Files in this item
Files  Description  Format 

application/pdf Qiang_Zeng.pdf (1MB)  (no description provided) 
Description
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, ZhongJin 
Doctoral Committee Member(s):  Song, Renming; Junge, Marius 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  noncommutative Lp spaces
Poincaré inequalities $\Gamma_2$criterion martingale inequalities Burkholder inequality spectral gap diffusion semigroups transportation cost inequalities law of the iterated logarithm subgaussian concentration 1cocycle 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 \[ \fE_{\mathrm{Fix}} f\_p \le C\sqrt{p} \max\{\\Gamma(f,f)^{1/2}\_p, \\Gamma(f^*,f^*)^{1/2}\_p\}, \] or a weaker version \[ \fE_{\mathrm{Fix}} f\_p \le C\sqrt{p} \max\{\\Gamma(f,f)^{1/2}\_\infty, \\Gamma(f^*,f^*)^{1/2}\_{\infty}\} \] 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:  20140916 
URI:  http://hdl.handle.net/2142/50527 
Rights Information:  Copyright 2014 Qiang Zeng 
Date Available in IDEALS:  20140916 
Date Deposited:  201408 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois