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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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