Noncommutative Riesz transforms - A probabilistic approach

For 2 ≤ p ≤ 8 we show the lower estimates ||A^1/2x||p ≤ c(p) max{||F(x,x)^1/2||p, ||F(x*, x*)^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 2F(x, y) = Ax*y + x*Ay - A(x*y). Among other hypotheses we assume that F² ≥ 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.