Let $\pi$ be an orthogonal representation of locally compact group $G$ on a real Hilbert space $H$. Recall that a continuous mapping $b \colon G \to H$ such that $b(gh) = b(g) + π(g)b(h)$, for all $g, h \in G$ is called a 1-cocycle with respect to $\pi$.
1) Suppose that $G$ is second countable. Is it possible to reduce the study of 1-cocycles to the study of 1-cocycles with a separable Hilbert space ?
2) Why the use of real Hilbert spaces is important (essential ?) in the theory ?