# Property A and ONL, after Kato

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

# Contract signed for “Winding Around”

So I signed the contract last week for “Winding Around”, my book based on the course I taught in the MASS geometry/topology track last year.  It will appear in the American Mathematical Society’s Student Mathematical Library series, and the manuscript is due to be delivered to them on April 1st – I leave it to you whether or not you think this is an auspicious day!   The book centers around the notion of “winding number” (hence “Winding Around”) and uses that as a peg on which to hang a variety of topics in geometry, topology and analysis — finishing up, in the final chapter, with the Bott periodicity theorem considered as one possible high-dimensional generalization of the winding number notion.

The intended audience is an undergraduate one (there was skepticism from some of the AMS readers about this, but I told them the MASS students made it through okay!) and the tone is, I hope, entertaining and discursive.  As I say in the introduction, “Winding around is a description of the book’s methodology as well as of its subject-matter.”

# Metric approach to limit operators VI

I thought I’d finished this sequence of posts with number five, but then I spent a little time talking with Jerry Kaminker and Rufus Willett and I think that I understood two things: first, how to formulate the limit point construction more cleanly and, second, the “symmetry breaking” role of the ultrafilters which is not clear in what I had written so far.  Read on. Continue reading

# Metric approach to limit operators V

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