I am looking for an example of a locally compact group $G$ and a continuous $G$ module $M$, which also is locally compact, such that the continuous cochain cohomology differs from group cohomology (ignoring the topologies). In the literature, I find many results, giving criteria when they agree, but I could not find an example, in which they don't.



Example of continuous cohomology vs cohomology
Consider the circle $G=\mathbb{R}/\mathbb{Z}$ and its trivial representation on $M=\mathbb{R}$.
Since $G$ is abelian, $H^1(G,M)\cong \text{Hom}(G,M)$ and $H^1\_c(G,M)\cong \text{Hom}\_c(G,M)$ (these are the continuous versions).
Clearly, $\text{Hom}\_c(G,M)=0$.
However, as abstract groups $\mathbb{R}\cong \mathbb{Q}\oplus\mathbb{R}/\mathbb{Q}$, thus $\mathbb{R}/\mathbb{Z}\cong \mathbb{Q}/\mathbb{Z}\oplus \mathbb{R}/\mathbb{Q}$, thus $\text{Hom}(G,M)\cong \text{Hom}(\mathbb{R}/\mathbb{Q},\mathbb{R})$, which is huge, as both arguments are high dimensional $\mathbb{Q}$-vector spaces.


