A state space of a Boolean algebra is a Choquet simplex but not all Choquet simplices can be viewed as state spaces of Boolean algebras. Is it known which Choquet simplices are precisely state spaces of Boolean algebras?
The state space of a unital Boolean algebra is characterized (as a Choquet simplex) by the extremal boundary (the set of extreme points) being both compact and totally disconnected. Once stated, this result is pretty obvious, as is the proof. [Just observe that continuous functions on the extremal boundary extend (uniquely) to affine functions on the whole C simplex; take the indicator functions of clopen sets in the ext boundary, extend these; the set of them recovers the boolean algebra; the reverse is based on the extremal traces of a lattice being compact, etc]