Gruenhage compacta and strictly convex dual norms

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 , 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 ϒ is a tree, then C0∗(ϒ) admits an equivalent, strictly convex dual norm if and only if ϒ is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images.