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! To get a feel for why, imagine restricting such a representation to the diagonal copy of \(\ell^\infty(X)\) inside the translation algebra; then we would obtain a representation of \(\ell^\infty(X)\) with the property that every element of \(c_0(X)\) maps to zero.  Such representations are built up from measures on the Stone-Cech corona \(\beta X \setminus X \) of \(X\).

What this suggests is that to obtain non-regular representations of \({\mathbb C}[X]\), we should look at constructions which are “supported near infinity”.  Here is a concrete example.  Suppose that \(X\) is a coarse disjoint union of finite subspaces \(X_i\) – for example, a “box space” – and that \(n_i\) is the number of elements in \(X_i\).  Each operator \(T\in {\mathbb C}[X]\) can be cut down to an operator \(T_i\) on \(X_i\), which can be regarded as an element of \(M_{n_i}({\mathbb C})\).  In this way we obtain a completely positive contraction

\[ {\mathbb C}[X] \to \prod_i M_{n_i}({\mathbb C}). \]

This is not a homomorphism because of the possibility that matrix elements of \(T\) may “tunnel” from one \(X_i\) to another.  But there can be only finitely many such “tunneling” elements.  Therefore the induced map

\[ {\mathbb C}[X] \to \prod_i M_{n_i}({\mathbb C})/ \bigoplus_i  M_{n_i}({\mathbb C}) \]

is a *-homomorphism.  It annihilates all the projections \(V_{x,x}\), so any representation of \({\mathbb C}[X] \) that factors through this map will  be disjoint from the regular representation.

Remark. Recall that, in the context of C*-algebras, the product \(\prod A_i\) refers to sequences of elements from \(A_i\) which are uniformly bounded in norm, and the sum \( \oplus A_i \) refers to such sequences which tend to zero in norm.  The “product modulo sum” construction above has appeared in many places, e.g. in Wassermann’s proof that quasidiagonality need not pass to the Calkin algebra, see Wassermann, Simon. 1991. “C*-Algebras Associated with Groups with Kazhdan’s Property T.” Annals of Mathematics 134(2):423–431. Retrieved March 26, 2013.