Poincar�� inequalities in noncommutative Lp spaces

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