Tag Archives: property A

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?

 

Permanence properties in coarse geometry

I hate to think how long it has been since I last posted here. My apologies – it has been a difficult summer for various non-mathematical reasons. Anyhow, trying to get back on track let me mention a survey article that Erik Guentner sent me called “Permanence properties in coarse geometry”. What Erik means by “permanence properties” is statements like “the property of having finite asymptotic dimension is closed under group extensions”. Many statements of this kind, for a variety of coarse properties (asymptotic dimension, embeddability in Hilbert space, property A/exactness, etc) have by now been proved and this is a very nice survey bringing together general techniques for obtaining such results with specific applications.

JohnR

More about characterizations of exactness

Following up an earlier post with some notes on the three papers below:

All of these papers focus on the question of characterizing in “homological” terms what it is for a discrete group \( G \) to be exact (or, more generally, to act amenably on some compact space — it is known that exactness is equivalent to the amenability of the action of \( G \) on its Stone-Cech compactification \( \beta G \)).

Continue reading

Various characterizations of exactness

This post is a place to list a number of papers that have recently appeared on the arXiv which reformulate the notion of exactness for groups (or property A for spaces or amenable actions) in different ways:

I’ll post more later about the relations between these.