# CONFLUENTES MATHEMATICI

Reversible part of quantum dynamical systems: A review
Confluentes Mathematici, Tome 10 (2018) no. 2, pp. 51-74.

In this work a quantum dynamical system $\left(𝔐,\Phi ,\varphi \right)$ is constituted by a von Neumann algebra $𝔐$, a unital Schwartz map $\Phi :𝔐\to 𝔐$ and a $\Phi$-invariant normal faithful state $\varphi$ on $𝔐$. We will prove that the ergodic properties of a quantum dynamical system are determined by its reversible part $\left({𝔇}_{\infty },{\Phi }_{\infty },{\varphi }_{\infty }\right)$; i.e. by a von Neumann sub-algebra ${𝔇}_{\infty }$ of $𝔐$, with an automorphism ${\Phi }_{\infty }$ and a normal state ${\varphi }_{\infty }$, as the restrictions on ${𝔇}_{\infty }$. Moreover, if ${𝔇}_{\infty }$ is a trivial algebra, then the quantum dynamical system is ergodic. Furthermore, we will show some properties of reversible part of the quantum dynamical system, finally we will study its relations with the canonical decomposition of Nagy-Fojas of linear contraction related to a quantum dynamical system.

Reçu le : 2017-09-25
Accepté le : 2018-07-26
Publié le : 2019-03-04
DOI : https://doi.org/10.5802/cml.50
Classification : 46L07,  46L55,  81S22
Mots clés: Quantum dynamical system, Multiplicative core, Algebra of effective observables.
@article{CML_2018__10_2_51_0,
author = {Carlo Pandiscia},
title = {Reversible part of quantum dynamical systems: A review},
journal = {Confluentes Mathematici},
publisher = {Institut Camille Jordan},
volume = {10},
number = {2},
year = {2018},
pages = {51-74},
doi = {10.5802/cml.50},
language = {en},
url = {cml.centre-mersenne.org/item/CML_2018__10_2_51_0/}
}
Pandiscia, Carlo. Reversible part of quantum dynamical systems: A review. Confluentes Mathematici, Tome 10 (2018) no. 2, pp. 51-74. doi : 10.5802/cml.50. https://cml.centre-mersenne.org/item/CML_2018__10_2_51_0/

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