Factorization and Boundedness for Representations of Locally Compact Groups on Topological Vector Spaces

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 G 0 < G of a locally compact group in terms of sufficiently discriminating families of continuous functions on G valued in Hausdorff spaces generalizing an analogous result by Kadison-Singer; (c) apply that characterization to recover the von Neumann kernel of G 0 as the joint kernel of all appropriately bounded and continuous 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 & eacute;chet 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.