Noncommutative Riesz transforms — a probabilistic approach

For $2\le p < \infty$ we show the lower estimates $$ \|A^{\frac 12}x\|_p \le c(p)\max\{\|\Gamma(x,x)^{\frac{1}{2}}\|_p, \|\Gamma(x^*,x^*)^{\frac{1}{2}}\|_p\} $$ for the Riesz transform associated to a semigroup $(T_t)$ of completely positive maps on a von Neumann algebra with negative generator $T_t=e^{-tA}$, and gradient form $$ 2\Gamma(x,y) = Ax^*y+x^*Ay-A(x^*y).$$ Among other hypotheses we assume that $\Gamma^2\ge 0$ and the existence of a Markov dilation for $(T_t)$. As an application we provide new examples of quantum metric spaces for discrete groups with rapid decay. In this context a compactness condition follows from Sobolev embedding results based on a notion of dimension due to Varopoulos.