Moscow Mathematical Journal
Volume 24, Issue 3, July–September 2024 pp. 407–425.
Non-Singular Actions of Infinite-Dimensional Groups and Polymorphisms
Authors:
Yury A. Neretin (1)
Author institution:(1) High School of Modern Mathematics MIPT;
Math. Dept., University of Vienna until 14.01.2024;
MechMath Dept., Moscow State University
Summary:
Let Z be a probability measure space with a measure ζ, R× be the multiplicative group of positive reals, let t be the coordinate on R×. A polymorphism of Z is a measure π on Z×Z×R× such that for any measurable A, B⊂Z we have π(A×Z×R×)=ζ(A) and the integral ∫tdπ(z,u,t) over Z×B×R× is ζ(B). The set of all polymorphisms has a natural semigroup structure, the group of all nonsingular transformations is dense in this semigroup. We discuss a problem of closure in polymorphisms for certain types of infinite dimensional (‘large’) groups and show that a non-singular action of an infinite-dimensional group generates a representation of its train (category of double cosets) by polymorphisms.
2020 Math. Subj. Class. 37A40, 37A15, 22F10.
Keywords: Measure preserving actions, nonsingular actions, polymorphisms, unitary representations, double cosets.
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