Hyper-Stonean envelopes of compact spaces

Let K be a compact space, and denote by (K) over tilde its hyper-Stonean envelope. We discuss the class of spaces K with the property that (K) over tilde is homeomorphic to (I) over tilde, the hyper-Stonean envelope of the closed unit interval d. Certainly each uncountable, compact, metrizable space K belongs to this class. We describe several further classes of compact spaces K for which (K) over tilde = (I) over tilde. In fact, (K) over tilde = (I) over tilde if and only if the Banach spaces M(K) and M(I) of measures on K and If are isometrically isomorphic