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Normal Structure in Dual Banach Spaces Associated With a Locally Compact Group

Anthony To-Ming Lau and Peter F. Mah
Transactions of the American Mathematical Society
Vol. 310, No. 1 (Nov., 1988), pp. 341-353
DOI: 10.2307/2001126
Stable URL: http://www.jstor.org/stable/2001126
Page Count: 13
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Normal Structure in Dual Banach Spaces Associated With a Locally Compact Group
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Abstract

In this paper we investigated when the dual of a certain function space defined on a locally compact group has certain geometric properties. More particularly, we asked when weak$^\ast$ compact convex subsets in these spaces have normal structure, and when the norm of these spaces satisfies one of several types of Kadec-Klee property. As samples of the results we have obtained, we have proved, among other things, the following two results: (1) The measure algebra of a locally compact group has weak$^\ast$-normal structure iff it has property $\mathrm{SUKK}^\ast \operatorname{iff}$ it has property $\mathrm{SKK}^\ast \operatorname{iff}$ the group is discrete; (2) Among amenable locally compact groups, the Fourier-Stieltjes algebra has property $\mathrm{SUKK}^\ast \operatorname{iff}$ it has property $\mathrm{SKK}^\ast \operatorname{iff}$ the group is compact. Consequently the Fourier-Stieltjes algebra has weak$^\ast$-normal structure when the group is compact.

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