Rosenthal type inequalities for free chaos

Let A denote the reduced amalgamated free product of a family A ₁, A₂,...., 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.