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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

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


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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