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

TL;DR

This paper characterizes the generators of GNS-symmetric quantum Markov semigroups as squares of derivations, introducing Tomita bimodules to handle the non-tracial case and connecting to free Araki-Woods factors.

Contribution

It generalizes the structure of generators from tracial to GNS-symmetric semigroups using Tomita bimodules and operator-valued free Araki-Woods factors.

Findings

Generators can be expressed as squares of derivations.

Introduction of Tomita bimodules to handle modular group twist.

Realization inside $L^2$ space of larger von Neumann algebra.

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 $L^2$ space of a bigger von Neumann algebra, whose construction is an operator-valued version of free Araki-Woods factors.