eBook ISBN:  9781470405670 
Product Code:  MEMO/203/953.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 
eBook ISBN:  9781470405670 
Product Code:  MEMO/203/953.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 203; 2009; 155 ppMSC: Primary 46
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

Chapters

Introduction

1. Noncommutative integration

2. Amalgamated $L_p$ spaces

3. An interpolation theorem

4. Conditional $L_p$ spaces

5. Intersections of $L_p$ spaces

6. Factorization of $\mathcal {J}_{p,q}^n(\mathcal {M}, \mathsf {E})$

7. Mixednorm inequalities

8. Operator space $L_p$ embeddings


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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.

Chapters

Introduction

1. Noncommutative integration

2. Amalgamated $L_p$ spaces

3. An interpolation theorem

4. Conditional $L_p$ spaces

5. Intersections of $L_p$ spaces

6. Factorization of $\mathcal {J}_{p,q}^n(\mathcal {M}, \mathsf {E})$

7. Mixednorm inequalities

8. Operator space $L_p$ embeddings