  We (a) prove that continuous morphisms from locally compact groups to locally
exponential (possibly infinite-dimensional) Lie groups factor through Lie
quotients, recovering a result of Shtern's on factoring norm-continuous
representations on Banach spaces; (b) characterize the maximal
almost-periodicity of the identity component $\mathbb{G}_0\le \mathbb{G}$ of a
locally compact group in terms of sufficiently discriminating families of
continuous functions on $\mathbb{G}$ valued in Hausdorff spaces generalizing an
analogous result by Kadison-Singer; (c) apply that characterization to recover
the von Neumann kernel of $\mathbb{G}_0$ as the joint kernel of all
appropriately bounded and continuous $\mathbb{G}$-representations on
topological vector spaces extending Kallman's parallel statement for unitary
representations, and (d) provide large classes of complete locally convex
topological vector spaces (e.g. arbitrary products of Fr\'echet spaces) with
the property that compact-group representations thereon whose vectors all have
finite-dimensional orbits decompose as finite sums of isotypic components. This
last result specializes to one of Hofmann-Morris on representations on products
of real lines.
