Wishart–Pickrell distributions and closures of group actions

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