On unitary representations of the group of diffeomorphisms of the space $R^n$, $n\geqslant2$

In this paper, one considers representations of the group $D^0(R^n)$, defined as the connected component of the identity in the group of all finitary diffeomorphisms of $R^n$, $n\geqslant2$, continuous with respect to natural convergence and containing a trivial subrepresentation of the subgroup $D_0^*(R^n)$, preserving volume in $R^n$. Under some additional assumptions there is a description of the irreducible unitary representations in Hilbert spaces. It is shown that each such representation is connected with some dynamical system by means of the standard construction of induced representations.
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