Matthijs Vernooij: Derivations and KMS-symmetric quantum Markov semigroups

Date: 2023-06-16

Time: 09:45 - 10:10

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Matthijs Vernooij, University of Delft


Given a von Neumann algebra with a fixed state, there exist multiple notions of symmetry for operators on this von Neumann algebra; the most notable ones are GNS-symmetry and KMS-symmetry. If the state is tracial, both symmetries coincide, and in this setting Cipriani and Sauvageot showed that the L^2 implementation of a generator of a symmetric quantum Markov semigroup can be written as the square of a derivation with values in a Hilbert bimodule. Wirth extended this result to GNS-symmetric quantum Markov semigroups, and in this talk I will present the extension to KMS-symmetric quantum Markov semigroups. The essential ingredient is a new completely positive map on the bounded operators on the GNS Hilbert space. It maps symmetric Markov operators to symmetric Markov operators and is essential to obtain the required inner product on the Hilbert bimodule. This is joint work with Wirth.