# The Big Question about Limit Operators I

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  operator spectrum of $$A$$ and denoted $$\sigma_{op}(A)$$. Continue reading