## Abstract

We investigate sectorial operators and semigroups acting on noncommutative L^{p}-spaces. We introduce new square functions in this context and study their connection with H^{∞} functional calculus, extending some famous work by Cowling, Doust, Mclntoch and Yagi concerning commutative L^{p}-spaces. This requires natural variants of Rademacher sectoriality and the use of the matriciel structure of noncommutative L^{p}-spaces. We mainly focus on noncommutative diffusion semigroups, that is, semigroups (T_{t})_{t≥o} of normal selfadjoint operators on a semifinite von Neumann algebra (M,τ) such that T_{t}: L^{p}(M) → L^{p}(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 g-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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Title of host publication | H Functional Calculus and Square Functions on Noncommutative L- Spaces |

Editors | Marius Junge, Christian Le Merdy, Quanhua Xu |

Pages | 1-138 |

Number of pages | 138 |

Edition | 305 |

State | Published - 2006 |

### Publication series

Name | Asterisque |
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Number | 305 |

ISSN (Print) | 0303-1179 |

## Keywords

- Completely bounded maps
- Diffusion semigroups
- H functional calculus
- Multipliers
- Noncommutative L -spaces
- Sectorial operators
- Square functions

## ASJC Scopus subject areas

- General Mathematics

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