Amenability of the second dual of hypergroup algebras. (English) Zbl 0970.46030

Let \(A\) be a Banach algebra \(A^{**}\) the second conjugate algebra of \(A\) under the Arens products. The paper considers the amenability for the group algebra \(A=L^1(G)\) as well as the measure algebra \(A=M(G)\), and their second conjugate algebra for a locally compact group \(G\). The following results are known:
1. \(L^1(G)^{**}\) is amenable if and only if \(G\) is finite.
2. If \(LUC(G)^*\), the dual space of left uniformly continuous function on \(G\), is amenable, then \(G\) is compact and \(M(G)\) is amenable.
3. If \(M(G)^{**}\) is amenable, then \(G\) is finite.
This paper generalizes all of the above results to the case of locally compact hypergroup \(X\).


46G12 Measures and integration on abstract linear spaces
46H15 Representations of topological algebras
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