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Mixed-Norm Inequalities and Operator Space \(L_p\) Embedding Theory
Marius Junge, University of Illinois at Urbana-Champaign, IL, and Javier Parcet, Instituto de Ciencias Mathemáticas CSIC-UAM-UC3M-UCM, Madrid, Spain
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Memoirs of the American Mathematical Society
2009; 155 pp; softcover
Volume: 203
ISBN-10: 0-8218-4655-8
ISBN-13: 978-0-8218-4655-1
List Price: US$74
Individual Members: US$44.40
Institutional Members: US$59.20
Order Code: MEMO/203/953
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Let \(f_1, f_2, \ldots, f_n\) be a family of independent copies of a given random variable \(f\) in a probability space \((\Omega, \mathcal{F}, \mu)\). Then, the following equivalence of norms holds whenever \(1 \le q \le p < \infty\), \[\left( \int_{\Omega} \left[ \sum_{k=1}^n |f_k|^q \right]^{p/q} d \mu \right)^{1/p} \sim \max_{r \in \{p,q\}} \left\{ n^{1/r} \left( \int_\Omega |f|^r d\mu \right)^{1/r} \right\}.\] The authors prove a noncommutative analogue of this inequality for sums of free random variables over a given von Neumann subalgebra. This formulation leads to new classes of noncommutative function spaces which appear in quantum probability as square functions, conditioned square functions and maximal functions.

Table of Contents

  • Introduction
  • Noncommutative integration
  • Amalgamated \(L_p\) spaces
  • An interpolation theorem
  • Conditional \(L_p\) spaces
  • Intersections of \(L_p\) spaces
  • Factorization of \(\mathcal {J}_{p,q}^n(\mathcal {M}, \mathsf {E})\)
  • Mixed-norm inequalities
  • Operator space \(L_p\) embeddings
  • Bibliography
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