On CL-spaces and almost CL-spaces
A complex Banach space \(X\) is said to be a \(CL\)-space (almost CL-space) if for every maximal face \(F\) of the unit sphere, the unit ball is the (norm closed) absolute convex hull of \(F\). These concepts were studied earlier by \textit{Å. Lima} [Ann. Inst. Fourier 28, No. 3, 35--65 (1978; Zbl 0347.46018)]. This paper contributes mainly to the study of complex CL (almost CL) spaces. Newer examples of spaces satisfying these properties are generated by considering direct sums and spaces of vector-valued continuous functions. For example, these properties are preserved by \(c_0\)-direct sums, but only by finite \(\ell^1\)-direct sums. In the case of \(C(K,X)\), the space of vector-valued continuous functions on a compact set \(K\), \(C(K,X)\) is an almost CL-space if and only if \(X\) is, and if \(C(K,X)\) is a CL-space, then so is \(X\).
scientific article; zbMATH DE number 2103841