In a couple of posts earlier this year (post I and post II) I started getting to grips with the paper An Affirmative Answer to the Big Question on Limit Operators by Lindner and Seidel. The first of these posts gives some background to limit operator theory and to what the big question is that Lindner and Seidel solved for the case of the group 
Now, I am reading the paper A metric approach to limit operators by Jan Spakula and Rufus WIllett, in which they carry out this kind of idea in much greater generality than I had been imagining. Following the limit operator literature, they don’t simply confine their attention to the Hilbert space as I did; their arguments work on \( \ell^p(X;E) \) with \( 1 < p < \infty \) and coefficients in an auxiliary Banach space \(E\). What’s more, their notion of limit operator does not even require an underlying group structure (and so the Fredholm theory that they develop will work for all bounded geometry discrete metric spaces that have property A). In this post I want to explain their generalized definition of “limit operator”, as a preliminary to getting into the analysis proper.
So, let \(X\) be a bg discrete metric space. Recall the notion of partial translation: a partial translation is a partially defined map \(t\) from \(X\) to \(X\), such that the distance \(d(x,t(x))\) is uniformly bounded. If \(\omega \in \beta X\) is a point of the Stone-Cech boundary – aka an unltrafilter, and if the domain of \(t\) has \(\omega\) in its closure (i.e., \(t\) is densely defined on \(\omega\) ), then the limit \(t(\omega)=\lim_{x\in\omega} t(x) \in \partial X \) exists. The authors call the collection of all points of \(\partial X\) that can be reached from \(\omega\) in this way the limit space at \(\omega\), and they denote it \(X(\omega)\). This is a metric space, with distance defined by the ultralimit:
\[ d(s(\omega),t(\omega)) = \lim_{x\in\omega} d(s(x),t(x)). \]
If \(X\) is a discrete group, the obvious construction using group elements as (partial) translations provides an isometry from \(X\) to \(X(\omega)\) (whatever \(\omega\) is).
Now given a finite propagation matrix \(T\) on \(X\) one can pass to ultralimits on the matrix entries also and obtain a finite propagation matrix \(T_\omega\) on \(X(\omega)\) for every \(\omega\). (If the coefficient space \(E\) is infinite dimensional one needs an auxiliary hypothesis (“richness”) at this point, but I will not go into this, which is standard in the limit operator business.) In Section 4 of the paper one finds the analysis which shows that this procedure is bounded in norm and defines a homomorphism, so that it passes to the closure: that is the algebra of “band dominated operators”. At this point therefore Willett and Spakula have defined a notion of limit operators or operator spectrum on any space \(X\), which reduces to the standard one when that space is a discrete group.
More next time…