A recent paper on the ArXiv (see bibliography below) is entitled “An Affirmative Answer to the Big Question about Limit Operators”. I want to do a series of posts about this paper. In this one I will try to explain the background, at least in the most elementary (Hilbert space) case. In this introduction I will begin by re-expressing matters in the language of coarse geometry, following my paper (also in the bibliography below).

The basic setting is a discrete group \(\Gamma\) with a (left-invariant) word metric; in the usual literature about limit operators this group is \(\mathbb Z\) or \({\mathbb Z}^n\), but there is no particular need for this restriction. Let \(A\) denote the rough algebra of \(\Gamma\), that is the C*-algebra of operators on \(H=\ell^2(\Gamma)\) generated by \(\ell^\infty(\Gamma)\) together with the right translations by elements of \(\Gamma\); equivalently, the closure of the bounded finite propagation operators on \(H\). If \(T\in A\), then all the translates \(L_\gamma T L_\gamma^*\) also belong to \(A\), and indeed it is easy to see that they form a precompact subset of \(A\) in the strong (or *-strong) operator topology. The set of *-strong limit points of this subset is called the \emph{operator spectrum} of \(A\) and denoted \(\sigma_{op}(A) \). Continue reading