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Concerning the second dual of the group algebra of a locally compact group. (English) Zbl 0667.43004

Let G be a locally compact group. This paper continues the study of \(L^ 1(G)^{**}\), the second conjugate algebra of the group algebra \(L^ 1(G)\) with the Arens multiplication. We introduce a subalgebra \(L_ 0^{\infty}(G)^*\) which is related to \(L^ 1(G)^{**}\) in much the same way as the measure algebra M(G) is related to the space of finitely additive measures. For \(L_ 0^{\infty}(G)^*\) we can recover most of the results obtained for \(L^ 1(G)^{**}\) in the compact case. For example, we find that the maximal ideal space of \(L_ 0^{\infty}(G)\) is a semigroup with a simple algebraic structure. We also consider operators from \(L^{\infty}(G)\) into certain of its subspaces commuting with convolution on the left by elements of \(L^ 1(G)\). Algebras we find in this way are subalgebras of \(LUC(G)^*\) and are related to the structure of \(L^ 1(G)^{**}\).
Reviewer: J.S.Pym

MSC:

43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A10 Measure algebras on groups, semigroups, etc.
22D15 Group algebras of locally compact groups
43A40 Character groups and dual objects
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