Quantitative K-theory for Banach Algebras and Its Applications

Abstract

This dissertation can be said to fall under the broad theme of computability of K-theory of Lp operator algebras (and perhaps more general Banach algebras).

The first part of the dissertation is about a variant of K-theory known as quantitative K-theory, which has been defined for C*_-algebras and applied in a number of situations. Our goal is to extend the theory to a larger class of Banach algebras so that it becomes applicable to Lp operator algebras and thus a tool for investigating an Lp version of the Baum-Connes conjecture. We develop the general framework for this theory, culminating in a version of the controlled Mayer-Vietoris sequence that has featured prominently in existing applications in the C*_-algebra setting.

In the second part of the dissertation, we study the Lp version of one of these applications.This application involves the notion of dynamic asymptotic dimension, which is a notion of dimension associated to group actions on spaces (and more generally to groupoids). In the C*_-algebra setting, the work of Guentner-Willett-Yu showed that when a group G acts on a compact space X with finite dynamic asymptotic dimension, the Baum-Connes conjecture with coefficients in C(X) holds for the group G. We will formulate an Lp version of the Baum-Connes conjecture with coefficients and show that under the same assumption, the Lp Baum-Connes conjecture with coefficients in C(X) holds for the group G. As a consequence, the K-theory of the Lp reduced crossed product of C(X) by G does not depend on p if the action has finite dynamic asymptotic dimension.