Locally uniformly rotund renormings of the spaces of continuous functions on Fedorchuk compacts (Q2217250)

A Banach space \((X,\|\cdot\|)\) is said to be {\em locally uniformly rotund} (LUR, for short) if \(\|x_n-x\|\rightarrow 0\) whenever \(\lim\|(x_n+x)/2\|=\lim\|x_n\|=\|x\|\). Techniques for getting an equivalent LUR norm are essentially of two sorts: Having enough convex functions on the space to apply the so-called Deville's Master Lemma or, alternatively, having a special mapping from \(X\) into some metric space \(M\) (the so-called ``transfer technique''). Quite often \(M\) is just \(c_0(\Gamma)\). The technique of projectional resolutions of the identity share elements of both approaches. The case when \(X\) is a \(C(K)\) space, for \(K\) a compact topological space, has been amply treated in the literature, looking for sufficient conditions on \(K\) for getting a pointwise lower semicontinuous LUR equivalent norm. In this paper, the authors find a new class of compact spaces \(K\) such that \(C(K)\) has such a renorming. The main result reads: \(C(K)\) has a pointwise lower semicontinuous LUR renorming whenever \(K\) admits a fully closed map \(f\) onto a metrizable compact space \(M\) such that the fibers \(f^{-1}(m)\) are metrizable for every \(m\in M\). The map \(f\) is said to be \textit{fully closed} if, for all disjoint closed subsets \(A\) and \(B\) of \(X\), the set \(f(A)\cap f(B)\) is finite. Spaces \(K\) with this property form a subclass of the class of Fedorchuk compact spaces (F-compacta, for short, a class of compact spaces defined by a well-ordered continuous inverse system). More precisely, F-compacta of spectral height \(3\) (a feature of the defining system) can be characterized as non-metrizable compacta that satisfy the condition in the main theorem. The authors raise the natural question whether \(C(K)\) is LUR renormable if \(K\) is an F-compact space of countable spectral height.