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Journal of Lie Theory 26 (2016), No. 3, 787--805
Copyright Heldermann Verlag 2016

On the Characterization of Trace Class Representations and Schwartz Operators

Gerrit van Dijk
Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands

Karl-Hermann Neeb
Department Mathematik, FAU Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany

Hadi Salmasian
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa Ont. K1N 6N5, Canada

Christoph Zellner
Department Mathematik, FAU Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany


\def\cH{{\cal H}} We collect several characterizations of unitary representations $(\pi, \cH)$ of a finite dimensional Lie group $G$ which are trace class, i.e., for each compactly supported smooth function $f$ on $G$, the operator $\pi(f)$ is trace class. In particular we derive the new result that, for some $m \in \mathbb{N}$, all operators $\pi(f)$, $f \in C^m_c(G)$, are trace class. As a consequence the corresponding distribution character $\theta_\pi$ is of finite order. We further show $\pi$ is trace class if and only if every operator $A$, which is smoothing in the sense that $A\cH\subseteq \cH^\infty$, is trace class and that this in turn is equivalent to the Fr\'echet space $\cH^\infty$ being nuclear, which in turn is equivalent to the realizability of the Gaussian measure of $\cH$ on the space $\cH^{-\infty}$ of distribution vectors. Finally we show that, even for infinite dimensional Fr\'echet-Lie groups, $A$ and $A^*$ are smoothing if and only if $A$ is a Schwartz operator, i.e., all products of $A$ with operators from the derived representation are bounded.

Keywords: Trace class representation, smoothing operator, Schwartz operator, unitary representation.

MSC: 22E45, 22E66

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