Abstract
We characterize when the reduced C ∗-algebra of a non-trivial group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group C ∗-algebra. We also give a simple proof of the recent result by Breuillard, Kalantar, Kennedy and Ozawa that the reduced C ∗-algebra of a group has unique tracial state if and only if the amenable radical of the group is trivial.
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Notes
- 1.
Uffe Haagerup tragically passed away on July 5, 2015. The results in this article were proven by Haagerup in the early Spring of 2015, and privately communicated to his (and Magdalena Musat’s) PhD student Kristian Knudsen Olesen in May 2015. Based on this, Kristian Knudsen Olesen, Magdalena Musat and Mikael Rørdam (University of Copenhagen, Denmark) have written up this paper.
- 2.
Throughout the paper, groups are assumed to be discrete.
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Haagerup, U. (2016). A New Look at C∗-Simplicity and the Unique Trace Property of a Group. In: Carlsen, T.M., Larsen, N.S., Neshveyev, S., Skau, C. (eds) Operator Algebras and Applications. Abel Symposia, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-39286-8_7
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DOI: https://doi.org/10.1007/978-3-319-39286-8_7
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