NONCOMMUTATIVE RIESZ TRANSFORMS — A PROBABILISTIC APPROACH

For 2 ≤ p < ∞ we show the lower estimates $\left\| {A\frac{1} {2}x\left\| {p \leqslant c(p)\max \{ \left\| {\Gamma (x,x)\frac{1} {2}\left\| {p,\left\| {\Gamma (x^* ,x^* )\frac{1} {2}\left\| {p\} } \right.} \right.} \right.} \right.} \right.} \right.$ 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 2Γ(x,y) = Ax*y + x*Ay - (x*y). Among other hypotheses we assume that Γ² ≥ 0 and the existence of a Markov dilation for (T₁). 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.