# 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