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Bulletin of the London Mathematical Society Advance Access originally published online on December 21, 2006
Bulletin of the London Mathematical Society 2007 39(1):1-26; doi:10.1112/blms/bdl014
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© 2006 London Mathematical Society

On simplicity of reduced C*-algebras of groups

Pierre de la Harpe

Section de Mathématiques
Université de Genève
C.P. 64
CH-1211 Genève 4
Switzerland
Pierre.delaHarpe{at}math.unige.ch

Received 14 September 2005. Revision received 24 April 2006.

A countable group is C*-simple if its reduced C*-algebra is a simple algebra. Since Powers recognised in 1975 that non-abelian free groups are C*-simple, large classes of C*-simple groups which appear naturally in geometry have been identified, including non-elementary Gromov hyperbolic groups and lattices in semisimple groups. In this exposition, C*-simplicity for countable groups is viewed as an extreme case of non-amenability. The basic examples are described, and several open problems are formulated.


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