A note on nonfragmentability of Banach spaces - S. Alireza Kamel Mirmostafaee (International Journal of Mathematics and Mathematical Sciences)
We use Kenderov-Moors characterization of fragmentability to show that if a compact Hausdorff space X with the tree-completeness property contains a disjoint sequences of clopen sets, then (C(X), weak) is not fragmented by any metric which is stronger than weak topology. In particular, C(X) does not admit any equivalent locally uniformly convex renorming.