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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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