# 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 $\zeta$, $\mathbb{R}^\times$ be the multiplicative group of positive reals, let $t$ be the coordinate on $\mathbb{R}^\times$. A polymorphism of $Z$ is a measure $\pi$ on $Z\times Z\times \mathbb{R}^\times$ such that for any measurable $A$, $B\subset Z$ we have $\pi(A\times Z\times \mathbb{R}^\times)=\zeta(A)$ and the integral $\int t\,d\pi(z,u,t)$ over $Z\times B\times \mathbb{R}^\times$ is $\zeta(B)$. 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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