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\(C^\ast\)-simplicity of \(n\)-periodic products. (English. Russian original) Zbl 1358.20033
Math. Notes 99, No. 5, 631-635 (2016); translation from Mat. Zametki 99, No. 5, 643-648 (2016).
Summary: The \(C^\ast\)-simplicity of \(n\)-periodic products is proved for a large class of groups. In particular, the \(n\)-periodic products of any finite or cyclic groups (including the free Burnside groups) are \(C^\ast\)-simple. Continuum-many nonisomorphic 3-generated nonsimple \(C^\ast\)-simple groups are constructed in each of which the identity \(x^n = 1\) holds, where \(n\geq1003\) is any odd number. The problem of the existence of \(C^\ast\)-simple groups without free subgroups of rank 2 was posed by P. de la Harpe [Bull. Lond. Math. Soc. 39, No. 1, 1–26 (2007; Zbl 1123.22004)].

MSC:
20F50 Periodic groups; locally finite groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F05 Generators, relations, and presentations of groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
43A07 Means on groups, semigroups, etc.; amenable groups
46L05 General theory of \(C^*\)-algebras
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