Noncommutative Riesz transforms -- a probabilistic approach

For $2\le p<\mathrm{\infty }$ we show the lower estimates $$\mathrm{\parallel }{A}^{\frac{1}{2}}x{\mathrm{\parallel }}_p\text{kl}c(p)\max \{\text{pl}\mathrm{\parallel }\mathrm{\Gamma }(x,x{)}^{1\mathrm{/}2}{\mathrm{\parallel }}_p,\text{pl}\mathrm{\parallel }\mathrm{\Gamma }(x^{\ast },x^{\ast }{)}^{1\mathrm{/}2}{\mathrm{\parallel }}_p\}$$ 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 $$2\mathrm{\Gamma }(x,y)\text{lel}A{x}^{\ast }y+x^{\ast }Ay-A(x^{\ast }y)\text{pl}\mathrm{.}$$ As additional hypothesis we assume that $\Gamma^2\gl 0$andtheexistenceofaMarkovdilationfor$(T_t)$. We give applications to quantum metric spaces and show the equivalence of semigroup Hardy norms and martingale Hardy norms derived from the Markov dilation. In the limiting case we obtain a viable definition of BMO spaces for general semigroups of completely positive maps which can be used as an endpoint for interpolation. For torsion free ordered groups we construct a connection between Riesz transforms and the Hilbert transform induced by the order.