## Abstract

We construct Markov semi-groups T and associated BMO-spaces on a finite von Neumann algebra (M,τ) and obtain results for perturbations of commutators and non-commutative Lipschitz estimates. In particular, we prove that for any A∈M self-adjoint and f:R→R Lipschitz there is a Markov semi-group T such that for x∈M, ‖[f(A),x]‖_{bmo(M,T)}≤c_{abs}‖f^{′}‖_{∞}‖[A,x]‖_{∞}. We obtain an analogue of this result for more general von Neumann valued-functions f:R^{n}→N by imposing Hörmander-Mikhlin type assumptions on f. In establishing these result we show that Markov dilations of Markov semi-groups have certain automatic continuity properties. We also show that Markov semi-groups of double operator integrals admit (standard and reversed) Markov dilations.

Original language | English |
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Article number | 108317 |

Pages (from-to) | 1-39 |

Number of pages | 39 |

Journal | Journal of Functional Analysis |

Volume | 278 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

- Commutator estimates
- Non-commutative BMO-spaces
- Non-commutative Lp-spaces
- Quantum Markov semi-groups