On non-fragmentability of Banach spaces

In this paper, we use the game characterization of Kenderov and Moors 11] to construct an example of a non-fragmentable Banach space. More precisely, we will show that ifX is the tree-complete Banach algebra of Haydon and Zizler 3], (X/c ₀, weak) is not fragmentable by any metric. In particular, this shows thatX/c ₀ cannot be equivalently renormed to be rotund.