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

TL;DR

This paper investigates the conditions under which modules of Lie group representations have continuous algebra actions, focusing on the topological properties of smooth and analytic vectors within locally convex modules.

Contribution

It establishes new conditions ensuring the continuity of module multiplication for D(G) and A(G) algebras in Lie group representations, extending previous results.

Findings

Analytic vectors form topological A(G)-modules under certain conditions.

Continuity of module multiplication is guaranteed if E embeds into a projective limit of Banach G-modules.

D(G) and A(G) are topological algebras under specific topological conditions.

Abstract

Let G be a Lie group and E be a locally convex topological G-module. If E is sequentially complete, then E and its space of smooth vectors are modules for the algebra D(G) of compactly supported smooth functions on G. However, 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 the space of analytic vectors is a module for the algebra A(G) of superdecaying analytic functions introduced by Gimperlein, Kroetz and Schlichtkrull. We prove that the space of analytic vectors is a topological A(G)-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 D(G) and A(G) are topological algebras is also addressed.