A NONCOMMUTATIVE MARTINGALE CONVEXITY INEQUALITY

Let M be a von Neumann algebra equipped with a faithful semifinite normal weight phi and N be a von Neumann subalgebra of M such that the restriction of phi to N is semifinite and such that N is invariant by the modular group of phi. Let epsilon be the weight preserving conditional expectation from M onto N. We prove the following inequality: parallel to x parallel to(2)(p) >= parallel to epsilon(x)parallel to(2)(p) + (p - 1)parallel to x - epsilon(x)parallel to(2)(p), x is an element of L-p(M), 1 < p <= 2, which extends the celebrated Ball-Carlen-Lieb convexity inequality. As an application we show that there exists epsilon(0) > 0 such that for any free group F-n and any q >= 4 - epsilon(0), parallel to Pt parallel to(2 -> q) <= 1 double left right arrow t >= log root q - 1, where (Pt) is the Poisson semigroup defined by the natural length function of F-n.