Moscow Mathematical Journal
Volume 24, Issue 3, July–September 2024 pp. 407–425.
Non-Singular Actions of Infinite-Dimensional Groups and Polymorphisms
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.
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:
Keywords: Measure preserving actions, nonsingular actions, polymorphisms, unitary representations, double cosets.
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