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