H functional calculus and square functions on noncommutative Lp-spaces

Marius Junge, Christian Le Merdy, Quanhua Xu

Research output: Book/Report/Conference proceedingBook

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 languageEnglish (US)
Place of PublicationParis
PublisherSociété mathématique de France
Number of pages138
ISBN (Print)9782856291894
DOIs
StatePublished - 2006

Publication series

NameAstérisque
Volume305
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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