Proposition 4

Let G be a locally compact group and H a closed normal subgroup of G such that K = G/H is compact. Then every continuous representation u of H in $\mathbf{R}$, such that $u(s\xi s^{-1}) = u(\xi)$ for all $\xi \in H$ and $s \in G$, may be extended to a continuous representation of G in $\mathbf{R}$.

Let $L = G \times \mathbf{R}$ and let M be the set of $(\xi, -u(\xi))$, where $\xi$ runs over H. It is clear that M is a closed normal subgroup of L. Let $L' = L/M$ and let $\pi$ be the canonical mapping of L onto $L'$. The subgroup of L generated by M and R is H × R, hence is closed; therefore π(R) is a closed subgroup N of L'. The restriction ρ of π to R is a bijective continuous representation of R onto N. Lemma 2 of Appendix 1 proves that ρ is bicontinuous. Moreover, L'/N is isomorphic to L/(H × R) = G/H, hence is compact. By Prop. 3, and taking into account the fact that N is in the center of L', L' is the product of N with another subgroup. Therefore there exists a continuous representation of L' onto N that reduces on N to the identity mapping. Therefore there exists a continuous representation v of L onto R that is trivial on M and reduces on R to the identity mapping. For ξ ∈ H, one has v((ξ, 0)) = v((ξ, −u(ξ))(e, u(ξ))) = u(ξ), which completes the proof.