Abstract
Let be a finite-dimensional Lie group, and let be a locally convex topological -module. If is sequentially complete, then and the space of smooth vectors are -modules, but the module multiplication need not be continuous. The pathology can be ruled out if is (or embeds into) a projective limit of Banach -modules. Moreover, in this case (the space of analytic vectors) is a module for the algebra of superdecaying analytic functions introduced by Gimperlein, Krötz, and Schlichtkrull. We prove that is a topological -module if is a Banach space or, more generally, if every countable set of continuous seminorms on has an upper bound. The same conclusion is obtained if has a compact Lie algebra. The question of whether and are topological algebras is also addressed.
Citation
Helge Glöckner. "Continuity of LF-algebra representations associated to representations of Lie groups." Kyoto J. Math. 53 (3) 567 - 595, Fall 2013. https://doi.org/10.1215/21562261-2265895
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