Ghost ideals in uniform Roe algebras of coarse spaces

Let C_u*(X) be the uniform Roe algebra of a coarse space X with uniformly locally finite coarse structure. We show that an operator G in C_u*(X) is a ghost element if and only if the finite propagation operators in the principal ideal 〈G〉 are all compact operators. In contrast, if X is a discrete metric space with Yu's property (A), then any ideal in C_u*(X) is the closure of the finite propagation operators in the ideal.