Lie groups of 𝐶𝑘-maps on non-compact manifolds and the fundamental theorem for Lie group-valued mappings

We study the existence of Lie group structures on groups of the form C k ⁢ ( M , K ) C^{k}(M,K) , where 𝑀 is a non-compact smooth manifold with rough boundary and 𝐾 is a, possibly infinite-dimensional, Lie group. Motivated by introducing this new class of infinite-dimensional Lie groups, we obtain a new version of the fundamental theorem for Lie algebra-valued functions.

Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

We study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient {M\simeq G/K}. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_{1}\subseteq G_{2}\subseteq \cdots \,$ of topological groups $G_{n}$ n such that $G_{n}$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on $G_{n}$, for each $n\in \mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each $G_{n}$ whenever $\cup _{n\in \mathbb{N}}V_{1}V_{2}\cdots V_{n}$ is an identity neighbourhood in $G$ for all identity neighbourhoods $V_{n}\subseteq G_{n}$. If, moreover, each $G_{n}$ is complete, then $G$ is complete. We also show that the weak direct product $\oplus _{j\in J}G_{j}$ is complete for each family $(G_{j})_{j\in J}$ of complete Lie groups $G_{j}$. As a consequence, every strict direct limit $G=\cup _{n\in \mathbb{N}}G_{n}$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\text{Diff}_{c}(M)$ of a paracompact finite-dimensional smooth manifold $M$ and the test function group $C_{c}^{k}(M,H)$, for each $k\in \mathbb{N}_{0}\cup \{\infty \}$ and complete Lie group $H$ modelled on a complete locally convex space.

Smoothing operators and C∗-algebras for infinite dimensional Lie groups

A smoothing operator for a unitary representation [Formula: see text] of a (possibly infinite dimensional) Lie group [Formula: see text] is a bounded operator [Formula: see text] whose range is contained in the space [Formula: see text] of smooth vectors of [Formula: see text]. Our first main result characterizes smoothing operators for Fréchet–Lie groups as those for which the orbit map [Formula: see text] is smooth. For unitary representations [Formula: see text] which are semibounded, i.e. there exists an element [Formula: see text] such that all operators [Formula: see text] from the derived representation, for [Formula: see text] in a neighborhood of [Formula: see text], are uniformly bounded from above, we show that [Formula: see text] coincides with the space of smooth vectors for the one-parameter group [Formula: see text]. As the main application of our results on smoothing operators, we present a new approach to host [Formula: see text]-algebras for infinite dimensional Lie groups, i.e. [Formula: see text]-algebras whose representations are in one-to-one correspondence with certain continuous unitary representations of [Formula: see text]. We show that smoothing operators can be used to obtain host algebras and that the class of semibounded representations can be covered completely by host algebras. In particular, the latter class permits direct integral decompositions.