Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers Introduction

In this chapter, we give an overview of the book and its contents. We start by giving a short introduction to Riesz transforms and their L-p-boundedness in various settings in that they have been studied in the literature. We also explain the emergence of noncommutative L-p-spaces and noncommutative geometry in this context. Furthermore, we describe our main results concerning Riesz transforms, functional calculus of Hodge-Dirac operators and spectral triples. We equally present examples that can used with our results and end with an overview of the contents of the other chapters.