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\(H^\infty\) Functional Calculus and Square Functions on Noncommutative \(L^p\)-Spaces
Marius Junge, University of Illinois, Urbana, IL, and Christian Le Merdy and Quanhua Xu, Université de Franche-Comté, Besançon, France
A publication of the Société Mathématique de France.
2005; 138 pp; softcover
Number: 305
ISBN-10: 2-85629-189-9
ISBN-13: 978-2-85629-189-4
List Price: US$38
Individual Members: US$34.20
Order Code: AST/305
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The authors investigate sectorial operators and semigroups acting on noncommutative \(L^p\)-spaces. They introduce new square functions in this context and study their connection with \(H^\infty\) functional calculus, extending some famous work by Cowling, Doust, McIntoch and Yagi concerning commutative \(L^p\)-spaces. This requires natural variants of Rademacher sectoriality and the use of the matricial structure of noncommutative \(L^p\)-spaces. They mainly focus on noncommutative diffusion semigroups, that is, semigroups \((T_t)_{t\geq 0}\) of normal selfadjoint operators on a semifinite von Neumann algebra \((\mathcal M,\tau )\) such that \(T_t\colon L^p(\mathcal M )\to L^p(\mathcal M )\) is a contraction for any \(p\geq 1\) and any \(t\geq 0\). They discuss several examples of such semigroups for which they establish bounded \(H^\infty\) 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.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicians interested in analysis.

Table of Contents

  • Introduction
  • Noncommutative Hilbert space valued \(L^p\)-spaces
  • Bounded and completely bounded \(H^\infty\) functional calculus
  • Rademacher boundedness and related notions
  • Noncommutative diffusion semigroups
  • Square functions on noncommutative \(L^p\)-spaces
  • \(H^\infty\) functional calculus and square function estimates
  • Various examples of multipliers
  • Semigroups on \(q\)-deformed von Neumann algebras
  • A noncommutative Poisson semigroup
  • The non tracial case
  • Comparing row and column square functions
  • Measurable functions in \(L^p(L^2)\)
  • Bibliography
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