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
In my book Elliptic operators, topology and asymptotic methods (both the first and the second editions) I give a discussion of the representation theory of the groups Spin and Pin which was based (as far as I can now remember) on some notes that I took when I attended Adams’ famous course on the exceptional Lie groups, as a Part III student in 1981. I no longer seem to have those, unfortunately (although meanwhile a version of Adams’ own notes on his approach has been published by University of Chicago Press). Meanwhile, in 2010 Darij Grinberg pointed out on Math Overflow that the argument I gave was garbled: see this link. In this post I want to explain what is garbled and how the useful part of the argument can be recovered. 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