Ghostbusting and Property a John Roe and Rufus Willett

Let X be a metric space (we may allow +∞ as a value for some distances in X). We say that X has bounded geometry if, for each R > 0, there is a natural number N such that every ball of radius R in X contains at most N points. (In particular, X is discrete.) In this paper, we will consider bounded geometry metric spaces in this sense. Let X be such a space, and let `(X) denote the usual Hilbert space of square summable functions on X with fixed orthonormal basis {δx | x ∈ X} of Dirac masses. Let B(`(X)) denote the C∗-algebra of bounded operators on `(X). If T is an element of B(`(X)), then T can be uniquely represented as an X-by-X matrix (Txy)x,y∈X , where