  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$.
