Astérisque 2005; 138 pp; softcover Number: 305 ISBN10: 2856291899 ISBN13: 9782856291894 List Price: US$38 Individual Members: US$34.20 Order Code: AST/305
 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, \(q\)OrnsteinUhlenbeck semigroups acting on the \(q\)deformed von Neumann algebras of BozejkoSpeicher, 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. Readership 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
