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.)