Category Archives: Research

The Big Question About Limit Operators II

In the first post in this series, I gave some background to the “Big Question” on limit operators which it appears that Lindner and Seidel have solved for the case of free abelian groups.  In the next couple of posts I want to sketch some of the key ideas of their proof and to explore to what extent it can also be generalized to all exact groups (in the same way that I generalized the basic theory of limit operators to all exact groups in my 2005 paper).

There are two components to the L-S argument, it seems to me.

  • a localization property for the “lower norm” of a finite propagation operators, and
  • a “condensation of singularities” argument.

In this post we’ll look at the first of those. Continue reading

The Big Question about Limit Operators I

A recent paper on the ArXiv (see bibliography below) is entitled “An Affirmative Answer to the Big Question about Limit Operators”.   I want to do a series of posts about this paper.  In this one I will try to explain the background, at least in the most elementary (Hilbert space) case.  In this introduction I will begin by re-expressing matters in the language of coarse geometry, following my paper (also in the bibliography below).

The basic setting is a discrete group \(\Gamma\) with a (left-invariant) word metric; in the usual literature about limit operators this group is \(\mathbb Z\) or \({\mathbb Z}^n\), but there is no particular need for this restriction.  Let \(A\) denote the rough algebra of \(\Gamma\), that is the C*-algebra of operators on \(H=\ell^2(\Gamma)\) generated by \(\ell^\infty(\Gamma)\) together with the right translations by elements of \(\Gamma\); equivalently, the closure of the bounded finite propagation operators on \(H\).  If \(T\in A\), then all the translates \(L_\gamma T L_\gamma^*\) also belong to \(A\), and indeed it is easy to see that they form a precompact subset of \(A\) in the strong (or *-strong) operator topology.  The set of *-strong limit points of this subset is called the \emph{operator spectrum} of \(A\) and denoted \(\sigma_{op}(A) \). Continue reading

Paper on sheaves of C*-algebras and K-homology published

After a busy day giving final exams in the MASS program it was nice to learn today that my paper with Paul Siegel about sheaves of C*-algebras has appeared in the Journal of K-Theory.  The link for the published version is

http://journals.cambridge.org/repo_A91rlWyM

This paper arose from some discussions when Paul was writing his thesis.  We were talking about the “lifting and controlling” arguments for Paschke duals that are used in the construction of various forms of operator-algebraic assembly maps (an early example is the one that appears in my paper with Nigel on the coarse Baum-Connes conjecture, which asserts that the quotient \( D^*(X)/C^*(X) \) of the “controlled” pseudolocal by the “controlled” locally compact operators does not depend on the assumed “control”).  At some point in these discussions I casually remarked that, “of course”, what is really going on is that the Paschke dual is a sheaf.  Some time later I realized that what I had said was, in fact, true.  There aren’t any new results here but I hope that there is some conceptual clarification.   (There is an interesting spectral sequence that I’ll try to write about another time, though.)

“Finite part of operator K-theory” V

This is the final one of a series of posts about the manuscript “Finite Part of Operator K-theory for Groups Finitely Embeddable into Hilbert Space and the Degree of Non-rigidity of Manifolds” (ArXiv e-print 1308.4744. http://arxiv.org/abs/1308.4744) by Guoliang Yu and Shmuel Weinberger. In previous posts (most recently this one) I’ve described their main result about the assembly map, what I call the Finite Part Conjecture, and explained some of the methodology of the proof for the large class of groups that they call “finitely embeddable in Hilbert space”.  Now I want to explain some of the consequences of the Finite Part Conjecture. Continue reading

“Finite part of operator K-theory” IV

Continuing this series (earlier posts here, here and here) on the paper of Weinberger and Yu, I’m expecting to make two more posts: this one, which will say something about the class of groups for which they can prove their Finite Part Conjecture, and one more, which will say something about what can be done with the conjecture once one knows it. Continue reading