# My talk at BIRS

I gave a talk yesterday (August 8th, 2013) on Ghostbusting and property A.  Thanks to the technology system at BIRS you can watch the talk on video here.

The paper has now been accepted for the Journal of Functional Analysis.

# Schur multipliers and ideals in the translation algebra

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

# 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?