# Ghostbusting and Property A

Let $$X$$ be a bounded geometry discrete metric space.  Guoliang Yu defined a ghost to be an element of the Roe algebra $$C^*_u(X)$$ that is given by a matrix $$T_{xy}$$ whose entries tend to zero as $$x,y\to\infty$$.

The original counterexamples of Higson to the coarse Baum-Connes conjecture were noncompact ghost projections on box spaces derived from property T groups.  On the other hand, all ghost operators on a property A space are compact.

In Ghostbusting and Property A, Rufus Willett and I show that all ghosts on $$X$$ are compact if and only if $$X$$ has property A.  (Appropriately enough, on a space without property A we construct ghosts using the spectral theorem.) The paper will appear in the Journal of Functional Analysis.

Question: To what characterization of ordinary amenability does this correspond?