The Differential Structure of Generators of GNS-symmetric Quantum Markov Semigroups

Understand

We show that the generator of a GNS-symmetric quantum Markov semigroup can be written as the square of a derivation.

• This generalizes a result of Cipriani and Sauvageot for tracially symmetric semigroups.
• Compared to the tracially symmetric case, the derivations in the general case satisfy a twisted product rule, reflecting the non-triviality of their modular group.
• This twist is captured by the new concept of Tomita bimodules we introduce.