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