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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. Ülger [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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