On Differentiable Vectors for Representations of Infinite Dimensional Lie Groups

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations $\pi G\to \text{GL}(V)$ of an infinite dimensional Lie group $G$ on a locally convex space $V$. The first class of results concerns the space $V^{\mathrm{\infty }}$ of smooth vectors. If $G$ is a Banach–Lie group, we define a topology on the space $V^{\mathrm{\infty }}$ of smooth vectors for which the action of $G$ on this space is smooth. If $V$ is a Banach space, then $V^{\mathrm{\infty }}$ is a Fr'echet space. This applies in particular to $C^{\ast }$-dynamical systems $(\text{cA},G,\alpha )$, where $G$ is a Banach–Lie group. For unitary representations we show that a vector $v$ is smooth if the corresponding positive definite function $\text{la}\pi (g)v,v\text{ra}$ 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 $k\in \mathbb{N}$ 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.