A noncommutative martingale convexity inequality

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful semifinite normal weight $\phi$ and $\mathcal{N}$ be a von Neumann subalgebra of $\mathcal{M}$ such that the restriction of $\phi$ to $\mathcal{N}$ is semifinite and such that $\mathcal{N}$ is invariant by the modular group of $\phi$. Let $\mathcal{E}$ be the weight preserving conditional expectation from $\mathcal{M}$ onto $\mathcal{N}$. We prove the following inequality: $$\|x\|_p^2\ge\bigl \|\mathcal{E}(x)\bigr\|_p^2+(p-1)\bigl\|x-\mathcal{E}(x)\bigr\|_p^2, \qquad x\in L_p(\mathcal{M}),1<p\le2,$$ which extends the celebrated Ball-Carlen-Lieb convexity inequality. As an application we show that there exists $\varepsilon_0>0$ such that for any free group $\mathbb{F}_n$ and any $q\ge4-\varepsilon_0$, $$\|P_t\|_{2\to q}\le1\quad\Leftrightarrow\quad t\ge\log{\sqrt{q-1}},$$ where $(P_t)$ is the Poisson semigroup defined by the natural length function of $\mathbb{F}_n$.