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L~∞(G)* as the Second Dual of the Group Algebra L1(G) with a Locally Convex Topology

机译:L〜∞(G)*是局部凸拓扑的群代数L1(G)的第二对偶

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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.
机译:Isik,Pym和Ulger [8]很好地说明了紧群G的群代数L1(G)的第二对偶L1(G)**的结构。Lau和Pym [10]研究了一般情况。他们引入了一个子代数Lg,即L1(G)**中元素的范数闭包为紧载波,并通过对子空间L〜∞的限制用L〜∞_0(G)*进行标识。 G上有界可测函数的_0(G)在无穷大处消失。对于L〜∞_0(G)*,在紧凑的情况下,它们能够覆盖从Li(G)**获得的大多数结果。因此,他们在[10]中建议,L1(G)`*的明智替代应为L〜∞_0(G)*。本文的目的是给出Li(G)上的局部凸拓扑t,其中L〜∞_0(G)*(||。|||∞)是其强对偶,因此存在L〜∞_0(G )*作为(L1(G),t)的第二对偶。我们证明,除了G有限的琐碎情况外,还有无数这样的拓扑,并且我们讨论了乘法连续性的各个级别。

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