Continuity of LF-algebra representations associated to representations of Lie groups

Abstract

Let $G$ be a finite-dimensional Lie group, and let $E$ be a locally convex topological $G$-module. If $E$ is sequentially complete, then $E$ and the space $E^{\infty }$ of smooth vectors are $C_c^{\infty }\left(G\right)$-modules, but the module multiplication need not be continuous. The pathology can be ruled out if $E$ is (or embeds into) a projective limit of Banach $G$-modules. Moreover, in this case $E^{\omega }$ (the space of analytic vectors) is a module for the algebra $\mathcal{A}\left(G\right)$ of superdecaying analytic functions introduced by Gimperlein, Krötz, and Schlichtkrull. We prove that $E^{\omega }$ is a topological $\mathcal{A}\left(G\right)$-module if $E$ is a Banach space or, more generally, if every countable set of continuous seminorms on $E$ has an upper bound. The same conclusion is obtained if $G$ has a compact Lie algebra. The question of whether $C_c^{\infty }\left(G\right)$ and $\mathcal{A}\left(G\right)$ are topological algebras is also addressed.