Category Archives: Research

Metric approach to limit operators II

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

Metric approach to limit operators

In a couple of posts earlier this year (post I and post II) I started getting to grips with the paper An Affirmative Answer to the Big Question on Limit Operators by Lindner and Seidel. The first of these posts gives some background to limit operator theory and to what the big question is that Lindner and Seidel solved for the case of the group  {\mathbb Z}^n .  Circumstances prevented me continuing the post series, but I suggested that the arguments should work just as well for any group whose underlying metric space has property A (that is, for any exact group).

Now, I am reading the paper A metric approach to limit operators by Jan Spakula and Rufus WIllett, in which they carry out this kind of idea in much greater generality than I had been imagining.   Following the limit operator literature, they don’t simply confine their attention to the Hilbert space as I did; their arguments work on \( \ell^p(X;E) \) with \( 1 < p < \infty \) and coefficients in an auxiliary Banach space \(E\).   What’s more, their notion of limit operator does not even require an underlying group structure (and so the Fredholm theory that they develop will work for all bounded geometry discrete metric spaces that have property A).   In this post I want to explain their generalized definition of “limit operator”, as a preliminary to getting into the analysis proper. Continue reading

Marc Rieffel 75th birthday proceedings

RieffelI just received a message about the publication of a set of conference proceedings honoring Marc Rieffel’s 75th birthday.  The proceedings originate from a program at the Fields Institute that was held last year (June, 2013).

To quote the web site of the proceedings: This special issue is a tribute to Marc A. Rieffel, marking (approximately) his 75th birthday. It is the outgrowth of a Focus Program on Noncommutative Geometry and Quantum Groups that took place at the Fields Institute for Research in Mathematical Sciences in Toronto, Canada, in June, 2013. Marc Rieffel has been one of the most influential researchers in the world in the areas of noncommutative geometry and quantum groups. He has had over 30 PhD students and over 80 “mathematical descendants”. Among his major contributions were the introduction of Morita equivalence as a fundamental notion in noncommutative geometry and the classification of C*-algebras, the introduction of stable rank as a basic invariant of C*-algebras, the introduction of strict deformation quantization to construct new examples of quantum groups, and the analysis of the metric structure of noncommutative geometries. The papers in this special issue reflect the wide range of his contributions to mathematics as well as the great esteem in which he is held by the world mathematical community.

The proceedings may be downloaded here.

Photo is copyright Mathematische Forschungsintitut Oberwolfach and licensed under Creative Commons.

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