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

Lau, Anthony To Ming; Mah, Peter F.

doi:10.1090/S0002-9947-1988-0937247-0

Normal structure in dual Banach spaces associated with a locally compact group
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by Anthony To Ming Lau and Peter F. Mah

Trans. Amer. Math. Soc. 310 (1988), 341-353

DOI: https://doi.org/10.1090/S0002-9947-1988-0937247-0

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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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