Smoothing operators and $C^*$-algebras for infinite dimensional Lie groups

Abstract

A host algebra of a (possibly infinite dimensional) Lie group $G$ is a $C^*$-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations $\pi \colon G \to \U(\cH)$. In this paper we present a new approach to host algebras for infinite dimensional Lie groups which is based on smoothing operators, i.e., operators whose range is contained in the space $\cH^\infty$ of smooth vectors. Our first major result is a characterization of smoothing operators $A$ that in particular implies smoothness of the maps $\pi^A \colon G \to B(\cH), g \mapsto \pi(g)A$. The concept of a smoothing operator is particularly powerful for representations $(\pi,\cH)$ which are semibounded, i.e., there exists an element $x_0 \in\g$ for which all operators $i\dd\pi(x)$, $x \in \g$, from the derived representation are uniformly bounded from above in some neighborhood of $x_0$. Our second main result asserts that this implies that $\cH^\infty$ coincides with the space of smooth vectors for the one-parameter group $\pi_{x_0}(t) = \pi(\exp tx_0)$. We then show that natural types of smoothing operators can be used to obtain host algebras and that, for every metrizable Lie group, the class of semibounded representations can be covered completely by host algebras. In particular, it permits direct integral decompositions.