Alicki's theorem (for semigroups on matrix algebras) and the results of Cipriani and Sauvageot (for tracially symmetric semigroups) show that Markov generators give rise to a first-order differential calculus. These results are crucial for the construction of the dynamical noncommutative transport distance by Carlen--Maas and the speaker. In this talk I will discuss an extension of these results to GNS-symmetric quantum Markov semigroups on arbitrary von Neumann algebras. Compared to the tracially symmetric case, the derivations satisfy a chain rule with a twist by the modular group, which is encapsulated in the new notion of Tomita bimodules. This could open up the way to extend the dynamical optimal transport methods to GNS-symmetric Markov semigroups on infinite-dimensional von Neumann algebras.
