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$.Comment: Published at http://dx.doi.org/10.1214/14-AOP990 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Hypercontractivity in finite-dimensional matrix algebras

“…More precisely, it is established in [26,Proposition 13] and in [16,Theorem B] that for any Poisson-like length function ψ with order of growth given by N R (ψ) ≤ Cρ R (recall the definition of N R (ψ) in the statement of Theorem A) and any 2 ≤ q < ∞ we have…”

“…Ricard and Xu's result [26] yields a better time t(q, C, ρ, σ) = max 1 σ 1 2 − 1 q log(2Cρ σ ) + 1 2 log(q − 1) , log ρ . Their estimate relies on a deep result, an extension of the Ball-Carlen-Lieb convexity inequality to any semifinite von Neumann algebra.…”

“…The key point to prove Theorem A is that the combinatorics involved in the problem are exactly the same as those appearing when one studies the Poisson semigroup on Z n × Z n associated to a function ψ, for which some hypercontractivity results are obtained in [16,26]. That is, even when no group von Neumann algebra is described by the matrix algebra M n , the results developed in [26] can be transfered to the matrix algebra setting whenever we consider q an even integer.…”

“…The aim of the present work is to transfer some recent results on the cyclic groups [16,26] to obtain hypercontractivity estimates for certain semigroups defined on finite-dimensional matrix algebras M n . More precisely, given n ≥ 1, consider the (n × n)-unitary matrices…”

“…That is, even when no group von Neumann algebra is described by the matrix algebra M n , the results developed in [26] can be transfered to the matrix algebra setting whenever we consider q an even integer.…”