Dimensional Analysis, Quantum Probability and Related Topics
Volume11
Issue number3
DOIs
StatePublished - 1 Sep 2008
Externally publishedYes

Keywords

  • CP-semigroup
  • Cocycle conjugacy
  • E-semigroup
  • Minimal dilation
  • Two-parameter semigroup

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