Roe algebras and quasi-local operators
Roe algebras are certain large C*-algebras which encode coarse information about a metric space, and which play a role in index theory. The uniform Roe algebra of a metric space X with bounded geometry (typically a finitely generated group with the word metric) consists of operators on l2(X) which can be approximated by those with finite propogation. Quasi-local operators are ones which approximately satisfy the relation of being approximated by finite propogation operators, and a natural question is whether each of these is in the Roe algebra (i.e., actually approximated by finite propogation operators). I will describe joint work with Ján Špakula where we settle this question for a large class of spaces X.