Isik, Pym and Ulger [8] give a good account of the structure of the second dual L1(G)** of the group algebra L1(G) of a compact group G. Lau and Pym [10] investigate the general case of a locally compact group G. They introduce a sub- algebra Lg , the norm closure of elements in L1(G)** with compact carriers, and identify it with L~∞ _0(G)* via restriction on the subspace L~∞_0(G) of bounded mea- surable functions on G that vanish at infinity. For L~∞_0(G)* , they are able to re- cover most of the results obtained for Li(G)** in the compact case. Therefore, they suggest in [10] that the sensible replacement for L1(G)`* should be L~∞_0(G)* . The purpose of this paper is to give a locally convex topology t on Li(G) under which L~ ∞_0(G)* (with || . ||∞) is its strong dual and thus present L~∞ _0(G)* as the sec- ond dual of (L1(G),t) . We show that, except for the trivial case of G finite, there are uncountably many such topologies, and we discuss various levels of continuity of multiplication.
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