A noncommutative martingale convexity inequality

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. We prove the following inequality: ||x||²_p ≥||ε(x)||²_p +(p - 1)||x - ε(x)||² _p, xε L_p(M), 1<p ≤ 2, which extends the celebrated Ball-Carlen-Lieb convexity inequality. As an application we show that there exists ε 0 > 0 such that for any free group Fn and any q ≥ 4 -ε₀, where (P_t) is the Poisson semigroup defined by the natural length function of F_n.