Is the left-regular representation of a locally compact group a homeomorphism onto its image? 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 $G$.
 A: Yes. It's more generally true for every faithful $C^0$ unitary representation $\pi$ of $G$. (Recall that a unitary representation $\pi$ is $C^0$ if for all $v,w$ in the Hilbert space, one has $\langle \pi(g)v,w\rangle\to 0$ when $g$ leaves compact subsets of $G$.)
Indeed if this is not a homeomorphism onto its image, there exists an ultrafilter $\eta$ on $G$, not converging to $1$, such that $\lim_{g\to\eta}\lambda_g=\mathrm{id}$ (for the strong topology, i.e., $\lim_{g\to\eta}\pi(g)v=v$ for every $v$ in the Hilbert space.
If $\eta$ is unbounded (i.e. no compact subset of $G$ is in $\eta$), then we get a contradiction, since the $C^0$ property implies $\lim_{g\to\eta}\langle\pi(g)v,v\rangle=0$ for every $v$ in the Hilbert space.
If $\eta$ is bounded, then $\eta$ has a limit $g_0$ (which by assumption is not $1_G$), and we deduce $\lim_{g\to\eta}\pi(g)=\pi(g_0)$. So $\pi(g_0)=\mathrm{id}$, contradicting the faithfulness assumption.
A: In an attempt of an alternative answer specifically for the left-regular representation (in addition to the very nice and general solution by @YCor):
Firstly, if $V$ is a neighbourhood of $1$ let $W$ be a symmetric neighbourhood of $1$ such that $W\cdot W\subset V$. If $g\not\in V$ then $gW\cap W=\varnothing$. Because if not, then there is $h\in W$ such that $gh\in W$, but since $W$ is symmetric we then have $g = (gh)h^{-1} \in W\cdot W\subset V$.
Now, we prove the contraposition of the question: Let $g_\alpha$ be a net in $G$ that does not converge to $1$. Hence, there exists a neighbourhood $V$ of $1$ and a subnet of $g_\alpha$ that never enters $V$, so we assume, without loss of generality, that $g_\alpha\not\in V$ for all $\alpha$. Choosing $W$ as before and setting $\psi = \chi_W$ to be the characteristic function gives $\|\lambda_{g_\alpha}\psi - \psi\|_2^2 = 2\mu(W)>0$.
