Noncommutative Poincaré recurrence theorem

Andrzej Łuczak¹

¹ Faculty of Mathematics Łódź University Stefana Banacha 22 90-238 Łódź Poland

Colloquium Mathematicum, Tome 89 (2001) no. 1, pp. 1-6

Résumé

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 $L^\infty $-space rather than the original measure space, thus allowing the replacement of the commutative von Neumann algebra $L^\infty $ by a noncommutative one.

Détail

DOI : 10.4064/cm89-1-1

Keywords: poincar classical recurrence theorem generalised noncommutative setup where measure space measure preserving transformation replaced von neumann algebra weight jordan morphism leaving weight invariant done suitable reformulation theorem language infty space rather original measure space allowing replacement commutative von neumann algebra infty noncommutative