# Matt Wiersma on exotic group C*-algebras

Recently Matt Wiersma from Waterloo spoke in our seminar about some of his work related to “exotic group C*-algebras”.  A more detailed account is on the arXiv.  I thought I would try to write up some of what I learned (probably, as usual, this is the most elementary points, but it was new to me).

What is an exotic group C*-algebra?  It is a completion of the group algebra which is different from the two standard examples (maximal and reduced) that we describe in C*-algebra courses.  Oversimplifying, we might make an analogy with compactifications of a locally compact Hausdorff space.   There is always a minimal one (one-point compactification) and a maximal (Stone-Cech), but there are also plenty of other things in between.  Analogously, in the case where a group $$\Gamma$$ is non amenable, one might imagine that there should be many other C*-completions of $${\mathbb C}\Gamma$$ lying between the maximal and the reduced C*-algebras.   (Whether, in fact, there exists any group for which $${\mathbb C}\Gamma$$ has exactly two distinct completions appears to be an open question.)

# Property A and ONL, after Kato

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

# 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:

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 $$\beta G$$).