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Involutions in algebras related to second duals of hypergroup algebras. (English) Zbl 1382.43011
Summary: Let \(K\) be a hypergroup. The purpose of this article is to study the question of involutions on algebras \(M(K)^{**}\), \(L(K)^{**}\), and \(L_{c}(K)^{**}\). We show that the natural involution of \(M(K)\) has the canonical extension to \(M(K)^{**}\) if and only if the natural involution of \(L(K)\) has the canonical extension to \(L(K)^{**}\). Also, we give necessary and sufficient conditions for \(M(K)^{**}\) and \(L(K)^{**}\) to admit an involution extending the natural involution of \(M(K)\) when \(K\) is left amenable. Finally, we find the necessary and sufficient conditions for \(L_{c}(K)^{**}\) to admit an involution.
MSC:
43A62 Harmonic analysis on hypergroups
46K05 General theory of topological algebras with involution
43A07 Means on groups, semigroups, etc.; amenable groups
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