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

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.