On quantitative operator K-theory

Abstract

In this paper, we develop a quantitative K-theory for filtered C * -algebras. Particularly interesting examples of filtered C * -algebras include group C * -algebras, crossed product C * -algebras and Roe algebras. We prove a quantitative version of the six term exact sequence and a quantitative Bott periodicity. We apply the quantitative K-theory to formulate a quantitative version of the Baum-Connes conjecture and prove that the quantitative Baum-Connes conjecture holds for a large class of groups.