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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 Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign 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 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 URI: http://hdl.handle.net/2142/50527 Rights Information: Copyright 2014 Qiang Zeng Date Available in IDEALS: 2014-09-16 Date Deposited: 2014-08
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