\(C^\ast\)-simplicity of free products with amalgamation and radical classes of groups. (English) Zbl 1373.46047
Summary: We give new characterizations to ensure that a free product of groups with amalgamation has a simple reduced group \(C^\ast\)-algebra, and provide a concrete example of an amalgam with trivial kernel, such that its reduced group \(C^\ast\)-algebra has a unique tracial state, but is not simple.
Moreover, we show that there is a radical class of groups for which the reduced group \(C^\ast\)-algebra of any group is simple precisely when the group has a trivial radical corresponding to this class.
Moreover, we show that there is a radical class of groups for which the reduced group \(C^\ast\)-algebra of any group is simple precisely when the group has a trivial radical corresponding to this class.
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 20M11 | Radical theory for semigroups |
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