Subjective expected utility with a spectral state space

An agent faces a decision under uncertainty with the following structure. There is a set $${\mathcal {A}}$$A of “acts”; each will yield an unknown real-valued payoff. Linear combinations of acts are feasible; thus, $${\mathcal {A}}$$A is a vector space. But there is no pre-specified set of states of nature. Instead, there is a Boolean algebra $${\mathfrak {I}}$$I describing information the agent could acquire. For each element of $${\mathfrak {I}}$$I, she has a conditional preference order on $${\mathcal {A}}$$A. I show that if these conditional preferences satisfy certain axioms, then there is a unique compact Hausdorff space $${\mathcal {S}}$$S such that elements of $${\mathcal {A}}$$A correspond to continuous real-valued functions on $${\mathcal {S}}$$S, elements of $${\mathfrak {I}}$$I correspond to regular closed subsets of $${\mathcal {S}}$$S, and the conditional preferences have a subjective expected utility (SEU) representation given by a Borel probability measure on $${\mathcal {S}}$$S and a continuous utility function. I consider two settings; in one, $${\mathcal {A}}$$A has a partial order making it a Riesz space or Banach lattice, and $${\mathfrak {I}}$$I is the Boolean algebra of bands in $${\mathcal {A}}$$A. In the other, $${\mathcal {A}}$$A has a multiplication operator making it a commutative Banach algebra, and $${\mathfrak {I}}$$I is the Boolean algebra of regular ideals in $${\mathcal {A}}$$A. Finally, given two such vector spaces $${\mathcal {A}}_1$$A1 and $${\mathcal {A}}_2$$A2 with SEU representations on topological spaces $${\mathcal {S}}_1$$S1 and $${\mathcal {S}}_2$$S2, I show that a preference-preserving homomorphism $${\mathcal {A}}_2{{\longrightarrow }}{\mathcal {A}}_1$$A2⟶A1 corresponds to a probability-preserving continuous function $${\mathcal {S}}_1{{\longrightarrow }}{\mathcal {S}}_2$$S1⟶S2. I interpret this as a model of changing awareness.