Twisted Roe algebras and their $K$-theory

Abstract: In this paper, we introduce a notion of twisted Roe algebra and a twisted coarse Baum-Connes conjecture with coefficients. We will study the basic properties of twisted Roe algebras, including a coarse analogue of the imprimitivity theorem for metric spaces with a structure of coarse fibrations. We show that the twisted coarse Baum-Connes conjecture with coefficients holds for a metric space with a coarse fibration structure when the base space and the fiber satisfy the twisted coarse Baum-Connes conjecture with coefficients. As an application, the coarse Baum-Connes conjecture holds for a finitely generated group which is an extension of coarsely embeddable groups.