# Maximal Roe algebras, part 2

Let $$X$$ be a bounded geometry metric space.  At the end of the previous post, we observed that if $$\pi \colon {\mathbb C}[X] \to {\mathfrak B}(H)$$ is a Hilbert space representation of the translation algebra of $$X$$, then any unit vector in the range of one of the projections $$\pi(V_{x,x})$$ corresponding to a point of $$X$$ generates a subrepresentation isomorphic to the regular one.  It follows that if $$\pi$$ does not contain a copy of the regular representation, then the projections $$\pi(V_{x,x})$$ must be zero for every $$x\in X$$.

Surprisingly enough, such representations do exist! Continue reading

# Thinking about “maximal Roe algebras”

One of the things that has happened in coarse geometry while I was busy being department chair is a bunch of papers about “maximal Roe algebras” (some references at the end). Of course these are objects that I feel I ought to understand, so I spent some time trying to figure out the basics.

Let $$X$$ be a bounded geometry uniformly discrete metric space.  (Something like bounded geometry seems to be necessary, for a reason that I’ll explain below.)  We know how to form the translation algebra $${\mathbb C}[X]$$ (the *-algebra of finite-propagation matrices on $$X$$ ), and this has an obvious representation (the regular representation) on $$\ell^2(X)$$.  Then the usual version of the (uniform) Roe algebra is just the C*-algebra obtained by completing $${\mathbb C}[X]$$ in this representation.  Because it involves only the regular representation we may call this the reduced Roe algebra (in analogy to the group case). Continue reading