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:

\[\|x\|_{p}^{2}\ge\|\mathcal{E}(x)\|_{p}^{2}+(p-1)\|x-\mathcal{E}(x)\|_{p}^{2},\qquad x\in L_{p}(\mathcal{M}),1<p\le2,\] 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 $\mathbb{F}_{n}$.