# Maximal Roe algebras, part 3

It is a well-known fact that if a group $$\Gamma$$ is amenable then the canonical map $$C^*_{\max}(\Gamma) \to C^*_r(\Gamma)$$ is an isomorphism.  In this post I’ll follow Spakula and Willett in proving the coarse analog of this statement: for a property A space the canonical map from the maximal to the reduced Roe algebra is an isomorphism.  (The converse is unknown, unlike in the group case.) Continue reading

# 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.

# Property A and large scale paracompactness

Jerzy Dydak and his collaborators have published a very interesting series of papers recently, whose overall theme is that property A is the large scale analog of paracompactness.  The papers that I have seen on the arXiv are these:

Cencelj, M., J. Dydak, and A. Vavpetič. “Asymptotic Dimension, Property A, and Lipschitz Maps.” Revista Matemática Complutense 26, no. 2 (July 1, 2013): 561–571. doi:10.1007/s13163-012-0102-2.

Cencelj, M., J. Dydak, and A. Vavpetič. Coarse Amenability Versus Paracompactness. ArXiv e-print, August 13, 2012. http://arxiv.org/abs/1208.2864.

Dydak, Jerzy. Coarse Amenability and Discreteness. ArXiv e-print, July 15, 2013. http://arxiv.org/abs/1307.3943.

Dydak, Jerzy, and Atish Mitra. “Large Scale Absolute Extensors.” arXiv:1304.5987 (April 22, 2013). http://arxiv.org/abs/1304.5987.

Recall that a Hausdorff topological space $$X$$ is paracompact if every open cover of $$X$$ has a refinement to a locally finite open cover.  (It is metacompact, or weakly paracompact, if every open cover has a refinement to a pointwise finite open cover.)  Most often though one applies paracompactness via the existence of partitions of unity: $$X$$ is paracompact if and only if there exists a (locally finite) partition of unity subordinate to any open cover.  The papers above elucidate what should be the “coarse version” of paracompactness both in the “covering” and the “partition of unity” interpretations, and in both cases they relate it to property A. Continue reading

# 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?