A noncommutative martingale convexity inequality

Abstract

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 \( \mathbb{F}_n\).