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 patrition 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.

 

 

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