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 X ∗ = span ¯ | | | ⋅ | | | ( 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 ϒ is a tree, then C 0 ( 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.