## Abstract

Let H be a separable Hilbert space, and let φ and θ be two strongly commuting CP_{0}-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 φ_{s} ̂ θ_{t} (P_{H} AP_{H}) =P_{H} α_{s}̂ β_{t}(A)P_{H} 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 E_{0}-semigroup.

Original language | English |
---|---|

Pages (from-to) | 393-403 |

Number of pages | 11 |

Journal | Infinite Dimensional Analysis, Quantum Probability and Related Topics |

Volume | 11 |

Issue number | 3 |

DOIs | |

State | Published - 1 Sep 2008 |

Externally published | Yes |

## Keywords

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