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Classical harmonic analysis viewed through the prism of noncommutative geometry

Cédric Arhancet

Abstract

The aim of this paper is to bridge noncommutative geometry with classical harmonic analysis on Banach spaces, focusing primarily on both classical and noncommutative $\mathrm{L}^p$ spaces. Introducing a notion of Banach Fredholm module, we define new abelian groups, $\mathrm{K}^{0}(\mathcal{A},\mathscr{B})$ and $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$, of $\mathrm{K}$-homology associated with an algebra $\mathcal{A}$ and a suitable class $\mathscr{B}$ of Banach spaces, such as the class of $\mathrm{L}^p$-spaces. We establish index pairings of these groups with the $\mathrm{K}$-theory groups of the algebra $\mathcal{A}$. Subsequently, by considering (noncommutative) Hardy spaces, we uncover the natural emergence of Hilbert transforms, leading to Banach Fredholm modules and culminating in new index theorems. Moreover, by associating each reasonable sub-Markovian semigroup of operators with a <<Banach noncommutative manifold>>, we explain how this leads to (possibly kernel-degenerate) Banach Fredholm modules, thereby revealing the role of vectorial Riesz transforms in this context. Overall, our approach significantly integrates the analysis of operators on $\mathrm{L}^p$-spaces into the expansive framework of noncommutative geometry, offering new perspectives.

Classical harmonic analysis viewed through the prism of noncommutative geometry

Abstract

The aim of this paper is to bridge noncommutative geometry with classical harmonic analysis on Banach spaces, focusing primarily on both classical and noncommutative spaces. Introducing a notion of Banach Fredholm module, we define new abelian groups, and , of -homology associated with an algebra and a suitable class of Banach spaces, such as the class of -spaces. We establish index pairings of these groups with the -theory groups of the algebra . Subsequently, by considering (noncommutative) Hardy spaces, we uncover the natural emergence of Hilbert transforms, leading to Banach Fredholm modules and culminating in new index theorems. Moreover, by associating each reasonable sub-Markovian semigroup of operators with a <<Banach noncommutative manifold>>, we explain how this leads to (possibly kernel-degenerate) Banach Fredholm modules, thereby revealing the role of vectorial Riesz transforms in this context. Overall, our approach significantly integrates the analysis of operators on -spaces into the expansive framework of noncommutative geometry, offering new perspectives.
Paper Structure (58 sections, 61 theorems, 403 equations, 1 figure)

This paper contains 58 sections, 61 theorems, 403 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the Paper
  3. Preliminaries
  4. Fredholm operators
  5. Unbounded operators
  6. $R$-boundedness
  7. Sectorial operators
  8. Fractional powers
  9. Bisectorial operators
  10. Functional calculus
  11. $\mathrm{K}$-theory
  12. $\mathrm{K}_0$-group
  13. $\mathrm{K}_1$-group
  14. Banach Fredholm modules and Banach $\mathrm{K}$-homology
  15. Coupling between Banach K-homology with K-theory
  16. ...and 43 more sections

Key Result

Theorem 2.1

A bounded operator $T \colon X \to Y$ between Banach spaces is a Fredholm operator if and only if there exists a bounded operator $R \colon Y \to X$ such that $RT-\mathrm{Id}_X$ and $TR-\mathrm{Id}_Y$ are compact operators. Moreover, we can replace <<compact>> by <<finite-rank projection>> in this a

Figures (1)

  • Figure :

Theorems & Definitions (118)

  • Theorem 2.1
  • Proposition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Proposition 2.10
  • ...and 108 more