Abstract
Several conditions for the countability of the weak spectrum of weakly almost periodicF: arbitrary semigroupS→ Banach space are given; especially, generalizing a recent result of Kadec and Kursten, the separability of the rangeF(S) is necessary and sufficient. Examples demonstrate that the classes of weakly a.p.F, weakly sequentially a.p.F, and weakly a.p.F with countable spectrum are all distinct, and that dual results for weak' a.p.F are false.
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