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On the second conjugate algebra of \(L_ 1(G)\) of a locally compact group. (English) Zbl 0608.43002
Let G be a locally compact group. On \(L_{\infty}(G)^ *=L_ 1(G)^{**}\) we have two Arens products mn and \(m\cdot n\). In this paper it is shown that \(L_ 1(G)=\{m\in L_{\infty}(G)^ *:\) \(mn=m\cdot n\) for all \(n\in L_{\infty}(G)^ *\}\). For G compact, this was shown independently by N. Isik, J. Pym and A. √úlger [ibid. 35, 135-148 (1987; Zbl 0585.43001)]. If G is abelian, it follows that \(L_ 1\)(G) is the center of \(L_{\infty}(G)^ *\). If \(G_ 1\) and \(G_ 2\) are abelian and the Banach algebras \(L_{\infty}(G_ 1)^ *\) and \(L_{\infty}(G_ 2)^ *\) are isometrically isomorphic, then it follows that \(G_ 1\) and \(G_ 2\) are isomorphic.

MSC:
43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A10 Measure algebras on groups, semigroups, etc.
46H99 Topological algebras, normed rings and algebras, Banach algebras
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