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

• URL: http://arxiv.org/abs/2207.09247v1
• Date: Tue, 19 Jul 2022 12:59:40 GMT
• Title: The Differential Structure of Generators of GNS-symmetric Quantum Markov Semigroups
• Authors: Melchior Wirth
• Abstract summary: 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.
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• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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