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$.
Categories: math.OA math.FA