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Normal structure in dual Banach spaces associated with a locally compact group. (English) Zbl 0706.43003

Summary: In this paper we investigate when the dual of a certain function space defined on a locally compact group has certain geometric properties. More particularly, we ask 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 prove, 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.

MSC:

43A10 Measure algebras on groups, semigroups, etc.
43A77 Harmonic analysis on general compact groups
22D15 Group algebras of locally compact groups
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
46B20 Geometry and structure of normed linear spaces
47L50 Dual spaces of operator algebras
43A07 Means on groups, semigroups, etc.; amenable groups
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