Geodesic
Wishart--Pickrell distributions and closures of group actions
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 236-245

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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. 
@article{ZNSL_2016_448_a14,
     author = {Yu. A. Neretin},
     title = {Wishart--Pickrell distributions and closures of group actions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {236--245},
     publisher = {mathdoc},
     volume = {448},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a14/}
}
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Yu. A. Neretin. Wishart--Pickrell distributions and closures of group actions. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 236-245. http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a14/