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July 2007 Rosenthal type inequalities for free chaos
Marius Junge, Javier Parcet, Quanhua Xu
Ann. Probab. 35(4): 1374-1437 (July 2007). DOI: 10.1214/009117906000000962

Abstract

Let $\mathscr{A}$ 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 $\mathscr {B}$ with respect to normal faithful conditional expectations $\mathsf {E}_{k}:\mathsf{A}_{k}\to \mathscr {B}$. We investigate the norm in $L_{p}(\mathscr {A})$ 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 $L_{\infty}(\mathscr {A})$. At the end of the paper we also consider Khintchine type inequalities for Shlyakhtenko’s generalized circular systems.

Citation

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Marius Junge. Javier Parcet. Quanhua Xu. "Rosenthal type inequalities for free chaos." Ann. Probab. 35 (4) 1374 - 1437, July 2007. https://doi.org/10.1214/009117906000000962

Information

Published: July 2007
First available in Project Euclid: 8 June 2007

zbMATH: 1125.46054
MathSciNet: MR2330976
Digital Object Identifier: 10.1214/009117906000000962

Subjects:
Primary: 42A61 , 46L07 , 46L52 , 46L54

Keywords: free random variables , homogeneous polynomial , Khintchine inequality , reduced amalgamated free product , Rosenthal inequality

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 4 • July 2007
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