A noncommutative martingale convexity inequality

Abstract

Let M be a von Neumann algebra equipped with a faithful semifinite normal weight ϕ and N be a von Neumann subalgebra of M such that the restriction of ϕ to N is semifinite and such that N is invariant by the modular group of ϕ. Let E be the weight preserving conditional expectation from M onto N. As an application we show that there exists e0>0 such that for any free group Fn and any q≥4−e0, ∥Pt∥2→q≤1⇔t≥logq−1−−−−√, where (Pt) is the Poisson semigroup defined by the natural length function of Fn.