In the previous post I sketched out the condensation of singularities argument which finishes the proof under the assumption that the underlying metric space \(X\) is a group. In this case all limit operators act on the same Hilbert space, namely \(\ell^2(X)\), and the *weak compactness* of the set of all limit operators plays a critical role.

In the more general situation described by Spakula and Willett, each limit operator (say at a boundary point \(\omega\)) acts on its *own* Hilbert space \(\ell^2(X(\omega))\). In order to bring this situation under sufficient control to continue to make the weak compactness argument, we are going to need some kind of bundle theory. Continue reading