Noncommutative Poincaré recurrence theorem
Volume 89 / 2001
Colloquium Mathematicum 89 (2001), 1-6 MSC: Primary 46L51; Secondary 28D05. DOI: 10.4064/cm89-1-1
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.