Wishart--Pickrell distributions and closures of group actions

TL;DR

This paper characterizes the closure of the unitary group acting on infinite Hermitian matrices within the space of spreading maps, revealing it as a semigroup of all contractive operators, thus advancing understanding of group actions in infinite-dimensional probability spaces.

Contribution

It provides a detailed description of the closure of the infinite unitary group in the space of polymorphisms, linking it to the semigroup of contractive operators, a novel structural insight.

Findings

Closure of $U( ext{infty})$ is a semigroup of all contractive operators.

Describes the structure of invariant distributions on $Herm( ext{infty})$.

Connects group actions with operator semigroups in infinite dimensions.

Abstract

Consider probabilistic distributions on the space of infinite Hermitian matrices $Herm(\infty)$ invariant with respect to the unitary group $U(\infty)$. We describe the closure of $U(\infty)$ in the space of spreading maps (polymorphisms) of $Herm(\infty)$, this closure is a semigroup isomorphic to the semigroup of all contractive operators.