## Abstract

Let A denote the reduced amalgamated free product of a family A _{1}, A_{2},...., A_{n} of von Neumann algebras over a von Neumann subalgebra ℬ with respect to normal faithful conditional expectations E_{k} : A_{k} → ℬ. We investigate the norm in L_{p}(script A sign) of homogeneous polynomials of a given degree d. We first generalize Voiculescu's inequality to arbitrary degree d ≥ 1 and indices 1 ≤ p ≤ ∞. This can be regarded as a free analogue of the classical Rosenthal inequality. Our second result is a length-reduction formula from, which we generalize recent results of Pisier, Ricard and the authors. All constants in our estimates are independent of n so that we may consider infinitely many free factors. As applications, we study square functions of free martingales. More precisely, we show that, in contrast with the Khintchine and Rosenthal inequalities, the free analogue of the Burkholder-Gundy inequalities does not hold in L_{∞}(script A sign). At the end of the paper we also consider Khintchine type inequalities for Shlyakhtenko's generalized circular systems.

Original language | English (US) |
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Pages (from-to) | 1374-1437 |

Number of pages | 64 |

Journal | Annals of Probability |

Volume | 35 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2007 |

## Keywords

- Free random variables
- Homogeneous polynomial
- Khintchine inequality
- Reduced amalgamated free product
- Rosenthal inequality

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty