Journal of Lie Theory 26 (2016), No. 1, 269--291
Copyright 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
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