The Differential Structure of Generators of GNS-symmetric Quantum Markov Semigroups
Operator Algebras
2022-07-20 v1 Mathematical Physics
Functional Analysis
math.MP
Quantum Physics
Abstract
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. If the quantum Markov semigroup satisfies a certain additional regularity condition, the associated Tomita bimodule can be realized inside the space of a bigger von Neumann algebra, whose construction is an operator-valued version of free Araki-Woods factors.
Cite
@article{arxiv.2207.09247,
title = {The Differential Structure of Generators of GNS-symmetric Quantum Markov Semigroups},
author = {Melchior Wirth},
journal= {arXiv preprint arXiv:2207.09247},
year = {2022}
}