## The Annals of Probability

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

#### Article information

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

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

https://projecteuclid.org/euclid.aop/1457960385

Digital Object Identifier
doi:10.1214/14-AOP990

Mathematical Reviews number (MathSciNet)
MR3474461

Zentralblatt MATH identifier
1345.46056

#### Citation

Ricard, Éric; Xu, Quanhua. A noncommutative martingale convexity inequality. Ann. Probab. 44 (2016), no. 2, 867--882. doi:10.1214/14-AOP990. https://projecteuclid.org/euclid.aop/1457960385

#### References

• [1] Albeverio, S. and Høegh-Krohn, R. (1977). Dirichlet forms and Markov semigroups on $C^{\ast}$-algebras. Comm. Math. Phys. 56 173–187.
• [2] Ball, K., Carlen, E. A. and Lieb, E. H. (1994). Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115 463–482.
• [3] Biane, P. (1997). Free hypercontractivity. Comm. Math. Phys. 184 457–474.
• [4] Carlen, E. A. and Lieb, E. H. (1993). Optimal hypercontractivity for Fermi fields and related noncommutative integration inequalities. Comm. Math. Phys. 155 27–46.
• [5] Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061–1083.
• [6] Haagerup, U. (1978/79). An example of a nonnuclear $C^{\ast}$-algebra, which has the metric approximation property. Invent. Math. 50 279–293.
• [7] Haagerup, U., Junge, M. and Xu, Q. (2010). A reduction method for noncommutative $L_{p}$-spaces and applications. Trans. Amer. Math. Soc. 362 2125–2165.
• [8] Junge, M., Palazuelos, C., Parcet, J. and Perrin, M. (2013). Hypercontractivity in group von Neumann algebras. Preprint.
• [9] Junge, M., Palazuelos, C., Parcet, J., Perrin, M. and Ricard, É. (2015). Hypercontractivity for free products. Ann. Sci. École Norm. Sup. (4) 48 861–889.
• [10] Olkiewicz, R. and Zegarlinski, B. (1999). Hypercontractivity in noncommutative $L_{p}$ spaces. J. Funct. Anal. 161 246–285.
• [11] Pisier, G. and Xu, Q. (2003). Non-commutative $L^{p}$-spaces. In Handbook of the Geometry of Banach Spaces, Vol. 2 (W. B. Johnson and J. Lindenstrauss, eds.) 1459–1517. North-Holland, Amsterdam.