Tag Archives: property A

Metric approach to limit operators II

Following on from my earlier post on the Spakula-Willett paper, let my try to summarize sections 5 and 6.  These parts produce, for their generalized notion of limit operator, an equivalent of how the classical limit operator theory looked prior to the Lindner-Seidel paper earlier this year.

Thus the main result of these parts is the following Theorem: A band dominated operator is Fredholm if and only if all its limit operators are invertible and there is a uniform bound on the norms of the inverses of all the limit operators. Continue reading

Metric approach to limit operators

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

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