Rosenthal type inequalities for free chaos

Marius Junge, Javier Parcet, Quanhua Xu

Research output: Contribution to journalArticlepeer-review


Let A denote the reduced amalgamated free product of a family A 1, A2,...., An of von Neumann algebras over a von Neumann subalgebra ℬ with respect to normal faithful conditional expectations Ek : Ak → ℬ. We investigate the norm in Lp(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 languageEnglish (US)
Pages (from-to)1374-1437
Number of pages64
JournalAnnals of Probability
Issue number4
StatePublished - Jul 2007


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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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