Ultraproducts of von Neumann algebras and ergodicity

S. G. Haliullin

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 2, pp. 287-292

Résumé

An ultraproduct of any linear spaces with respect of a non-trivial ultrafilter in an index set is generalization of the non-standard expansion $^{*}\mathbb {R}$ of the set of real numbers $\mathbb{R}$. The non-standard mathematical analysis has the objects and methods of a research, which only to some extent depend on laws of the standard mathematical analysis. In this work, non-standard objects — ultraproducts of von Neumann algebras — have been studied from the point of view of the standard analysis. This approach allows to receive, in particular, a criterion of contiguity of sequences of normal faithful states in terms of the equivalence of states on the corresponding ultraproducts. We note that the classical ultraproduct of von Neumann algebras, generally speaking, is not a von Neumann algebra. Therefore, in accordance with A. Ocneanu's work, we have considered the changed construction of the ultraproduct of von Neumann algebras. We have introduced the concept of ergodic action with respect to the normal state of group on an abelian von Neumann algebra. Its properties have been studied. In particular, we have provided the example showing that the ultraproduct of ergodic states is not, generally speaking, ergodic.

Détail

Keywords: ultraproducts, ergodicity, states on von Neumann algebra.
Mots-clés : actions of group