Factorization and boundedness for representations of locally compact groups on topological vector spaces (Q6654890)

The paper is devoted to some representation theory of locally compact groups in (in)finite dimensional locally compact topological vector spaces through the exponential Lie group quotients.\N\NThe main results of the paper are as follows:\N\begin{itemize}\N\item[(1)] Continuous morphisms from locally compact groups to locally exponential (possibly infinite-dimensional) Lie groups factor through Lie quotients (Theorems 1.1, 2.9); \N\item[(2)] The maximal almost-periodicity of the identity component \(G_0 \leq G\) of a locally compact group are characterized in terms of sufficiently discriminating families of continuous functions on \(G\) with values in Hausdorff spaces (Theorem 1.2); \N\item[(3)] The von Neumann kernel of \(G_0\) as the joint kernel of all appropriately bounded and continuous \(G\)-representations on topological vector spaces is described (Theorems 1.3, 2.9, Corollary 2.13); \N\item[(4)] A locally \(G\)-finite representation of a compact group \(G\) on a product of Fréchet spaces is a finite direct sum of isotypic components (Theorems 1.4, 3.15).\N\end{itemize}