A new look at \( C^{\ast}\)-simplicity and the unique trace property of a group. (English) Zbl 1376.46046
Carlsen, Toke M. (ed.) et al., Operator algebras and applications. The Abel symposium 2015 took place on the ship Finnmarken, part of the Coastal Express Line (the Norwegian Hurtigruten), from Bergen to the Lofoten Islands, Norway, August 7–11, 2015. Cham: Springer (ISBN 978-3-319-39284-4/hbk; 978-3-319-39286-8/ebook). Abel Symposia 12, 161-170 (2016).
Summary: We characterize when the reduced \( C^{\ast}\)-algebra of a non-trivial group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group \( C^{\ast}\)-algebra. We also give a simple proof of the recent result by E. Breuillard et al. [“\( C^{\ast}\)-simplicity and the unique trace property for discrete groups”, arXiv:1410.2518] that the reduced \( C^{\ast}\)-algebra of a group has unique tracial state if and only if the amenable radical of the group is trivial.
For the entire collection see [Zbl 1350.47002].
For the entire collection see [Zbl 1350.47002].
MSC:
| 46L30 | States of selfadjoint operator algebras |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
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