Tag Archives: property A

Metric approach to limit operators

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In a couple of posts earlier this year (post I and post II) I started getting to grips with the paper An Affirmative Answer to the Big Question on Limit Operators by Lindner and Seidel. The first of these posts gives some background to limit operator theory and to what the big question is that Lindner and Seidel solved for the case of the group {\mathbb Z}^n .  Circumstances prevented me continuing the post series, but I suggested that the arguments should work just as well for any group whose underlying metric space has property A (that is, for any exact group).

Now, I am reading the paper A metric approach to limit operators by Jan Spakula and Rufus WIllett, in which they carry out this kind of idea in much greater generality than I had been imagining.   Following the limit operator literature, they don’t simply confine their attention to the Hilbert space as I did; their arguments work on \( \ell^p(X;E) \) with \( 1 < p < \infty \) and coefficients in an auxiliary Banach space \(E\).   What’s more, their notion of limit operator does not even require an underlying group structure (and so the Fredholm theory that they develop will work for all bounded geometry discrete metric spaces that have property A).   In this post I want to explain their generalized definition of “limit operator”, as a preliminary to getting into the analysis proper. Continue reading →

The Big Question About Limit Operators II

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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

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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

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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 →