Poincaré’s classical recurrence theorem is generalised to the noncommutative setup where a measure space with a measure-preserving transformation is replaced by a von Neumann algebra with a weight and a Jordan morphism leaving the weight invariant. This is done by a suitable reformulation of the theorem in the language of -space rather than the original measure space, thus allowing the replacement of the commutative von Neumann algebra by a noncommutative one.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-1,
author = {Andrzej \L uczak},
title = {Noncommutative Poincar\'e recurrence theorem},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {1-6},
zbl = {0979.46042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-1}
}
Andrzej Łuczak. Noncommutative Poincaré recurrence theorem. Colloquium Mathematicae, Tome 89 (2001) pp. 1-6. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-1/