Von Neumann kernels of connected lie groups,revisited and refined

In the note, a 30-year old result concerning the von Neumann kernel of a connected Lie group (the intersection of kernels of all irreducible finite-dimensional continuous complex unitary representations of the group) is corrected and the smallest von Neumann kernel of a connected Lie group (the intersection of kernels of all irreducible finite-dimensional (not necessarily continuous) complex unitary representations of the group) is described.