H[∞]functional calculus and square functions on noncommutative L[p]-spaces

— We investigate sectorial operators and semigroups acting on noncommutative L-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 L-spaces. This requires natural variants of Rademacher sectoriality and the use of the matricial structure of noncommutative L-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. c © Astérisque 305, SMF 2006