Journal of Lie Theory

no. 1
Trace Class Groups
Journal of Lie Theory, Volume 26 (2016) no. 1, pp. 269-291
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.
DOI: 10.5802/jolt.891
Classification: 22D10, 11F72, 22D30, 43A65
Keywords: Trace class operator, type I group, unitary representation 
@article{JOLT_2016_26_1_a13,
     author = {A. Deitmar and G. van Dijk},
     title = {Trace {Class} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {269--291},
     year = {2016},
     volume = {26},
     number = {1},
     doi = {10.5802/jolt.891},
     zbl = {1342.22009},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.891/}
}
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A. Deitmar; G. van Dijk. Trace Class Groups. Journal of Lie Theory, Volume 26 (2016) no. 1, pp. 269-291. doi: 10.5802/jolt.891

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