Ghost ideals in uniform Roe algebras of coarse spaces

Abstract.

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.