Writing the Ghostbusting paper sent me back to the literature on “ideals in the Roe algebra” and in particular to this paper
Chen, Xiaoman, and Qin Wang. “Ideal Structure of Uniform Roe Algebras of Coarse Spaces.” Journal of Functional Analysis 216, no. 1 (November 1, 2004): 191–211. doi:10.1016/j.jfa.2003.11.015.
which contains (among other things) the following pretty theorem: Let \(X\) be a (bounded geometry discrete) coarse space, and let \(\phi\in\ell^\infty(X\times X)\) be a function with controlled support. Then the Schur multiplier
\[ S_\phi\colon C^*_u(X) \to C^*_u(X) \]
maps any (closed, two-sided) ideal of \(C^*_u(X)\) into itself. Continue reading