Twisted Roe algebras and \(K\)-theory

In this paper, we introduce a notion of twisted Roe algebras and a twisted coarse Baum-Connes conjecture. We 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 coarse Baum-Connes conjecture 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.