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.

What I write below is not in the paper but I think is equivalent to it. Let \(W\) be a compact Hausdorff space. Suppose that for each \(w\in W\) there is assigned a uniformly discrete (all nonzero distances \(\ge 1\)) metric space \(X_w\) such that the following is true:

- for each \(w\in W\) and each natural number \(n\) there are a neighborhood \(U\) of \(w\), a finite metric space \(F\), and a real \(r>n\), such that there are points \(x(w)\in X_w\) and bijections \(\phi_x\colon B_{X_w}(x(w);r)\to F \) which are within \(2^{-n}\) of being isometries (in the Gromov-Hausdorff sense), and moreover
- these “trivializations” are compatible for increasing \(n\), in the sense that if \(n’>n\) as above, then \(x'(w) = x(w)\) for all \(w\in U\cap U’\) and there is an injection \(F\to F’\) which is within \(2^{-n}\) of being an isometry and within \(2^{-n}\) of making the obvious diagram of maps \(B_{X_w}(x(w);\rho)\to F’ \) (defined for \(x\in U\cap U’\) and \(\rho=\min\{r,r’\}\)) commute.

Then we will say that the \(X_w\) define a *metric bundle* over \(W\). The corresponding collection of Hilbert spaces, \(\ell^2(X_w) \), will then be a continuous field of Hilbert spaces over \(W\) (we use the finite “local trivializations” to say what the continuous sections are).

My understanding of the SW paper, at least in the Hilbert space case, is that they now prove the following: Suppose that \(X\) is a uniformly discrete metric space and let \(\partial X\) be its Stone-Cech boundary. To each \(\omega\in\partial X\) is then assigned the *limit space* \(X_\omega\) (see post I of this series) and to each \(T\in C^*(X)\) is assigned the family \((T_\omega)_{\omega\in\partial X} \) of *limit operators* on \(\ell^2(X_\omega) \).

**Lemma** (I think): In the above situation:

- The family of limit spaces \(X_\omega\) constitute a metric bundle.
- If \(X\) has metric sparsification, the \(X_\omega\) all have metric sparsification uniformly (i.e., their coarse disjoint union has MS)
- The family of limit operators \(T_\omega\) is a strongly continuous family of operators on the Hilbert bundle \(\ell^2(X_\omega\)), which are uniformly band dominated.

We have now replaced the usual operator spectrum for a group (the range of a strongly continuous map from \(\partial\Gamma\) to the algebra of BDOs on a *fixed* space) with the range of a strongly continuous *section* of a bundle of algebras of BDOs. This change having been accomplished, the rest of the argument proceeds as before. Just a word about the role of the *translations* which were crucial in the previous argument. In the above definition of a metric bundle let \(U_n(\omega)\) be a neighborhood of \(\omega\) corresponding to the natural number \(n\) in the definition. The intersection \(G_\omega\) of all the \(U_n\) will then be a \(G_\delta\) set containing \(\omega\) (in fact it turns out to be closed in the example at hand) and for all \(\alpha\in G(\omega) \) the spaces \(X_\alpha\) are *isometric* – this is the content of Lemma 7.11 in the paper. But these isometries do not respect the choice of basepoint \(x(\omega)\) and the role of “translation” is now played by looking at the same space with a different basepoint.