Rank distributions on coarse spaces and ideal structure of Roe algebras

The notions of controlled truncations for operators in the Roe algebras C*(X) of a coarse space (X, ε) with uniformly locally finite coarse structure, and rank distributions on (X, ε) are introduced. It is shown that the controlled propagation operators in an ideal I of C*(X) are exactly the controlled truncations of elements in I. It follows that the lattice of the ideals of C*(X) in which controlled propagation operators are dense is isomorphic to the lattice of all rank distributions on (X, ε). If X is a discrete metric space with Yu's property A, then the ideal structure of the Roe algebra C*(X) is completely determined by the rank distributions on X.