2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94705A 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 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 shown to be an extreme case of non-amenability. The basic examples are described and several open problems are formulated.23 pagesOperator AlgebrasRepresentation Theory22D25; 46L05text