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arXiv · 2207.09247

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

Melchior Wirth
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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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Melchior Wirth. 2022-07-19. The Differential Structure of Generators of GNS-symmetric Quantum Markov Semigroups. https://arxiv.org/abs/2207.09247

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