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

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

2012-03-15

arxiv.org/abs/1203.3418v1

Mathematics

Functional Analysis

31 pages, LaTeX

Scientific paper

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 question of whether D(G) and A(G) are topological algebras is also addressed.

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