Smooth point measures and diffeomorphism groups (Q792681)

Let \(X={\mathbb{R}}^ d (d>1)\). The author considers the unitary representations of the non-locally compact topological group Diff(X) given by quasi-invariant measures under the action of Diff(X). Explicitly these representations are defined by \[ U^{\mu}_{\psi}(F)=F(\psi^{- 1}r)(d\psi \mu /d\mu(r))^{1/2},\quad F\in L^ 2_{\mu}(\Delta(X)),\quad \psi \in Diff(X) \] where (\(\Delta\) (X),\({\mathcal F}_ X)\) is the configuration space with Diff(X). The author proposes smooth point measures as generalizations of Poisson point measures and proves that every smooth point measure is quasi-invariant under the action of Diff(X) and if \(\{U_{\psi}^{\mu_ i}\}\), \(i=1,2\) are unitary representations of Diff(X) given by the smooth point measures \(\mu_ i\), \(i=1,2\), then \(\{U_{\psi}^{\mu_ 1}\}\) is unitarily equivalent to \(\{U_{\psi}^{\mu_ 2}\}\) iff \(\mu_ 1\sim \mu_ 2\). In particular if \(P_{m_ i}\) is a Poisson measure of smooth measures \(m_ i\), \(i=1,2\), then \(\{U_{\psi}^{m_ 1}\}\) is unitarily equivalent to \(\{U_{\psi}^{m_ 2}\}\) iff its Kakutani distance \((\int(\sqrt{dm_ i}-\sqrt{dm_ 2})^ 2)^{1/2}\) is finite.