On Banach algebras of band-dominated operators and their order structure

Abstract

The goal of this paper is to study band-dominated operators on Banach spaces with Schauder basis with respect to uniformly locally finite metric spaces as well as the Banach algebras generated by them: the so called uniform Roe algebras. We investigate several kinds of isomorphisms between those Banach spaces (e.g., isomorphisms preserving norm, order, algebraic structure, etc) and prove several rigidity results on when a certain kind of isomorphism between the uniform Roe algebras implies that the base metric spaces are (bijectively) coarsely equivalent. Contents1. Introduction 1 2. Preliminaries 5 3. Properties of different bases in uniform Roe algebras 8 4. Regular uniform Roe algebra 12 5. Rigidity for order preserving equivalences 15 6. Property A and ghost operators 20 7. Rigidity for Banach algebra isomorphisms 25 8. Open questions 33 9. Appendix 34 References 36