Tag Archives: property A

The Big Question About Limit Operators II

In the first post in this series, I gave some background to the “Big Question” on limit operators which it appears that Lindner and Seidel have solved for the case of free abelian groups.  In the next couple of posts I want to sketch some of the key ideas of their proof and to explore to what extent it can also be generalized to all exact groups (in the same way that I generalized the basic theory of limit operators to all exact groups in my 2005 paper).

There are two components to the L-S argument, it seems to me.

  • a localization property for the “lower norm” of a finite propagation operators, and
  • a “condensation of singularities” argument.

In this post we’ll look at the first of those. Continue reading

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

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