# Metric approach to limit operators V

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