**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

Individual 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.

#### Table of Contents

# Table of Contents

## $H^{}$ Functional Calculus and Square Functions on Noncommutative $L^{p}$-Spaces

#### Readership

Graduate students and research mathematicians interested in analysis.