Consider the left-regular representation $\lambda : G \to B(L^2(G))$, $\lambda_g f(h) = f(g^{-1}h)$, for a locally compact group.

It is well-known that this is a unitary faithful and strongly-continuous representation, but is it also a homeomorphism onto its image $\lambda(G)$ (equipped with the strong-operator topology?

It would suffice to show that for any net $g_\alpha$, convergence in the strong operator topology of $\lambda_{g_\alpha}$ to the identity $I_{L^2(G)}$ implies convergence of $g_\alpha$ to the neutral element $e$ in $G$.