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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.

A weak$^*$ separable $C(K)^*$ space whose unit ball is not weak$^*$ separable

A weak$^$ separable $C(K)^$ space whose unit ball is not weak$^*$ separable