Noncommutative Riesz transforms -- a probabilistic approach

TL;DR

This paper develops lower estimates for noncommutative Riesz transforms associated with semigroups on von Neumann algebras, introducing new tools for quantum metric spaces and BMO spaces, with applications to harmonic analysis.

Contribution

It provides the first lower bounds for noncommutative Riesz transforms under certain conditions, linking them to quantum metric spaces and establishing a new framework for BMO spaces in this setting.

Findings

Established lower estimates for noncommutative Riesz transforms.

Connected Riesz transforms to quantum metric space structures.

Proposed a new definition of BMO spaces for semigroups of completely positive maps.

Abstract

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.