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.