Power's property and simple \(C^*\)-algebras (Q1057456)

Consider a \(C^*\)-algebra A with unit, a group \(\Gamma\), an action \(\alpha: \Gamma \to Aut(A),\) and the \(C^*\)-algebra B which is the corresponding reduced crossed product. Power's poperty is a combinatorial property of \(\Gamma\), which holds for example if \(\Gamma\) is a non- amenable free product, a non-amenable subgroup of \(PSL(2,{\mathbb{C}}),\) or the group \(SL(3,{\mathbb{Z}}).\) If \(\Gamma\) has this property, we show in particular the following two facts, without any assumption on the action: if A is simple then B is simple; if A has a unique trace then B has a unique trace. For further information about Powers' groups, see by the first author, Lecture Notes Math. 1132, 230-253 (1985).