On the quotients of coarse structures and Roe algebras

We study the quotients of the Roe algebras of uniformly locally finite coarse spaces which are not necessarily metrizable. We introduce the coarse quotient structure of a coarse structure associated to an ideal and prove that in certain cases, the quotient of the Roe algebra by the ideal induced by the ideal coarse structure is isomorphic to the Roe corona of the coarse quotient structure. We establish that the functor which maps a coarse subspace to the quotient of Roe algebra associated to any ideal actually forms a cosheaf with values in the category of C^⁎-algebras. Moreover, we prove a coarse Mayer-Vietories theorem for the relative Roe algebra associated to a coarse subspace.