Consider an irreducible unitary representation $\rho: SE_3 \to U$ where $SE_3 = \mathbb{R}^3 \rtimes SO(3)$ is the (special) Euclidean group and $U$ is the (infinite dimensional) Lie group of unitary operators on the Hilbert space $L^2(S^2)$ where $S^2$ is the 2-sphere.
I am interested in the generalization of Burnside's theorem (If $(\pi,V)$ is irreducible, then $\pi(G)$ spans $\operatorname{End}(V)$) from finite dim irreps to infinite dim irreps . In other words, is it true that the image $\rho(SE_3)$ "spans" $U$ in the sense that we can write any element in $U$ as a linear combination of elements from $\rho(SE_3)$?
Related: Is this a basis for the bounded operators on $ L^2(\mathbb{R}) $?
