**Astérisque**

Volume: 305;
2005;
138 pp;
Softcover

MSC: Primary 47; 46;
**Print ISBN: 978-2-85629-189-4
Product Code: AST/305**

List Price: $38.00

AMS Member Price: $34.20

# \(H^{∞}\) Functional Calculus and Square Functions on Noncommutative \(L^{p}\)-Spaces

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*Marius Junge; Christian Le Merdy; Quanhua Xu*

A publication of the Société Mathématique de France

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.

#### Readership

Graduate students and research mathematicians interested in analysis.