Hiroki Sato’s paper on the equivalence of property A and operator norm localization was recently published in *Crelle* ( “Property A and the Operator Norm Localization Property for Discrete Metric Spaces.” *Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)* 2014 (690): 207–16. doi:10.1515/crelle-2012-0065.) and I wanted to write up my understanding of this result. It completes a circle of proofs that various forms of “coarse amenability” are equivalent to one another, thus underlining the significance and naturalness of the “property A” idea that Guoliang came up with twenty years ago. Continue reading

# Tag Archives: amenable

# 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?

# Around soficity

Andreas Thom just posted the article [1005.0823] About the metric approximation of Higman’s group on the arXiv today. It is quite short with a specific result about Higman’s group, but the introduction was most helpful to me in learning a bit about the ideas related to “soficity” of groups. It refers to another interesting paper: Elek, Gábor, and Endre Szabó. “Hyperlinearity, essentially free actions and L2-invariants. The sofic property.” Mathematische Annalen 332, no. 2 (4, 2005): 421-441.

It seems that these authors use some words like “hyperlinear” and “amenable action” in a sense different to that which is common to us in Baum-Connes land. for instance, for Elek-Szabo, the trivial action of a group on a point is *always* amenable.

# More about characterizations of exactness

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

- Invariant expectations and vanishing of bounded cohomology for exact groups by Douglas and Nowak [DN]
- Amenable actions, invariant means and bounded cohomology by Brodzki, Niblo, Nowak and Wright [BNNW]
- A note on topological amenability by Monod. [M]

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 \( \beta G \)).

# 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:

- A cohomological characterisation of Yu’s Property A for metric spaces by Brodzki, Niblo and Wright.
- Invariant expectations and vanishing of bounded cohomology for exact groups by Douglas and Nowak.
- Amenable actions, invariant means and bounded cohomology by Brodzki, Niblo, Nowak and Wright.
- A note on topological amenability by Monod.

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