1-cocycles for rotationally invariant measures (Q1265295)

Summary: Let \(H\) be a separable Hilbert space over \(\mathbb{R}\) \((\dim(H)\) is finite or infinite), \(H^a\) be the algebraic dual space of \(H,{\mathfrak B}\) be the cylindrical \(\sigma\)-algebra on \(H^a\) and \(\mu\) be a rotationally invariant probability measure on \((H^a, {\mathfrak B})\). Further let \(\theta= \theta (x,U)\) be a 1-cocycle defined on \((x,U)\in H^a \times O(H)\), where \(O(H)\) is the rotation group on \(H\). That is, (c.1) for any fixed \(U\in O(H)\), \(\theta (x,U)\) is a \({\mathfrak B}\)-measurable function on \(x\), (c.2) \(| \theta (x,U)| \equiv 1\), and (c.3) \(\forall U_1\), \(\forall U_2\in O(H)\), \(\theta (x,U_1) \theta (^tU_1x,U_2) =\theta (x,U_1U_2)\) for \(\mu\)-a.e. \(x\), where \(^tU\) is the algebraic transpose of \(U\). Moreover it is said to be continuous, if the following condition holds for \(\theta\): (c.4) \(\theta (x,U)\to 1\) in \(\mu\), if \(U\to \text{Id}\) in the strong operator topology. Our main result is as follows: Assume that \(\dim(H)\neq 3\). Then for any continuous 1-cocycle \(\theta\), there exists a \({\mathfrak B}\)-measurable function \(\varphi\) with modulus 1 such that for any fixed \(U\in O(H)\), \(\theta (x,U)=\varphi (^tUx)/ \varphi(x)\) for \(\mu\)-a.e. \(x\).