“Operator K-theory” has appeared on AMS Open Math Notes

My final Penn State course (Spring 2017) was about K-theory and operator algebras – the connection between these two has been central to my mathematical life.  I wrote up lecture notes for this course, as has become usual for me.  I’m pleased to report that these have now appeared on the AMS Open Math Notes page.

The American Mathematical Society hosts AMS Open Math Notes,  which is “a repository of freely downloadable mathematical works in progress hosted by the AMS as a service to researchers, teachers and students.”

The Open Math Notes homepage continues  “These draft works include course notes, textbooks, and research expositions in progress. They have not been published elsewhere, and, as works in progress, are subject to significant revision.  Visitors are encouraged to download and use these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.”

 

 

Author interview on AMS “Book Ends”

Eriko Hironaka was kind enough to interview me for the AMS blog “Book Ends” whose logo is above. The interview focuses on Winding Around, but I suppose appropriately enough, it winds about a bit too, from “what got you started on writing” to “do you have advice for new authors”.  I enjoyed being able to share a bit through this piece. It begins:

What made you decide to write the book “Winding Around”? The spark for Winding Around was lit when I was about nine. My dad drew an incredibly convoluted simple closed curve (something like Figure 4.3 in the book), made a dot on the paper somewhere in the midst of the convolutions, and asked me, “Is that inside or outside the curve?”

If you want to read more, the whole piece is here.

 

Epsilon propagation and the Roe algebra

I remember with great pleasure giving the CBMS lectures that became the book Index Theory, Coarse Geometry and the Topology of Manifolds. I was at the beginning of a six month visiting appointment, delighting in the change to be with my family in this beautiful city, and energized by coming up with two lectures every day (and I mostly wrote the book as we went along – how nice to be young and energetic).

However this speedy process meant that one (or maybe more) over-optimistic statements slipped by. One in particular that has caused trouble over the years is the Remark after Lemma 3.5  In this Remark I define a notion of \(\epsilon\)-propagation (notation being as in the usual setup for Roe algebras):

Definition: The \(\epsilon\)-propagation of an operator \(T\) is the infimal \(R\) such that

\[ \|f\|\le 1, \ \|g\|\le 1, \ d(\text{Support}(f),\text{Support}(g))\le R \Longrightarrow  \|fTg\| \le \epsilon. \]

It is clear that an operator in the Roe algebra has finite \(\epsilon\)-propagation for all \(\epsilon\). It is also clear (though I didn’t point this out at the time) that the collection of operators having finite \(\epsilon\)-propagation for all \(\epsilon\), which some people call quasi-local operators, is a \(C^*\)-algebra. So the question arises: is it the same as the Roe algebra?

So here’s where my optimism came in. I causally remarked, “If \(X\) is large scale finite dimensional—by which I meant what we would now call finite asymptotic dimension—then the converse is the case”  (“the converse” being the statement that every quasi-local operator is in the Roe algebra). I thought that there would be a fairly obvious “large scale partition of unity” proof—and somehow never checked. There isn’t.  Mea culpa.  Over the years people proved the statement for \(\mathbb Z\) and then \({\mathbb Z}^n\), but these proofs used Fourier series and clearly didn’t generalize.

Until this year.  Back last summer I received a note from Aaron Tikuisis of Aberdeen asking about the status of the statement and saying that he had a proof.  This has now led to a preprint by Tikuisis and Spakula showing the equivalence not just for spaces of finite asymptotic dimension but also of “finite decomposition complexity” (see Guentner, Tessera and Yu, A notion of geometric complexity and its application to topological rigidity, Inventiones 189 (1012), 315-37).  They say it will be on the arXiv soon.  I just had another inquiry from a grad student about this misleading remark of mine and was happy, this time, to be able to point him to an honest proof.

If only I’d called it a “conjecture”!  Still, I am very happy that the question is resolved in such a nice way.  The paper uses some of the new ideas in classification, like nuclear dimension, though I have not read it in enough detail to be able to explain how.