Wishart--Pickrell distributions and closures of group actions

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.

8pp, typos were corrected, minor additions