It is a well-known fact that if a group \(\Gamma\) is amenable then the canonical map \(C^*_{\max}(\Gamma) \to C^*_r(\Gamma) \) is an isomorphism.  In this post I’ll follow Spakula and Willett in proving the coarse analog of this statement: for a property A space the canonical map from the maximal to the reduced Roe algebra is an isomorphism.  (The converse is unknown, unlike in the group case.)

Let’s remember how the group version is proved (see the discussion in Nate and Taka’s book referenced below).  Amenability gives a sequence of finitely supported positive-definite functions on \(\Gamma\) tending pointwise to 1. These define a sequence of multipliers \( \{M_n\} \) that form an approximate identity, both on the maximal and the reduced algebras.  Now consider the commutative diagram

\[\begin{array}{ccc} C^*_{\max}(\Gamma)  & \to & C^*_{\max}(\Gamma)\\\downarrow &&\downarrow\\ C^*_{r}(\Gamma)  & \to & C^*_{r}(\Gamma)  \end{array} \]

where the horizontal arrows are the multipliers \( M_n\) and the vertical ones are the reduction homomorphism \(\lambda\).  If \(a \in C^*_{\max}(\Gamma) \) has \(\lambda(a)=0\), then (using the commutativity of the diagram and the fact that \(\lambda\) is injective on \({\mathbb C}[\Gamma]\)) one sees that \( M_na = 0 \in C^*_{\max}(\Gamma) \) for all \(n\), and therefore \(a=0\) since the \(M_n\) form an approximate unit.

Spakula and Willett point out that one can deploy basically the same argument in the coarse case, using the multipliers which come from an approximate unit of controlled-support normalized positive type kernels (which is one version of the definition of property A, see Lemma 11.37 in Lectures on Coarse Geometry.)  The only thing one needs to show is that these multipliers act on the maximal Roe algebra (as completely positive contractions). To see this, recall that if one thinks of property A in terms of controlled partitions of unity \(\{\phi_i\} \) with \(\sum \phi_i^2=1\), then the corresponding multipliers \(M\) can be written

\[ M(a) = \sum_i \phi_i a \phi_i. \]

Using the bounded geometry, one can group together the \(\phi_i\) into finitely many batches with disjoint support, in such a way as to yield the following: for any fixed \(S>0\) there exist finitely many \(\ell^\infty\) functions \(\psi_j\) such that

\[M(a) = \sum_j \psi_j a \psi_j \]

for any \(a\) of propagation at most \(S\).  This formula now shows that the multipliers \(M\) define unital completely positive maps from \( {\mathbb C}[X] \) to \(C^*_{\max}(X)\); such maps automatically extend to ucp maps from \(C^*_{\max}(X) \) to itself.

References
Brown, Nathanial P., and Narutaka Ozawa. C*-Algebras and Finite-Dimensional Approximations. American Mathematical Society, 2008.

Roe, John. Lectures on Coarse Geometry. Vol. 31. University Lecture Series. Providence, RI: American Mathematical Society, 2003. http://www.ams.org/mathscinet-getitem?mr=2761858.

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.