A note on the hyperstonian \(\beta\) X (Q1077671)

The Dixmier-Grothendieck characterization of spaces C(K) that are dual spaces has a lattice version given by Schaefer and Lotz: if K is a compact Hausdorff space, the C(K) is the dual of a Banach lattice if and only if K is hyperstonian. The corresponding vector lattice theorem for completely regular Hausdorff spaces X that are not necessarily compact has been obtained by \textit{Hong-Yun Xiong} [Math. Z. 183, 413-418 (1983; Zbl 0498.46019)]. From this result it follows that if C(X) is Riesz isomorphic to the order dual space of some Riesz space, then \(\beta\) X is hyperstonian, and the author asks if the converse is true. In this note we give a negative answer of this question.