Unitary representations of finite abelian groups realizable by an action (Q2435285)

With \(H\) a given infinite dimensional separable complex Hilbert space, \(\mathcal U(H)\) the unitary group of \(H\), consisting of all unitary operators on \(H\) endowed with the strong topology (a Polish group), Rep\((\Gamma,H)\) the set of all group homomorphisms from a countable group \(\Gamma\) to the unitary group \(\mathcal U(H)\), every element of Rep\((\Gamma,H)\) is called a unitary representation of \(\Gamma\) on \(H\). A measurable space \((X,s)\) is called a standard Borel space if there is a Polish space \(Y\) such that \((X,s)\) is isomorphic to \((Y,\mathcal B(Y))\), where \(Y\) is endowed with the \(\sigma\)-algebra \(\mathcal{B}(Y)\) of its Borel subsets. A standard probability space is a standard Borel space \((X,s)\) together with a non-atomic probability measure \(\mu\) defined on the \(\sigma\)-algebra \(s\). Two unitary representations \(\Pi,\rho\) of a countable group \(\Gamma\) on the Hilbert spaces \(H,K\), respectively, are called unitary equivalent if there exists a unitary operator \(\mathcal U\) from \(H\) onto \(K\) such that we have \(\mathcal U\Pi(\gamma)=\rho(\gamma)\mathcal U\), \(\gamma\in\Gamma\). A unitary representation \(\Pi\) of a countable group \(\Gamma\) on a Hilbert space \(H\) is realizable by an action if there is a standard probability space \((X,\mu)\) and \(a\in A(\Gamma,X,\mu)\) such that \(\Pi\) is unitarily equivalent to the Koopman representation \(x^a_0\) of \(\Gamma\) on \(L^2_0(X,\mu)\) associated with \(a\) \(\left(L^2_0(X,\mu)=\displaystyle\left\{f\in L^2(X,\mu):\int_Xfd\mu=0\right\}\right)\); here \(A(\Gamma,X,\mu)\) denotes the space of all measure preserving actions of \(\Gamma\) on \((X,\mu)\). The main result proved here is as follows: Let \(\Gamma\) be a finite abelian group and let \(H\) be an infinite dimensional separable complex Hilbert space. Then the set \(\big\{\Pi\in\text{Rep}(\Gamma,H): \Pi\text{ is realizable by an action}\big\}\) is comeagre in Rep\((\Gamma,H)\).