arXiv subjects
Publications and source records attributed to Todor Manev.
We prove that consistently $\omega_1<\mathfrak{c}$ and there exists a compact space $K$ whose Banach space $C(K)$ of continuous real-valued functions is Grothendieck, has density $\omega_1$, and admits no renorming which is strictly convex or sequentially Kadets--Klee.
We introduce the concept of F-decomposable systems, well-ordered inverse systems of Hausdorff compacta with fully closed bonding mappings. A continuous mapping between Hausdorff compacta is called fully closed if the intersection of the images of any two closed disjoint subsets is finite. We give a characterization of such systems in terms of a property of the continuous functions on their limit. When, moreover, the fibers of neighboring bonding mappings are metrizable, we call the limit of such a system an F_d-compact, a particular case of a Fedorchuk compact. The stated property allows us to obtain a locally uniformly rotund renorming on the space C(K), where K is an F_d-compact of countable spectral height.
We show that the space of continuous functions over a compact space X admits an equivalent pointwise-lowersemicontinuous locally uniformly rotund norm whenever X admits a fully closed map onto a compact Y such that C(Y) and the spaces of continuous functions over the fibers all admit such norms. A map is called fully closed if the intersection of the images of any two closed disjoint sets is finite. As a main corollary we obtain that C(X) is LUR renormable whenever X is a Fedorchuk compact of finite spectral height.