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.

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