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

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.

33 pages, LaTeX; v3: update of references

References in corpus (3)

• Lie groups over non-discrete topological fields
• Differentiable mappings on products with different degrees of differentiability in the two factors
• Continuity of convolution of test functions on Lie groups

Cited by in corpus (4)

• Differentiable mappings on products with different degrees of differentiability in the two factors
• Factorization in Denjoy-Carleman classes associated to representations of $({\mathbb{R}}^d,+)$
• Continuity of convolution of test functions on Lie groups
• Strong factorization of ultradifferentiable vectors associated with compact Lie group representations