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

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 sev...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1988, Vol.310 (1), p.341-353
Main Authors: Lau, Anthony To Ming, Mah, Peter F.
Format: Article
Language:English
Subjects:
Algebra
Banach space
Continuous functions
Fixed point property
Mathematical theorems
Periodic functions
Research article
Topological theorems
Topology
Unit ball
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-1988-0937247-0