Ideal structure of uniform Roe algebras of coarse spaces

Let Cu∗(X,E) be the uniform Roe algebra of a coarse space (X,E) with uniformly locally finite coarse structure. By a controlled truncation technique, we show that the controlled propagation operators in an ideal I of Cu∗(X,E) are exactly the controlled truncations of elements in I. It follows that the lattice of the ideals of the uniform Roe algebra Cu∗(X,E) in which controlled propagation operators are dense, the lattice of the invariant open subsets in the unit space of the groupoid G(X) introduced by Skandalis, Tu and Yu, the lattice of the ideals of the coarse structure E, and the lattice of the ideals of the coarse space X are mutually isomorphic. These lattices also give rise to a type of classification for the ideals of Cu∗(X,E).