Twisted Roe algebras and their $K$-theory

TL;DR

This paper introduces twisted Roe algebras and a twisted coarse Baum-Connes conjecture, establishing their properties and proving the conjecture for certain metric spaces and groups with coarse fibrations.

Contribution

It defines twisted Roe algebras, studies their properties, and proves the twisted coarse Baum-Connes conjecture for spaces with coarse fibrations and certain group extensions.

Findings

Twisted Roe algebras have well-defined basic properties.

The twisted coarse Baum-Connes conjecture holds for spaces with coarse fibrations.

The conjecture is verified for groups that are extensions of coarsely embeddable groups.

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.