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Title:Poincar�� inequalities in noncommutative Lp spaces
Author(s):Zeng, Qiang
Director of Research:Song, Renming; Junge, Marius
Doctoral Committee Chair(s):Kirkpatrick, Kay; Ruan, Zhong-Jin
Doctoral Committee Member(s):Song, Renming; Junge, Marius
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):noncommutative Lp spaces
Poincaré inequalities
martingale inequalities
Burkholder inequality
spectral gap
diffusion semigroups
transportation cost inequalities
law of the iterated logarithm
subgaussian concentration
1-cocycle on groups
finite von Neumann algebras
Abstract:Let $(\mathcal{N},\tau)$ be a noncommutative $W^*$ probability space, where $\mathcal{N}$ is a finite von Neumann algebra and $\tau$ is a normal faithful tracial state. Let $(T_t)_{\ge 0}$ be a normal, unital, completely positive, and symmetric semigroup acting on $(\mathcal{N},\tau)$, which is also pointwise weak* continuous. Denote by $\Gamma$ the ``carr\'e du champ'' associated to $T_t$. Let $\mathrm{Fix}$ be the fixed point algebra of $T_t$ and $E_{\mathrm{Fix}}: \mathcal{N}\to \mathrm{Fix}$ the corresponding conditional expectation. We are interested in the following $L_p$ Poincar\'e inequalities \[ \|f-E_{\mathrm{Fix}} f\|_p \le C\sqrt{p} \max\{\|\Gamma(f,f)^{1/2}\|_p, \|\Gamma(f^*,f^*)^{1/2}\|_p\}, \] or a weaker version \[ \|f-E_{\mathrm{Fix}} f\|_p \le C\sqrt{p} \max\{\|\Gamma(f,f)^{1/2}\|_\infty, \|\Gamma(f^*,f^*)^{1/2}\|_{\infty}\} \] for $p\ge 2$ and $f\in \mathcal{N}$. We study when such inequalities hold as well as their consequences. A crucial condition is the $\Gamma_2$-criterion of Bakry and Emery. These inequalities lead to (noncommutative) transportation cost inequalities and concentration inequalities. Our approaches to prove such Poincar\'e inequalities are based on martingale inequalities and Pisier's method on the boundedness of Riesz transforms.
Issue Date:2014-09-16
Rights Information:Copyright 2014 Qiang Zeng
Date Available in IDEALS:2014-09-16
Date Deposited:2014-08

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