On a quantitative operator K-theory

TL;DR

This paper develops a quantitative K-theory framework for filtered C*-algebras, including key theorems and applications to the Baum-Connes conjecture, with implications for group C*-algebras and related structures.

Contribution

It introduces a novel quantitative K-theory approach, establishing a quantitative six-term exact sequence and Bott periodicity, and applies these to the Baum-Connes conjecture.

Findings

Quantitative six-term exact sequence proved.

Quantitative Bott periodicity established.

Baum-Connes conjecture shown to hold for many groups.

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.