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?


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