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March 2016 A noncommutative martingale convexity inequality
Éric Ricard, Quanhua Xu
Ann. Probab. 44(2): 867-882 (March 2016). DOI: 10.1214/14-AOP990


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


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Éric Ricard. Quanhua Xu. "A noncommutative martingale convexity inequality." Ann. Probab. 44 (2) 867 - 882, March 2016.


Received: 1 May 2014; Revised: 1 November 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1345.46056
MathSciNet: MR3474461
Digital Object Identifier: 10.1214/14-AOP990

Primary: 46L51 , 47A30
Secondary: 60G42 , 81S25

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

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 2 • March 2016
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