On scattered Eberlein compact spaces (Q995364)

A compact space is called Eberlein (resp. uniform Eberlein) if it is homeomorphic to a weakly compact subset of a Banach space (resp. of a Hilbert space). The authors prove that every scattered Eberlein compact space of height less than or equal \(\omega +1\) is uniform Eberlein. For a set \(\Gamma\) and a natural number \(n\), let \(\sigma_n(\Gamma)\) denote the the subspace of the product space \(2^\Gamma\) consisting of the characteristic functions of subsets of cardinality at most \(n\). The authors also construct an example of a uniform Eberlein compact space \(K\) of height 3 which cannot be embedded into any space of the form \(\sigma_n(\Gamma)\). This space has weight \(\aleph_\omega\), while Argyros and Godefroy had proved that every Eberlein compact space of finite height and weight less than \(\aleph_\omega\) can be embedded into some \(\sigma_n(\Gamma)\). The space of continuous functions \(C(K)\) over this compact space has some interesting properties. \(C(K)\) is Lipschitz isomorphic to \(c_0(\aleph_\omega)\) but it is not isomorphic to it. Also, \(C(K)\) contains an uncomplemented copy of \(c_0(\alpha_\omega)\) (formed by the functions vanishing out of the isolated points) while \textit{S. A. Argyros, J. F. Castillo, A. S. Granero, M. Jimenez} and \textit{J. P. Moreno} [Proc. Lond. Math. Soc. (3) 85, No. 3, 742--768 (2002; Zbl 1017.46011)] proved that every copy of \(c_0(\aleph_n)\), \(n<\omega\), inside a WCG Banach space is complemented.