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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Normal structure in dual Banach spaces associated with a locally compact group


Authors: Anthony To Ming Lau and Peter F. Mah
Journal: Trans. Amer. Math. Soc. 310 (1988), 341-353
MSC: Primary 43A10; Secondary 43A15, 46B20
MathSciNet review: 937247
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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$ ^{*}$ 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$ ^{*}$-normal structure iff it has property SUKK$ ^{*}$ iff it has property SKK$ ^{*}$ iff the group is discrete; (2) Among amenable locally compact groups, the Fourier-Stieltjes algebra has property SUKK$ ^{*}$ iff it has property SKK$ ^{*}$ iff the group is compact. Consequently the Fourier-Stieltjes algebra has weak$ ^{*}$-normal structure when the group is compact.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0937247-0
PII: S 0002-9947(1988)0937247-0
Keywords: Locally compact group, amenable group, measure algebra, left uniformly continuous functions, (weakly) almost periodic functions, Fourier-Stieltjes algebra, weak$ ^{*}$-normal structure, Kadec-Klee property, uniformly Kadec-Klee property
Article copyright: © Copyright 1988 American Mathematical Society