  For $2\le p<\infty$ we show the lower estimates
  \[ \|A^{\frac 12}x\|_p \kl c(p)\max\{\pl \|\Gamma(x,x)^{{1/2}}\|_p,\pl
\|\Gamma(x^*,x^*)^{{1/2}}\|_p\} \] 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 \[ 2\Gamma(x,y)\lel
Ax^*y+x^*Ay-A(x^*y)\pl .\] As additional hypothesis we assume that $\Gamma^2\gl
0$ and the existence of a Markov dilation for $(T_t)$. 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.
