# 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.)

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.