Hiroki Sato’s paper on the equivalence of property A and operator norm localization was recently published in Crelle ( “Property A and the Operator Norm Localization Property for Discrete Metric Spaces.” Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal) 2014 (690): 207–16. doi:10.1515/crelle-2012-0065.) and I wanted to write up my understanding of this result. It completes a circle of proofs that various forms of “coarse amenability” are equivalent to one another, thus underlining the significance and naturalness of the “property A” idea that Guoliang came up with twenty years ago. Continue reading
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