---
_id: '13319'
abstract:
- lang: eng
  text: We prove that the generator of the L2 implementation of a KMS-symmetric quantum
    Markov semigroup can be expressed as the square of a derivation with values in
    a Hilbert bimodule, extending earlier results by Cipriani and Sauvageot for tracially
    symmetric semigroups and the second-named author for GNS-symmetric semigroups.
    This result hinges on the introduction of a new completely positive map on the
    algebra of bounded operators on the GNS Hilbert space. This transformation maps
    symmetric Markov operators to symmetric Markov operators and is essential to obtain
    the required inner product on the Hilbert bimodule.
acknowledgement: The authors are grateful to Martijn Caspers for helpful comments
  on a preliminary version of this manuscript. M. V. was supported by the NWO Vidi
  grant VI.Vidi.192.018 ‘Non-commutative harmonic analysis and rigidity of operator
  algebras’. M. W. was funded by the Austrian Science Fund (FWF) under the Esprit
  Programme [ESP 156]. For the purpose of Open Access, the authors have applied a
  CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising
  from this submission. Open access funding provided by Austrian Science Fund (FWF).
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Matthijs
  full_name: Vernooij, Matthijs
  last_name: Vernooij
- first_name: Melchior
  full_name: Wirth, Melchior
  id: 88644358-0A0E-11EA-8FA5-49A33DDC885E
  last_name: Wirth
  orcid: 0000-0002-0519-4241
citation:
  ama: Vernooij M, Wirth M. Derivations and KMS-symmetric quantum Markov semigroups.
    <i>Communications in Mathematical Physics</i>. 2023;403:381-416. doi:<a href="https://doi.org/10.1007/s00220-023-04795-6">10.1007/s00220-023-04795-6</a>
  apa: Vernooij, M., &#38; Wirth, M. (2023). Derivations and KMS-symmetric quantum
    Markov semigroups. <i>Communications in Mathematical Physics</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s00220-023-04795-6">https://doi.org/10.1007/s00220-023-04795-6</a>
  chicago: Vernooij, Matthijs, and Melchior Wirth. “Derivations and KMS-Symmetric
    Quantum Markov Semigroups.” <i>Communications in Mathematical Physics</i>. Springer
    Nature, 2023. <a href="https://doi.org/10.1007/s00220-023-04795-6">https://doi.org/10.1007/s00220-023-04795-6</a>.
  ieee: M. Vernooij and M. Wirth, “Derivations and KMS-symmetric quantum Markov semigroups,”
    <i>Communications in Mathematical Physics</i>, vol. 403. Springer Nature, pp.
    381–416, 2023.
  ista: Vernooij M, Wirth M. 2023. Derivations and KMS-symmetric quantum Markov semigroups.
    Communications in Mathematical Physics. 403, 381–416.
  mla: Vernooij, Matthijs, and Melchior Wirth. “Derivations and KMS-Symmetric Quantum
    Markov Semigroups.” <i>Communications in Mathematical Physics</i>, vol. 403, Springer
    Nature, 2023, pp. 381–416, doi:<a href="https://doi.org/10.1007/s00220-023-04795-6">10.1007/s00220-023-04795-6</a>.
  short: M. Vernooij, M. Wirth, Communications in Mathematical Physics 403 (2023)
    381–416.
date_created: 2023-07-30T22:01:03Z
date_published: 2023-10-01T00:00:00Z
date_updated: 2024-01-30T12:16:32Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00220-023-04795-6
external_id:
  arxiv:
  - '2303.15949'
  isi:
  - '001033655400002'
file:
- access_level: open_access
  checksum: cca204e81891270216a0c84eb8bcd398
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  creator: dernst
  date_created: 2024-01-30T12:15:11Z
  date_updated: 2024-01-30T12:15:11Z
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language:
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month: '10'
oa: 1
oa_version: Published Version
page: 381-416
project:
- _id: 34c6ea2d-11ca-11ed-8bc3-c04f3c502833
  grant_number: ESP156_N
  name: Gradient flow techniques for quantum Markov semigroups
publication: Communications in Mathematical Physics
publication_identifier:
  eissn:
  - 1432-0916
  issn:
  - 0010-3616
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Derivations and KMS-symmetric quantum Markov semigroups
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type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 403
year: '2023'
...
---
_id: '12104'
abstract:
- lang: eng
  text: We study ergodic decompositions of Dirichlet spaces under intertwining via
    unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular
    Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore,
    every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces
    is decomposable over their ergodic decompositions up to conjugation via an isomorphism
    of the corresponding indexing spaces.
acknowledgement: Research supported by the Austrian Science Fund (FWF) grant F65 at
  the Institute of Science and Technology Austria and by the European Research Council
  (ERC) (Grant agreement No. 716117 awarded to Prof. Dr. Jan Maas). L.D.S. gratefully
  acknowledges funding of his current position by the Austrian Science Fund (FWF)
  through the ESPRIT Programme (Grant No. 208). M.W. gratefully acknowledges funding
  of his current position by the Austrian Science Fund (FWF) through the ESPRIT Programme
  (Grant No. 156).
article_number: '9'
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Lorenzo
  full_name: Dello Schiavo, Lorenzo
  id: ECEBF480-9E4F-11EA-B557-B0823DDC885E
  last_name: Dello Schiavo
  orcid: 0000-0002-9881-6870
- first_name: Melchior
  full_name: Wirth, Melchior
  id: 88644358-0A0E-11EA-8FA5-49A33DDC885E
  last_name: Wirth
  orcid: 0000-0002-0519-4241
citation:
  ama: Dello Schiavo L, Wirth M. Ergodic decompositions of Dirichlet forms under order
    isomorphisms. <i>Journal of Evolution Equations</i>. 2023;23(1). doi:<a href="https://doi.org/10.1007/s00028-022-00859-7">10.1007/s00028-022-00859-7</a>
  apa: Dello Schiavo, L., &#38; Wirth, M. (2023). Ergodic decompositions of Dirichlet
    forms under order isomorphisms. <i>Journal of Evolution Equations</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s00028-022-00859-7">https://doi.org/10.1007/s00028-022-00859-7</a>
  chicago: Dello Schiavo, Lorenzo, and Melchior Wirth. “Ergodic Decompositions of
    Dirichlet Forms under Order Isomorphisms.” <i>Journal of Evolution Equations</i>.
    Springer Nature, 2023. <a href="https://doi.org/10.1007/s00028-022-00859-7">https://doi.org/10.1007/s00028-022-00859-7</a>.
  ieee: L. Dello Schiavo and M. Wirth, “Ergodic decompositions of Dirichlet forms
    under order isomorphisms,” <i>Journal of Evolution Equations</i>, vol. 23, no.
    1. Springer Nature, 2023.
  ista: Dello Schiavo L, Wirth M. 2023. Ergodic decompositions of Dirichlet forms
    under order isomorphisms. Journal of Evolution Equations. 23(1), 9.
  mla: Dello Schiavo, Lorenzo, and Melchior Wirth. “Ergodic Decompositions of Dirichlet
    Forms under Order Isomorphisms.” <i>Journal of Evolution Equations</i>, vol. 23,
    no. 1, 9, Springer Nature, 2023, doi:<a href="https://doi.org/10.1007/s00028-022-00859-7">10.1007/s00028-022-00859-7</a>.
  short: L. Dello Schiavo, M. Wirth, Journal of Evolution Equations 23 (2023).
date_created: 2023-01-08T23:00:53Z
date_published: 2023-01-01T00:00:00Z
date_updated: 2023-06-28T11:54:35Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00028-022-00859-7
ec_funded: 1
external_id:
  isi:
  - '000906214600004'
file:
- access_level: open_access
  checksum: 1f34f3e2cb521033de6154f274ea3a4e
  content_type: application/pdf
  creator: dernst
  date_created: 2023-01-20T10:45:06Z
  date_updated: 2023-01-20T10:45:06Z
  file_id: '12325'
  file_name: 2023_JourEvolutionEquations_DelloSchiavo.pdf
  file_size: 422612
  relation: main_file
  success: 1
file_date_updated: 2023-01-20T10:45:06Z
has_accepted_license: '1'
intvolume: '        23'
isi: 1
issue: '1'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
project:
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '716117'
  name: Optimal Transport and Stochastic Dynamics
- _id: 34dbf174-11ca-11ed-8bc3-afe9d43d4b9c
  grant_number: E208
  name: Configuration Spaces over Non-Smooth Spaces
- _id: 34c6ea2d-11ca-11ed-8bc3-c04f3c502833
  grant_number: ESP156_N
  name: Gradient flow techniques for quantum Markov semigroups
publication: Journal of Evolution Equations
publication_identifier:
  eissn:
  - 1424-3202
  issn:
  - 1424-3199
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Ergodic decompositions of Dirichlet forms under order isomorphisms
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 23
year: '2023'
...