Noncommutative Riesz transforms -- a probabilistic approach

Abstract

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