The Annals of Probability

Rosenthal type inequalities for free chaos

Marius Junge, Javier Parcet, and Quanhua Xu

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Let $\mathcal{A}$ denote the reduced amalgamated free product of a family $\mathsf{A}_{1},\mathsf{A}_{2},\ldots,\mathsf{A}_{n}$ of von Neumann algebras over a von Neumann subalgebra ℬ with respect to normal faithful conditional expectations $\mathsf {E}_{k}\dvtx\mathsf{A}_{k}\to \mathcal {B}$. We investigate the norm in $L_{p}(\mathcal {A})$ 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 $L_{\infty}(\mathcal {A})$. At the end of the paper we also consider Khintchine type inequalities for Shlyakhtenko’s generalized circular systems.

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Ann. Probab. Volume 35, Number 4 (2007), 1374-1437.

First available: 8 June 2007

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Primary: 46L54: Free probability and free operator algebras 42A61: Probabilistic methods 46L07: Operator spaces and completely bounded maps [See also 47L25] 46L52: Noncommutative function spaces

Khintchine inequality Rosenthal inequality reduced amalgamated free product free random variables homogeneous polynomial


Junge, Marius; Parcet, Javier; Xu, Quanhua. Rosenthal type inequalities for free chaos. The Annals of Probability 35 (2007), no. 4, 1374--1437. doi:10.1214/009117906000000962.

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