A weak* separable C(K)* space whose unit ball is not weak* separable
  We provide a ZFC example of a compact space K such that C(K)* is w*-separable but its closed unit ball is not w*-separable. All previous examples of such kind had been constructed under CH. We also discuss the measurability of the supremum norm on that C(K) equipped with its weak Baire sigma-algebra. 
Antonio Avil\'es, Grzegorz Plebanek and Jos\'e Rodr\'iguez
math.FA math.GN