## Abstract

We investigate sectorial operators and semigroups acting on noncommutative Lp-spaces. We introduce new square functions in this context and study their connection with H∞ functional calculus, extending some famous work by Cowling, Doust, McIntoch and Yagi concerning commutative Lp-spaces. This requires natural variants of Rademacher sectoriality and the use of the matricial structure of noncommutative Lp-spaces. We mainly focus on noncommutative diﬀusion semigroups, that is, semigroups (Tt)t≥0 of normal selfadjoint operators on a semiﬁnite von Neumann algebra (M,τ) such that Tt:Lp(M)→Lp(M) is a contraction for any p≥1 and any t≥0. We discuss several examples of such semigroups for which we establish bounded H∞ functional calculus and square function estimates. This includes semigroups generated by certain Hamiltonians or Schur multipliers, q-Ornstein-Uhlenbeck semigroups acting on the q-deformed von Neumann algebras of Bozejko-Speicher, and the noncommutative Poisson semigroup acting on the group von Neumann algebra of a free group.

Original language | English (US) |
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Place of Publication | Paris |

Publisher | Société mathématique de France |

Number of pages | 138 |

ISBN (Print) | 9782856291894 |

DOIs | |

State | Published - 2006 |

### Publication series

Name | Astérisque |
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Volume | 305 |

ISSN (Print) | 0303-1179 |

## Keywords

- H∞ functional calculus
- noncommutative Lp-spaces
- square functions
- sectorial operators
- diﬀusion semigroups
- completely bounded maps
- multipliers

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