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

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.Comment: 33 pages, LaTeX; v3: update of reference

Tensor products in the category of topological vector spaces are not associative

We show by example that the associative law does not hold for tensor products in the category of general (not necessarily locally convex) topological vector spaces. The same pathology occurs for tensor products of Hausdorff abelian topological groups.Comment: 7 pages; v2: new version focusses on the real cas

Conveniently Hoelder homomorphisms are smooth in the convenient sense

We show that every 'conveniently Hoelder' homomorphism between Lie groups in the sense of convenient differential calculus is smooth (in the convenient sense). In particular, every Lip^0 homomorphism is smooth.Comment: 25 pages, LaTeX; v2: minor improvements, typos correcte

Differentiable mappings between spaces of sections

In this work, various versions of the so-called Omega-Lemma are provided, which ensure differentiability properties of pushforwrds between spaces of C^r-sections (or compactly supported C^r-sections) in vector bundles over finite-dimensional base manifolds whose fibres are (possibly infinite-dimensional) locally convex spaces. Applications are given, including the proof of continuity for some natural module multiplications on spaces of sections and the construction of certain infinite-dimensional Lie groups of Lie group-valued maps.Comment: 31 pages, LaTeX. Up to minor updates, this is an unpublished manuscript from 2001/200

Every smooth p-adic Lie group admits a compatible analytic structure

We show that every finite-dimensional p-adic Lie group of class C^k admits a C^k-compatible analytic Lie group structure.Comment: LaTeX, 37 page