The e-Property for Quantum Markov Semigroups
Abstract
Let (𝒯t)tgeqslant0 be a quantum Markov semigroup on a von Neumann algebra and let 𝒜 be an invariant Feller C*-algebra. We formulate the e-property on the full state space and show that its quantitative defect is equivalent, within an optimal factor two, to the Hausdorff measure of non-compactness of observable orbits. We develop Cesàro and Lip-norm criteria and a generalised Doeblin minorisation by a state-space contracting positive map, including finite-dimensional and measure-prepare factorisations. When every observable has the e-property, the Jacobs--de Leeuw--Glicksberg decomposition identifies the reversible Choi--Effros algebra and the norm-stable sector. Expected and decoherence-free subalgebras transfer the property under norm decay of the complementary dynamics, including infinite-dimensional ultracontractive and finite-index Pimsner--Popa regimes. Finally, we establish the full comparison hierarchy among ordinary, eventual, Cesàro and pointwise eventual equicontinuity notions, reinforced by matrix Jamison and homoclinic flow counterexamples. Projection-face criteria are treated in a companion paper.
Related
- Quantum Cheeger Inequalities for KMS-Symmetric Quantum Markov Semigroups
- Second quantization and compactness of induced Gaussian quantum Markov semigroups
- Some entropic inequalities for primitive KMS symmetric quantum Markov semigroups
- Quantum Reverse Hypercontractivity for KMS-Symmetric Quantum Markov Semigroups
- The Differential Structure of Generators of KMS-Symmetric Quantum Markov Semigroups
BibTeX
@misc{kania2026eproperty,
title = {{The e-Property for Quantum Markov Semigroups}},
author = {Tomasz Kania and Mariusz Niwiński},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00703},
eprint = {2611.00703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00703},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The e-Property for Quantum Markov Semigroups
AU - Tomasz Kania
AU - Mariusz Niwiński
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00703
UR - https://arxiv.org/abs/2611.00703
AB - Let (𝒯_t)_tgeqslant0 be a quantum Markov semigroup on a von Neumann algebra and let 𝒜 be an invariant Feller C^*-algebra. We formulate the e-property on the full state space and show that its quantitative defect is equivalent, within an optimal factor two, to the Hausdorff measure of non-compactness of observable orbits. We develop Cesàro and Lip-norm criteria and a generalised Doeblin minorisation by a state-space contracting positive map, including finite-dimensional and measure-prepare factorisations. When every observable has the e-property, the Jacobs--de Leeuw--Glicksberg decomposition identifies the reversible Choi--Effros algebra and the norm-stable sector. Expected and decoherence-free subalgebras transfer the property under norm decay of the complementary dynamics, including infinite-dimensional ultracontractive and finite-index Pimsner--Popa regimes. Finally, we establish the full comparison hierarchy among ordinary, eventual, Cesàro and pointwise eventual equicontinuity notions, reinforced by matrix Jamison and homoclinic flow counterexamples. Projection-face criteria are treated in a companion paper.
ER -