ON UNIFORMLY GATEAUX DIFFERENTIABLE NORMS IN C(K)

It is proved that C(K) has no equivalent uniformly Gateaux differentia ble norm (UGD) when K is an uncountable separable scattered compact sp ace. This result is applied to obtain an example of scattered compact K such that K-m = null set and C(K) has no UGD renorming. In the last few years lots of remarkable results concerning renormings of C(K), wh en K is a scattered compact have been obtained. These results are pres ented in [DGZ, Chapter VIII and [H]. We will only mention that it foll ows from results of Deville [D] and Haydon-Rogers [HR] that if K is sc attered compact and K-(omega 1)= null set then C(K) admits an equivale nt locally uniformly rotund norm whose dual norm is locally uniformly rotund. The aim of this paper is to characterize the separable scatter ed compact spaces K for which C(K) has an equivalent uniformly Gateaux differentiable (UGD for short) norm.