
eBook ISBN: | 978-1-4704-0567-0 |
Product Code: | MEMO/203/953.E |
List Price: | $78.00 |
MAA Member Price: | $70.20 |
AMS Member Price: | $46.80 |

eBook ISBN: | 978-1-4704-0567-0 |
Product Code: | MEMO/203/953.E |
List Price: | $78.00 |
MAA Member Price: | $70.20 |
AMS Member Price: | $46.80 |
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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.
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Table of Contents
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Chapters
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Introduction
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1. Noncommutative integration
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2. Amalgamated $L_p$ spaces
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3. An interpolation theorem
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4. Conditional $L_p$ spaces
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5. Intersections of $L_p$ spaces
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6. Factorization of $\mathcal {J}_{p,q}^n(\mathcal {M}, \mathsf {E})$
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7. Mixed-norm inequalities
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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. Mixed-norm inequalities
-
8. Operator space $L_p$ embeddings