Structure spaces of approximately finite-dimensional C∗-algebras

AbstractA necessary and sufficient condition for a topological space Y to be the primitive ideal space of an approximately finite dimensional C∗-algebra is proved. The condition is that Y can be obtained from a second countable, totally disconnected, compact Hausdorff space X by equipping X with a topology τ coarser than the original topology and setting Y = (X, τ)/~. ~ is the equivalence relation on X defined by x ~ y iff all sets o ϵ τ that contain x also contain y, and conversely. The topology τ must satisfy some additional requirements