Non-singular actions of infinite-dimensional groups and polymorphisms

TL;DR

This paper explores the structure of polymorphisms in infinite-dimensional groups, demonstrating how non-singular actions generate representations through polymorphisms and discussing closure properties within this framework.

Contribution

It introduces a new perspective on non-singular actions of infinite-dimensional groups, connecting them with polymorphisms and their semigroup structures.

Findings

Polymorphisms form a natural semigroup with a dense group of nonsingular transformations.

Non-singular actions generate representations of the train (category of double cosets) by polymorphisms.

The paper discusses closure properties of polymorphisms for large infinite-dimensional groups.

Abstract

Let $Z$ be a probabilistic 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.