{1}-cocycles for rotationally invariant measures

Let H be a separable Hilbert space over R (dim (H) is finite or infinite) , H be the algebraic dual space of H, %$ be the cylindrical d-algebra on H and n be a rotationally invariant probability measure on (H, SB). Further let 6=6(x, U) be a 1-cocycle defined on Or, [/) e/fxo(#), where 0 (//) is the rotation group on H. That is, (c.l) for any fixed U^O(H), 6(x, U) is a ^-measurable function of x, (c.2) |0 Or, C/) | = 1. and (c.3) for [/i, [/2eO(tf), 0(r. UjOCUix, U2}=6(x, UiU2) for 0-a.e. *. where '£/ is the algebraic transpose of [/. Moreover it is said to be continuous, if the following condition holds for 6. (c.4) 6(x, U)— »1 in fi, if U —* Id in the strong operator topology. Our main result is as follows. Assume that dim(//) =£3. Then for any continuous 1-cocycle 0, there exists a SS-measurable function 0 with modulus 1 such that for any fixed U^O(H), 6(x, U) =(/>(Ux)/ (x) for fjt-a.e.x.