The Annals of Probability

A noncommutative martingale convexity inequality

Éric Ricard and Quanhua Xu

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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 $\mathbb{F}_{n}$.

Article information

Ann. Probab., Volume 44, Number 2 (2016), 867-882.

Received: May 2014
Revised: November 2014
First available in Project Euclid: 14 March 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L51: Noncommutative measure and integration 47A30: Norms (inequalities, more than one norm, etc.)
Secondary: 60G42: Martingales with discrete parameter 81S25: Quantum stochastic calculus

Noncommutative $L_{p}$-spaces martingale convexity inequality hypercontractivity free groups


Ricard, Éric; Xu, Quanhua. A noncommutative martingale convexity inequality. Ann. Probab. 44 (2016), no. 2, 867--882. doi:10.1214/14-AOP990.

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