Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 26 (2016), No. 1, 269--291Copyright Heldermann Verlag 2016 Trace Class Groups Anton Deitmar Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany deitmar@uni-tuebingen.de Gerrit van Dijk Mathematical Institute, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands dijk@math.leidenuniv.nl [Abstract-pdf] A representation $\pi$ of a locally compact group $G$ is called {\it trace class}, if for every test function $f$ the induced operator $\pi(f)$ is a trace class operator. The group $G$ is called {\it trace class}, if every $\pi\in\widehat G$ is trace class. In this paper we give a survey of what is known about trace class groups and ask for a simple criterion to decide whether a given group is trace class. We show that trace class groups are type I and give a criterion for semi-direct products to be trace class and show that a representation $\pi$ is trace class if and only if $\pi\otimes\pi'$ can be realized in the space of distributions. Keywords: Trace class operator, type I group, unitary representation. MSC: 22D10, 11F72, 22D30, 43A65 [ Fulltext-pdf  (373  KB)] for subscribers only.