Gruenhage compacta and strictly convex dual norms

TL;DR

This paper establishes that certain compact spaces called Gruenhage spaces have dual Banach spaces with strictly convex norms, and explores conditions under which this property holds, including stability under perfect images.

Contribution

It proves that Gruenhage compact spaces lead to dual Banach spaces with strictly convex norms and characterizes this property for spaces of continuous functions on trees.

Findings

C(K)* admits a strictly convex dual norm for Gruenhage compact K

Characterization of strict convexity for C(T)* when T is a tree

Stability of Gruenhage spaces under perfect images

Abstract

We prove that if K is a Gruenhage compact space then C(K)* admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X* is the |.|-closed linear span of K, where K is a Gruenhage compact in the w*-topology and |.| is equivalent to a coarser, w*-lower semicontinuous norm on X*, then X* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if T is a tree, then C(T)* admits an equivalent, strictly convex dual norm if and only if T is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images.