In this last post in this series, I want to explain the simplest example of a coarse space for which the maximal and reduced Roe algebras are not the same. Again, this follows the papers referenced below.
The example is again a box space \(X \require{AMSsymbols} = \square \Gamma = \bigsqcup \Gamma/\Gamma_n \) of a certain residually finite group with respect to a family of finite index normal subgroups. For any such space one can construct a homomorphism
\[ C^*_{\max}(\Gamma) \to C^*_? (\square\Gamma) \]
from the maximal group \(C^*\)-algebra of \(\Gamma\) to the maximal or reduced Roe algebras of the box space. The aim here is to prove that, for a suitable \(X\), the homomorphism into the maximal Roe algebra is injective, whereas the one into the reduced Roe algebra isn’t. Thus, the two Roe algebras cannot be the same.
Lemma 1: For any box space the homomorphism \(\alpha\colon C^*_{\max}(\Gamma) \to C^*_\max (\square\Gamma) \) is injective.
Proof Let \(X=\square\Gamma\). Following Oyono-Oyono and Yu, there is a short exact sequence
\[ 0 \to {\mathfrak K} \to C^*_{\max}(X) \to (\ell^\infty(X)/c_0(X))\rtimes_\max\Gamma \to 0. \]
This is because a given element of \({\mathbb C}[X]\) is nonzero only on finitely many “diagonals” of \(X\), and such “diagonals” are labeled by group elements modulo a finite amount of ambiguity at the “small end” of the box space. Clearly, now, the image \(\alpha(C^*_{\max}(\Gamma)\) is just \( {\mathbb C}\rtimes_\max\Gamma\) inside \( (\ell^\infty(X)/c_0(X))\rtimes_\max\Gamma \), and so \(\alpha \) is injective.
Lemma 2. Consider the special case where \(\Gamma=SL(2,{\mathbb Z} )\) and \(\{\Gamma_n\}\) is the family of congruence subgroups. Then the homomorphism \(\beta\colon C^*_{\max}(\Gamma) \to C^*_r (\square\Gamma) \) is not injective.
Proof The relevant point is that \(\Gamma\) has property \( (\tau)\) with respect to this family of subgroups, but does not have property T. It follows that the family of irreducible unitary representations of \(\Gamma\) that factor through one of the quotients \(\Gamma/\Gamma_n\) is not dense in the unitary dual \(\widehat{\Gamma}\). But the representation of \({\mathbb C}[\Gamma] \) that one gets by composing \(\beta\) with the regular representation of \(X\) is just the direct sum of the regular representations of \(\Gamma/\Gamma_n\). Hence it cannot be a faithful representation of \(C^*_\max(\Gamma)\).
References
Lubotzky, Alex. “What Is Property \((\tau)\)?” Notices of the American Mathematical Society 52, no. 6 (2005): 626–627.
Oyono-Oyono, Hervé, and Guoliang Yu. “K-theory for the Maximal Roe Algebra of Certain Expanders.” Journal of Functional Analysis 257, no. 10 (November 15, 2009): 3239–3292. doi:10.1016/j.jfa.2009.04.017.
Špakula, Ján, and Rufus Willett. “Maximal and Reduced Roe Algebras of Coarsely Embeddable Spaces.” Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal) 2013, no. 678 (2013): 35–68.