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

Abstract

We show that $C(X)$ admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided $X$ is Fedorchuk compact of spectral height 3. In other words $X$ admits a fully closed map $f$ onto a metric compact $Y$ such that $f^{-1}(y)$ is metrizable for all $y\in Y$ . A continuous map of compacts $f : X \to Y$ is said to be fully closed if for any disjoint closed subsets $A;B \subset X$ the intersection $f(A) \cap f(B)$ is finite. For instance the projection of the lexicographic square onto the first factor is fully closed and all its fibers are homeomorphic to the closed interval.