Non-singular actions of infinite-dimensional groups and polymorphisms

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 $\int td\pi (z,u,t)$ 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.

arXiv comment17p, corrected and extended version