Exercise 3, Appendix 0

Let G be a topological group and $\pi$ a continuous unitary representation of G on a complex hilbertian space E. A continuous linear mapping $c : G \to E$ which satisfies the relation

for every s, t in G, is called a continuous 1-cocycle. Let $Z^1(G; E)$ denote the complex vector space of continuous 1-cocycles. For every $a \in E$, the mapping $\delta(a) : s \mapsto \pi(s).a - a$ is a continuous 1-cocycle, called the cobord of a. Let $B^1(G; E)$ denote the image of the linear mapping $\delta : E \to Z^1(G; E)$; put $H^1(G; E) = Z^1(G; E)/B^1(G; E)$ ("first continuous cohomology group of G with values in E"). a) Show that $B^1(G; E)$ is composed of continuous and bounded 1-cocycles. (For every continuous 1-cocycle c and every $s \in G$, we define an affine transformation $\lambda_s$ on E by $\lambda_s.x = \pi(s).x + c(s)$; then $\lambda_{st} = \lambda_s.\lambda_t$ for all s, t in G. Let K be the closed convex envelope of $c(G)$; then $\lambda_s(K) = K$ for all $s \in G$, and $\lambda_s$ induces an isometry of K onto itself. If c is bounded, show that the Ryll-Nardzewski theorem applies to $\lambda_s$, and if $\lambda_s.a = a$ for all $s \in G$, then $c = -\delta(a)$. b) If G is compact, show that $H^1(G; E) = \{0\}$.