Abstract:We present an example of a Banach space $E$ admitting an equivalent weakly uniformly rotund norm and such that there is no $\Phi :E\to c_0(\Gamma )$, for any set $\Gamma $, linear, one-to-one and bounded. This answers a problem posed by Fabian, Godefroy, H\'ajek and Zizler. The space $E$ is actually the dual space $Y^*$ of a space $Y$ which is a subspace of a WCG space.
Keywords: WCG Banach space, weakly uniformly rotund norms, tree
AMS Subject Classification: 46B20, 46B26, 03E05