In the previous post I sketched out the condensation of singularities argument which finishes the proof under the assumption that the underlying metric space \(X\) is a group. In this case all limit operators act on the same Hilbert space, namely \(\ell^2(X)\), and the weak compactness of the set of all limit operators plays a critical role.
In the more general situation described by Spakula and Willett, each limit operator (say at a boundary point \(\omega\)) acts on its own Hilbert space \(\ell^2(X(\omega))\). In order to bring this situation under sufficient control to continue to make the weak compactness argument, we are going to need some kind of bundle theory. Continue reading
In this post I’ll finally get to the “condensation of singularities” argument that was invented by Lindner and Seidel in the (free abelian) group context and generalized by Spakula and Willett to metric spaces. (Calling this “condensation of singularities” is my idea, but it does seem to me to get at what is going on. I can’t help feeling that there should be a way of replacing some of the explicit constructions with an abstract argument involving the Baire category theorem. But I have not yet been able to come up with one.) Continue reading
This is a continuation of my posts on the Spakula-Willett paper Metric approach to limit operators (see part I and part II). In this post I will talk about “lower norm witnesses” on spaces with property A. (This is quite close to what is done in my earlier post here, though using direct geometric tools rather than the functional analysis tricks I suggested, which only work in the Hilbert space case.) Then in the next post I will talk about the “condensation of singularities” argument that completes the proof. Continue reading
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
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 . 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