Tag Archives: ghosts

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  operator spectrum of \(A\) and denoted \(\sigma_{op}(A) \). Continue reading

Schur multipliers and ideals in the translation algebra

Writing the Ghostbusting paper sent me back to the literature on “ideals in the Roe algebra” and in particular to this paper

Chen, Xiaoman, and Qin Wang. “Ideal Structure of Uniform Roe Algebras of Coarse Spaces.” Journal of Functional Analysis 216, no. 1 (November 1, 2004): 191–211. doi:10.1016/j.jfa.2003.11.015.

which contains (among other things) the following pretty theorem: Let \(X\) be a (bounded geometry discrete) coarse space, and let \(\phi\in\ell^\infty(X\times X)\) be a function with controlled support.  Then the Schur multiplier

\[ S_\phi\colon C^*_u(X) \to C^*_u(X) \]

maps any (closed, two-sided) ideal of \(C^*_u(X)\) into itself. Continue reading

Ghostbusting and Property A

Let \(X\) be a bounded geometry discrete metric space.  Guoliang Yu defined a ghost to be an element of the Roe algebra \( C^*_u(X) \) that is given by a matrix \(T_{xy}\) whose entries tend to zero as \(x,y\to\infty\).

The original counterexamples of Higson to the coarse Baum-Connes conjecture were noncompact ghost projections on box spaces derived from property T groups.  On the other hand, all ghost operators on a property A space are compact.

In Ghostbusting and Property A, Rufus Willett and I show that all ghosts on \(X\) are compact if and only if \(X\) has property A.  (Appropriately enough, on a space without property A we construct ghosts using the spectral theorem.) The paper will appear in the Journal of Functional Analysis.

Question: To what characterization of ordinary amenability does this correspond?