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

TL;DR

This paper constructs a ZFC example of a compact space K where the dual space C(K)* is weak*-separable but its unit ball is not, challenging previous assumptions under CH.

Contribution

It provides the first ZFC example of a compact space with a weak*-separable dual but non-separable unit ball, expanding understanding of dual space properties.

Findings

C(K)* is weak*-separable while its unit ball is not

Previous examples required CH, this one does not

Discusses measurability of the supremum norm

Abstract

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.