Rosenthal type inequalities for free chaos

• 成果类型: Article
• 署名作者: Junge, Marius; Parcet, Javier; Xu, Quanhua
• 署名单位: University of Illinois System; University of Illinois Urbana-Champaign; Autonomous University of Madrid; Consejo Superior de Investigaciones Cientificas (CSIC); CSIC - Instituto de Ciencias Matematicas (ICMAT); Universite Marie et Louis Pasteur
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000962
• 发表日期: 2007
• 页码: 1374-1437
• 关键词: khintchine type inequalities; noncommutative martingales; vonneumann algebra; linear-operators; random-variables; star-algebras; SPACES; interpolation; convolution; MODULES

摘要

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 (A) of homogeneous polynomials of a given degree d. We first generalize Voiculescu's inequality to arbitrary degree d >= 1 and indices 1 <= p <= infinity. 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 infinity (A). At the end of the paper we also consider Khintchine type inequalities for Shlyakhtenko's generalized circular systems.