S.G. Haliullin
Kazan Federal University, Kazan, 420008 Russia
For citation: Haliullin S.G. Ultraproducts of von Neumann algebras and ergodicity. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 287–292.
Для цитирования: Haliullin S.G. Ultraproducts of von Neumann algebras and ergodicity // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. – 2018. – Т. 160, кн. 2. – С. 287–292.
Abstract
An ultraproduct of any linear spaces with respect of a non-trivial ultrafilter in an index set is generalization of the non-standard expansion *R of the set of real numbers 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.
Keywords: ultraproducts, actions of group, ergodicity, states on von Neumann algebra
References
1. Mushtari D.H., Haliullin S.G. Linear spaces with a probability measure, ultraproducts and contiguity. Lobachevskii J. Math., 2014, vol. 35, no. 2, pp. 138–146. doi: 10.1134/S1995080214020097.
2. Haliullin S. Orthogonal decomposition of the Gaussian measure. Lobachevskii J. Math., 2016, vol. 37, no. 4, pp. 436–438. doi: 10.1134/S1995080216040090.
3. Heinrich S. Ultraproducts in Banach space theory. J. Reine Angew. Math., 1980, vol. 313, pp. 72–104.
4. Ocneanu A. Actions of Discrete Amenable Groups on von Neumann Algebras. Berlin, Heidelberg, Springer, 1985. vii, 114 p. doi: 10.1007/BFb0098579.
5. Ando H., Haagerup U. Ultraproducts of von Neumann algebras. J. Funct. Anal., 2014, vol. 266, no. 12, pp. 6842–6913. doi: 10.1016/j.jfa.2014.03.013.
6. Gudder S.P. A Radon-Nikodym theorem for 
-algebras. Pac. J. Math., 1979, vol. 80, no. 1, pp. 141–149.
7. Chetcutti E., Hamhalter J. Vitali-Hahn-Saks theorem for vector measures on operator algebras. Q. J. Math., 2006, vol. 57, pp. 479-493. doi: 10.1093/qmath/hal006.
8. Haliullin S. Contiguity and entire separability of states on von Neumann Algebras. Int. J. Theor. Phys., 2017, vol. 56, no. 12, pp. 3889–3894. doi: 10.1007/s10773-017-3373-z.
9. Takesaki M. Theory of Operator Algebras III. Berlin, Heidelberg, Springer, 2003. xxii, 548 p. doi: 10.1007/978-3-662-10453-8.
Received
November 13, 2017
Haliullin Samigulla Garifullovich, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematical Statistics
Kazan Federal University
ul. Kremlevskaya, 18, Kazan, 420008 Russia
E-mail: Samig.Haliullin@kpfu.ru
Контент доступен под лицензией Creative Commons Attribution 4.0 License.