On Differentiable Vectors for Representations of Infinite Dimensional Lie Groups

TL;DR

This paper develops tools to analyze differentiability of vectors in infinite dimensional Lie group representations, establishing smoothness criteria and topologies for spaces of smooth vectors, with applications to Banach--Lie groups and $C^*$-dynamical systems.

Contribution

It introduces a topology on smooth vectors for Banach--Lie groups, characterizes smooth vectors via positive definite functions, and provides criteria for $C^k$-vectors in Banach space representations.

Findings

Smooth vectors form a Fréchet space for Banach--Lie groups.

Smoothness of vectors can be characterized by positive definite functions.

Examples show the density and triviality of $C^k$-vector spaces in certain representations.

Abstract

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations $\pi \: G \to \GL(V)$ of an infinite dimensional Lie group $G$ on a locally convex space $V$. The first class of results concerns the space $V^\infty$ of smooth vectors. If $G$ is a Banach--Lie group, we define a topology on the space $V^\infty$ of smooth vectors for which the action of $G$ on this space is smooth. If $V$ is a Banach space, then $V^\infty$ is a Fr\'echet space. This applies in particular to $C^*$-dynamical systems $(\cA,G, \alpha)$, where $G$ is a Banach--Lie group. For unitary representations we show that a vector $v$ is smooth if the corresponding positive definite function $\la \pi(g)v,v\ra$ is smooth. The second class of results concerns criteria for $C^k$-vectors in terms of operators of the derived representation for a Banach--Lie group $G$ acting on a Banach space $V$. In particular, we provide for each $k \in \N$ examples of continuous unitary representations for which the space of $C^{k+1}$-vectors is trivial and the space of $C^k$-vectors is dense.