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.)

At any rate, various constructions for specific examples of exotic group C*-algebras have appeared recently, and Matt’s talk reviewed several of these and then discussed some theorems about them.

#### Construction 1

These are the examples due to Nate Brown and Erik Guentner. Let \(\Gamma\) be a discrete group (standing assumption) and let \(D\) be an ideal (it does not have to be closed) in \(\ell^\infty(\Gamma)\). A representation \(\pi\colon \Gamma \to {\mathfrak B}(H)\) is called a \(D\)-representation if there is a dense set of \(\xi\in H\) for which the matrix coefficient

\[ g \mapsto \langle \xi,\pi(g)\xi \rangle \]

belongs to \(D\). (Remark – of course if \(D\) happens to be a norm closed ideal that is equivalent to saying that *all* matrix coefficients belong to \(D\), but that is not the most interesting case.) Then we define \(C^*_D(\Gamma)\) by completing in the norm given by the universal \(D\)-representation (ie the sup over all such norms)

**Example ** The regular representation is a \(D\)-representation for \(D=c_0(\Gamma)\) or for \(D=\ell^p(\Gamma) \), \(1\le p < \infty\).

**Theorem** (Brown-Guentner) We have \(C^*_{\max}(\Gamma) = C^*_{c_0}(\Gamma) \) if and only if \(\Gamma\) has the Haagerup property.

**Theorem** (Otayasu) For the free group on \(d\ge 2\) generators, the algebras \(C^*_{\ell^p}(F_d)\) are all distinct for \(2\le p < \infty \). Every group with a nonabelian free subgroup also has this property.

#### Construction 2

A C*-algebra is called *residually finite dimensional* if it has a separating family of representations on finite-dimensional Hilbert spaces. A group \(\Gamma\) is *maximally almost periodic *if there is a family of finite-dimensional representations that separate points of \(\Gamma\) (it is equivalent to say that \(\Gamma\) embeds in a compact group). Clearly either of the conditions

- \(\Gamma\) is residually finite
- \(C^*(\Gamma)\) is residually finite-dimensional

implies that \(\Gamma\) is MAP, but neither of the bulleted conditions implies the other in general. Bekka proved that for *amenable* groups, MAP for the group implies RFD for the C*-algebra; but he also gave many examples of non amenable groups that are residually finite but whose maximal C*-algebras do not have MAP; for instance \(SL_n({\mathbb Z})\) for \(n\ge 3\) and \(SP_n({\mathbb Z})\) for \(n\ge 2\). For such a group \(C^*_{\mathcal F}(\Gamma)\) is an exotic C*-algebra, where the subscript denotes completing relative to the family of all *finite dimensional * representations of \(\Gamma\). (This differs from the reduced C*-algebra because the reduced C*-algebra for a non-amenable group has no finite-dimensional representations.)

**Reference: **Bekka, Mohammed E. B. “Amenable Unitary Representations of Locally Compact Groups.” *Inventiones Mathematicae* 100, no. 1 (December 1990): 383–401. doi:10.1007/BF01231192.

#### Construction 3

This one is also due to Bekka et al. Let \(G\) be a locally compact group. Then \(L^2(G)\) can also be considered as a representation of \(G_\delta\), the discrete group obtained by forgetting the topology of \(G\). We can define an algebra \(C^*_\delta(G_\delta)\) by completing the group algebra of \(G_\delta\) in this representation.

**Theorem** If \(G\) is a non solvable connected Lie group, then \(C^*_\delta(G_\delta)\) is an exotic group C*-algebra.

#### Wiersma’s results

The above represented a good part of the talk, but it is really just an introduction to Wiersma’s results. He has constructed examples of exotic group C*-algebras which are extremely far from being nuclear: they do not even have the *local lifting property* (Chapter 13 of Nate and Taka’s book). Notice that the maximal C*-algebra of e.g. the free group *does* have this property (a result of Kirchberg). The requirements are that \(\Gamma\) be an algebraic subgroup of an amenable locally compact \(G\), and that \(\pi\) be a representation of \(\Gamma\) generating an exotic C*-algebra, such that \(\pi\otimes\lambda_G \) is weakly contained in \(\pi\). Examples satisfying these conditions can be found among all three families of exotic C*-algebra constructions listed above.