The structure of homomorphisms of connected locally compact groups into compact groups

We obtain consequences of the theorem concerning the automatic continuity of locally bounded finite-dimensional representations of connected Lie groups on the commutator subgroup of the group and also of an analogue of Lie's theorem for (not necessarily continuous) finite-dimensional representations of soluble Lie groups. In particular, we prove that an almost connected locally compact group admitting a (not necessarily continuous) injective homomorphism into a compact group also admits a continuous injective homomorphism into a compact group, and thus the given group is a finite extension of the direct product of a compact group and a vector group. We solve the related problem of describing the images of (not necessarily continuous) homomorphisms of connected locally compact groups into compact groups. Moreover, we refine the description of the von Neumann kernel of a connected locally compact group and describe the intersection of the kernels of all (not necessarily continuous) finite-dimensional unitary representations of a given connected locally compact group. Some applications are mentioned. We also show that every almost connected locally compact group admitting a (not necessarily continuous) locally bounded injective homomorphism into an amenable almost connected locally compact group is amenable.