The uniform Roe algebra of a discrete metric space is a $C^*$-algebra that analytically encodes the coarse geometry of the underlying space. It is known that counter-examples to the coarse Baum-Connes conjecture can be constructed by using expander graphs, the crucial point of which lies in the fact that the uniform Roe algebra of the metric space of a sequence of expander graphs contains a noncompact projection $P$, called the ghost projection, which is locally invisible at infinity. In terms of ideal structure, accordingly, the finite propagation operators in the principal ideal generated by $P$, are not dense in the ideal. We will first focus on the class of ideals of a uniform Roe algebra in which finite propagation operators are dense, and show that these ideals can be described geometrically in terms of coarse structure and invariant open subsets of the unit space of the Skandalis-Tu-Yu groupoid. We show that if the metric space has Yu’s property A, then all ideals are geometric. We introduce a notion of ghostly ideal and partial property A to investigate the ideal structure of the uniform Roe algebra for a general metric space beyond the scope of Yu’s property A. This talk is based on joint works with Xiaoman Chen and Jiawen Zhang.