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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 \(X\).

MSC:

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