# Maximal Roe algebras, part 4

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.