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 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.

⁰⁰footnotetext: 2020 Mathematics subject classification: 58B34, 47D03, 46L80, 47B90
Key words:

\mathrm{K}

-homology,

\mathrm{K}

-theory, spectral triples, Riesz transforms, Hilbert transforms,

\mathrm{L}^{p}

-spaces, Fredholm modules, noncommutative geometry.

1 Introduction

The aim of this paper is to provide a coherent framework that encompasses various aspects of harmonic analysis on $\mathrm{L}^{p}$ -spaces within the context of noncommutative geometry. We seek to elucidate connections that appear sporadically throughout the literature, clarifying these overlaps and coincidences. It is crucial to emphasize that noncommutative geometry is not merely a ¡¡generalization¿¿ of classical geometry. While it subsumes known spaces as particular cases, it offers a radically different approach to some classical spaces, such as fractals or leaf spaces of foliations, by introducing powerful analytical points of view and tools. This work can be seen as the next step in our ongoing research program, which began in our previous studies [ArK22], [Arh24a], [Arh24b], and [Arh24c].

In noncommutative geometry, the starting point is an algebra $\mathcal{A}$ (which may be commutative or not), representing the space, with its elements acting as bounded operators on a complex Hilbert space $H$ via a representation $\pi\colon\mathcal{A}\to\mathrm{B}(H)$ . This algebra $\mathcal{A}$ replaces the algebra $\mathrm{C}(K)$ of continuous functions on a classical compact space $K$ endowed with a finite Borel measure, which acts on the complex Hilbert space $\mathrm{L}^{2}(K)$ by multiplication operators.

A fundamental concept in this framework is that of a Fredholm module. If $\mathcal{A}$ is unital, a Fredholm module over $\mathcal{A}$ is a bounded operator $F\colon H\to H$ satisfying the three relations $F^{2}=\mathrm{Id}_{H}$ , $F=F^{*}$ and $[F,\pi(a)]=0$ for any $a\in\mathcal{A}$ modulo a compact operator. If $D$ is the Dirac operator on a compact Riemannian spin manifold $M$ , the Fredholm module $\operatorname{\mathrm{sign}}D$ canonically associated to $M$ encodes the conformal structure of the manifold, as discussed in [Bar07, Theorem 3.1 p. 388]. These operators enable the definition of $\mathrm{K}$ -homology groups $\mathrm{K}^{0}(\mathcal{A})$ and $\mathrm{K}^{1}(\mathcal{A})$ , which are linked to the $\mathrm{K}$ -theory groups $\mathrm{K}_{0}(\mathcal{A})$ and $\mathrm{K}_{1}(\mathcal{A})$ of the algebra $\mathcal{A}$ through a pairing that leads to index theorems. The culmination of this theory is perhaps the index theorem of Connes-Moscovici itself [CoM95], [Hig02], and its generalization to the locally compact case [CGRS14], drawing inspiration from the Atiyah-Singer index theorem, whose applications are thoroughly covered in [BlB13]. For more information on noncommutative geometry, we refer to the books [GVF01], [EcI18] and the survey article [CPR11], as well as specific applications to solid state physics in [PSB16] and related works on $\mathrm{K}$ -theory and $\mathrm{K}$ -homology [Pus11], [Had03], [Had04], [NeT11], [EmN18], [FGMR19], [Ger22] all of which are grounded in the classical text [HiR00].

In this paper, we define a notion of Banach Fredholm module $F\colon X\to X$ on an arbitrary Banach space $X$ over an algebra $\mathcal{A}$ , where elements act on the space $X$ via a representation $\pi\colon\mathcal{A}\to\mathrm{B}(X)$ . In Connes’ foundational work [Con94], an illustrative example of a Fredholm module is given by $F\overset{\mathrm{def}}{=}\mathrm{i}\mathcal{H}$ where $\mathcal{H}\colon\mathrm{L}^{2}(\mathbb{R})\to\mathrm{L}^{2}(\mathbb{R})$ is the Hilbert transform. This example is sometimes seen as an ¡¡exotic¿¿ Fredholm module. Here, we show that various Hilbert transforms on $\mathrm{L}^{p}$ -spaces satisfying $F^{2}=\mathrm{Id}_{X}$ modulo a compact operator naturally arise as examples. Actually, we introduce huge classes of Banach Fredholm modules over group $\mathrm{C}^{*}$ -algebras, which have not been previously considered even in the Hilbertian case. If $M_{f}\colon\mathrm{L}^{p}(\mathbb{R}^{n})\to\mathrm{L}^{p}(\mathbb{R}^{n})$ , $g\mapsto fg$ is the multiplication operator by a function $f$ , the commutators $[\mathcal{H},M_{f}]$ of the Hilbert transform $\mathcal{H}$ (with $n=1$ ) and more generally commutators $[\mathcal{R}_{j},M_{f}]$ of Riesz transforms $\mathcal{R}_{j}\colon\mathrm{L}^{p}(\mathbb{R}^{n})\to\mathrm{L}^{p}(\mathbb{R}^{n})$ where $1\leqslant j\leqslant n$ , on $\mathrm{L}^{p}$ -spaces or even similar operators in other contexts is a classical topic in analysis, see the survey [Wic20] and references therein, connected to Hardy spaces, spaces of functions of bounded/vanishing mean oscillation and factorization of functions, and we demonstrate that they also emerge naturally in the commutators of Banach Fredholm modules. Such a commutator is given by $[F,\pi(a)]$ for some $a\in\mathcal{A}$ . In this framework, we can introduce the Connes quantized differential

(1.1)

\,{\raisebox{-0.56905pt}{$\mathchar 22\relax$}\mkern-12.0mu\mathrm{d}}a\overset{\mathrm{def}}{=}\mathrm{i}[F,\pi(a)],\quad a\in\mathcal{A}

of $a$ . In the case $F=\mathrm{i}\mathcal{H}$ , we have

(1.2)

\big{(}[F,M_{a}]f\big{)}(x)=\operatorname{p.v.}\int_{-\infty}^{\infty}\frac{a(x)-a(y)}{x-y}f(y)\mathop{}\mathopen{}\mathrm{d}y.

A well-known phenomenon in the Hilbertian context is the interplay between the differentiability of $a$ and the ¡¡degree of compactness¿¿ of the quantized differential $\,{\raisebox{-0.56905pt}{$\mathchar 22\relax$}\mkern-12.0mu\mathrm{d}}a$ . In the Banach space context, we use some suitable quasi-Banach ideal $S^{q}_{\mathrm{app}}(X)$ , where $0<q<\infty$ , relying on the concept of $s$ -numbers introduced by Pietsch, as a substitute of the Schatten space $S^{q}(H)\overset{\mathrm{def}}{=}\{T\colon H\to H:\operatorname{Tr}|T|^{q}<\infty\}$ , allowing us to define a notion of finitely summable Banach Fredholm module. The notion of $s$ -numbers is a generalization of the notion of singular value of operators acting on Hilbert spaces.

We equally introduce $\mathrm{K}$ -homology groups $\mathrm{K}^{0}(\mathcal{A},\mathscr{B})$ and $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$ associated to the algebra $\mathcal{A}$ and with a class $\mathscr{B}$ of Banach spaces, as the class of (noncommutative) $\mathrm{L}^{p}$ -spaces. We explore their pairings with the $\mathrm{K}$ -theory groups $\mathrm{K}_{0}(\mathcal{A})$ and $\mathrm{K}_{1}(\mathcal{A})$ of the algebra $\mathcal{A}$ . In the simplest case, the pairing gives the index theorem of Gohberg-Krein for a Toeplitz operator $T_{a}\colon\mathrm{H}^{p}(\mathbb{T})\to\mathrm{H}^{p}(\mathbb{T})$ acting on the classical Hardy space $\mathrm{H}^{p}(\mathbb{T})$ on the circle for any $1<p<\infty$ , generalizing the well-known case $p=2$ .

Connes demonstrated how to associate a canonical cyclic cocycle, known as the Chern character, with a finitely summable Fredholm module, and how this character can be used to compute the index pairing between the $\mathrm{K}$ -theory of $\mathcal{A}$ and the $\mathrm{K}$ -homology class of the Fredholm module. We show that our framework admits a similar Chern character. The (odd) Chern character in K-homology is defined from the quantized calculus (1.1) associated to a Banach Fredholm module by

\mathrm{Ch}_{n}^{F}(a_{0},a_{1},\dots,a_{n})\overset{\mathrm{def}}{=}c_{n}\operatorname{Tr}(F\,{\raisebox{-0.56905pt}{$\mathchar 22\relax$}\mkern-12.0mu\mathrm{d}}a_{0}\,{\raisebox{-0.56905pt}{$\mathchar 22\relax$}\mkern-12.0mu\mathrm{d}}a_{1}\cdots\,{\raisebox{-0.56905pt}{$\mathchar 22\relax$}\mkern-12.0mu\mathrm{d}}a_{n}),\qquad a_{0},a_{1},\dots,a_{n}\in\mathcal{A},

for some sufficiently large odd integer $n$ and where $c_{n}$ is some suitable constant. Here, we use the trace $\operatorname{Tr}$ on the space $S^{1}_{\mathrm{app}}(X)$ which is the unique continuous extension of the trace defined on the space of finite-rank operators acting on the Banach space $X$ .

Within the Hilbertian setting of Connes, it is well-known that a Fredholm module can be constructed from a compact spectral triple (i.e. a noncommutative compact Riemannian spin manifold). Such a compact spectral triple $(\mathcal{A},H,D)$ consists of a selfadjoint operator $D$ , defined on a dense subspace of the Hilbert space $H$ , and satisfying certain axioms. Specifically, $D^{-1}$ must be compact on $\operatorname{Ker}D^{\perp}$ , often referred to as the ¡¡unit length¿¿ or ¡¡line element¿¿, denoted by $\mathop{}\mathopen{}\mathrm{d}s$ since we can introduce the noncommutative integral $\int$ with a Dixmier trace.

The Dirac operator $D$ on a spin Riemannian compact manifold $M$ can be used for constructing a classical example of spectral triple. Several examples in different contexts are discussed in the survey [CoM08]. The Fredholm module obtained from a spectral triple $(\mathcal{A},H,D)$ is built using the bounded operator

(1.3)

\operatorname{\mathrm{sign}}D\colon H\to H,

constructed with the spectral theorem.

In this paper, we replace Connes’ spectral triples by the Banach compact spectral triples of [ArK22]. Such a triple $(\mathcal{A},X,D)$ is composed of a Banach space $X$ , a representation $\pi\colon\mathcal{A}\to\mathrm{B}(X)$ and a bisectorial operator $D$ on a Banach space $X$ that admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}})$ functional calculus on the open bisector $\Sigma_{\theta}^{\mathrm{bi}}\overset{\mathrm{def}}{=}\Sigma_{\theta}\cup(-\Sigma_{\theta})$ where $\Sigma_{\theta}\overset{\mathrm{def}}{=}\big{\{}z\in\mathbb{C}\backslash\{0\}:\>|\arg z|<\theta\big{\}}$ , with $0<\theta<\frac{\pi}{2}$ . We refer to Definition 7.4 for a precise definition with an assumption on some commutators. Roughly speaking, this means that the spectrum $\sigma(D)$ of $D$ is a subset of the closed bisector $\overline{\Sigma^{\mathrm{bi}}_{\omega}}$ for some $\omega\in(0,\theta)$ as in Figure 2, that we have an appropriate ¡¡resolvent estimate¿¿ and that

(1.4)

\left\|f(D)\right\|_{X\to X}\lesssim\left\|f\right\|_{\mathrm{H}^{\infty}(\Sigma^{\mathrm{bi}}_{\theta})}

for any suitable function $f$ of the algebra $\mathrm{H}^{\infty}(\Sigma^{\mathrm{bi}}_{\theta})$ of all bounded holomorphic functions defined on the bisector $\Sigma^{\mathrm{bi}}_{\sigma}$ . Here ¡¡suitable¿¿ means regularly decaying at 0 and at $\infty$ . In broad terms, the operator $f(D)$ is defined by a ¡¡Cauchy integral¿¿

(1.5)

f(D)=\frac{1}{2\pi\mathrm{i}}\int_{\partial\Sigma^{\mathrm{bi}}_{\nu}}f(z)R(z,D)\mathop{}\mathopen{}\mathrm{d}z

by integrating over the boundary of a larger bisector $\Sigma^{\mathrm{bi}}_{\nu}$ using the resolvent operator $R(z,D)\overset{\mathrm{def}}{=}(z-D)^{-1}$ , where $\omega<\nu<\theta$ . Note that the boundedness of such a functional calculus is not free, contrary to the case of the functional calculus of a selfadjoint operator. We refer to our paper [Arh24c] for concrete examples of operator with such a bounded functional calculus, where we use a notion of curvature for obtaining it. Using the function $\operatorname{\mathrm{sign}}$ defined by $\operatorname{\mathrm{sign}}(z)\overset{\mathrm{def}}{=}1_{\Sigma_{\theta}}(z)-1_{-\Sigma_{\theta}}(z)$ , we show as the hilbertian case that $\operatorname{\mathrm{sign}}D\colon X\to X$ is a Fredholm module. This notion of functional calculus was popularized in the paper [AKM06], which contains a (second) solution to famous Kato’s square root problem solved in [AHLMT02] and in [AKM06] (see also [HLM02] and [Tch01]).

Let us explain how this approach integrates with ours, starting with the simplest case, the one-dimensional scenario. Consider a function $a\in\mathrm{L}^{\infty}(\mathbb{R})$ such that $\operatorname{Re}a(x)\geqslant\kappa>0$ for almost all $x\in\mathbb{R}$ and the multiplication operator $M_{a}\colon\mathrm{L}^{2}(\mathbb{R})\to\mathrm{L}^{2}(\mathbb{R})$ , $f\mapsto af$ . Following the approach of [AKM06], we can consider the unbounded operator

(1.6)

D\overset{\mathrm{def}}{=}\begin{bmatrix}0&-\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}M_{a}\\ \frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}&0\end{bmatrix},

acting on the complex Hilbert space $\mathrm{L}^{2}(\mathbb{R})\oplus\mathrm{L}^{2}(\mathbb{R})$ , where $\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}M_{a}$ denotes the composition $\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}\circ M_{a}$ . Note that the operator $D$ is not selfadjoint in general. However, by [AKM06, Theorem 3.1 (i) p. 465], $D$ admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}})$ functional calculus for some angle $0<\theta<\frac{\pi}{2}$ . Using $D^{2}=\begin{bmatrix}-\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}M_{a}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}&0\\ 0&-\frac{\mathop{}\mathopen{}\mathrm{d}^{2}}{\mathop{}\mathopen{}\mathrm{d}x^{2}}M_{a}\end{bmatrix}$ , we see that formally we have a bounded operator

(1.7)

\operatorname{\mathrm{sign}}D=D(D^{2})^{-\frac{1}{2}}=\begin{bmatrix}0&*\\ \frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}(-\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}M_{a}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x})^{-\frac{1}{2}}&0\end{bmatrix}.

It is immediate to obtain the obtain the estimate $\left\|\frac{\mathop{}\mathopen{}\mathrm{d}f}{\mathop{}\mathopen{}\mathrm{d}x}\right\|_{\mathrm{L}^{2}(\mathbb{R})}\lesssim\left\|\big{(}-\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}M_{a}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}\big{)}^{\frac{1}{2}}f\right\|_{\mathrm{L}^{2}(\mathbb{R})}$ . Actually, a slightly more elaborate argument gives the Kato square root estimate in one dimension

(1.8)

\left\|\bigg{(}-\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}M_{a}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}\bigg{)}^{\frac{1}{2}}f\right\|_{\mathrm{L}^{2}(\mathbb{R})}\approx\left\|\frac{\mathop{}\mathopen{}\mathrm{d}f}{\mathop{}\mathopen{}\mathrm{d}x}\right\|_{\mathrm{L}^{2}(\mathbb{R})},\quad f\in\mathrm{W}^{1,2}(\mathbb{R}).

Using the homomorphism $\pi\colon\mathrm{C}_{c}^{\infty}(\mathbb{R})\mapsto\mathrm{B}(\mathrm{L}^{2}(\mathbb{R})\oplus\mathrm{L}^{2}(\mathbb{R}))$ , $a\mapsto M_{a}\oplus M_{a}$ . We will easily show that $(\mathrm{C}_{c}^{\infty}(\mathbb{R}),\mathrm{L}^{2}(\mathbb{R})\oplus\mathrm{L}^{2}(\mathbb{R}),D)$ is a Banach locally compact spectral triple, which is not a locally compact spectral triple in the classical hilbertian sense.

It is important to realize that all the theory of sub-Markovian semigroups acting on $\mathrm{L}^{p}$ -spaces can be integrated into the notion of Banach spectral triples. Indeed the $\mathrm{L}^{2}$ -generator $-A_{2}$ of such semigroup $(T_{t})_{t\geqslant 0}$ acting on the Hilbert space $\mathrm{L}^{2}(\Omega)$ can be written $A_{2}=\partial^{*}\partial$ where $\partial$ is a (unbounded) closed derivation defined on a dense subspace of $\mathrm{L}^{2}(\Omega)$ with values in a Hilbert $\mathrm{L}^{\infty}(\Omega)$ -bimodule $\mathcal{H}$ . Here $T_{t}=\mathrm{e}^{-tA_{2}}$ for any $t\geqslant 0$ . The map $\partial$ can be seen as an ¡¡abstract¿¿ analogue of the gradient operator $\nabla$ of a smooth Riemannian manifold $M$ , which is a closed operator defined on a subspace of $\mathrm{L}^{2}(M)$ into the space $\mathrm{L}^{2}(M,\mathrm{T}M)$ satisfying the relation $-\Delta=\nabla^{*}\nabla$ where $\Delta$ is the Laplace-Beltrami operator and where $\nabla^{*}=-\mathrm{div}$ .

This fundamental result allows anyone to introduce a triple $(\mathrm{L}^{\infty}(\Omega),\mathrm{L}^{2}(\Omega)\oplus_{2}\mathcal{H},D)$ associated to the semigroup in the spirit of the previous Banach spectral triples. Here $D$ is the unbounded selfadjoint operator acting on a dense subspace of the Hilbert space $\mathrm{L}^{2}(\Omega)\oplus_{2}\mathcal{H}$ defined by

(1.9)

D\overset{\mathrm{def}}{=}\begin{bmatrix}0&\partial^{*}\\ \partial&0\end{bmatrix}.

It is possible in this context to introduce a homomorphism $\pi\colon\mathrm{L}^{\infty}(\Omega)\to\mathrm{B}(\mathrm{L}^{2}(\Omega)\oplus_{2}\mathcal{H})$ , see (11.8). Now, suppose that $1<p<\infty$ . Sometimes, the map $\partial$ induces a closable unbounded operator $\partial\colon\operatorname{dom}\partial\subset\mathrm{L}^{p}(\mathcal{M})\to\mathcal{X}_{p}$ for some Banach space $\mathcal{X}_{p}$ . So we can consider the $\mathrm{L}^{p}$ -realization of the previous operator $D$ as acting on a dense subspace of the Banach space $\mathrm{L}^{p}(\Omega)\oplus_{p}\mathcal{X}_{p}$ . If $D$ is bisectorial and admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}})$ functional calculus then using the equalities $D^{2}=\begin{bmatrix}\partial^{*}\partial&0\\ 0&\partial\partial^{*}\end{bmatrix}$ and $A_{2}=\partial^{*}\partial$ , we see that formally we have a bounded operator

(1.10)

\operatorname{\mathrm{sign}}D=D(D^{2})^{-\frac{1}{2}}=\begin{bmatrix}0&*\\ \partial A^{-\frac{1}{2}}&0\end{bmatrix}.

From this, it is apparent that the vectorial Riesz transform $\partial A^{-\frac{1}{2}}\colon\mathrm{L}^{p}(\Omega)\to\mathcal{X}_{p}$ is bounded and appears in the operator $F\overset{\mathrm{def}}{=}\operatorname{\mathrm{sign}}D$ and particularly with the possible pairing with the group $\mathrm{K}_{0}(\mathcal{A})$ of $\mathrm{K}$ -theory for a suitable subalgebra $\mathcal{A}$ of the algebra $\mathrm{L}^{\infty}(\Omega)$ , as we will see. Of course, we can consider more generally sub-Markovian semigroups acting on the noncommutative $\mathrm{L}^{p}$ -space $\mathrm{L}^{p}(\mathcal{M})$ of a von Neumann algebra $\mathcal{M}$ in the previous discussion.

Structure of the Paper

This paper is structured as follows. Section 2 provides the necessary background and revisits key notations, as well as essential results required for our work. In Section 3, we introduce the concept of (odd or even) Banach Fredholm modules and define new groups, $\mathrm{K}^{0}(\mathcal{A},\mathscr{B})$ and $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$ , in $\mathrm{K}$ -homology, associated with an algebra $\mathcal{A}$ and a suitable class $\mathscr{B}$ of Banach spaces. Section 4 focuses on establishing pairings with the $\mathrm{K}$ -theory groups of the algebra $\mathcal{A}$ , which is central to our approach. In Section 5, we introduce a suitable notion of summability for Banach Fredholm modules, revisiting key concepts related to approximation numbers of operators acting on Banach spaces. Section 6 is dedicated to the Chern character associated with finitely summable Banach Fredholm modules. Following the classical Hilbert space approach, we demonstrate how the pairing between $\mathrm{K}$ -homology and $\mathrm{K}$ -theory groups can be described using the Chern character. In Section 7, we revisit the notion of Banach spectral triples and explain how Banach Fredholm modules can be constructed from these triples. Section 8 explores concrete examples of triples constructed using Dirac operators, leading to Banach Fredholm modules, where we compute the pairings with $\mathrm{K}$ -theory in specific cases. We also reveal the connection between the summability of Banach spectral triples and Sobolev embedding theorems. In Section 9, we present examples of Banach Fredholm modules in the context of noncommutative Hardy spaces, leading to new Banach Fredholm modules on group $\mathrm{C}^{*}$ -algebras, even within the Hilbert space framework. Section 10 present a Banach Fredholm module on the reduced $\mathrm{C}^{*}$ -algebra of the free group $\mathbb{F}_{\infty}$ . Finally, in Section 11 discusses how to associate a Banach spectral triple with each reasonable sub-Markovian semigroup. We also observe that the corresponding Banach Fredholm module, which may be kernel-degenerate, is closely linked to the vectorial Riesz transform associated to the semigroup.

2 Preliminaries

$\mathrm{K}$ -theory

We refer to the books [Bl98], [GVF01], [RLL00] and [WeO93] for more information. Two idempotents $e,f$ in a $\mathbb{C}$ -algebra $\mathcal{A}$ are Murray von Neumann equivalent, written $e\sim f$ , if there are $v,w\in S$ with $vw=e$ and $wv=f$ . Let $\mathcal{A}$ be a unital $\mathbb{C}$ -algebra, and let $\mathrm{V}(\mathcal{A})$ denote the collection of all Murray-von Neumann equivalence classes of idempotents in $\mathrm{M}_{\infty}(\mathcal{A})$ . Equip $\mathrm{V}(\mathcal{A})$ with the addition operation defined by

[e]+[f]=\begin{bmatrix}e&0\\ 0&f\\ \end{bmatrix}

Then $\mathrm{V}(\mathcal{A})$ becomes a commutative monoid. Let $\mathcal{A}$ be a unital complex algebra. Then idempotents $e,f\in\mathrm{M}_{\infty}(\mathcal{A})$ are Murray-von Neumann equivalent if and only if there is an invertible element $u$ in some $\mathrm{M}_{n}(\mathcal{A})$ such that $ueu^{-1}=f$ .

Let $A$ be a unital $\mathbb{C}$ -algebra. The group $\mathrm{K}_{0}(\mathcal{A})$ is the Grothendieck group of $\mathrm{V}(\mathcal{A})$ . Now, note that $\mathrm{K}_{0}$ is functorial for unital $\mathrm{C}^{*}$ -algebra homomorphisms in a natural way: if $\phi\colon\mathcal{A}\to B$ is a unital ring homomorphism, and $e\in\mathrm{M}_{\infty}(\mathcal{A})$ is an idempotent, then $\phi(e)$ is an idempotent in $\mathrm{M}_{\infty}(\mathcal{B})$ , and the map $[e]\mapsto[\phi(e)]$ is well-defined as a map $\mathrm{V}(\mathcal{A})\to\mathrm{V}(\mathcal{B})$ . Hence it induces a map $\phi_{*}\colon\mathrm{K}_{0}(\mathcal{A})\to\mathrm{K}_{0}(\mathcal{B})$ by the universal properties of this group

Let $\mathcal{A}$ be a not-necessarily unital $\mathbb{C}$ -algebra, and let $\tilde{A}$ be its unitisation, which is equipped with a canonical unital $\mathbb{C}$ -algebra homomorphism $\phi\colon\tilde{\mathcal{A}}\to\mathbb{C}$ with kernel $\mathcal{A}$ . Then the group $\mathrm{K}_{0}(\mathcal{A})$ is defined by

\mathrm{K}_{0}(\mathcal{A})\overset{\mathrm{def}}{=}\operatorname{Ker}\phi_{*}\colon\mathrm{K}_{0}(\tilde{\mathcal{A}})\to\mathrm{K}_{0}(\mathbb{C}).

A local Banach algebra is a normed algebra $\mathcal{A}$ which is closed under holomorphic functional calculus (i.e. if $x\in\mathcal{A}$ and $f$ is an analytic function on a neighborhood of the spectrum of $x$ in the completion of $\mathcal{A}$ , with $f(0)=0$ if $\mathcal{A}$ is nonunital, then $f(x)\in\mathcal{A}$ .) For technical reasons we will also require that all matrix algebras over $A$ have the same property. If $\mathcal{A}$ is a $*$ -algebra, it will be called a local Banach $*$ -algebra; if the norm is a pre- $\mathrm{C}^{*}$ -norm, $\mathcal{A}$ will be called a local $\mathrm{C}^{*}$ -algebra.

Recall that $\mathrm{GL}_{\infty}(\mathcal{A})\overset{\mathrm{def}}{=}\varinjlim\mathrm{GL}_{n}(\mathcal{A})$ and that $\mathrm{GL}_{\infty}(\mathcal{A})_{0}\overset{\mathrm{def}}{=}\varinjlim\mathrm{GL}_{n}(\mathcal{A})_{0}$ , where the group $\mathrm{GL}_{n}(\mathcal{A})_{0}$ is the connected component of the identity. The group $\mathrm{K}_{1}(\mathcal{A})$ is defined by $\mathrm{K}_{1}(\mathcal{A})\overset{\mathrm{def}}{=}\mathrm{GL}_{\infty}(\mathcal{A})/\mathrm{GL}_{\infty}(\mathcal{A})_{0}$ . It is known [Bl98, p. 59] that the group $\mathrm{K}_{1}(\mathcal{A})$ is countable if the algebra $\mathcal{A}$ is separable.

Example 2.1

If $\mathcal{A}=\mathrm{C}(X)$ is the algebra of a compact Hausdorff space $X$ , then by [GVF01, Corollary 3.21 p. 101] we have an isomorphism $\mathrm{K}_{0}(\mathrm{C}(X))=\mathrm{K}^{0}(X)$ where $\mathrm{K}^{0}(X)$ is the topological $\mathrm{K}$ -theory group defined by vector bundles.

Fredholm operators

Following [AbA02, Definition 4.37 p. 156], we say that a bounded linear operator $T\colon X\to Y$ , acting between complex Banach spaces $X$ and $Y$ , is a Fredholm operator if the subspaces $\operatorname{Ker}T$ and $Y/\operatorname{Ran}T$ are finite-dimensional. In this case, we introduce the index

(2.1)

\operatorname{Index}T\overset{\mathrm{def}}{=}\dim\operatorname{Ker}T-\dim Y/\operatorname{Ran}T.

Every Fredholm operator has a closed range by [AbA02, Lemma 4.38 p. 156]. Recall the Banach version of Atkinson’s theorem [AbA02, Theorem 4.46 p. 161].

Theorem 2.2

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

Note that [AbA02, Theorem 4.48 p. 163] the set $\mathrm{Fred}(X,Y)$ of all Fredholm operators from $X$ into $Y$ is an open subset of $\mathrm{B}(X,Y)$ and the index function $\operatorname{Index}\colon\mathrm{Fred}(X,Y)\to\mathbb{Z}$ is continuous (hence locally constant). By [AbA02, Corollaey 4.47 p. 162], if $T\colon X\to Y$ is a Fredholm operator and if $K\colon X\to Y$ is a compact operator, then $A+K$ is a Fredholm operator and

(2.2)

\operatorname{Ind}A=\operatorname{Ind}(A+K)

(invariance under compact perturbations). If $T\colon X\to Y$ and $S\colon Y\to Z$ are Fredholm operators then $ST$ is also a Fredholm operator and $\operatorname{Index}ST=\operatorname{Index}S+\operatorname{Index}T$ , see [AbA02, Theorem 4.43 p. 158]. By [AbA02, Theorem 4.42 p. 157], a bounded operator is a Fredholm operator if and only if its adjoint is also a Fredholm operator. In this case, we have $\operatorname{Index}T^{*}=-\operatorname{Index}T$ .

Recall that by [AbA02, Theorem 4.54 p. 167] any Fredholm operator $T\colon X\to Y$ admits a generalized inverse (or pseudo-inverse), i.e. an operator $U\colon Y\to X$ such that

(2.3)

TUT=T.

By the way, the same result says that every generalized inverse $U$ of $T$ is also a Fredholm operator and satisfies $\operatorname{Index}U=-\operatorname{Index}T$ . In this case, the proof of [AbA02, Theorem 4.32 p. 166] shows that the operators $TU\colon Y\to Y$ and $\mathrm{Id}-UT\colon X\to X$ are bounded projections on the subspaces $\operatorname{Ran}T$ and $\operatorname{Ker}T$ .

Unbounded operators

The following result is [HvNVW16, Theorem 1.2.4 p. 15].

Proposition 2.3

Let $f\colon\Omega\to X$ be a Bochner integrable function and let $T$ be a closed linear operator with domain $\operatorname{dom}T$ in $X$ and with values in a Banach space $Y$ . Suppose that $f$ takes its values in $\operatorname{dom}T$ almost everywhere and the almost everywhere defined function $Tf\colon\Omega\to Y$ is Bochner integrable. Then $f$ is Bochner integrable as a $\operatorname{dom}(T)$ -valued function, $\int_{\Omega}f\mathop{}\mathopen{}\mathrm{d}\mu$ belongs to $\operatorname{dom}(T)$ and

(2.4)

T\bigg{(}\int_{\Omega}f\mathop{}\mathopen{}\mathrm{d}\mu\bigg{)}=\int_{\Omega}T\circ f\mathop{}\mathopen{}\mathrm{d}\mu.

Bisectorial operators

We refer to [Ege15] and to the books [HvNVW18] and [HvNVW23] for more information on bisectorial operators. For any angle $\omega\in(0,\frac{\pi}{2})$ , we consider the open bisector $\Sigma_{\omega}^{\mathrm{bi}}\overset{\mathrm{def}}{=}\Sigma_{\omega}\cup(-\Sigma_{\omega})$ where $\Sigma_{\omega}\overset{\mathrm{def}}{=}\big{\{}z\in\mathbb{C}\backslash\{0\}:\>|\arg z|<\omega\big{\}}$ and $\Sigma_{0}^{\mathrm{bi}}=(-\infty,\infty)$ . Following [HvNVW18, Definition 10.6.1 p. 447] , we say that a closed densely defined operator $D$ on a Banach space $X$ is bisectorial of type $\omega\in(0,\frac{\pi}{2})$ if its spectrum $\sigma(D)$ is a subset of the closed bisector $\overline{\Sigma^{\mathrm{bi}}_{\omega}}$ and if the subset $\big{\{}zR(z,D):z\not\in\overline{\Sigma_{\omega}^{\mathrm{bi}}}\big{\}}$ is bounded in $\mathrm{B}(X)$ . See Figure 1. The infimum of all $\omega\in(0,\frac{\pi}{2})$ such that $D$ is bisectorial is called the angle of bisectoriality of $D$ . The definition of an $R$ -bisectorial operator is obtained by replacing ¡¡bounded¿¿ by ¡¡ $R$ -bounded¿¿.

Figure 1: the spectrum of a bisectorial operator $D$

By [HvNVW18, p. 447], the operator $D$ is bisectorial if and only if

(2.5)

\mathrm{i}\mathbb{R}^{*}\subset\rho(D)\quad\text{and}\quad\sup_{t\in\mathbb{R}_{*}^{+}}\left\|tR(\mathrm{i}t,D)\right\|_{X\to X}<\infty.

Similarly, the operator $D$ is $R$ -bisectorial if and only if

(2.6)

\mathrm{i}\mathbb{R}^{*}\subset\rho(D)\quad\text{and if the set}\quad\{tR(\mathrm{i}t,D):t\in\mathbb{R}_{+}^{*}\}\text{ is $R$-bounded}.

Example 2.4

If the operator $\mathrm{i}D$ generates a strongly continuous group of operators on a Banach space $X$ then [HvNVW18, Example 10.6.3 p. 448] the operator $D$ is bisectorial of angle 0. Combined with Stone’s theorem [HvNVW18, Theorem G.6.2 p. 544], we see in particular that unbounded selfadjoint operators on Hilbert spaces are bisectorial of angle 0.

If the operator $D$ is bisectorial of type $\omega$ then by [HvNVW18, Proposition 10.6.2 (2) p. 448] its square $D^{2}$ is sectorial of type $2\omega$ and we have

(2.7)

\overline{\operatorname{Ran}D^{2}}=\overline{\operatorname{Ran}D}\quad\text{and}\quad\operatorname{Ker}D^{2}=\operatorname{Ker}D.

Functional calculus

Consider a bisectorial operator $D$ on a Banach space $X$ of type $\omega\in(0,\frac{\pi}{2})$ . For any $\theta\in(\omega,\frac{\pi}{2})$ and any function $f$ in the space

\mathrm{H}^{\infty}_{0}(\Sigma_{\theta}^{\mathrm{bi}})\overset{\mathrm{def}}{=}\left\{f\in\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}}):\>\exists C,s>0\>\forall\>z\in\Sigma_{\theta}^{\mathrm{bi}}:\>|f(z)|\leqslant C\min\{|z|^{s},|z|^{-s}\}\right\},

we can define an operator $f(D)$ acting on the space $X$ by integrating on the boundary of a bisector

(2.8)

f(D)\overset{\mathrm{def}}{=}\frac{1}{2\pi\mathrm{i}}\int_{\partial\Sigma^{\mathrm{bi}}_{\nu}}f(z)R(z,D)\mathop{}\mathopen{}\mathrm{d}z,

using the resolvent operator $R(z,D)$ where $\omega<\nu<\theta$ . See [Ege15, Section 3.2.1] for a more precise explanation. The operator $D$ is said to have a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}})$ functional calculus, if there exists a constant $C\geqslant 0$ such that

(2.9)

\big{\|}f(D)\big{\|}_{X\to X}\leqslant C\left\|f\right\|_{\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}})}

for any function $f$ in the space $\mathrm{H}^{\infty}_{0}(\Sigma_{\theta}^{\mathrm{bi}})$ .

Example 2.5

By [HvNVW18, Theorem 10.7.10 p. 461], if the operator $\mathrm{i}D$ is the generator of a bounded strongly continuous group on a $\mathrm{UMD}$ Banach space $X$ then the operator $D$ admits a $\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}})$ functional calculus for any angle $\theta>0$ .

$R$ -boundedness

Suppose that $1<p<\infty$ . Following [HvNVW18, Definition 8.1.1, Remark 8.1.2 p. 165], we say that a set $\mathcal{F}$ of bounded operators on a Banach space $X$ is $R$ -bounded provided that there exists a constant $C\geqslant 0$ such that for any finite families $T_{1},\ldots,T_{n}$ in $\mathcal{F}$ and $x_{1},\ldots,x_{n}$ in $X$ , we have

(2.10)

\Bigg{\|}\sum_{k=1}^{n}\varepsilon_{k}\otimes T_{k}(x_{k})\Bigg{\|}_{\mathrm{L}^{p}(\Omega,X)}\leqslant C\Bigg{\|}\sum_{k=1}^{n}\varepsilon_{k}\otimes x_{k}\Bigg{\|}_{\mathrm{L}^{p}(\Omega,X)},

where $(\varepsilon_{k})_{k\geqslant 1}$ is a sequence of independent Rademacher variables on some probability space $\Omega$ . This property is independent of $p$ .

3 Banach Fredholm modules and Banach $\mathrm{K}$ -homology

In this section, we define a Banach space variant of the theory of $\mathrm{K}$ -homology, relying on the notion of Fredholm module. We want replace Hilbert spaces by Banach spaces. Note that the classical notion of Fredholm module admit different variations in the literature (compare the references [HiR00, Definition 8.1.1 p. 199], [Con94, Definition 1 p. 293] and [CGIS14, Definition 2.2]). We start to introduce the following definition.

Definition 3.1

Let $\mathcal{A}$ be an algebra. An odd Banach Fredholm module $(X,F)$ over $\mathcal{A}$ on $X$ consists of a Banach space $X$ endowed with a representation $\pi\colon\mathcal{A}\to\mathrm{B}(X)$ , and a bounded operator $F\colon X\to X$ such that

1.

$F^{2}-\mathrm{Id}_{X}$ is a compact operator on $X$ ,
2.

for any $a\in\mathcal{A}$ the commutator $[F,\pi(a)]$ is a compact operator on $X$ .

Sometimes, we will use the notation $[F,a]$ for $[F,\pi(a)]$ . We also introduce a natural notion of even Fredholm module.

Definition 3.2

Let $\mathcal{A}$ be an algebra. An even Banach Fredholm module $(X,F,\gamma)$ over $\mathcal{A}$ on $X$ consists of a Fredholm module $(X,F)$ endowed with a bounded operator $\gamma\colon X\to X$ with $\gamma^{2}=\mathrm{Id}_{X}$ and such that

(3.1)

F\gamma=-\gamma F\quad\text{and}\quad[\pi(a),\gamma]=0

for any $a\in\mathcal{A}$ . We say that $\gamma$ is the grading operator.

In this case, $P\overset{\mathrm{def}}{=}\frac{\mathrm{Id}_{X}+\gamma}{2}$ is a bounded projection and we can write $X=X_{+}\oplus X_{-}$ where $X_{+}\overset{\mathrm{def}}{=}\operatorname{Ran}P$ and $X_{-}\overset{\mathrm{def}}{=}\operatorname{Ran}(\mathrm{Id}_{X}-P)$ . With respect to this decomposition, the equations of (3.1) implies that we can write

(3.2)

\pi(a)=\begin{bmatrix}\pi_{+}(a)&0\\ 0&\pi_{-}(a)\\ \end{bmatrix}\quad\text{and}\quad F=\begin{bmatrix}0&F_{-}\\ F_{+}&0\\ \end{bmatrix},

where $\pi_{+}\colon\mathcal{A}\to\mathrm{B}(X_{+})$ and $\pi_{-}\colon\mathcal{A}\to\mathrm{B}(X_{-})$ are representations of $\mathcal{A}$ . In particular, we have

(3.3)

\left[F,\pi(a)\right]=\begin{bmatrix}0&F_{-}a_{-}-a_{+}F_{+}\\ F_{+}a_{+}-a_{-}F_{+}&0\\ \end{bmatrix}.

Remark 3.3

Note that if $X$ is a Hilbert space, these definitions are weaker generalizations of the notions of Fredholm modules of the previous references since the assumption of selfadjointness is not required.

Example 3.4 (from odd to even Fredholm Banach modules)

Let $(X,F)$ be an odd Banach Fredholm module over $\mathcal{A}$ . It is possible to construct an even Banach Fredholm module $(X^{\prime},F^{\prime},\gamma)$ by letting $X^{\prime}\overset{\mathrm{def}}{=}X\oplus X$ , $\pi^{\prime}\overset{\mathrm{def}}{=}\pi\oplus\pi$ , $F^{\prime}\overset{\mathrm{def}}{=}\begin{bmatrix}0&F\\ F&0\\ \end{bmatrix}$ and $\gamma\overset{\mathrm{def}}{=}\begin{bmatrix}-\mathrm{Id}_{X}&0\\ 0&\mathrm{Id}_{X}\\ \end{bmatrix}$ . Indeed, we have $(F^{\prime})^{2}=\begin{bmatrix}F^{2}&0\\ 0&F^{2}\\ \end{bmatrix}\sim\begin{bmatrix}\mathrm{Id}_{X}&0\\ 0&\mathrm{Id}_{X}\\ \end{bmatrix}$ . Moreover, the commutator

\left[F^{\prime},\pi^{\prime}(a)\right]=F^{\prime}\pi^{\prime}(a)-\pi^{\prime}(a)F^{\prime}=\begin{bmatrix}0&[F,\pi(a)]\\ [F,\pi(a)]&0\\ \end{bmatrix}

is compact. Furthermore, we observe that $\gamma^{2}=\begin{bmatrix}\mathrm{Id}_{X}&0\\ 0&\mathrm{Id}_{X}\\ \end{bmatrix}$ ,

F^{\prime}\gamma=\begin{bmatrix}0&F\\ F&0\\ \end{bmatrix}\begin{bmatrix}-\mathrm{Id}_{X}&0\\ 0&\mathrm{Id}_{X}\\ \end{bmatrix}=\begin{bmatrix}0&F\\ -F&0\\ \end{bmatrix}=-\begin{bmatrix}-\mathrm{Id}_{X}&0\\ 0&\mathrm{Id}_{X}\\ \end{bmatrix}\begin{bmatrix}0&F\\ F&0\\ \end{bmatrix}=-\gamma F^{\prime}

and finally for any $a\in\mathcal{A}$

\left[\pi^{\prime}(a),\gamma\right]=\begin{bmatrix}\pi(a)&0\\ 0&\pi(a)\\ \end{bmatrix}\begin{bmatrix}-\mathrm{Id}_{X}&0\\ 0&\mathrm{Id}_{X}\\ \end{bmatrix}-\begin{bmatrix}-\mathrm{Id}_{X}&0\\ 0&\mathrm{Id}_{X}\\ \end{bmatrix}\begin{bmatrix}\pi(a)&0\\ 0&\pi(a)\\ \end{bmatrix}=0.

There is a natural notion of direct sum for Banach Fredholm modules. One takes the direct sum $\oplus_{2}$ of the Banach spaces, of the representations, and of the operators. We define the zero Banach Fredholm module with the zero Banach space, the zero representation and the zero operator.

The next definition is straightforward variation of [HiR00, Definition 8.2.1 p. 204].

Definition 3.5

Let $(X,F)$ be an odd Banach Fredholm module and let $U\colon Y\to X$ be an isometric isomorphism. Then $(U^{-1}\pi U,Y,U^{-1}FU)$ is also a Banach Fredholm module, and we say that it is isometrically equivalent to $(X,F)$ .

Similarly to [HiR00, Definition 8.2.2 p. 204], we introduce the following notion of equivalence.

Definition 3.6

Suppose that $(X,F_{t})$ is a family of odd Banach Fredholm modules parametrized by $t\in[0,1]$ , in which the representation and the Banach space remain constant but the operator $F_{t}$ varies with $t$ . If the function $[0,1]\to\mathrm{B}(X)$ , $t\mapsto F_{t}$ is norm continuous, then we say that the family defines an operator homotopy between the odd Banach Fredholm modules $(X,F_{0})$ and $(X,F_{1})$ , and that these two odd Fredholm modules are operator homotopic.

Definition 3.7 (compact perturbation)

Suppose that $(X,F)$ and $(X,F^{\prime})$ are Banach Fredholm modules on the same Banach space $X$ , and that $(F-F^{\prime})\pi(a)$ is compact for all $a\in\mathcal{A}$ . In this case, we say that $(X,F)$ is a compact perturbation of $(X,F^{\prime})$ .

Compact perturbation implies operator homotopy since the linear path from $F$ to $F^{\prime}$ defines an operator homotopy.

Similar definitions for the even case are left to the reader. In the spirit of [HiR00, Definition 8.2.5 p. 205], we introduce the next definition.

Definition 3.8 (odd Banach $\mathrm{K}$ -homology group)

Let $\mathscr{B}$ be a class of Banach spaces stable under countable sums containing the zero space. The Banach $\mathrm{K}$ -homology group $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$ is the abelian group with one generator for each isometric equivalence class $[(X,F)]$ of Banach Fredholm modules over $\mathcal{A}$ on any $X$ of $\mathscr{B}$ subject only to the relations:

1.

if $(X,F_{0})$ and $(X,F_{1})$ are operator homotopic Fredholm modules then $[(X,F_{0})]=[(X,F_{1})]$ in $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$ ,
2.

if $(X,F_{0})$ and $(X,F_{1})$ are any two Fredholm modules then

$[(X,F_{0})\oplus(X,F_{1})]=[(X,F_{0})]+[(X,F_{1})]$

in $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$ .

Similarly, we introduce a group $\mathrm{K}^{0}(\mathcal{A},\mathscr{B})$ in the even case.

Example 3.9

If $\mathscr{B}$ is the class of Hilbert spaces, the previous groups seems different from the K-homology groups $\mathrm{K}^{0}(\mathcal{A})$ and $\mathrm{K}^{1}(\mathcal{A})$ of [HiR00, Definition 8.2.5 p. 205] (defined for a separable $\mathrm{C}^{*}$ -algebra $\mathcal{A}$ ) since the bounded operator $F$ of a Banach Fredholm module $(H,F)$ is not necessarily selfadjoint.

Example 3.10

For any $1\leqslant p<\infty$ , it is natural to consider the class $\mathscr{L}^{p}$ of $\mathrm{L}^{p}$ -spaces and the class $\mathscr{L}^{p}_{\mathrm{nc}}$ of noncommutative $\mathrm{L}^{p}$ -spaces. We could also consider the class $\mathscr{S}\mathscr{L}_{p}$ of subspaces of $\mathrm{L}^{p}$ -spaces and the class $\mathscr{S}\mathscr{Q}\mathscr{L}_{p}$ of subspaces of quotients of $\mathrm{L}^{p}$ -spaces.

Definition 3.11 (degenerated Fredholm module)

Let $\mathcal{A}$ be an algebra. A odd Fredholm module $(X,F)$ over $\mathcal{A}$ on $X$ is said to be degenerate if for any $a\in\mathcal{A}$ we have $[F,\pi(a)]=0$ .

The interest is the following result.

Proposition 3.12

The class defined by a degenerate odd Banach Fredholm module $(X,F)$ in the group $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$ is zero.

Proof : We introduce $(Y,F^{\prime})$ where $Y$ is the direct sum $\oplus_{i=1}^{\infty}X$ of infinitely many copies of $X$ , and where $F^{\prime}$ and $\pi^{\prime}$ are infinite direct sums of copies of $F$ and $\pi$ . For any $a\in\mathcal{A}$ , we have $[F^{\prime},\pi(a)]=0$ . Consequently $(Y,F^{\prime})$ is a Banach Fredholm module. But clearly $(X,F)\oplus(Y,F^{\prime})$ is isometrically equivalent to $(Y,F^{\prime})$ . Hence we have $[(X,F)]+[(Y,F^{\prime})]=[(Y,F^{\prime})]$ in K-homology. We conclude that $[(X,F)]=0$ .

Proposition 3.13 (even to odd Fredholm module)

Let $\mathcal{A}$ be an algebra. Consider an even Banach Fredholm module $(X,F,\gamma)$ . Let $V\colon X_{-}\to X_{+}$ be a surjective isomorphism such that

(3.4)

VF_{+}V=F_{-},\quad V\pi_{-}(a)=\pi_{+}(a)V,\quad a\in\mathcal{A}.

With the notations of (3.2) and $G\overset{\mathrm{def}}{=}VF_{+}\colon X_{+}\to X_{+}$ , the pair $(X_{+},G)$ is an odd Banach Fredholm module on $\mathcal{A}$ , where the Banach space $X_{+}$ is endowed with the homomorphism $\pi_{+}\colon\mathcal{A}\to\mathrm{B}(X_{+})$ .

Proof : Since $F^{2}=\mathrm{Id}_{X}$ , we have $F_{-}F_{+}=\mathrm{Id}_{X_{+}}$ and $F_{+}F_{-}=\mathrm{Id}_{X_{-}}$ . We obtain that the operator

G^{2}-\mathrm{Id}_{X_{+}}=VF_{+}VF_{+}-\mathrm{Id}_{X_{+}}\overset{\eqref{div-986}}{=}F_{-}F_{+}-\mathrm{Id}_{X_{+}}

is compact. Moreover, we have

	$\displaystyle\big{[}VF_{+},\pi_{+}(a)\big{]}=VF_{+}\pi_{+}(a)-\pi_{+}(a)VF_{+}\overset{\eqref{div-986}}{=}VF_{+}\pi_{+}(a)VV^{-1}-\pi_{+}(a)F_{-}V^{-1}$
	$\displaystyle\overset{\eqref{div-986}}{=}VF_{+}V\pi_{-}(a)V^{-1}-\pi_{+}(a)F_{-}V^{-1}\overset{\eqref{div-986}}{=}F_{-}\pi_{-}(a)V^{-1}-\pi_{+}(a)F_{-}V^{-1}$
	$\displaystyle=(F_{-}\pi_{-}(a)-\pi_{+}(a)F_{-})V^{-1},$

which is a compact operator by (3.3) and since the space of compact operators is an ideal.

For the proof of Theorem 6.6, we introduce the notation

(3.5)

\mathop{}\mathopen{}\mathrm{d}x\overset{\mathrm{def}}{=}[F,x]

(Note that $\mathop{}\mathopen{}\mathrm{d}x\overset{\mathrm{def}}{=}\mathrm{i}[F,x]$ is better for some topics but we does not need it in this paper). A simple computation shows that

(3.6)

\mathop{}\mathopen{}\mathrm{d}(ab)=a\mathop{}\mathopen{}\mathrm{d}b+\mathop{}\mathopen{}\mathrm{d}a\cdot b\quad\text{and}\quad F\mathop{}\mathopen{}\mathrm{d}a=-\mathop{}\mathopen{}\mathrm{d}aF.

We also need a notion of Banach Fredholm module such that the relation $F^{2}-\mathrm{Id}$ is compact is ¡¡almost satisfied¿¿.

Definition 3.14

Let $\mathcal{A}$ be an algebra. A possibly kernel-degenerate Banach Fredholm module $(X,F)$ over $\mathcal{A}$ on $X$ consists of a Banach space $X$ endowed with a representation $\pi\colon\mathcal{A}\to\mathrm{B}(X)$ , and a bounded operator $F\colon X\to X$ such that

1.

$X$ can be written $X=Y\oplus\operatorname{Ker}F$ for some Banach space $Y$ , i.e. the subspace $\operatorname{Ker}F$ is complemented in $X$ by a bounded projection $Q\colon X\to X$ ,
2.

$F^{2}-i_{Y}$ is compact from $Y$ into $X$ where $i_{Y}\colon Y\to X$ is the canonical injection,
3.

for any $a\in\mathcal{A}$ the commutator $[F,\pi(a)]$ is a compact operator on $Y$ .

4 Coupling between Banach K-homology with K-theory

We start with the odd case.

Theorem 4.1 (coupling, odd case)

Let $\mathcal{A}$ be a unital algebra and let $\mathscr{B}$ be a class of Banach spaces stable under countable sums and containing the zero space. Consider an invertible $u$ of the matrix algebra $\mathrm{M}_{n}(\mathcal{A})$ and an odd Banach Fredholm module $(X,F)$ over $\mathcal{A}$ with $\pi(1)=\mathrm{Id}_{X}$ . We introduce the bounded operators $P_{n}\overset{\mathrm{def}}{=}\mathrm{Id}\otimes\frac{\mathrm{Id}+F}{2}\colon\ell^{2}_{n}(X)\to\ell^{2}_{n}(X)$ and $u_{n}\overset{\mathrm{def}}{=}(\mathrm{Id}\otimes\pi)(u)\colon\ell^{2}_{n}(X)\to\ell^{2}_{n}(X)$ . Then the bounded operator

(4.1)

P_{n}u_{n}P_{n}-(\mathrm{Id}-P_{n})\colon\ell^{2}_{n}(X)\to\ell^{2}_{n}(X)

is Fredholm. If $X$ is in $\mathscr{B}$ , its Fredholm index depends only on $[u]\in\mathrm{K}_{1}(\mathcal{A})$ and on $[F]\in\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$ .

Proof : We only do the case $n=1$ and we use the shorthand notation $P$ for the map $P_{1}=\frac{\mathrm{Id}+F}{2}$ . The case $n>1$ is left to the reader. Since

(4.2)

P^{2}=\frac{\mathrm{Id}+2F+F^{2}}{4}=\frac{\mathrm{Id}+2F}{4}+\frac{1}{4}F^{2}\sim\frac{\mathrm{Id}+2F}{4}+\frac{1}{4}\mathrm{Id}=\frac{\mathrm{Id}+F}{2}=P.

the map $P$ is a projection modulo a compact operator. If $u\in\mathcal{A}$ is invertible, we have

	$\displaystyle\big{[}P\pi(u)P-(\mathrm{Id}-P)\big{]}\big{[}P\pi(u^{-1})P-(\mathrm{Id}-P)\big{]}$
	$\displaystyle=P\pi(u)\underbrace{P^{2}}_{\sim P}\pi(u^{-1})P-P\pi(u)\underbrace{P(\mathrm{Id}-P)}_{\sim 0}-\underbrace{(\mathrm{Id}-P)P}_{\sim 0}\pi(u^{-1})P+\underbrace{(\mathrm{Id}-P)^{2}}_{\mathrm{I}-P}$
	$\displaystyle\sim P\pi(u)P\pi(u^{-1})P+(\mathrm{Id}-P)=P\pi(u)\big{[}P\pi(u^{-1})-\pi(u^{-1})P+\pi(u^{-1})P\big{]}P+(\mathrm{Id}-P)$
	$\displaystyle=P\pi(u)\big{(}[P,\pi(u^{-1})]+\pi(u^{-1})P\big{)}P+(\mathrm{Id}-P)$
	$\displaystyle=P\pi(u)\big{(}\tfrac{1}{2}[F,\pi(u^{-1})]+\pi(u^{-1})P\big{)}P+(\mathrm{Id}-P)$
	$\displaystyle\overset{\eqref{P2=P}}{\sim}\tfrac{1}{2}P\pi(u)[F,\pi(u^{-1})]P+P\pi(uu^{-1})P+(\mathrm{Id}-P)$
	$\displaystyle=\tfrac{1}{2}P\pi(u)\underbrace{[F,\pi(u^{-1})]}_{\textrm{compact}}P+P^{2}+(\mathrm{Id}-P)\sim\mathrm{Id}.$

Similarly, we have $\big{[}P\pi(u^{-1})P-(\mathrm{Id}-P)\big{]}\big{[}P\pi(u)P-(\mathrm{Id}-P)\big{]}\sim\mathrm{Id}$ . By Atkinson’s theorem (Theorem 2.2), we deduce that the bounded operator $P\pi(u)P-(\mathrm{Id}-P)$ is Fredholm. The other verifications are routine.

So we have a pairing $\langle\cdot,\cdot\rangle_{\mathrm{K}_{1}(\mathcal{A}),\mathrm{K}^{1}(\mathcal{A},\mathscr{B})}\colon\mathrm{K}_{1}(\mathcal{A})\times\mathrm{K}^{1}(\mathcal{A},\mathscr{B})\to\mathbb{Z}$ defined by

(4.3)

\big{\langle}[u],[(X,F)]\big{\rangle}_{\mathrm{K}_{1}(\mathcal{A}),\mathrm{K}^{1}(\mathcal{A},\mathscr{B})}=\operatorname{Index}P_{n}u_{n}P_{n}-(\mathrm{Id}-P_{n})=\operatorname{Index}PuP-(\mathrm{Id}-P).

If $P^{2}=P$ , a similar proof shows that

(4.4)

\big{\langle}[u],[(X,F)]\big{\rangle}_{\mathrm{K}_{1}(\mathcal{A}),\mathrm{K}^{1}(\mathcal{A},\mathscr{B})}=\operatorname{Index}P_{n}u_{n}P_{n}\colon P_{n}(\ell^{2}_{n}(X)))\to P_{n}(\ell^{2}_{n}(X)).

Similarly, we prove the even case.

Theorem 4.2 (coupling, even case)

Let $\mathcal{A}$ be a unital algebra and let $\mathscr{B}$ be a class of Banach spaces stable under countable sums and containing the zero space. Consider a projection $e$ of the matrix algebra $\mathrm{M}_{n}(\mathcal{A})$ and an even Banach Fredholm module $\bigg{(}X_{+}\oplus X_{-},\begin{bmatrix}0&F_{-}\\ F_{+}&0\\ \end{bmatrix},\gamma\bigg{)}$ over $\mathcal{A}$ with $\pi(1)=\mathrm{Id}_{X}$ . We introduce the bounded operator $e_{n}\overset{\mathrm{def}}{=}(\mathrm{Id}\otimes\pi)(e)\colon\ell^{2}_{n}(X)\to\ell^{2}_{n}(X)$ . Then the bounded operator

(4.5)

e_{n}(\mathrm{Id}\otimes F_{+})e_{n}\colon e_{n}(\ell^{2}_{n}(X_{+}))\to e_{n}(\ell^{2}_{n}(X_{-}))

is Fredholm. If $X$ is in $\mathscr{B}$ , its Fredholm index depends only on $[e]\in\mathrm{K}_{0}(\mathcal{A})$ and on $[F]\in\mathrm{K}^{0}(\mathcal{A},\mathscr{B})$ .

Proof : We only do the case $n=1$ and we use the shorthand notation $e=e_{1}$ . The case $n>1$ is left to the reader. Note that the operators $F_{-}F_{+}-\mathrm{Id}_{X_{+}}$ and $F_{+}F_{-}-\mathrm{Id}_{X_{-}}$ are compact since $\begin{bmatrix}F_{-}F_{+}&0\\ 0&F_{+}F_{-}\\ \end{bmatrix}-\mathrm{Id}_{X}=F^{2}-\mathrm{Id}_{X}$ is compact. We have

	$\displaystyle(eF_{+}e)(eF_{-}e)=eF_{+}eF_{-}e=e(F_{+}e-eF_{+}+eF_{+})F_{-}e$
	$\displaystyle=e([F_{+},e]+eF_{+})F_{-}e=e\underbrace{[F_{+},e]}_{\textrm{compact}}F_{-}e+eF_{+}F_{-}e\sim e.$

Similarly, we have $(eF_{-}e)(eF_{+}e)\sim e$ . By Atkinson’s theorem (Theorem 2.2), we deduce that the bounded operator $eF_{+}e\colon e(X_{+})\to e_{n}(X_{-})$ is Fredholm. The proof is complete.

Consequently, we have a pairing $\langle\cdot,\cdot\rangle_{\mathrm{K}_{0}(\mathcal{A}),\mathrm{K}^{0}(\mathcal{A},\mathscr{B})}\colon\mathrm{K}_{0}(\mathcal{A})\times\mathrm{K}^{0}(\mathcal{A},\mathscr{B})\to\mathbb{Z}$ defined by

(4.6)

\big{\langle}[e],[(X,F,\gamma)]\big{\rangle}_{\mathrm{K}_{0}(\mathcal{A}),\mathrm{K}^{0}(\mathcal{A},\mathscr{B})}=\operatorname{Index}e_{n}(\mathrm{Id}\otimes F_{+})e_{n}.

Remark 4.3

Similarly to [Con94, Proposition 2 p. 289], it is easy to extend these result to the non-unital case. For example, in the odd case, it suffices to replace $\mathrm{M}_{n}(\mathcal{A})$ by $\mathrm{M}_{n}(\tilde{\mathcal{A}})$ where $\tilde{\mathcal{A}}$ is the unitization of the non-unital algebra $\mathcal{A}$ .

5 Summability of Banach Fredholm modules

Approximation numbers

Following [Kon86, Definition p. 68], we say that a map assigning to any bounded operator $T$ between Banach spaces a sequence $((s_{n}(T))$ of real numbers is an $s$ -number function if the following conditions are satisfied.

1.

$\left\|T\right\|\geqslant s_{1}(T)\geqslant s_{2}(T)\geqslant\cdots\geqslant 0$ .
2.

If $R,T\colon X\to Y$ and if $n,m\geqslant 1$ then $s_{n+m-1}(R+T)\leqslant s_{n}(R)+s_{n}(T)$ .
3.

If $T\colon X_{0}\to X$ , $S\colon X\to Y$ and $R\colon Y\to Y_{0}$ we have

(5.1) $s_{n}(RST)\leqslant\left\|R\right\|s_{n}(S)\left\|T\right\|.$
4.

We have $s_{n}(T)=0$ if $\operatorname{rank}T<n$ and $s_{n}(\mathrm{Id}_{\ell^{2}_{n}})=1$ .

This concept was introduced by Pietsch in [Pie74]. If $X$ and $Y$ are Hilbert spaces and if $T\colon X\to Y$ is compact then by [Kon01, p. 69] these numbers coincide with the singular numbers of the operator $T$ . We refer to the books [Pi80], [Pie87], [Kon01] and to the survey papers [Kon01] and [Pie23] for more information. The history of this topic is equally described in [Pie07, Chapter 6].

Example 5.1

The approximation numbers [Kon86, Definition I.d.14 p. 69] of a bounded linear operator $T\colon X\to Y$ are defined by

a_{n}(T)\overset{\mathrm{def}}{=}\inf\{\left\|T-R\right\|:R\colon X\to Y,\operatorname{rank}R<n\},\quad n\geqslant 1.

The Weyl numbers are defined by

(5.2)

x_{n}(T)=\sup\{a_{n}(TA):A\colon\ell^{2}\to X\left\|A\right\|=1\},\quad n\geqslant 1.

These sequences are example of $s$ -number sequences by [Kon86, Lemma p. 69]. Moreover, we have $x_{n}(T)\leqslant a_{n}(T)$ for any integer $n\geqslant 1$ . Actually, the approximation numbers are the largest $s$ -numbers by [Kon86, Lemma p. 69].

Quasi-Banach ideals

Following [Kon86, Definition I.d.1 p. 56], a quasi-Banach ideal of operators $(\mathfrak{A},\alpha)$ is a subclass $\mathfrak{A}$ of all bounded linear operators between Banach spaces together with $\alpha\colon\mathfrak{A}\to\mathbb{R}^{+}$ such that for all Banach spaces $X,Y$ the sets $\mathfrak{A}(X,Y)\overset{\mathrm{def}}{=}\mathfrak{A}\cap\mathrm{B}(X,Y)$ satisfy:

1.

$\mathfrak{A}(X,Y)$ contains all finite-rank operators from $X$ to $Y$ and $\alpha(\mathrm{Id}_{\mathbb{C}})=1$ .
2.

$(\mathfrak{A}(X,Y),\alpha)$ is a quasi-Banach space with quasi-triangle constant $K$ independent of $X$ and $Y$ , i.e.

$\alpha(R+T)\leqslant K(\alpha(R)+\alpha(T)),\quad R,T\in\mathfrak{A}(X,Y)$

If $R\in\mathrm{B}(X_{0},X)$ , $T\in\mathfrak{A}(X,Y)$ , $S\in\mathrm{B}(Y,Y_{0})$ for some Banach spaces $X_{0}$ , $Y_{0}$ , then

(5.3)

STR\in\mathfrak{A}(X_{0},Y_{0})\quad\text{and}\quad\alpha(STR)\leqslant\left\|S\right\|\alpha(T)\left\|R\right\|.

If $\alpha$ is a norm on each $\mathfrak{A}(X,Y)$ , i.e. $K=1$ , then $(\mathfrak{A},\alpha)$ is called Banach ideal of operators, $\alpha$ the ideal norm. We also define $\mathfrak{A}(X)\overset{\mathrm{def}}{=}\mathfrak{A}(X,X)$ .

Quasi-Banach ideals associated to $s$ -numbers

Suppose that $0<q<\infty$ . Let $T\mapsto(s_{n}(T))_{n\geqslant 0}$ be an $s$ -number sequence. For any Banach spaces $X$ and $Y$ , following [Kon86, Definition 1.d.18 p. 72] we define the class

(5.4)

S^{q}_{s}(X,Y)\overset{\mathrm{def}}{=}\big{\{}T\colon X\to Y:(s_{n}(T))\in\ell^{q}\big{\}}.

Moreover, if $T\in S^{q}_{s}(X,Y)$ , we let

\left\|T\right\|_{S^{q}_{s}(X,Y)}\overset{\mathrm{def}}{=}\left\|(s_{n}(T))\right\|_{\ell^{q}}.

By [Kon86, Lemma p. 72], $(S^{q}_{s},\left\|\cdot\right\|_{S^{q}_{s}})$ is a quasi-Banach ideal. For the cases $s_{n}=a_{n}$ or if $s_{n}=x_{n}$ we will use the notations $S^{q}_{\mathrm{app}}$ and $S^{q}_{\mathrm{weyl}}$ . If $X=Y$ , we use the notation $S^{q}_{s}(X)$ . If in addition $X$ is equal to a Hilbert space $H$ , we recover the Schatten class $S^{q}(H)$ . If $0<p<q<\infty$ , we have $S^{p}_{s}(X)\subset S^{q}_{s}(X)$ . Furthermore, if $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$ , we have by [Kon86, Proposition 1.d.19 p. 73]

(5.5)

S^{p}_{s}(X)\circ S^{q}_{s}(X)\subset S^{r}_{s}(X).

By [Kon80, p. 220], the trace $\operatorname{Tr}$ on the space of finite-rank operators acting on $X$ admits a unique continuous extension on $S^{1}_{\mathrm{app}}(X)$ , again denoted $\operatorname{Tr}$ . The same thing is true for $S^{1}_{\mathrm{weyl}}$ for the class of Banach spaces with the bounded approximation property. Moreover, by [Kon86, pp. 224-225]), the previously trace coincide with the sum of eigenvalues of T, i.e. $\operatorname{Tr}T=\sum_{n=0}^{\infty}\lambda_{n}(T)$ . Furthermore, if $T\in S^{1}_{\mathrm{app}}(X)$ and if $R\colon X\to X$ is a bounded operator then by [Kon86, Corollary 2 p. 228]

(5.6)

\operatorname{Tr}(TR)=\operatorname{Tr}(RT).

It is worth noting that by [Kon86, Theorem 4.b.12 p. 245] a Banach space $X$ is isomorphic to a Hilbert space if and only if the space $\mathcal{N}(X)$ of nuclear operators coincide with the space $S^{1}_{\mathrm{app}}(X)$ .

Recall the Weyl’s inequality for operators acting on Banach spaces. For that recall that a bounded operator $T\colon X\to X$ is a Riesz operator if

1.

for all $\lambda\in\mathbb{C}-\{0\}$ , $T-\lambda\mathrm{Id}$ is a Fredholm operator and has finite ascent and finite descent,
2.

all non-zero spectral values $\lambda\in\sigma(T)$ are eigenvalues of finite multiplicity and have no accumulation point except possibly zero.

Then by [Kon86, Theorem 2.a.6 p. 85] any bounded operator $T\in S^{q}_{\mathrm{weyl}}(X)$ is a Riesz operator such that its sequence $(\lambda_{n})$ of eigenvalues belongs to the space $\ell^{q}$ and we have

(5.7)

\left\|(\lambda_{n})\right\|_{\ell^{q}}\leqslant 2^{\frac{1}{q}}\sqrt{2\mathrm{e}}\left\|T\right\|_{S^{q}_{\mathrm{weyl}}(X)}.

Summability of Banach Fredholm modules

Here, we give a generalization of [Con94, Definition 3 p. 290].

Definition 5.2

Suppose that $0<q<\infty$ . We say an odd Banach Fredholm module $(X,F)$ or an even Banach Fredhlom $(X,F,\gamma)$ is $q$ -summable if the commutator $[F,\pi(a)]$ belongs to the space $S^{q}_{\mathrm{app}}(X)$ for any $a\in\mathcal{A}$ .

By (5.5), this implies that if $n+1\geqslant q$ every product $[F,a_{0}][F,a_{1}]\cdots[F,a_{n}]$ of commutators belongs to the space $S^{1}_{\mathrm{app}}(X)$ .

6 The Chern character of a $q$ -summable Banach Fredhlom module

In this section, we extend the notion of Chern character introduced by Connes to our setting. In the spirit of [Con94, Definition 3 p. 295] (see also [CPR11, Definition 4.13 p. 34]), we introduce the following definition.

Definition 6.1 (odd case)

Consider a $q$ -summable odd Banach Fredholm module $(X,F)$ for some $q>0$ over an algebra $\mathcal{A}$ . Let $n$ be an odd integer with $n+1\geqslant q$ . We define the Chern character by the formula

(6.1)

\mathrm{Ch}_{n}^{F}(a_{0},a_{1},\dots,a_{n})\overset{\mathrm{def}}{=}c_{n}\operatorname{Tr}(F[F,a_{0}][F,a_{1}]\cdots[F,a_{n}]),\qquad a_{0},a_{1},\dots,a_{n}\in\mathcal{A},

where $c_{n}$ is a constant (but the exact value is not useful for this paper).

In the even case, the definition is slightly different.

Definition 6.2 (even case)

Consider a $q$ -summable even Banach Fredholm module $(X,F,\gamma)$ for some $q>0$ over an algebra $\mathcal{A}$ . Let $n$ be an even integer with $n+1\geqslant q$ . We define the Chern character by the formula

(6.2)

\mathrm{Ch}_{n}^{F}(a_{0},a_{1},\dots,a_{n})\overset{\mathrm{def}}{=}c_{n}\operatorname{Tr}(\gamma F[F,a_{0}][F,a_{1}]\cdots[F,a_{n}]),\qquad a_{0},a_{1},\dots,a_{n}\in\mathcal{A},

where we use a constant $c_{n}$ (but the exact value is not useful for this paper).

We need the following Banach space generalization of [GVF01, p. 143].

Proposition 6.3

Let $X$ be a Banach space. Let $T\colon X\mapsto X$ be a Fredholm operator and let $R\colon X\mapsto X$ be a bounded operator such that the operators $\mathrm{Id}-TR$ and $\mathrm{Id}-RT$ belong to the space $S^{1}_{\mathrm{app}}(X)$ . Then

\operatorname{Ind}T=\operatorname{Tr}(\mathrm{Id}-TR)-\operatorname{Tr}(\mathrm{Id}-RT).

Proof : Let $U$ be a generalized inverse of $T$ as in (2.3). Then

R=U+\underbrace{(\mathrm{Id}-UT)R-U(\mathrm{Id}-TR)}_{\overset{\mathrm{def}}{=}G}.

Since $\mathrm{Id}-UT$ has finite rank and $\mathrm{Id}-TR$ belongs to the space $S^{1}_{\mathrm{app}}(X)$ , it follows that $G$ also belongs to the space $S^{1}_{\mathrm{app}}(X)$ . Observe that

RT-TR=(U+G)T-T(U+G)=UT-TU+GT-TG.

We note that $\operatorname{Tr}GT\overset{\eqref{trace-prop}}{=}\operatorname{Tr}TG$ . Consequently, it suffices to prove the result for $R=U$ .

Recall that $\mathrm{Id}-TU$ and $\mathrm{Id}-UT$ are projections of finite rank. Thus $\mathrm{Id}-TU$ and $\mathrm{Id}-UT$ belong to $S^{1}_{\mathrm{app}}(X)$ . Hence $TU-UT$ belongs to $S^{1}_{\mathrm{app}}(X)$ and recalling that the trace of a finite rank projection is equal to the rank of the projection, we obtain

	$\displaystyle\operatorname{Tr}(\mathrm{Id}-UT)-\operatorname{Tr}(\mathrm{Id}-TU)=\operatorname{rank}(\mathrm{Id}-UT)-\operatorname{rank}(\mathrm{Id}-TU)$
	$\displaystyle=\dim\operatorname{Ker}T-\dim(Y/\operatorname{Ran}T)=\operatorname{Ind}T.$

Remark 6.4

Let $T\colon X\to X$ be a bounded operator. At the time of writing, it is unclear if the existence of a bounded operator $R\colon X\mapsto X$ be a bounded operator such that the operators $\mathrm{Id}-TR$ and $\mathrm{Id}-RT$ belong to the space $S^{1}_{\mathrm{app}}(X)$ implies that $T$ is a Fredholm operator.

The following is a Banach space generalization of [GVF01, Proposition 4.2 p. 144].

Corollary 6.5

Let $X$ be a Banach space. Let $T\colon X\mapsto X$ be a Fredholm operator and let $R\colon X\mapsto X$ be a bounded operator such that the operators $(\mathrm{Id}-TR)^{n}$ and $(\mathrm{Id}-RT)^{n}$ belong to the space $S^{1}_{\mathrm{app}}(X)$ for some integer $n\geqslant 1$ . Then the index of the Fredholm operator $T$ is given by

(6.3)

\operatorname{Ind}T=\operatorname{Tr}(\mathrm{Id}-RT)^{n}-\operatorname{Tr}(\mathrm{Id}-TR)^{n}.

Proof : We let $K\overset{\mathrm{def}}{=}\mathrm{Id}-RT$ and $L\overset{\mathrm{def}}{=}\mathrm{Id}-TR$ . Note that $TK=T-TRT=LT$ . We introduce the sum $S_{n}=\sum_{j=0}^{n-1}K^{j}R$ . We have

S_{n}T=\sum_{j=0}^{n-1}K^{j}RT=\sum_{j=0}^{n-1}K^{j}(\mathrm{Id}-K)=\mathrm{Id}-K^{n}.

Moreover, using $TK=LT$ in the second equality, we see that

TS_{n}=T\sum_{j=0}^{n-1}K^{j}R=\sum_{j=0}^{n-1}L^{j}TR=\sum_{j=0}^{n-1}L^{j}(\mathrm{Id}-L)=\mathrm{Id}-L^{n}.

Hence the operators $\mathrm{Id}-S_{n}T=K^{n}=(\mathrm{Id}-RT)^{n}$ and $\mathrm{Id}-TS_{n}=L^{n}=(\mathrm{Id}-TR)^{n}$ belong to the space $S^{1}_{\mathrm{app}}$ . Consequently, using Proposition 6.3 in the first equality, we obtain

\operatorname{Ind}T=\operatorname{Tr}(\mathrm{Id}-S_{n}T)-\operatorname{Tr}(\mathrm{Id}-TS_{n})=\operatorname{Tr}K^{n}-\operatorname{Tr}L^{n}=\operatorname{Tr}(\mathrm{Id}-RT)^{n}-\operatorname{Tr}(\mathrm{Id}-TR)^{n}.

Now we describe the pairing (4.4) in the odd case.

Theorem 6.6 (odd case)

Let $(X,F)$ be a $q$ -summable odd Banach Fredholm module over the algebra $\mathcal{A}$ for some $q>0$ with $F^{2}=\mathrm{Id}$ . Consider a class $\mathscr{B}$ of Banach spaces stable under countable sums and containing the zero space. For any element $[u]$ of the group $\mathrm{K}_{1}(\mathcal{A})$ and any odd integer $n$ with $n+1\geqslant q$ , we have

(6.4)

\big{\langle}[u],[(X,F)]\big{\rangle}_{\mathrm{K}_{1}(\mathcal{A}),\mathrm{K}^{1}(\mathcal{A},\mathscr{B})}=\frac{(-1)^{\frac{n+1}{2}}}{4^{\frac{n+1}{2}}c_{n}}\mathrm{Ch}_{n}^{F}(u^{-1},u,\ldots,u^{-1},u).

Proof : We let $N\overset{\mathrm{def}}{=}\frac{n+1}{2}$ and $P\overset{\mathrm{def}}{=}\frac{\mathrm{I}+F}{2}$ . Note that

	$\displaystyle P[P,u^{-1}]P=P(Pu^{-1}-u^{-1}P)P=P(Pu^{-1}+Pu^{-1}P-Pu^{-1}P-u^{-1}P)P$
	$\displaystyle=P(Pu^{-1}+u^{-1}P-Pu^{-1}-u^{-1}P)P=P[P,u^{-1}]P+P[P,u^{-1}]P.$

Hence $P[P,u^{-1}]P=0$ which is equivalent to $P\mathop{}\mathopen{}\mathrm{d}u^{-1}P=0$ . Using this equality, in the last equality, we obtain

(6.5)		$\displaystyle P-Pu^{-1}PuP=Pu^{-1}uP-Pu^{-1}PuP=-P(u^{-1}P-Pu^{-1})uP$
	$\displaystyle=P([P,u^{-1}])uP=\frac{1}{2}P\mathop{}\mathopen{}\mathrm{d}u^{-1}uP=\frac{1}{2}P\mathop{}\mathopen{}\mathrm{d}u^{-1}(uP-Pu+Pu)$
	$\displaystyle=-\frac{1}{4}P\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u+\frac{1}{2}P\mathop{}\mathopen{}\mathrm{d}u^{-1}Pu=-\frac{1}{4}P\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u.$

Since the the Banach Fredholm module is $q$ -summable, the elements $\mathop{}\mathopen{}\mathrm{d}u$ and $\mathop{}\mathopen{}\mathrm{d}u^{-1}$ belongs to the space $S^{q}_{\mathrm{app}}(X)$ . We deduce that $P-Pu^{-1}PuP$ belongs to $S^{\frac{q}{2}}_{\mathrm{app}}(X)$ , hence to $S^{N}_{\mathrm{app}}(X)$ since $n+1\geqslant q$ . Similarly, we can prove that

(6.6)

P-PuPu^{-1}P=-\frac{1}{4}P\mathop{}\mathopen{}\mathrm{d}u\mathop{}\mathopen{}\mathrm{d}u^{-1}.

Consequently, this operator also belongs to the space $S^{N}_{\mathrm{app}}(X)$ . So we use Corollary 6.5 with the operators $PuP\colon P(X)\to P(X)$ and $Pu^{-1}P\colon P(X)\to P(X)$ replacing $T$ and $R$ . Using the equalities $\operatorname{Tr}(\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u)^{N}=\operatorname{Tr}(\mathop{}\mathopen{}\mathrm{d}u\mathop{}\mathopen{}\mathrm{d}u^{-1})^{N}$ and $\operatorname{Tr}F(\mathop{}\mathopen{}\mathrm{d}u\mathop{}\mathopen{}\mathrm{d}u^{-1})^{N}\overset{\eqref{prop-quantized-deriv}}{=}-\operatorname{Tr}F(\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u)^{N}$ , we obtain

	$\displaystyle\operatorname{Index}(PuP)\overset{\eqref{index-power-N}}{=}\operatorname{Tr}(P-Pu^{-1}PuP)^{N}-\operatorname{Tr}(P-PuPu^{-1}P)^{N}$
	$\displaystyle\overset{\eqref{div96}\eqref{div90}}{=}\operatorname{Tr}(-\tfrac{1}{4}P\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u)^{N}-\operatorname{Tr}(-\tfrac{1}{4}P\mathop{}\mathopen{}\mathrm{d}u\mathop{}\mathopen{}\mathrm{d}u^{-1})^{N}$
	$\displaystyle=\frac{(-1)^{N}}{4^{N}}\big{[}\operatorname{Tr}(P\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u)^{N}-\operatorname{Tr}(P\mathop{}\mathopen{}\mathrm{d}u\mathop{}\mathopen{}\mathrm{d}u^{-1})^{N}\big{]}=\frac{(-1)^{N}}{4^{N}}\big{[}\operatorname{Tr}P(\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u)^{N}-\operatorname{Tr}P(\mathop{}\mathopen{}\mathrm{d}u\mathop{}\mathopen{}\mathrm{d}u^{-1})^{N}\big{]}$
	$\displaystyle=\frac{(-1)^{N}}{4^{N}}\bigg{[}\operatorname{Tr}\tfrac{\mathrm{I}+F}{2}(\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u)^{N}-\operatorname{Tr}\tfrac{\mathrm{I}+F}{2}(\mathop{}\mathopen{}\mathrm{d}u\mathop{}\mathopen{}\mathrm{d}u^{-1})^{N}\bigg{]}$
	$\displaystyle=\frac{(-1)^{N}}{2\cdot 4^{n}}\bigg{[}\operatorname{Tr}(\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u)^{N}+\operatorname{Tr}F(\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u)^{N}-\operatorname{Tr}(\mathop{}\mathopen{}\mathrm{d}u\mathop{}\mathopen{}\mathrm{d}u^{-1})^{N}-\operatorname{Tr}F(\mathop{}\mathopen{}\mathrm{d}u\mathop{}\mathopen{}\mathrm{d}u^{-1})^{N}\bigg{]}$
	$\displaystyle=\frac{(-1)^{N}}{4^{N}}\operatorname{Tr}\big{(}F(\mathop{}\mathopen{}\mathrm{d}u^{-1}\mathop{}\mathopen{}\mathrm{d}u)^{N}\big{)}=\frac{(-1)^{N}}{4^{N}}\operatorname{Tr}\big{(}F\underbrace{[F,u^{-1}][F,u]\cdots[F,u^{-1}]}_{2N=n+1}\big{)}$
	$\displaystyle\overset{\eqref{Chern-odd}}{=}\frac{(-1)^{\frac{n+1}{2}}}{4^{\frac{n+1}{2}}c_{n}}\mathrm{Ch}_{n}^{F}(u^{-1},u,\ldots,u^{-1},u).$

Now, we describe the pairing (4.6) in the even case. The proof is left to the reader as an easy exercise.

Theorem 6.7 (even case)

Let $(X,F,\gamma)$ be a $q$ -summable even Banach Fredholm module over the algebra $\mathcal{A}$ for some $q>0$ with $F^{2}=\mathrm{Id}$ . Consider a class $\mathscr{B}$ of Banach spaces stable under countable sums and containing the zero space. For any element $[e]$ of the group $\mathrm{K}_{0}(\mathcal{A})$ and any even integer $n\geqslant q$ , we have

(6.7)

\big{\langle}[e],[(X,F,\gamma)]\big{\rangle}_{\mathrm{K}_{0}(\mathcal{A}),\mathrm{K}^{0}(\mathcal{A},\mathscr{B})}=\frac{(-1)^{\frac{n}{2}}}{2c_{n}}\mathrm{Ch}_{n}^{F}(e,e,\ldots,e).

7 From Banach spectral triples to Banach Fredhlom modules

The following definition is extracted of [ArK22].

Definition 7.1 (Lipschitz algebra)

Consider a triple $(\mathcal{A},X,D)$ constituted of the following data: a Banach space $X$ , a closed unbounded operator $D$ on $X$ with dense domain $\operatorname{dom}D\subset X$ , and an algebra $\mathcal{A}$ endowed with a homomorphism $\pi\colon\mathcal{A}\to\mathrm{B}(X)$ . In this case, we define the Lipschitz algebra

(7.1)			$\displaystyle\operatorname{\mathrm{Lip}}_{D}(\mathcal{A})\overset{\mathrm{def}}{=}\big{\{}a\in\mathcal{A}:\pi(a)\cdot\operatorname{dom}D\subset\operatorname{dom}D\text{ and the unbounded operator }$
		$\displaystyle\qquad\qquad[D,\pi(a)]\colon\operatorname{dom}D\subset X\to X\text{ extends to an element of }\mathrm{B}(X)\big{\}}.$

We refer to [ArK22, Section 5.7] for some properties of $\operatorname{\mathrm{Lip}}_{D}(\mathcal{A})$ . The following definition is essentially [ArK22, Definition 5.10 p. 218].

Definition 7.2 (compact Banach spectral triple)

Consider a triple $(\mathcal{A},X,D)$ constituted of the following data: a reflexive¹¹10. It may perhaps be possible to replace the reflexivity by an assumption of weak compactness, see [HvNVW18, p. 361]. Banach space $X$ , a closed unbounded bisectorial operator $D$ on $X$ with dense domain $\operatorname{dom}D\subset X$ , and a Banach algebra $\mathcal{A}$ equipped with a homomorphism $\pi\colon\mathcal{A}\to\mathrm{B}(X)$ . We say that $(\mathcal{A},X,D)$ is a possibly kernel-degenerate compact Banach spectral triple if

1.

$D$ admits a bounded $\mathrm{H}^{\infty}(\Sigma^{\mathrm{bi}}_{\omega})$ functional calculus on a bisector $\Sigma^{\mathrm{bi}}_{\omega}$ .
2.

$D^{-1}$ is a compact operator on $\overline{\operatorname{Ran}D}$ .
3.

The subset $\operatorname{\mathrm{Lip}}_{D}(A)$ is dense in $\mathcal{A}$ .

If $\operatorname{Ker}D$ is in addition finite-dimensional, we say that $(\mathcal{A},X,D)$ is a compact Banach spectral triple.

In this situation, we have by [HvNVW18, p. 448] a direct sum decomposition $X=\overline{\operatorname{Ran}D}\oplus\operatorname{Ker}D$ . Now, we introduce of summability similar to the hilbertian context. Recall that the space $S^{q}_{\mathrm{app}}(X)$ is defined in (5.4). The operator $|D|^{-1}$ is well-defined on $\overline{\operatorname{Ran}D}$ . Furthermore, we can extend it by letting $|D|^{-1}=0$ on $\operatorname{Ker}D$ .

Definition 7.3 (summability)

Suppose that $1<q<\infty$ . We say that a possibly kernel-degenerate Banach spectral triple $(\mathcal{A},X,D)$ is $q$ -summable if the operator $|D|^{-1}$ is bounded and belongs to the space $S^{q}_{\mathrm{app}}(X)$ .

The following definition is essentially [ArK22, Definition 5.11 p. 2123]. Compare to [CGRS14, Definition 2.1 p. 33]. We can also introduce variants for algebras which are not Banach algebras. These extensions are left to the reader.

Definition 7.4 (locally compact Banach spectral triple)

Consider a triple $(\mathcal{A},X,D)$ constituted of the following data: a reflexive Banach space $X$ , a closed unbounded bisectorial operator $D$ on $X$ with dense domain $\operatorname{dom}D\subset X$ , and a Banach algebra $\mathcal{A}$ equipped with a homomorphism $\pi\colon\mathcal{A}\to\mathrm{B}(X)$ . We say that $(\mathcal{A},X,D)$ is a possibly kernel-degenerate locally compact Banach spectral triple if

1.

$D$ admits a bounded $\mathrm{H}^{\infty}(\Sigma^{\mathrm{bi}}_{\omega})$ functional calculus on a bisector $\Sigma^{\mathrm{bi}}_{\omega}$ .
2.

$\pi(a)|D|^{-1}|_{\overline{\operatorname{Ran}D}}$ is a compact operator on $\overline{\operatorname{Ran}D}\to X$ .
3.

The subset $\operatorname{\mathrm{Lip}}_{D}(A)$ is dense in $\mathcal{A}$ .

Strong convex compactness property

Recall that compactness is preserved under strong integrals by [Voi92, Theorem 1.3 p. 260]. More precisely, for any measure space $(\Omega,\mu)$ and any strongly measurable function $T\colon\Omega\to\mathcal{K}(X,Y)$ , with values in the space $\mathcal{K}(X,Y)$ of compact operators from $X$ into $Y$ , such that $\int_{\Omega}\left\|T(\omega)\right\|\mathop{}\mathopen{}\mathrm{d}\mu(\omega)$ is finite, the strong integral $\int_{\Omega}T(\omega)\mathop{}\mathopen{}\mathrm{d}\mu(\omega)$ is a compact operator.

Fractional powers

We need some background on fractional powers for a sectorial operator $A$ acting on a Banach space $X$ . If $0<\operatorname{Re}\alpha<1$ , we have by [Haa06, Proposition 3.2.1 p. 70] or [HvNVW23, p. 449]

(7.2)

A^{-\alpha}=\frac{\sin\pi\alpha}{\pi}\int_{0}^{\infty}t^{-\alpha}(t+A)^{-1}x\mathop{}\mathopen{}\mathrm{d}t,\quad x\in\operatorname{Ran}A.

Proposition 7.5

Let $(\mathcal{A},X,D)$ be a compact Banach spectral triple over an algebra $\mathcal{A}$ . Then $(X,\operatorname{\mathrm{sign}}D)$ is a Banach Fredholm module over $\mathcal{A}$ (where $X$ is endowed with the same homomorphism $\pi$ ).

Proof : We let $F\overset{\mathrm{def}}{=}\operatorname{\mathrm{sign}}D$ . Since $F^{2}-I$ vanishes on $\overline{\operatorname{Ran}D}$ by construction, the first requirement in the Definition 3.1 of a Banach Fredholm module hold true.

The operator $D$ is bisectorial. Consequently, by [HvNVW18, Proposition 10.6.2 p. 448] the operator $D^{2}$ is sectorial. Moreover, it admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta})$ functional calculus by [HvNVW18]. Recall that the functional calculus of $D$ and $D^{2}$ are compatible. So we have

(7.3)

|D|^{-1}x=(D^{2})^{-\frac{1}{2}}x\overset{\eqref{for-frac-powers}}{=}\frac{1}{\pi}\int_{0}^{\infty}(t+D^{2})^{-1}x\frac{\mathop{}\mathopen{}\mathrm{d}t}{\sqrt{t}},\quad x\in\operatorname{Ran}D.

The end of the proof is similar to the hilbertian one using that the compactness is preserved under strong integrals, Proposition 2.3 and [Haa06, Example 2.2.5 p. 29].

Remark 7.6

A more simple proof can be given with the additional assumption that $[|D|,a]$ defines a bounded operator on $X$ for all $a\in\mathcal{A}$ . Indeed, Let $a\in\mathcal{A}$ . Note that

(7.4)		$\displaystyle\left[\operatorname{\mathrm{sign}}D,a\right]=D\|D\|^{-1}a-aD\|D\|^{-1}=(Da-aD)\|D\|^{-1}+D(\|D\|^{-1}a-a\|D\|^{-1})$
	$\displaystyle=[D,a]\|D\|^{-1}+D[\|D\|^{-1},a]=[D,a]\|D\|^{-1}-D\|D\|^{-1}[\|D\|,a]\|D\|^{-1}$
	$\displaystyle=\big{(}\underbrace{[D,a]-\operatorname{\mathrm{sign}}(D)[\|D\|,a]\|}_{\overset{\mathrm{def}}{=}R}\big{)}\|D\|^{-1}.$

This operator is compact. Let $a\in\overline{\pi(\mathcal{A})}$ . There exists a sequence $(a_{k})$ of elements of $\mathcal{A}$ such that $\pi(a_{k})\to a$ . Each operator $[\operatorname{\mathrm{sign}}D,a_{k}]$ is compact by the first part of the proof. We have $[\operatorname{\mathrm{sign}}D,a_{k}]\to\left[\operatorname{\mathrm{sign}}D,a\right]$ since the operator product is continuous in $\mathrm{B}(X)$ . We conclude that the operator $\left[\operatorname{\mathrm{sign}}D,a\right]$ is compact.

Proposition 7.7

Suppose that $(\mathcal{A},X,D)$ is a $q$ -summable Banach spectral triple. Assume that $[|D|,a]$ is a well-defined bounded operator for any $a\in\mathcal{A}$ . Then $(X,\operatorname{\mathrm{sign}}D)$ is a $q$ -summable Banach Fredholm module for $\mathcal{A}$ .

Proof : Let $a\in\mathcal{A}$ . Recall that

\left[\operatorname{\mathrm{sign}}D,a\right]\overset{\eqref{formula-simple}}{=}R|D|^{-1}.

By the ideal property (5.3), the operator $R|D|^{-1}$ belongs to the space $S^{q}_{\mathrm{app}}(X)$ .

8 Examples of triples and Banach Fredholm modules

8.1 The Dirac operator $\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}$ on the space $\mathrm{L}^{p}(\mathbb{T})$ and the periodic Hilbert transform

Suppose that $1<p<\infty$ . Here $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$ is the one-dimensional torus. The functions defined on $\mathbb{T}$ can be identified with periodic functions on $\mathbb{R}$ with period $2\pi$ . In particular, we have

\mathrm{C}(\mathbb{T})=\{g\in\mathrm{C}(\mathbb{R}):f(\theta+2\pi)=f(\theta)\}

Recall that any multiplication operator $M_{f}\colon\mathrm{L}^{p}(\mathbb{T})\to\mathrm{L}^{p}(\mathbb{T})$ , $fg\mapsto fg$ by a function $f$ of $\mathrm{L}^{\infty}(\mathbb{T})$ is bounded, see [EnN00, Proposition 4.10 p. 31]. Consequently, we can consider the homomorphism $\pi\colon\mathrm{L}^{\infty}(\mathbb{T})\to\mathrm{B}(\mathrm{L}^{p}(\mathbb{T}))$ , $f\mapsto M_{f}$ . It is well-known [Kha13, Exercise 4.2.6 p. 194] [GVF01, p. 390] that $(\mathrm{C}^{\infty}(\mathbb{T}),\mathrm{L}^{2}(\mathbb{T}),\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta})$ is a compact spectral triple, where the used homomorphism is the restriction of $\pi$ on the algebra $\mathrm{C}^{\infty}(\mathbb{T})$ . First, we prove an $\mathrm{L}^{p}$ -generalization of this classical fact. Here, we consider the closure of the unbounded operator $\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}\colon\mathrm{C}^{\infty}(\mathbb{T})\subset\mathrm{L}^{p}(\mathbb{T})\to\mathrm{L}^{p}(\mathbb{T})$ . We denote again $\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}$ its closure whose the domain $\operatorname{dom}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}$ is the Sobolev space

\mathrm{W}^{1,p}(\mathbb{T})\overset{\mathrm{def}}{=}\{f\in\mathrm{L}^{p}(\mathbb{R}):f(\theta+2\pi)=f(\theta),f\text{ is absolutely continuous},f^{\prime}\in\mathrm{L}^{p}(\mathbb{R})\}.

Suppose that $1<p<\infty$ and $\alpha\geqslant 0$ . We define the subspace

(8.1)

\mathrm{L}^{p}_{\alpha}(\mathbb{T})\overset{\mathrm{def}}{=}\operatorname{dom}\Delta^{\frac{\alpha}{2}}_{p}

of $\mathrm{L}^{p}(\mathbb{T})$ . If $f\in\mathrm{L}^{p}_{\alpha}(\mathbb{T})$ , we will use the notation

(8.2)

\left\|f\right\|_{\mathrm{L}^{p}_{\alpha}(\mathbb{T})}\overset{\mathrm{def}}{=}\big{\|}(\mathrm{Id}+\Delta_{p})^{\frac{\alpha}{2}}(f)\big{\|}_{\mathrm{L}^{p}(\mathbb{T})}.

Suppose $1<q<p<\infty$ and $\alpha>0$ such that $\frac{\alpha}{d}\geqslant\frac{1}{q}-\frac{1}{p}$ . We will use the Sobolev inequality

(8.3)

\left\|f\right\|_{\mathrm{L}^{p}(\mathbb{T}^{d})}\lesssim\left\|f\right\|_{\mathrm{L}^{q}_{\alpha}(\mathbb{T}^{d})}.

of [BEO13, Corollary 1.2], which gives a map $J\colon\mathrm{L}^{q}_{\alpha}(\mathbb{T}^{d})\to\mathrm{L}^{p}(\mathbb{T}^{d})$ . Moreover, it is known [Kon86, Theorem p. 187] that if $\frac{\alpha}{d}>\frac{1}{q}-\frac{1}{p}$

(8.4)

a_{n}(J)\lesssim\frac{1}{n^{\frac{\alpha}{d}}},\quad n\geqslant 0.

The main point of the following result is the new connection between Sobolev embedding and summability in the proof.

Theorem 8.1

Suppose that $1<p<\infty$ . The triple $(\mathrm{C}^{\infty}(\mathbb{T}),\mathrm{L}^{p}(\mathbb{T}),\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta})$ is a Banach compact spectral triple which is $q$ -summable for any $q>1$ .

Proof : Note that it is well-known that the operator $\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}$ generate a strongly continuous group of operators acting on $\mathrm{L}^{p}(\mathbb{T})$ , namely the group of translations. By Example 2.4 and Example 2.5, we deduce that the unbounded operator $\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}$ is bisectorial of angle 0 and admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\omega}^{\mathrm{bi}})$ functional calculus for any angle $\omega>0$ on the Banach space $\mathrm{L}^{p}(\mathbb{T})$ . For any function $f\in\mathrm{C}^{\infty}(\mathbb{T})$ , an elementary computation reveals that $M_{f}\cdot\operatorname{dom}\tfrac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}\subset\operatorname{dom}\tfrac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}$ and that

(8.5)

\big{[}\tfrac{1}{\mathrm{i}}\tfrac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta},f\big{]}=\frac{1}{\mathrm{i}}M_{f^{\prime}}.

So the commutator $[\frac{1}{\mathrm{i}}\tfrac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta},f]$ defines a bounded operator on the Banach space $\mathrm{L}^{p}(\mathbb{T})$ . Finally, it is stated (without proof; the end of the proof below is stronger than this fact) in [Kat76, Example 6.31 p. 187] the operator $\frac{1}{\mathrm{i}}\tfrac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}$ has compact resolvent. So Definition 7.4 is satisfied.

Suppose that $1<q<p<2$ . Let $\alpha>0$ . For any function $f\in\mathrm{L}^{p}(\mathbb{T})$ , we have

\left\|(\mathrm{Id}+\Delta)^{-\frac{\alpha}{2}}(f)\right\|_{\mathrm{L}^{q}_{\alpha}(\mathbb{T})}\overset{\eqref{Def-Lpalpha-bis}}{=}\left\|(\mathrm{Id}+\Delta)^{\frac{\alpha}{2}}(\mathrm{Id}+\Delta)^{-\frac{\alpha}{2}}f\right\|_{\mathrm{L}^{q}(\mathbb{T})}=\left\|f\right\|_{\mathrm{L}^{q}(\mathbb{T})}\lesssim\left\|x\right\|_{\mathrm{L}^{p}(\mathbb{T})}.

If $\alpha>\frac{1}{q}-\frac{1}{p}$ , using the Sobolev inequality (8.3), we deduce the factorization

(8.6)

(\mathrm{Id}+\Delta)^{-\frac{\alpha}{2}}\colon\mathrm{L}^{p}(\mathbb{T})\xrightarrow{(\mathrm{Id}+\Delta)^{-\frac{\alpha}{2}}}\mathrm{L}^{q}_{\alpha}(\mathbb{T})\xrightarrow{J}\mathrm{L}^{p}(\mathbb{T}).

With [Kon86, Theorem p. 187], we obtain for any integer $n\geqslant 1$

\displaystyle a_{n}((\mathrm{Id}+\Delta)^{-\frac{\alpha}{2}})\overset{\eqref{facto-1}}{=}a_{n}(J(\mathrm{Id}+\Delta)^{-\frac{\alpha}{2}})\overset{\eqref{majo-sn-2}}{\leqslant}\left\|(\mathrm{Id}+\Delta)^{-\frac{\alpha}{2}}\right\|a_{n}(J)\overset{\eqref{sn-de-Sobolev}}{\lesssim}\frac{1}{n^{\alpha}}.

The series $\sum a_{n}((\mathrm{Id}+\Delta)^{-\frac{\alpha}{2}})$ converges if $\alpha>1$ . We deduce that the operator $(\mathrm{Id}+\Delta)^{-\frac{\alpha}{2}}\colon\mathrm{L}^{p}(\mathbb{T})\to\mathrm{L}^{p}(\mathbb{T})$ belongs to the space $S^{1}_{\mathrm{app}}(\mathrm{L}^{p}(\mathbb{T}))$ . Thus the triple $(\mathrm{C}^{\infty}(\mathbb{T}),\mathrm{L}^{p}(\mathbb{R}),\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x})$ is $q$ -summable for any $q>1$ . If $p>2$ , we reason by duality.

Remark 8.2

Suppose that $1<p<\infty$ . In this remark, we sketch a proof explaing why the previous result is optimal. If $p=2$ it is well-known. The Banach compact spectral triple $(\mathrm{C}^{\infty}(\mathbb{T}),\mathrm{L}^{p}(\mathbb{T}),\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta})$ is not $1$ -summable. Indeed, by [AmG16, Remark B.7], the spectrum of the Dirac operator $D\overset{\mathrm{def}}{=}\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}$ is independent of $p$ and the Laplacian has the same property by [Dav89, Theorem 1.6.3 p. 36]. In the case $p=2$ , it is known that $\operatorname{Sp}D=\mathbb{Z}$ . Using Weyl’s inequality (5.7) and a ¡¡bisectorial version¿¿ (left to the reader) of the spectral theorem of [Haa06, p. 56] [Haa05], we see that for any $\mu\not\in\operatorname{Sp}D$

\displaystyle\sum_{n\in\mathbb{Z}}\frac{1}{|\mu-n|}=\left\|\big{(}\lambda_{n}(R_{\mu}(D))\big{)}\right\|_{\ell^{1}}\overset{\eqref{Weyl-inequality}}{\lesssim}\left\|R_{\mu}(D)\right\|_{S^{1}_{\mathrm{weyl}}(\mathrm{L}^{p}(\mathbb{T}))}\leqslant\left\|R_{\mu}(D)\right\|_{S^{1}_{\mathrm{app}}(\mathrm{L}^{p}(\mathbb{T}))},

where we use the resolvent operator $R_{\mu}(D)$ . We conclude that the triple is not $1$ -summable according to Definition 7.3.

Recall that the periodic Hilbert transform $\mathcal{H}\colon\mathrm{L}^{p}(\mathbb{T})\to\mathrm{L}^{p}(\mathbb{T})$ is defined by the principal value [Kin09a, (3.285) p. 132] [Cas22, p. 181]

(8.7)

(\mathcal{H}f)(\theta)\overset{\mathrm{def}}{=}\frac{1}{2\pi}\lim_{\varepsilon\to 0^{+}}\int_{\varepsilon\leqslant|s|\leqslant\pi}f(\theta-s)\cot(\tfrac{s}{2})\mathop{}\mathopen{}\mathrm{d}s.

and can be seen as a Fourier multiplier with symbol $-\mathrm{i}\,\operatorname{\mathrm{sign}}(n)$ , i.e. we have

\widehat{\mathcal{H}(f)}(n)=-\mathrm{i}\operatorname{\mathrm{sign}}(n)\hat{f}(n),\quad n\in\mathbb{Z},

where $\operatorname{\mathrm{sign}}(0)\overset{\mathrm{def}}{=}0$ . The boundedness of the periodic Hilbert transform $\mathcal{H}$ on the space $\mathrm{L}^{p}(\mathbb{T})$ is proved in [Kin09a, Section 6.17] or [HvNVW16, Proposition 5.2.5 p. 391] (by transference from the Hilbert transform). Another proof is provided by the next result. Now, we determine the index pairing of the associated Banach Fredholm module by Proposition 7.5.

Recall that the winding number of a a function $f\colon\mathbb{T}\to\mathbb{C}-\{0\}$ is the number of turns of the point $f(e^{\mathrm{i}t})$ around the origin when $t$ runs from $0$ to $2\pi$ . More precisely, consider a continuous branch of the argument $\arg_{f}$ of the function $[0,2\pi]\to\mathbb{C}-\{0\}$ , $t\mapsto f(\mathrm{e}^{\mathrm{i}t})$ , i.e. $\arg_{f}$ is continuous on $[0,2\pi]$ and

\frac{f(\mathrm{e}^{\mathrm{i}t})}{|f(\mathrm{e}^{\mathrm{i}t})|}=\mathrm{e}^{\mathrm{i}\arg_{f}(t)},\quad t\in[0,2\pi]

The winding number of $f$ is defined by

(8.8)

\operatorname{wind}f\overset{\mathrm{def}}{=}\frac{1}{2\pi}(\arg_{f}(2\pi)-\arg_{f}(0)).

It is worth noting that if $f\in\mathrm{C}^{1}(\mathbb{T})$ and does not vanish on $\mathbb{T}$ then

\operatorname{wind}f=\frac{1}{2\pi\mathrm{i}}\int_{0}^{2\pi}\frac{f^{\prime}(\mathrm{e}^{\mathrm{i}\theta})}{f(\mathrm{e}^{\mathrm{i}\theta})}\mathop{}\mathopen{}\mathrm{d}\theta.

Now, we describe the index pairing. The case $p=2$ is folklore.

Proposition 8.3

Suppose that $1<p<\infty$ . The kernel-degenerate odd Banach Fredholm module over the algebra $\mathrm{C}(\mathbb{T})$ associated to the Banach compact spectral triple $(\mathrm{C}^{\infty}(\mathbb{T}),\mathrm{L}^{p}(\mathbb{T}),\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta})$ is $(\mathrm{L}^{p}(\mathbb{T}),\mathrm{i}\mathcal{H})$ . For any function $f\in\mathrm{C}(\mathbb{T})$ which does not vanish on $\mathbb{T}$ , we have

(8.9)

\big{\langle}[f],(\mathrm{L}^{p}(\mathbb{T}),\mathrm{i}\mathcal{H})\big{\rangle}_{\mathrm{K}_{1}(\mathrm{C}(\mathbb{T})),\mathrm{K}^{1}(\mathrm{C}(\mathbb{T}),\mathscr{L}^{p})}=-\operatorname{wind}f.

Proof : The first sentence is obvious. We let $F\overset{\mathrm{def}}{=}\mathrm{i}\mathcal{H}$ . Note that the map $Q\colon\mathrm{L}^{p}(\mathbb{T})\to\mathrm{L}^{p}(\mathbb{T})$ , $\mathrm{e}^{\mathrm{i}n\theta}\to\delta_{n=0}$ (which maps a function on the constant term in its Fourier series) is a finite-rank bounded map. So $F^{\prime}\overset{\mathrm{def}}{=}F+Q$ is clearly a Banach Fredholm module since $(F^{\prime})^{2}=\mathrm{Id}_{\mathrm{L}^{p}(\mathbb{T})}$ and a compact perturbation of $F$ . Moreover, observe that the map $P\overset{\mathrm{def}}{=}\frac{\mathrm{Id}+F^{\prime}}{2}$ identifies to the Riesz projection $\mathrm{L}^{p}(\mathbb{T})\to\mathrm{L}^{p}(\mathbb{T})$ , $\mathrm{e}^{\mathrm{i}n\theta}\mapsto\delta_{n\geqslant 0}\mathrm{e}^{\mathrm{i}n\theta}$ . The range of this map is the Hardy space $\mathrm{H}^{p}(\mathbb{T})$ . Let $f\in\mathrm{C}(\mathbb{T})$ which does not vanish on $\mathbb{T}$ . This means that $f$ is an invertible element of the unital algebra $\mathrm{C}(\mathbb{T})$ . The operator $PM_{f}P\colon P(\mathrm{L}^{p}(\mathbb{T}))\to P(\mathrm{L}^{p}(\mathbb{T}))$ of (4.1) (with $n=1$ ) identifies to the Toeplitz operator $T_{f}\colon\mathrm{H}^{p}(\mathbb{T})\to\mathrm{H}^{p}(\mathbb{T})$ , $g\mapsto P(fg)$ with symbol $f\in\mathrm{L}^{\infty}(\mathbb{T})$ , where we identify the map $P$ with its corestriction $P|^{\mathrm{H}^{p}(\mathbb{T})}$ . So the operator $T_{f}$ is Fredholm. Using the classical Gohberg-Krein index theorem [BoS06, Theorem 2.42 p. 74], we deduce that

\displaystyle\big{\langle}[f],(\mathrm{L}^{p}(\mathbb{T}),\mathrm{i}\mathcal{H})\big{\rangle}_{\mathrm{K}_{1}(\mathrm{C}(\mathbb{T})),\mathrm{K}^{1}(\mathrm{C}(\mathbb{T}),\mathscr{L}^{p})}\overset{\eqref{pairing-odd-2}}{=}\operatorname{Index}PM_{f}P=\operatorname{Index}T_{f}=-\operatorname{wind}f.

Recall that the first group of $\mathrm{K}$ -theory is given by $\mathrm{K}_{1}(\mathrm{C}(\mathbb{T}))=\mathbb{Z}$ . By the way, we refer to [ScS23] for a nice study of the K-groups of spheres.

Remark 8.4

It is folklore that for any function $f\in\mathrm{L}^{\infty}(\mathbb{T})$ the commutator $[F,f]$ is compact on $\mathrm{L}^{p}(\mathbb{T})$ if and only if $f$ belongs to the space $\mathrm{VMO}(\mathbb{T})$ of functions of vanishing mean oscillation. In the case $p=2$ , it is also known (essentially in [JaW82]) that the commutator $[F,f]$ belongs to the Schatten space $S^{q}(\mathrm{L}^{2}(\mathbb{T}))$ if and only if $f$ belongs to the Besov space $\mathrm{B}^{\frac{1}{q}}_{q,q}(\mathbb{T})$ .

Remark 8.5

It is easy to compute the Chern character in the case $1<p<\infty$ . The case $p=2$ is folklore.

8.2 The Dirac operator $\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}$ on the space $\mathrm{L}^{p}(\mathbb{R})$ and the Hilbert transform

Suppose that $1<p<\infty$ . Recall that any multiplication operator $M_{f}\colon\mathrm{L}^{p}(\mathbb{R})\to\mathrm{L}^{p}(\mathbb{R})$ , $fg\mapsto fg$ by a function $f$ of $\mathrm{L}^{\infty}(\mathbb{R})$ is bounded, see [EnN00, Proposition 4.10 p. 31]. Consequently, we can consider the homomorphism $\pi\colon\mathrm{L}^{\infty}(\mathbb{R})\to\mathrm{B}(\mathrm{L}^{p}(\mathbb{R}))$ , $f\mapsto M_{f}$ . It is well-known [Ren04, Corollary 14 p. 92] that $(\mathrm{C}^{\infty}_{0}(\mathbb{R}),\mathrm{L}^{2}(\mathbb{R}),\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x})$ is a locally compact spectral triple, where the used homomorphism is the restriction of $\pi$ on the algebra $\mathrm{C}^{\infty}_{0}(\mathbb{R})$ . First, we prove a weaker $\mathrm{L}^{p}$ -variant of this classical fact. Here, we consider the closure of the unbounded operator $\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}\colon\mathrm{C}^{\infty}_{0}(\mathbb{R})\subset\mathrm{L}^{p}(\mathbb{R})\to\mathrm{L}^{p}(\mathbb{R})$ . We denote again $\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}$ its closure whose the domain $\operatorname{dom}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}$ is the Sobolev space

\mathrm{W}^{1,p}(\mathbb{R})\overset{\mathrm{def}}{=}\{f\in\mathrm{L}^{p}(\mathbb{R}):f\text{ is absolutely continuous},f^{\prime}\in\mathrm{L}^{p}(\mathbb{R})\}.

Proposition 8.6

Suppose that $2\leqslant p<\infty$ . Consider the triple $(\mathrm{C}_{0}^{\infty}(\mathbb{R}),\mathrm{L}^{p}(\mathbb{R}),\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x})$ . The operator $\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}$ is bisectorial of angle 0 and admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\omega}^{\mathrm{bi}})$ functional calculus for any angle $\omega>0$ and the commutator $\big{[}\tfrac{1}{\mathrm{i}}\tfrac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x},f\big{]}$ is a well-defined bounded operator for any function $f\in\mathrm{C}^{\infty}_{0}(\mathbb{R})$ .

Proof : Note that it is well-known [EnN00, Proposition 1 p. 66] that the operator $\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}$ generate a strongly continuous group of operators acting on $\mathrm{L}^{p}(\mathbb{R})$ , namely the group of translations. By Example 2.4 and Example 2.5, we deduce that the unbounded operator $\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}\theta}$ is bisectorial of angle 0 and admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\omega}^{\mathrm{bi}})$ functional calculus for any angle $\omega>0$ on the Banach space $\mathrm{L}^{p}(\mathbb{R})$ . For any function $f\in\mathrm{C}_{c}^{\infty}(\mathbb{R})$ , an elementary computation reveals that $M_{f}\cdot\operatorname{dom}\tfrac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}\subset\operatorname{dom}\tfrac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}$ and that

(8.10)

\big{[}\tfrac{1}{\mathrm{i}}\tfrac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x},f\big{]}=\frac{1}{\mathrm{i}}M_{f^{\prime}}.

So $[\frac{1}{\mathrm{i}}\tfrac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x},f]$ defines a bounded operator on the Banach space $\mathrm{L}^{p}(\mathbb{R})$ .

Now, we consider the Hilbert transform $\mathcal{H}\colon\mathrm{L}^{p}(\mathbb{R})\to\mathrm{L}^{p}(\mathbb{R})$ . This transformation is a bounded operator by [HvNVW16, Theorem 5.1.1 p. 374] defined by the principal value

(8.11)

(\mathcal{H}f)(x)\overset{\mathrm{def}}{=}\frac{1}{\pi}\operatorname{p.v.}\int_{\mathbb{R}}\frac{f(y)}{x-y}\mathop{}\mathopen{}\mathrm{d}y,\quad f\in\mathcal{S}(\mathbb{R}),\text{ for almost every }x\in\mathbb{R}

and can be seen by [HvNVW16, Proposition 5.2.2 p. 389] as a Fourier multiplier with symbol $-\mathrm{i}\,\operatorname{\mathrm{sign}}(\xi)$ , i.e. we have

(8.12)

\widehat{\mathcal{H}(f)}(\xi)=-\mathrm{i}\,\operatorname{\mathrm{sign}}(\xi)\hat{f}(\xi),\quad f\in\mathcal{S}(\mathbb{R}),\xi\in\mathbb{R}.

Note that it was observed in [Con94, p. 314] that $\mathrm{i}\mathcal{H}$ is a Fredholm module on the Hilbert space $\mathrm{L}^{2}(\mathbb{R})$ . Note that the commutator is defined by (1.2)

For the computation of the index pairing, we need some information on Hardy spaces. Recall the notation $\mathbb{C}_{+}\overset{\mathrm{def}}{=}\{z\in\mathbb{C}:\mathrm{Im}z>0\}$ . Following [Nik02, Definition 6.3.2 p. 145], we denote by $\mathrm{H}^{p}(\mathbb{C}_{+})$ the space of all functions $F$ which are analytic in the upper half plane $\mathbb{C}_{+}$ such that

\left\|F\right\|_{\mathrm{H}^{p}(\mathbb{C}_{+})}\overset{\mathrm{def}}{=}\sup_{y>0}\left(\int_{\mathbb{R}}|F(x+\mathrm{i}y)|^{p}\mathop{}\mathopen{}\mathrm{d}x\right)^{\frac{1}{p}}<\infty,

and $\sup_{z\in\mathbb{C}_{+}}|F(z)|<\infty$ for $p=\infty$ . We also introduce the closed subspace

\mathrm{H}^{p}(\mathbb{R})=\left\{f\in\mathrm{L}^{p}(\mathbb{R}):\int_{\mathbb{R}}\frac{f(t)}{t-\overline{z}}\mathop{}\mathopen{}\mathrm{d}t=0\text{ for all }z\in\mathbb{C}_{+}\right\}

of the Banach space $\mathrm{L}^{p}(\mathbb{R})$ . Cauchy’s representation theorem in Hardy spaces, e.g. [Mas09, Chapter 13], can be formulated as follows. Consider some analytic function $F\colon\mathbb{C}_{+}\to\mathbb{C}$ . Then the following assertions are equivalent.

(i)

The function $F$ belongs to the space $\mathrm{H}^{p}(\mathbb{C}_{+})$ .
(ii)

There exists a unique function $f\in\mathrm{H}^{p}(\mathbb{R})$ such that

(8.13) $F(z)=\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}}\frac{f(t)}{t-z}\mathop{}\mathopen{}\mathrm{d}t,\quad z\in\mathbb{C_{+}}.$

In this case, we have

\left\|F\right\|_{\mathrm{H}^{p}(\mathbb{C}_{+})}=\left\|f\right\|_{\mathrm{L}^{p}(\mathbb{R})}.

and the nontangential boundary function of $F$ is equal to $f$ , i.e. $f(x)=\lim_{z\to x,z\in\mathbb{C}_{+}}F(z)$ . Consequently, we have an isometric isomorphism of the Hardy space $\mathrm{H}^{p}(\mathbb{C}_{+})$ onto the space $\mathrm{H}^{p}(\mathbb{R})$ . Thus the Hardy space $\mathrm{H}^{p}(\mathbb{C}_{+})$ can be identified with a closed subspace of the Banach space $\mathrm{L}^{p}(\mathbb{R})$ .

The Toeplitz operator $T_{g}\colon\mathrm{H}^{p}(\mathbb{R})\to\mathrm{H}^{p}(\mathbb{R})$ with symbol $g\in\mathrm{L}^{\infty}(\mathbb{R})$ is defined by

(8.14)

T_{g}(f)\overset{\mathrm{def}}{=}P_{+}(gf),\quad f\in\mathrm{H}^{p}(\mathbb{R}).

If $f\colon\mathbb{R}\cup\{\infty\}\to\mathbb{C}$ is a continuous function then the image of $f$ in the complex plane is a closed curve $\Gamma$ . If

(8.15)

\inf\limits_{x\in\mathbb{R}\cup\{\infty\}}|f(x)|>0,

we define the winding number $\operatorname{wind}f$ as the winding number of $\Gamma$ around the origin. The condition (8.15) is equivalent to say that $f$ is invertible in the algebra $\mathrm{C}(\mathbb{R}\cup\{\infty\})$ .

We will use the following result [Cam17, Theorem 6.2].

Theorem 8.7

Let $f\in\mathrm{C}(\mathbb{R}\cup\{\infty\})$ . The operator $T_{f}\colon\mathrm{H}^{p}(\mathbb{R})\to\mathrm{H}^{p}(\mathbb{R})$ has closed range if and only if (8.15) is satisfied. In this case, the operator $T_{f}$ is Fredholm and its Fredholm index is $\operatorname{Index}T_{f}=-\operatorname{wind}f$ .

Remark 8.8

It is known that the condition (8.15) is equivalent to the existence of a Wiener-Hopf $p$ -factorization of $f$ . In this case, $f=f_{-}r^{k}f_{+}$ with $k=\operatorname{wind}f$ .

The following observation is new even if $p=2$ . Recall that $\mathrm{K}_{1}(\mathrm{C}_{0}(\mathbb{R}))=\mathbb{Z}$ , see e.g. [WeO93].

Proposition 8.9

Suppose that $1<p<\infty$ . The Banach odd Fredholm module over the algebra $\mathrm{C}_{0}(\mathbb{R})$ associated to the triple $(\mathrm{C}^{\infty}_{0}(\mathbb{R}),\mathrm{L}^{p}(\mathbb{R}),\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x})$ is $(\mathrm{L}^{p}(\mathbb{R}),\mathrm{i}\mathcal{H})$ . For any function $f\in\mathrm{C}_{0}^{\infty}(\mathbb{R})$ satisfying (8.15), we have

(8.16)

\big{\langle}[f],(\mathrm{L}^{p}(\mathbb{R}),\mathrm{i}\mathcal{H})\big{\rangle}_{\mathrm{K}_{1}(\mathrm{C}_{0}(\mathbb{R})),\mathrm{K}^{1}(\mathrm{C}_{0}(\mathbb{R}),\mathscr{L}^{p})}=-\operatorname{wind}f,

where is the winding number of $f$ .

Proof : We let $F\overset{\mathrm{def}}{=}\mathrm{i}\mathcal{H}$ . We have $F^{2}=\mathrm{Id}$ . It is known [Uch78, Theorem 2 p. 17] that a function $f\in\mathrm{L}^{\infty}(\mathbb{R})$ belongs to the space $\mathrm{VMO}(\mathbb{R})$ of functions of vanishing mean oscillation if and only if the commutator $[F,M_{f}]$ is a compact operator acting on the Banach space $\mathrm{L}^{p}(\mathbb{R})$ . We infer that $(\mathrm{L}^{p}(\mathbb{R}),F)$ is a odd Fredhlom module over the algebra $\mathrm{C}_{0}(\mathbb{R})$ of continuous functions that vanish at infinity, since $\mathrm{C}_{0}(\mathbb{R})$ is a subspace of $\mathrm{VMO}(\mathbb{R})$ .

Moreover, observe that the map $P\overset{\mathrm{def}}{=}\frac{\mathrm{Id}+F}{2}$ identifies to the bounded projection $\mathrm{L}^{p}(\mathbb{R})\to\mathrm{L}^{p}(\mathbb{R})$ on the subspace $\mathrm{H}^{p}(\mathbb{R})$ . Let $f\in\mathrm{C}_{0}(\mathbb{R})$ satisfying (8.15). This means that $f$ is an invertible element of the unitization $\mathrm{C}(X\cup\{\infty\})$ of the algebra $\mathrm{C}_{0}(X)$ . The operator $PM_{f}P\colon P(\mathrm{L}^{p}(\mathbb{R}))\to P(\mathrm{L}^{p}(\mathbb{R}))$ of (4.1) (with $n=1$ ) identifies to the Toeplitz operator $T_{f}\colon\mathrm{H}^{p}(\mathbb{R})\to\mathrm{H}^{p}(\mathbb{R})$ , $g\mapsto P(fg)$ with symbol $f$ , where we identify the map $P$ with its corestriction $P|^{\mathrm{H}^{p}(\mathbb{R})}$ . So the operator $T_{f}$ is Fredholm. Using Theorem 8.7, we deduce that

\displaystyle\big{\langle}[f],(\mathrm{L}^{p}(\mathbb{R}),\mathrm{i}\mathcal{H})\big{\rangle}_{\mathrm{K}_{1}(\mathrm{C}_{0}(\mathbb{R})),\mathrm{K}^{1}(\mathrm{C}_{0}(\mathbb{R}),\mathscr{L}^{p})}\overset{\eqref{pairing-odd-2}}{=}\operatorname{Index}PM_{f}P=\operatorname{Index}T_{f}=-\operatorname{wind}f.

Remark 8.10

Our approach is flexible. We can state variations with a possible weight, matricial versions and $\mathrm{UMD}$ -vector-valued variants.

Remark 8.11

Suppose that $p=2$ . If $0<q<\infty$ , It is known that the commutator $[F,M_{f}]$ is in the Schatten class $S^{q}(\mathrm{L}^{2}(\mathbb{R}))$ if and only if the function $f$ is in the Besov space $\mathrm{B}^{q}_{q,q}(\mathbb{R})$ .

Remark 8.12

It should be possible to prove the case $p=2$ of Theorem 8.7 using Connes character formula for locally compact spectral triples. It suffices to find the result in the literature on the noncommutative measure (using the Dixmier trace) and to check the assumptions of [SuZ23, Theorem 1.2.5].

8.3 The Dirac operator on the space $\mathrm{L}^{p}(\mathbb{C})\oplus\mathrm{L}^{p}(\mathbb{C})$ and the complex Riesz transform

In this section, we will use the classical operators

\partial_{\overline{z}}\overset{\mathrm{def}}{=}\frac{1}{2}\bigg{(}\frac{\partial}{\partial x}+\mathrm{i}\frac{\partial}{\partial y}\bigg{)}\quad\text{and}\quad\partial_{z}\overset{\mathrm{def}}{=}\frac{1}{2}\bigg{(}\frac{\partial}{\partial x}-\mathrm{i}\frac{\partial}{\partial y}\bigg{)}.

Recall that $\Delta=4\partial_{\overline{z}}\partial_{z}$ The operators $\partial_{\overline{z}}$ , $\partial_{z}$ and $\Delta$ can be seen as Fourier multipliers on with symbol $-\mathrm{i}\pi\zeta$ , $-\mathrm{i}\pi\overline{\zeta}$ and $|\zeta|^{2}$ , see [AIM09, p. 99]. In the sequel, we consider the Dirac operator

(8.17)

D\overset{\mathrm{def}}{=}\begin{bmatrix}0&-2\partial_{\overline{z}}\\ 2\partial_{z}&0\\ \end{bmatrix}.

acting on a subspace of the Banach space $\mathrm{L}^{p}(\mathbb{C})\oplus\mathrm{L}^{p}(\mathbb{C})$ . The square of this operator is given by $D^{2}=\begin{bmatrix}-4\partial_{\overline{z}}\partial_{z}&0\\ 0&-4\partial_{z}\partial_{\overline{z}}\\ \end{bmatrix}=\begin{bmatrix}-\Delta&0\\ 0&-\Delta\\ \end{bmatrix}$ . In the sequel, we use the notations $\partial\overset{\mathrm{def}}{=}2\partial_{z}$ and $\partial^{*}\overset{\mathrm{def}}{=}-2\partial_{\overline{z}}$ . We start with a technical result.

Proposition 8.13

Suppose that $1<p<\infty$ . The family

(8.18)

\Big{\{}t\partial(\mathrm{Id}-t^{2}\Delta)^{-1}:t>0\Big{\}}

of operators of $\mathrm{B}(\mathrm{L}^{p}(\mathbb{C}))$ is $R$ -bounded.

Proof : Note that the Riesz transform $\partial(-\Delta)^{-\frac{1}{2}}\colon\mathrm{L}^{p}(\mathbb{C})\to\mathrm{L}^{p}(\mathbb{C})$ is a well-defined bounded operator by [AIM09, Corollary 4.5.1 p. 127]. Suppose that $t>0$ . A standard functional calculus argument gives

(8.19)

\displaystyle t\partial(\mathrm{Id}-t^{2}\Delta)^{-1}=\partial(-\Delta)^{-\frac{1}{2}}\Big{(}(-t^{2}\Delta)^{\frac{1}{2}}(\mathrm{Id}-t^{2}\Delta)^{-1}\Big{)}.

By [HvNVW18, Theorem 10.2.25 p. 391], note that the Laplacian $-\Delta$ has a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta})$ functional calculus for any angle $\theta>0$ . Moreover, the Banach space $\mathrm{L}^{p}(\mathbb{C})$ is $\mathrm{UMD}$ by [HvNVW16, Proposition 4.2.15 p. 291], hence has the triangular contraction property $(\Delta)$ by [HvNVW18, Theorem 7.5.9 p. 137]. We deduce by [HvNVW18, Theorem 10.3.4 (2) p. 401] that the operator $-\Delta$ is $R$ -sectorial. By [HvNVW18, Example 10.3.5 p. 402] applied with $\alpha=\frac{1}{2}$ and $\beta=1$ , we infer that the set

\big{\{}(-t^{2}\Delta)^{\frac{1}{2}}(\mathrm{Id}-t^{2}\Delta)^{-1}:t>0\big{\}}

of operators of $\mathrm{B}(\mathrm{L}^{p}(\mathbb{C}))$ is $R$ -bounded. Recalling that a singleton is $R$ -bounded by [HvNVW18, Example 8.1.7 p. 170], we obtain by composition [HvNVW18, Proposition 8.1.19 (3) p. 178] that the set

\Big{\{}\partial(-\Delta)^{-\frac{1}{2}}\Big{(}(-t^{2}\Delta)^{\frac{1}{2}}(\mathrm{Id}-t^{2}\Delta)^{-1}\Big{)}:t>0\Big{\}}

of operators of $\mathrm{B}(\mathrm{L}^{p}(\mathbb{C}))$ is $R$ -bounded. Hence with (8.19) we conclude that the subset (8.27) is $R$ -bounded.

Theorem 8.14

Suppose that $1<p<\infty$ . The unbounded operator $D$ admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}})$ functional calculus on the Banach space $\mathrm{L}^{p}(\mathbb{C})\oplus\mathrm{L}^{p}(\mathbb{C})$ for some $0<\theta<\frac{\pi}{2}$ .

Proof : We will start by showing that the set $\{\mathrm{i}t:t\in\mathbb{R},t\not=0\}$ is contained in the resolvent set of $D$ . We will do this by showing that $\mathrm{Id}-\mathrm{i}tD$ has a two-sided bounded inverse $(\mathrm{Id}-\mathrm{i}tD)^{-1}$ given by

(8.20)

\begin{bmatrix}(\mathrm{Id}-t^{2}\Delta)^{-1}&\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{*}\\ \mathrm{i}t\partial(\mathrm{Id}-t^{2}\Delta)^{-1}&(\mathrm{Id}-t^{2}\Delta)^{-1}\end{bmatrix}

acting on $\mathrm{L}^{p}(\mathbb{C})\oplus_{p}\mathrm{L}^{p}(\mathbb{C})$ . By Proposition 8.13 and since the operator $-\Delta$ satisfy the property (2.6) of $R$ -sectoriality, the four entries are bounded. It only remains to check that this matrix defines a two-sided inverse of $\mathrm{Id}-\mathrm{i}tD$ . We have the following equalities of operators acting on $\operatorname{dom}D$ .

	$\displaystyle\begin{bmatrix}(\mathrm{Id}-t^{2}\Delta)^{-1}&\mathrm{i}t(\mathrm{Id}_{\mathrm{L}^{p}}-t^{2}\Delta)^{-1}\partial^{*}\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial&(\mathrm{Id}-t^{2}\Delta)^{-1}\end{bmatrix}(\mathrm{Id}-\mathrm{i}tD)$
	$\displaystyle=\begin{bmatrix}(\mathrm{Id}-t^{2}\Delta)^{-1}&\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial&(\mathrm{Id}-t^{2}\Delta)^{-1}\end{bmatrix}\begin{bmatrix}\mathrm{Id}&-\mathrm{i}t\partial^{}\\ -\mathrm{i}t\partial&\mathrm{Id}\end{bmatrix}$
	$\displaystyle=\left[\begin{matrix}(\mathrm{Id}-t^{2}\Delta)^{-1}+t^{2}(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}\partial&-\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}+\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial-\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial&t^{2}(\mathrm{Id}-t^{2}\Delta)^{-1}\partial\partial^{}+(\mathrm{Id}-t^{2}\Delta)^{-1}\end{matrix}\right]$
	$\displaystyle=\left[\begin{matrix}(\mathrm{Id}-t^{2}\Delta)^{-1}+t^{2}(\mathrm{Id}-t^{2}\Delta)^{-1}\Delta&0\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial-\mathrm{i}t\big{(}\mathrm{Id}-t^{2}\Delta\big{)}^{-1}\partial&(\mathrm{Id}-t^{2}\Delta)^{-1}(t^{2}\partial\partial^{*}+\mathrm{Id})\end{matrix}\right]=\begin{bmatrix}\mathrm{Id}&0\\ 0&\mathrm{Id}\end{bmatrix}$

and similarly

	$\displaystyle(\mathrm{Id}-\mathrm{i}tD)\begin{bmatrix}(\mathrm{Id}-t^{2}\Delta)^{-1}&\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{*}\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial&(\mathrm{Id}-t^{2}\Delta)^{-1}\end{bmatrix}$
	$\displaystyle=\begin{bmatrix}\mathrm{Id}&-\mathrm{i}t\partial^{}\\ -\mathrm{i}t\partial&\mathrm{Id}\end{bmatrix}\begin{bmatrix}(\mathrm{Id}-t^{2}\Delta)^{-1}&\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial&(\mathrm{Id}-t^{2}\Delta)^{-1}\end{bmatrix}$
	$\displaystyle=\left[\begin{matrix}(\mathrm{Id}-t^{2}\Delta)^{-1}+t^{2}\partial^{}(\mathrm{Id}-t^{2}\Delta)^{-1}\partial&\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}-\mathrm{i}t\partial^{}\big{(}(\mathrm{Id}-t^{2}\Delta)^{-1}\big{)}\\ -\mathrm{i}t\partial(\mathrm{Id}-t^{2}\Delta)^{-1}+\mathrm{i}t\partial(\mathrm{Id}-t^{2}\Delta)^{-1}&t^{2}\partial(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}+(\mathrm{Id}-t^{2}\Delta)^{-1}\end{matrix}\right]$
	$\displaystyle=\begin{bmatrix}\mathrm{Id}&0\\ 0&\mathrm{Id}\end{bmatrix}.$

It remains to show that the set $\{\mathrm{i}t(\mathrm{i}t-D)^{-1}:t\not=0\}=\{(\mathrm{Id}-\mathrm{i}tD)^{-1}:t\not=0\}$ is $R$ -bounded. For this, observe that the diagonal entries of (8.20) are $R$ -bounded by the $R$ -sectoriality of $\Delta$ . The $R$ -boundedness of the other entries follows from the $R$ -gradient bounds of Proposition 8.13. Since a set of operator matrices is $R$ -bounded precisely when each entry is $R$ -bounded, we conclude that (2.6) is satisfied, i.e. that the operator $D$ is $R$ -bisectorial.

Remark 8.15

We can introduce the homomorphism $\pi\colon\mathrm{C}_{0}^{\infty}(\mathbb{C})\to\mathrm{B}(\mathrm{L}^{p}(\mathbb{C})\oplus\mathrm{L}^{p}(\mathbb{C}))$ , $f\mapsto\begin{bmatrix}M_{f}&0\\ 0&M_{f}\\ \end{bmatrix}$ . It is easy to check that the commutators $[D,\pi(f)]$ are well-defined bounded operators.

Note that we have

\operatorname{\mathrm{sign}}D\overset{\eqref{Dirac-operator-complex}}{=}D|D|^{-1}=\begin{bmatrix}0&-2\partial_{\overline{z}}\\ 2\partial_{z}&0\\ \end{bmatrix}\begin{bmatrix}-\Delta&0\\ 0&-\Delta\\ \end{bmatrix}^{-\frac{1}{2}}=\begin{bmatrix}0&*\\ 2\partial_{z}(-\Delta)^{-\frac{1}{2}}&0\\ \end{bmatrix}

Note that the operator $\partial_{z}(-\Delta)^{-\frac{1}{2}}$ is the complex Riesz transform $R\colon\mathrm{L}^{p}(\mathbb{C})\to\mathrm{L}^{p}(\mathbb{C})$ (or complex Hilbert transform) considered in [AIM09, pp. 102-103]. It is a Fourier multiplier with symbol $\frac{\overline{\zeta}}{|\zeta|}$ . We have an even Banach Fredholm module $\bigg{(}\mathrm{L}^{p}(\mathbb{C})\oplus\mathrm{L}^{p}(\mathbb{C}),\operatorname{\mathrm{sign}}D,\begin{bmatrix}-\mathrm{Id}&0\\ 0&\mathrm{Id}\\ \end{bmatrix}\bigg{)}$ .

Remark 8.16

By [WeO93, Exercise 9.C p. 172], we have $\mathrm{K}_{0}(\mathrm{C}_{0}(\mathbb{R}^{2}))=\mathbb{Z}$ and $\mathrm{K}_{1}(\mathrm{C}_{0}(\mathbb{R}^{2}))=0$ . The index pairing is related to the Bott projector. We skip the well-known explanation.

Remark 8.17

It is worth noting that by [AIM09, p. 102], the (planar) Beurling-Ahlfors operator $\mathcal{S}\colon\mathrm{L}^{p}(\mathbb{C})\to\mathrm{L}^{p}(\mathbb{C})$ defined by

(8.21)

(\mathcal{S}f)(z)\overset{\mathrm{def}}{=}-\frac{1}{\pi}\operatorname{p.v.}\int_{\mathbb{C}}\frac{f(w)}{(w-z)^{2}}\mathop{}\mathopen{}\mathrm{d}A(w).

is equal to $R^{2}$ , with symbol $\big{(}\frac{\overline{\zeta}}{\zeta}\big{)}^{2}=\frac{\overline{\zeta}}{\zeta}$ , see [AIM09, Corollary 4.1.1 p. 102]. This operator has the property that it turns $\bar{z}$ -derivatives into $z$ -derivatives, i.e. we have $\mathcal{T}(\partial_{\overline{z}}f)=\partial_{z}f$ for any function $f\in\mathrm{C}^{\infty}_{0}(\mathbb{C})$ . Note that $\bigg{(}\mathrm{L}^{p}(\mathbb{C})\oplus\mathrm{L}^{p}(\mathbb{C}),\begin{bmatrix}0&\mathcal{S}^{-1}\\ \mathcal{S}&0\\ \end{bmatrix},\begin{bmatrix}-\mathrm{Id}&0\\ 0&\mathrm{Id}\\ \end{bmatrix}\bigg{)}$ is clearly an even Banach Fredholm module, since by [AIM09, Theorem 4.6.14 p. 145] the commutators $[\mathcal{T},M_{f}]$ are compact on $\mathrm{L}^{p}(\mathbb{C})$ for any function $f$ of $\mathrm{VMO}(\mathbb{C})$ and since . By the way, it is worth noting that by [AIM09, Theorem 4.6.13 p. 143] the commutators $[\mathcal{T},M_{f}]$ are bounded on the Banach space $\mathrm{L}^{p}(\mathbb{C})$ for any function $f$ belonging to the space $\mathrm{BMO}(\mathbb{C})$ .

8.4 The Dirac operator on the space $\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})\oplus\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})$

Quantum tori

We will use standard notations and we refer to the papers [CXY13], [FXZ23], [EcI18], [MSX19] and [XXY18] for more information. Let $d\geqslant 2$ . To each $d\times d$ real skew-symmetric matrix $\theta$ , one may associate a 2-cocycle $\sigma_{\theta}\colon\mathbb{Z}^{d}\times\mathbb{Z}^{d}\to\mathbb{T}$ of the group $\mathbb{Z}^{d}$ defined by $\sigma_{\theta}(m,n)\overset{\mathrm{def}}{=}\mathrm{e}^{\frac{\mathrm{i}}{2}\langle m,\theta n\rangle}$ where $m,n\in\mathbb{Z}^{d}$ . We have $\sigma(m,-m)=\sigma(-m,m)$ for any $m\in\mathbb{Z}^{d}$ .

We define the $d$ -dimensional noncommutative torus $\mathrm{L}^{\infty}(\mathbb{T}_{\theta}^{d})$ as the twisted group von Neumann algebra $\mathrm{VN}(\mathbb{Z}^{d},\sigma_{\theta})$ . One can provide a concrete realization in the following manner. If $(\varepsilon_{n})_{n\in\mathbb{Z}^{d}}$ is the canonical basis of the Hilbert space $\ell^{2}_{\mathbb{Z}^{d}}$ and if $m\in\mathbb{Z}^{d}$ , we can consider the bounded operator $U_{m}\colon\ell^{2}_{\mathbb{Z}^{d}}\to\ell^{2}_{\mathbb{Z}^{d}}$ defined by

(8.22)

U_{m}(\varepsilon_{n})\overset{\mathrm{def}}{=}\sigma_{\theta}(m,n)\varepsilon_{m+n},\quad n\in\mathbb{Z}.

The $d$ -dimensional noncommutative torus $\mathrm{L}^{\infty}(\mathbb{T}_{\theta}^{d})$ is the von Neumann subalgebra of $\mathrm{B}(\ell^{2}_{\mathbb{Z}^{d}})$ generated by the $*$ -algebra $\mathcal{P}_{\theta}\overset{\mathrm{def}}{=}\mathrm{span}\big{\{}U^{m}\ :\ m\in\mathbb{Z}^{d}\big{\}}$ . Recall that for any $m,n\in\mathbb{Z}^{d}$ we have

(8.23)

U_{m}U_{n}=\sigma_{\theta}(m,n)U_{m+n}\quad\text{and}\quad\big{(}U_{m}\big{)}^{*}=\overline{\sigma_{\theta}(m,-m)}U_{-m}.

The von Neumann algebra $\mathrm{L}^{\infty}(\mathbb{T}_{\theta}^{d})$ is finite with normalized trace given by $\tau(x)\overset{\mathrm{def}}{=}\langle\varepsilon_{0},x(\varepsilon_{0})\rangle_{\ell^{2}_{\mathbb{Z}^{d}}}$ where $x\in\mathrm{L}^{\infty}(\mathbb{T}_{\theta}^{d})$ . In particular, we have $\tau(U_{m})=\delta_{m=0}$ for any $m\in\mathbb{Z}^{d}$ .

Let $\Delta$ be the unbounded operator acting on $\mathrm{L}^{\infty}(\mathbb{T}_{\theta}^{d})$ defined on the weak* dense subspace $\mathcal{P}_{\theta}$ by $\Delta(U_{m})\overset{\mathrm{def}}{=}4\pi^{2}|m|^{2}U_{m}$ where $|m|\overset{\mathrm{def}}{=}m_{1}^{2}+\cdots+m_{d}^{2}$ . Then this operator is weak* closable and its weak* closure is the opposite of a weak* generator of a symmetric Markovian semigroup $(T_{t})_{t\geqslant 0}$ of operators acting on $\mathrm{L}^{\infty}(\mathbb{T}_{\theta}^{d})$ , called the noncommutative heat semigroup on the noncommutative torus.

For any $j=1,\ldots,d$ , we may define the partial differentiation operators $\partial_{j}$ by

\partial_{j}(U^{n})\overset{\mathrm{def}}{=}2\pi\mathrm{i}n_{j}U^{n},\quad n=(n_{1},\ldots,n_{d})\in\mathbb{Z}^{d}.

Every partial derivation $\partial_{j}$ can be viewed a densely defined closed unbounded operator acting on the Hilbert space $\mathrm{L}^{2}(\mathbb{T}^{d}_{\theta})$ .

The Dirac operator

The Dirac operator $D$ is defined in terms of $\gamma$ matrices in direct analogy to commutative tori. Define $N\overset{\mathrm{def}}{=}2^{\lfloor\frac{d}{2}\rfloor}$ and select $N\times N$ complex selfadjoint matrices $\{\gamma_{1},\ldots,\gamma_{d}\}$ satisfying $\gamma_{j}\gamma_{k}+\gamma_{k}\gamma_{j}=2\delta_{j,k}1$ . Following [EcI18, (B.6) p. 147] and [GVF01, Definition 12.14 p. 545], we define the unbounded densely defined linear operator

(8.24)

D\overset{\mathrm{def}}{=}-\mathrm{i}\sum_{j=1}^{d}\gamma_{j}\otimes\partial_{j}

acting on the complex Hilbert space $\ell^{2}_{N}(\mathrm{L}^{2}(\mathbb{T}^{d}_{\theta}))$ . The operator $D$ is selfadjoint. By [GVF01, p. 545], we have

(8.25)

D^{2}=-\mathrm{Id}_{\ell^{2}_{N}}\otimes\Delta.

Example 8.18

If $d=2$ then $N=2$ and by [GVF01, p. 545] we have

(8.26)

D=-\mathrm{i}\begin{bmatrix}0&\partial_{1}-\mathrm{i}\partial_{2}\\ \partial_{1}+\mathrm{i}\partial_{2}&0\\ \end{bmatrix}.

Boundedness of the functional calculus

We will use the following result which says that some Riesz transforms are bounded. It will be proved in a companion paper [Arh24d].

Theorem 8.19

Suppose that $1<p<\infty$ . Consider an integer $d\geqslant 2$ and a real skew-symmetric matrix $\theta\in\mathrm{M}_{d}(\mathbb{R})$ . For any $1\leqslant j\leqslant d$ , the linear map $\partial_{j}\Delta_{p}^{-\frac{1}{2}}$ is bounded from the subspace $\operatorname{Ran}\Delta_{p}^{\frac{1}{2}}$ into the Banach space $\mathrm{L}^{p}(\mathbb{T}_{\theta}^{d})$ .

We introduce the map $\partial\overset{\mathrm{def}}{=}\partial_{1}+\mathrm{i}\partial_{2}$ . First, we prove a technical result, similar to Proposition 8.13.

Proposition 8.20

Suppose that $1<p<\infty$ . The family

(8.27)

\Big{\{}t\partial(\mathrm{Id}-t^{2}\Delta)^{-1}:t>0\Big{\}}

of operators of $\mathrm{B}(\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta}))$ is $R$ -bounded.

Proof : Since we have the equality $\partial(-\Delta)^{-\frac{1}{2}}=\partial_{1}(-\Delta)^{-\frac{1}{2}}+\mathrm{i}\partial_{2}(-\Delta)^{-\frac{1}{2}}$ , the Riesz transform $\partial(-\Delta)^{-\frac{1}{2}}\colon\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})\to\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})$ is a well-defined bounded operator by Theorem 8.19. Suppose that $t>0$ . A standard functional calculus argument gives

(8.28)

\displaystyle t\partial(\mathrm{Id}-t^{2}\Delta)^{-1}=\partial(-\Delta)^{-\frac{1}{2}}\Big{(}(-t^{2}\Delta)^{\frac{1}{2}}(\mathrm{Id}-t^{2}\Delta)^{-1}\Big{)}.

By transference, note that the Laplacian $-\Delta$ has a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta})$ functional calculus for any angle $\theta>0$ . Moreover, the noncommutative $\mathrm{L}^{p}$ -space $\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})$ is a $\mathrm{UMD}$ Banach space by [PiX03, Corollary 7.7 p. 1494], hence has the triangular contraction property $(\Delta)$ by [HvNVW18, Theorem 7.5.9 p. 137]. We deduce by [HvNVW18, Theorem 10.3.4 (2) p. 401] that the operator $-\Delta$ is $R$ -sectorial. By [HvNVW18, Example 10.3.5 p. 402] applied with $\alpha=\frac{1}{2}$ and $\beta=1$ , we infer that the set

\big{\{}(-t^{2}\Delta)^{\frac{1}{2}}(\mathrm{Id}-t^{2}\Delta)^{-1}:t>0\big{\}}

of operators of $\mathrm{B}(\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta}))$ is $R$ -bounded. Recalling that a singleton is $R$ -bounded by [HvNVW18, Example 8.1.7 p. 170], we obtain by composition [HvNVW18, Proposition 8.1.19 (3) p. 178] that the set

\Big{\{}\partial(-\Delta)^{-\frac{1}{2}}\Big{(}(-t^{2}\Delta)^{\frac{1}{2}}(\mathrm{Id}-t^{2}\Delta)^{-1}\Big{)}:t>0\Big{\}}

of operators of $\mathrm{B}(\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta}))$ is $R$ -bounded. Hence with (8.28) we conclude that the subset (8.27) is $R$ -bounded.

Proposition 8.21

Suppose that $1<p<\infty$ . The operator $D$ is bisectorial and admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}}\big{)}$ functional calculus on the Banach space $\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})\oplus_{p}\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})$ for some angle $0<\theta<\frac{\pi}{2}$ .

(8.29)

\begin{bmatrix}(\mathrm{Id}-t^{2}\Delta)^{-1}&\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{*}\\ \mathrm{i}t\partial(\mathrm{Id}-t^{2}\Delta)^{-1}&(\mathrm{Id}-t^{2}\Delta)^{-1}\end{bmatrix}

acting on the Banach space $\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})\oplus_{p}\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})$ . By Proposition 8.13 and since the operator $-\Delta$ satisfy the property (2.6) of $R$ -sectoriality, the four entries are bounded. It only remains to check that this matrix defines a two-sided inverse of $\mathrm{Id}-\mathrm{i}tD$ . We have the following equalities of operators acting on $\operatorname{dom}D$ .

	$\displaystyle\begin{bmatrix}(\mathrm{Id}-t^{2}\Delta)^{-1}&\mathrm{i}t(\mathrm{Id}_{\mathrm{L}^{p}}-t^{2}\Delta)^{-1}\partial^{*}\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial&(\mathrm{Id}-t^{2}\Delta)^{-1}\end{bmatrix}(\mathrm{Id}-\mathrm{i}tD)$
	$\displaystyle\overset{\eqref{Dirac-dim-2}}{=}\begin{bmatrix}(\mathrm{Id}-t^{2}\Delta)^{-1}&\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial&(\mathrm{Id}-t^{2}\Delta)^{-1}\end{bmatrix}\begin{bmatrix}\mathrm{Id}&-\mathrm{i}t\partial^{}\\ -\mathrm{i}t\partial&\mathrm{Id}\end{bmatrix}$
	$\displaystyle=\left[\begin{matrix}(\mathrm{Id}-t^{2}\Delta)^{-1}+t^{2}(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}\partial&-\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}+\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{}\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial-\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial&t^{2}(\mathrm{Id}-t^{2}\Delta)^{-1}\partial\partial^{}+(\mathrm{Id}-t^{2}\Delta)^{-1}\end{matrix}\right]$
	$\displaystyle=\left[\begin{matrix}(\mathrm{Id}-t^{2}\Delta)^{-1}+t^{2}(\mathrm{Id}-t^{2}\Delta)^{-1}\Delta&0\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial-\mathrm{i}t\big{(}\mathrm{Id}-t^{2}\Delta\big{)}^{-1}\partial&(\mathrm{Id}-t^{2}\Delta)^{-1}(t^{2}\partial\partial^{*}+\mathrm{Id})\end{matrix}\right]=\begin{bmatrix}\mathrm{Id}&0\\ 0&\mathrm{Id}\end{bmatrix}$

and similarly

\displaystyle(\mathrm{Id}-\mathrm{i}tD)\begin{bmatrix}(\mathrm{Id}-t^{2}\Delta)^{-1}&\mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial^{*}\\ \mathrm{i}t(\mathrm{Id}-t^{2}\Delta)^{-1}\partial&(\mathrm{Id}-t^{2}\Delta)^{-1}\end{bmatrix}=\begin{bmatrix}\mathrm{Id}&0\\ 0&\mathrm{Id}\end{bmatrix}.

For any $f\in\mathrm{L}^{\infty}(\mathbb{T}^{d}_{\theta})$ , we denote $M_{f}\colon\mathrm{L}^{p}(\mathbb{T}^{d}_{\theta})\to\mathrm{L}^{p}(\mathbb{T}^{d}_{\theta})$ , $g\mapsto fg$ the operator of left multiplication on the Banach space $\mathrm{L}^{p}(\mathbb{T}^{d}_{\theta})$ . Consequently, we can consider the homomorphism $\pi\colon\mathrm{L}^{\infty}(\mathbb{T}^{2}_{\theta})\to\mathrm{B}(\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})\oplus\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta}))$ , $f\mapsto M_{f}\oplus M_{f}$ .

Theorem 8.22

Suppose that $1<p<\infty$ . Consider an integer $d\geqslant 2$ and a real skew-symmetric matrix $\theta\in\mathrm{M}_{d}(\mathbb{R})$ . The triple $(\mathcal{A}_{\theta},\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})\oplus\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta}),D)$ is a compact Banach spectral triple.

Proof : For any function $f\in\mathcal{A}_{\theta}$ , an elementary computation reveals that $M_{f}\cdot\operatorname{dom}D\subset\operatorname{dom}D$ and that

(8.30)

[D,f]=\frac{1}{\mathrm{i}}\begin{bmatrix}0&M_{\partial f}\\ M_{\partial f}&0\end{bmatrix}.

So the commutator $[D,f]$ defines a bounded operator on the Banach space $\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})\oplus_{p}\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})$ . It is clear that $\Delta$ has compact resolvent on $\mathrm{L}^{2}(\mathbb{T}^{2}_{\theta})$ . By a standard interpolation argument (adapt [Dav89, Section 1.6]), we deduce that $\Delta$ has compact resolvent on $\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})$ for any $1<p<\infty$ . By (8.25), we conclude that $D$ has compact resolvent on the space $\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})\oplus\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})$ . So Definition 7.4 is satisfied.

Fredholm module

Using the functional calculus, we can define the operator $\operatorname{\mathrm{sign}}D$ acting on the Banach space $\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})\oplus\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})$ . This operator ca be seen as ¡¡Riesz-Clifford transform¿¿

\operatorname{\mathrm{sign}}(D)=\sum_{j=1}^{2}\gamma_{j}\otimes\partial_{j}\Delta^{-\frac{1}{2}}.

Using Proposition 7.5, we obtain the following consequence of Theorem 8.22.

Corollary 8.23

Suppose that $1<p<\infty$ . The triple $(\mathcal{A}_{\theta},\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta})\oplus\mathrm{L}^{p}(\mathbb{T}^{2}_{\theta}),\operatorname{\mathrm{sign}}D)$ is a Banach Fredholm module.

Remark 8.24

Suppose that $p=2$ . If $f\in\mathrm{L}^{2}(\mathbb{T}^{d}_{\theta})$ satisfies $\,{\raisebox{-0.56905pt}{$\mathchar 22\relax$}\mkern-12.0mu\mathrm{d}}f\in S^{q}$ , for some $q\in(0,d]$ , then by [MSX19, Corollary 1.5] $f$ is a constant. Moreover, if $f$ belongs to the noncommutative homogeneous Sobolev space $\dot{\mathrm{H}}^{1}_{d}(\mathbb{T}^{d}_{\theta})$ , then by [MSX19, Theorem 1.1] $\,{\raisebox{-0.56905pt}{$\mathchar 22\relax$}\mkern-12.0mu\mathrm{d}}f$ admits a bounded extension, and the extension belongs to the weak Schatten space $S^{d,\infty}(\ell^{2}_{N}(\mathrm{L}^{2}(\mathbb{T}^{2}_{\theta})))$ . In particular, the Fredholm module $(\mathcal{A}_{\theta},\mathrm{L}^{2}(\mathbb{T}^{2}_{\theta})\oplus\mathrm{L}^{2}(\mathbb{T}^{2}_{\theta}),\operatorname{\mathrm{sign}}D)$ is $q$ -summable for any $q>2$ .

8.5 Perturbed Dirac operators and the Kato square root problem

Let $A\colon\mathbb{R}^{n}\to\mathrm{M}_{n}(\mathbb{C})$ be a measurable bounded function satisfying for some constant $\kappa>0$ the ellipticity condition

\operatorname{Re}\langle A_{x}\xi,\xi\rangle_{\mathbb{C}^{n}}\geqslant\kappa|\xi|^{2}

for almost all $x\in\mathbb{R}^{n}$ and all $\xi\in\mathbb{C}^{n}$ . We can see $A$ as an element of the von Neumann algebra $\mathrm{L}^{\infty}(\mathbb{R}^{n},\mathrm{M}_{n}(\mathbb{C}))$ . We denote the angle of accretivity by $\omega\overset{\mathrm{def}}{=}\operatorname*{esssup}_{x,\xi}|\arg\langle A_{x}\xi,\xi\rangle|$ . Consider the multiplication operator $M_{A}\colon\mathrm{L}^{2}(\mathbb{R}^{n},\mathbb{C}^{n})\to\mathrm{L}^{2}(\mathbb{R}^{n},\mathbb{C}^{n})$ , $u\mapsto Au$ and the unbounded operator

(8.31)

D\overset{\mathrm{def}}{=}\begin{bmatrix}0&-\mathrm{div}M_{A}\\ \nabla&0\end{bmatrix},

acting on the complex Hilbert space $\mathrm{L}^{2}(\mathbb{R})\oplus\mathrm{L}^{2}(\mathbb{R},\mathbb{C}^{n})$ , where $\mathrm{div}M_{a}$ denotes the composition $\mathrm{div}\circ M_{A}$ . This operator generalizes the one defined in (1.6). We can see this operator as a deformation of the ¡¡Hodge-Dirac operator¿¿

(8.32)

\mathscr{D}\overset{\mathrm{def}}{=}\begin{bmatrix}0&-\mathrm{div}\\ \nabla&0\end{bmatrix}.

By [AKM06, Theorem 3.1 (i) p. ], the operator is bisectorial and admits a bounded $\mathrm{H}^{\infty}(\Sigma^{\mathrm{bi}}_{\theta})$ functional calculus for any $\theta\in(\omega,\frac{\pi}{2})$ . So, we can consider the bounded operator $\operatorname{\mathrm{sign}}D\colon\mathrm{L}^{2}(\mathbb{R}^{n})\oplus\mathrm{L}^{2}(\mathbb{R}^{n},\mathbb{C}^{n})\to\mathrm{L}^{2}(\mathbb{R}^{n})\oplus\mathrm{L}^{2}(\mathbb{R}^{n},\mathbb{C}^{n})$ . As explained in [AKM06] the boundedness of the operator $\operatorname{\mathrm{sign}}D$ allows to obtain the Kato square root estimate

\big{\|}(-\mathrm{div}A\nabla)^{\frac{1}{2}}f\big{\|}_{\mathrm{L}^{2}(\mathbb{R}^{n})}\approx\left\|\nabla f\right\|_{\mathrm{L}^{2}(\mathbb{R}^{n},\ell^{2}_{n})},\quad u\in\mathrm{W}^{1,2}(\mathbb{R}^{n}).

We refer to [HvNVW23, pp. 509-513], [Gra14b, Section 4.7] and [Ouh05, Chapter 8] for more information on this famous estimate.

Note that $\mathrm{L}^{2}(\mathbb{R}^{n},\mathbb{C}^{n})=\ell^{2}_{n}(\mathrm{L}^{2}(\mathbb{R}^{n}))$ . If $f\in\mathrm{L}^{\infty}(\mathbb{R}^{n})$ , we define the bounded operator $\pi_{f}\colon\mathrm{L}^{2}(\mathbb{R}^{n})\oplus\mathrm{L}^{2}(\mathbb{R}^{n},\mathbb{C}^{n})\to\mathrm{L}^{2}(\mathbb{R}^{n})\oplus\mathrm{L}^{2}(\mathbb{R}^{n},\mathbb{C}^{n})$ by

(8.33)

\pi_{f}\overset{\mathrm{def}}{=}\begin{bmatrix}\mathrm{M}_{f}&0\\ 0&\tilde{\mathrm{M}}_{f}\\ \end{bmatrix},\quad f\in\mathrm{L}^{\infty}(\mathbb{R}^{n})

where the linear map $\mathrm{M}_{f}\colon\mathrm{L}^{2}(\mathbb{R}^{n})\to\mathrm{L}^{2}(\mathbb{R}^{n})$ , $g\mapsto fg$ is the multiplication operator by the function $f$ and where

\tilde{\mathrm{M}}_{f}\overset{\mathrm{def}}{=}\mathrm{Id}_{\ell^{2}_{n}}\otimes\mathrm{M}_{f}\colon\ell^{2}_{n}(\mathrm{L}^{2}(\mathbb{R}^{n}))\to\ell^{2}_{n}(\mathrm{L}^{2}(\mathbb{R}^{n})),\,(h_{1},\ldots,h_{n})\mapsto(fh_{1},\ldots,fh_{n})

is also a multiplication operator (by the function $(f,\ldots,f)$ of $\ell^{\infty}_{m}(\mathrm{L}^{\infty}(\mathbb{R}^{n}))$ ). Using [EnN00, Proposition 4.10 p. 31], it is (really) easy to check that $\pi\colon\mathrm{L}^{\infty}(\mathbb{R}^{n})\to\mathrm{B}(\mathrm{L}^{2}(\mathbb{R}^{n})\oplus\mathrm{L}^{2}(\mathbb{R}^{n},\mathbb{C}^{n}))$ is an isometric homomorphism.

Proposition 8.25

The triple $(\mathrm{C}_{c}^{\infty}(\mathbb{R}),\mathrm{L}^{2}(\mathbb{R})\oplus\mathrm{L}^{2}(\mathbb{R},\mathbb{C}^{n}),D)$ is a locally compact Banach spectral triple.

Proof : We already said that the operator $D$ is bisectorial and admits a bounded $\mathrm{H}^{\infty}(\Sigma^{\mathrm{bi}}_{\theta})$ functional calculus. Note that $D=\begin{bmatrix}0&-\mathrm{div}\\ \nabla&0\end{bmatrix}\begin{bmatrix}\mathrm{I}&0\\ 0&M_{A}\end{bmatrix}\overset{\eqref{D-456}}{=}\mathscr{D}B$ where $B\overset{\mathrm{def}}{=}\begin{bmatrix}\mathrm{I}&0\\ 0&M_{A}\end{bmatrix}$ . It is easy to see that $\pi_{f}B=B\pi_{f}$ . Consequently, we have

[D,\pi_{f}]=D\pi_{f}-\pi_{f}D=\mathscr{D}B\pi_{f}-\pi_{f}\mathscr{D}B=\mathscr{D}\pi_{f}B-\pi_{f}\mathscr{D}B=(\mathscr{D}\pi_{f}B-\pi_{f}\mathscr{D})B=[\mathscr{D},\pi_{f}]B.

Since it is well-known that the commutator $[\mathscr{D},\pi_{f}]$ is bounded, we infer that the commutator $[D,\pi_{f}]$ is bounded. We have

\pi_{f}D^{-1}=\pi_{f}(\mathscr{D}B)^{-1}=\pi_{f}B^{-1}\mathscr{D}^{-1}=B^{-1}\pi_{f}\mathscr{D}^{-1}.

Since it is well-known that $\pi_{f}\mathscr{D}^{-1}$ is compact, we conclude that the operator $\pi_{f}D^{-1}\colon\overline{\operatorname{Ran}D}\to X$ is also compact.

Remark 8.26

It is very easy to introduce variants or generalization of this result. We refer to [AKM06] and [AMN97] for the considered operators.

8.6 Perturbed Dirac operators and the Cauchy singular integral operator

Recall that by Rademacher’s theorem, a function $g\colon\mathbb{R}\to\mathbb{R}$ is Lipschitz if and only if $g$ is differentiable almost everywhere on $\mathbb{R}$ and $g^{\prime}\in\mathrm{L}^{\infty}(\mathbb{R})$ . In this case, the Lipschitz constant

\operatorname{\mathrm{Lip}}g\overset{\mathrm{def}}{=}\sup\bigg{\{}\frac{|g(x)-g(y)|}{|x-y|}:x,y\in\mathbb{R},x\not=y\bigg{\}}.

is equal to $\left\|g^{\prime}\right\|_{\mathrm{L}^{\infty}(\mathbb{R})}$ . In the sequel, we fix a Lipschitz function $g\colon\mathbb{R}\to\mathbb{R}$ and we consider the associated Lipschitz curve in the complex plane, which is the graph $\gamma\overset{\mathrm{def}}{=}\{z=x+\mathrm{i}g(x):x\in\mathbb{R}\}$ .

Suppose that $1<p<\infty$ . Consider the unbounded operator $D_{\gamma}\overset{\mathrm{def}}{=}-B\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}$ acting on a suitable dense domain of $\mathrm{L}^{p}(\mathbb{R})$ where $B$ is the multiplication operator by the function $(1+\mathrm{i}g^{\prime})^{-1}$ . By [AKM06, Theorem 3.1 (iii) p. 465] and [AKM06, Consequence 3.2 p. 466], for any angle $\theta\in(\arctan\operatorname{\mathrm{Lip}}g,\frac{\pi}{2})$ , this operator is bisectorial and admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}})$ functional calculus. Note that this result is also proved in [QiL19, Theorem 1.4.1 p. 25] and [McQ91, Theorem 7.1 p. 159] using a different but equivalent formulation.

Consequently, the operator $\operatorname{\mathrm{sign}}D_{\gamma}\colon\mathrm{L}^{p}(\mathbb{R})\to\mathrm{L}^{p}(\mathbb{R})$ is bounded. Moreover, according to [AKM06, Consequence 3.2 p. 466], we have

\operatorname{\mathrm{sign}}(D_{\gamma})=C_{\gamma},

where $C_{\gamma}$ is the ¡¡Cauchy singular integral operator on the real line¿¿ defined by

(8.34)

(C_{\gamma}f)(x)\overset{\mathrm{def}}{=}\lim_{\varepsilon\to 0}\frac{\mathrm{i}}{\pi}\int_{y\in{\mathbb{R}}\setminus(x-\varepsilon,x+\varepsilon)}\frac{f(y)}{y-x+\mathrm{i}(g(y)-g(x))}(1+\mathrm{i}g^{\prime}(y))\mathop{}\mathopen{}\mathrm{d}y.

Here the integral exists for almost all $x\in\mathbb{R}$ and define an element of the Banach space $\mathrm{L}^{p}(\mathbb{R})$ . We refer to [McQ91, Theorem 1 p. 142] and [McQ91, (b) p. 157] for a proof of this formula stated in a different language. We direct the reader to the excellent paper [Ver21] for a detailed description of numerous applications of this operator and to [ADM96, Section 8], [Gra14b, Section 4.6] and [Jou83, Chapter 7] for more information.

Remark 8.27

If $g=0$ , then we recover the operator $\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}$ .

Remark 8.28

Actually, the boundedness of the operator $C_{\gamma}$ was first proved by Calderón in [Cal77, Theorem 1, p. 1324] in the case where $\operatorname{\mathrm{Lip}}g$ is small enough and in the general case in the famous paper [CMM82, Théorème I]. The used definition is

(8.35)

(\mathcal{C}_{\gamma}u)(z)\overset{\mathrm{def}}{=}\lim_{\varepsilon\to 0}\frac{1}{2\pi\mathrm{i}}\int\limits_{\zeta\in\gamma\setminus B(z,\varepsilon)}\frac{u(\zeta)}{\zeta-z}\mathop{}\mathopen{}\mathrm{d}\zeta,\qquad z\in\gamma.

Making the bi-Lipschitz change of variables $\mathbb{R}\to\gamma$ , $y\mapsto y+\mathrm{i}g(y)$ and identifying $f$ with the function $h(y)\overset{\mathrm{def}}{=}u(y+\mathrm{i}g(y))$ for $x\in\mathbb{R}$ , this becomes the Cauchy singular integral operator on the real line, up to a multiplicative constant.

As noted in [Gra14b, p. 289], the $\mathrm{L}^{p}$ -boundedness of the operator $C_{\gamma}\colon\mathrm{L}^{p}(\mathbb{R})\to\mathrm{L}^{p}(\mathbb{R})$ is equivalent to that of the related operator $\mathscr{C}_{\gamma}\colon\mathrm{L}^{p}(\mathbb{R})\to\mathrm{L}^{p}(\mathbb{R})$ defined by

(8.36)

\mathscr{C}_{\gamma}(f)(x)\overset{\mathrm{def}}{=}\lim_{\varepsilon\to 0}\frac{\mathrm{i}}{\pi}\int_{y\in{\mathbb{R}}\setminus(x-\varepsilon,x+\varepsilon)}\frac{f(y)}{y-x+\mathrm{i}(g(y)-g(x))}\mathop{}\mathopen{}\mathrm{d}y,\quad f\in\mathrm{L}^{p}(\mathbb{R}).

Consider a function $f\in\cup_{1<q<\infty}\mathrm{L}^{q}_{\textup{loc}}(\mathbb{R})$ and suppose that $1<p<\infty$ . It is proved in [LNWW20, Theorem 1] that the function $f$ belongs to the space $\mathrm{BMO}(\mathbb{R})$ if and only if the commutator $[M_{f},\mathscr{C}_{\gamma}]$ is bounded on the Banach space $\mathrm{L}^{p}(\mathbb{R})$ . In this case, we have

(8.37)

\left\|[M_{f},\mathscr{C}_{\gamma}]\right\|_{\mathrm{L}^{p}(\mathbb{R})\to\mathrm{L}^{p}(\mathbb{R})}\approx\left\|f\right\|_{\mathrm{BMO}(\mathbb{R})}.

We denote by $\mathrm{VMO}(\mathbb{R})$ the space of functions of vanishing mean oscillation, defined to be the $\mathrm{BMO}(\mathbb{R}$ )-closure of the set $\mathrm{C}^{\infty}_{c}(\mathbb{R})$ of functions of class $\mathrm{C}^{\infty}$ with compact support. Furthermore, if $f\in\mathrm{BMO}(\mathbb{R})$ it is proved in [LNWW20, Theorem 2] that $f$ belongs to $\mathrm{VMO}(\mathbb{R})$ if and only if the commutator $[M_{f},\mathscr{C}_{\gamma}]$ is a compact operator on the Banach space $\mathrm{L}^{p}(\mathbb{R})$ . It is obvious that this entails similar results for the operator $C_{\gamma}$ .

Proposition 8.29

Then $(\mathrm{C}_{0}^{\infty}(\mathbb{R}),\mathrm{L}^{2}(\mathbb{R}),D_{\gamma})$ is a locally compact Banach spectral triple over the algebra $\mathrm{C}_{0}^{\infty}(\mathbb{R})$ .

Proof : Note that the function $1+\mathrm{i}g^{\prime}\colon\mathbb{R}\to\mathbb{R}$ is bounded above and below. Using the notation $D\overset{\mathrm{def}}{=}\frac{1}{\mathrm{i}}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}x}$ , for any function $f\in\mathrm{C}_{0}^{\infty}(\mathbb{R})$ we have

[D_{\gamma},\pi_{f}]=D_{\gamma}\pi_{f}-\pi_{f}D_{\gamma}=BD\pi_{f}-\pi_{f}BD=BD\pi_{f}-B\pi_{f}D=B(D\pi_{f}-\pi_{f}D)=B[D,\pi_{f}].

Since it is well-known that the commutator $[D,\pi_{f}]$ is bounded, we infer that the commutator $[D_{\gamma},\pi_{f}]$ is bounded. We have

\pi_{f}D_{\gamma}^{-1}=\pi_{f}(BD)^{-1}=\pi_{f}D^{-1}B^{-1}.

Since it is well-known that $\pi_{f}D^{-1}$ is compact, we conclude that the operator $\pi_{f}D_{\gamma}^{-1}\colon\overline{\operatorname{Ran}D}\to\mathrm{L}^{2}(\mathbb{R})$ is also compact.

Proposition 8.30

Suppose that $1<p<\infty$ . Then $(\mathrm{L}^{p}(\mathbb{R}),C_{\gamma})$ is an odd Banach Fredholm module over the algebra $\mathrm{C}_{0}(\mathbb{R})$ .

Proof : If $p=2$ , we could use a possible (easy) locally compact extension of Proposition 7.5. We could also use the following reasoning. Recall that the algebra $\mathrm{C}_{0}(\mathbb{R})$ of continuous functions that vanish at infinity, since $\mathrm{C}_{0}(\mathbb{R})$ is a subspace of $\mathrm{VMO}(\mathbb{R})$ . So if $f\in\mathrm{C}_{0}(\mathbb{R})$ by [LNWW20, Theorem 2] each commutator $[M_{f},\mathscr{C}_{\gamma}]$ is a compact operator on the Banach space $\mathrm{L}^{p}(\mathbb{R})$ . So Definition 3.1 is satisfied.

Remark 8.31

It is possible to make a similar analysis with Clifford-Cauchy singular integrals on Lipschitz surfaces.

Remark 8.32

If $A$ and $B$ are some bounded operators on a complex Hilbert space $H$ , the previous section section and this lead us to consider ¡¡deformed¿¿ triples $(\mathcal{A},BDA,H)$ of a spectral triple $(\mathcal{A},D,H)$ where the selfadjoint operator $D$ is replaced by the deformed (not necessarily selfadjoint) operator $BDA$ , possibly bisectorial and admitting a bounded $\mathrm{H}^{\infty}(\Sigma^{\mathrm{bi}}_{\theta})$ functional calculus for some angle $\theta\in(0,\frac{\pi}{2})$ .

9 Banach Fredholm modules arising from noncommutative Hardy spaces

9.1 Noncommutative Hardy spaces associated to subdiagonal algebras

First, we explain how introduce Hilbert transform associated to subdiagonal algebras of von Neumann algebras. Such algebras were introduced by Arveson in [Arv67] in the case of finite von Neumann algebras and in [Bek15] for the semifinite case. We refer also to [BlL07], [Ji14], [LaX13], [MaW98], [Ran98], [Ran02] and [PiX03] for more information on Hardy spaces associated with subdiagonal algebras of von Neumann algebras. Let $\mathcal{M}$ be a von Neumann algebra endowed with a normal semifinite faithful trace and let $\mathrm{E}\colon\mathcal{M}\to\mathcal{M}$ be a trace preserving conditional expectation onto a von Neumann subalgebra $\mathcal{D}$ of $\mathcal{M}$ . Following [Bek15, Definition 2.1 p. 1350], a subdiagonal algebra $\mathrm{H}^{\infty}(\mathcal{M})$ with respect to $\mathrm{E}$ is a weak* closed subalgebra of $\mathcal{M}$ such that

1.

$\mathrm{H}^{\infty}(\mathcal{M})\cap(\mathrm{H}^{\infty}(\mathcal{M}))^{*}=\mathcal{D}$ ,
2.

the trace preserving conditional expectation $\mathrm{E}\colon\mathrm{L}^{\infty}(\mathcal{M})\to\mathrm{L}^{\infty}(\mathcal{M})$ is multiplicative on $\mathrm{H}^{\infty}(\mathcal{M})$ , i.e. we have $\mathrm{E}(fg)=\mathrm{E}(f)\mathrm{E}(g)$ for any $f,g\in\mathrm{H}^{\infty}(\mathcal{M})$ ,
3.

$\mathrm{H}^{\infty}(\mathcal{M})+(\mathrm{H}^{\infty}(\mathcal{M}))^{*}$ is weak* dense in $\mathrm{L}^{\infty}(\mathcal{M})$ .

Here, we warn the reader that $(\mathrm{H}^{\infty}(\mathcal{M}))^{*}$ is not the dual of $\mathrm{H}^{\infty}(\mathcal{M})$ , the family of the adjoints of the elements of $\mathrm{H}^{\infty}(\mathcal{M})$ . In this case, $\mathcal{D}$ is called the ¡¡diagonal¿¿ of the algebra $\mathrm{H}^{\infty}(\mathcal{M})$ . We also define $\mathrm{H}_{0}^{\infty}(\mathcal{M})\overset{\mathrm{def}}{=}\{f\in\mathrm{H}^{\infty}(\mathcal{M}):\mathrm{E}(f)=0\}$ . Suppose that $1<p<\infty$ . The closures of the subspaces $\mathrm{H}^{\infty}(\mathcal{M})\cap\mathrm{L}^{p}(\mathcal{M})$ and $\mathrm{H}_{0}^{\infty}(\mathcal{M})\cap\mathrm{L}^{p}(\mathcal{M})$ in $\mathrm{L}^{p}(\mathcal{M})$ will be denoted by $\mathrm{H}^{p}(\mathcal{M})$ and $\mathrm{H}^{p}_{0}(\mathcal{M})$ . Note that $\mathrm{E}$ extends to a contractive projection $\mathrm{E}\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{L}^{p}(\mathcal{M})$ onto the subspace $\mathrm{L}^{p}(\mathcal{D})$ with kernel $\mathrm{H}^{p}_{0}(\mathcal{M})$ . By [Bek15, Theorem 4.2 p. 1356], we have a canonical topological direct sum decomposition

(9.1)

\mathrm{L}^{p}(\mathcal{M})=(\mathrm{H}_{0}^{p}(\mathcal{M}))^{*}\oplus\mathrm{L}^{p}(\mathcal{D})\oplus\mathrm{H}_{0}^{p}(\mathcal{M}).

In this setting any $f\in\mathrm{L}^{p}(\mathcal{M})$ admits a unique decomposition

f=h^{*}+\delta+g,\quad\text{ with }g,h\in{\mathrm{H}_{0}^{p}(\mathcal{M})},\delta\in\mathrm{H}^{p}(\mathcal{D}).

The Riesz projection $P\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{L}^{p}(\mathcal{M})$ , defined by

(9.2)

P(h^{*}+\delta+g)=\delta+g,

is a bounded projection on the subspace $\mathrm{H}^{p}(\mathcal{M})$ for any $1<p<\infty$ . The Hilbert transform $\mathcal{H}\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{L}^{p}(\mathcal{M})$ associated to the subdiagonal algebra $\mathrm{H}^{\infty}(\mathcal{M})$ is the bounded operator defined by

(9.3)

\mathcal{H}(f)\overset{\mathrm{def}}{=}-\mathrm{i}(-h^{*}+g).

Toeplitz operators

Given $a\in\mathrm{L}^{\infty}(\mathcal{M})$ , the Toeplitz operator $T_{a}$ with symbol $a$ is defined to be the map

(9.4)

T_{a}\colon\mathrm{H}^{p}(\mathcal{M})\to\mathrm{H}^{p}(\mathcal{M}),b\mapsto P_{+}(ab),

where $P_{+}\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{H}^{p}(\mathcal{M})$ is the corestriction of the Riesz projection $P\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{L}^{p}(\mathcal{M})$ onto the subspace $\mathrm{H}^{p}(\mathcal{M})$ .

Banach Fredholm module and pairing with the $\mathrm{K}$ -theory

Let $\mathcal{A}$ be a unital subalgebra of $\mathcal{M}$ . Under suitable assumption, we introduce a Banach Fredholm module and we describe the pairing. We consider the homomorphism $\pi\colon\mathcal{A}\to\mathrm{B}(\mathrm{L}^{p}(\mathcal{M}))$ , $a\mapsto M_{a}$ , where $M_{a}\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{L}^{p}(\mathcal{M})$ , $y\mapsto ay$ is the multiplication operator by $a$ . We consider the bounded operator

(9.5)

F\overset{\mathrm{def}}{=}\mathrm{i}\mathcal{H}+\mathrm{E}\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{L}^{p}(\mathcal{M}).

Proposition 9.1

Suppose that $1<p<\infty$ . Assume that the commutator $[\mathcal{H},M_{a}]$ is a compact operator acting on $\mathrm{L}^{p}(\mathcal{M})$ for any $a\in\mathcal{A}$ and that the algebra $\mathcal{D}$ is finite-dimensional. Then $(\mathrm{L}^{p}(\mathcal{M}),F)$ is an odd Banach Fredholm module. For any invertible element $a\in\mathcal{A}$ , the Toeplitz operator $T_{a}$ is Fredholm and we have

(9.6)

\big{\langle}[a],(\mathrm{L}^{p}(\mathcal{M}),F)\big{\rangle}_{\mathrm{K}_{1}(\mathcal{A}),\mathrm{K}^{1}(\mathcal{A},\mathscr{L}^{p}_{\mathrm{nc}})}=\operatorname{Index}T_{a}.

Proof : Since the map $\mathrm{E}\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{L}^{p}(\mathcal{M})$ is a projection on the subspace $\mathrm{L}^{p}(\mathcal{D})$ , we have with (9.3)

F^{2}=(\mathrm{i}\mathcal{H}+\mathrm{E})^{2}=\mathrm{Id}.

Moreover, for any $a\in\mathcal{A}$ we see that

[F,M_{a}]=[\mathrm{i}\mathcal{H}+\mathrm{E},M_{a}]=\mathrm{i}[\mathcal{H},M_{a}]+[\mathrm{E},M_{a}]

is a compact operator, since the latter commutator is finite-rank operator. Hence $(\mathrm{L}^{p}(\mathcal{M}),F)$ is an odd Banach Fredholm module. Now, observe that the map $P\overset{\mathrm{def}}{=}\frac{\mathrm{Id}+F}{2}\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{L}^{p}(\mathcal{M})$ is the Riesz projection. Let $a\in\mathcal{A}$ be an invertible element. We deduce that $PaP\colon P(\mathrm{L}^{p}(\mathcal{M}))\to P(\mathrm{L}^{p}(\mathcal{M}))$ is a Fredholm operator. This operator identifies to the Toeplitz operator $T_{a}\colon\mathrm{H}^{p}(\mathcal{M})\to\mathrm{H}^{p}(\mathcal{M})$ and that

\displaystyle\big{\langle}[a],(\mathrm{L}^{p}(\mathcal{M}),F)\big{\rangle}_{\mathrm{K}_{1}(\mathcal{A}),\mathrm{K}^{1}(\mathcal{A},\mathscr{L}^{p}_{\mathrm{nc}})}\overset{\eqref{pairing-odd-2}}{=}\operatorname{Index}PaP\colon P(\mathrm{L}^{p}(\mathcal{M}))\to P(\mathrm{L}^{p}(\mathcal{M}))\overset{\eqref{Toeplitz}}{=}T_{a}.

9.2 Classical Hardy spaces

The fundamental example of subdiagonal algebras is the algebra $\mathrm{H}^{\infty}(\mathbb{T})$ on the unit circle. In this case the index pairing is described in Proposition 8.3. We can also state a similar result for the matrix-valued Hardy space $\mathrm{H}^{\infty}(\mathbb{T},\mathrm{M}_{n})$ , which be of interest for the multivariate prediction theory. This algebra is a subalgebra of the von Neumann algebra $\mathrm{L}^{\infty}(\mathbb{T},\mathrm{M}_{n})$ .

9.3 Fredholm module associated to the discrete Schur-Hilbert transform on $S^{p}_{n}$

Consider the trace preserving conditional expectation $\mathrm{E}\colon\mathrm{M}_{n}\to\mathrm{M}_{n}$ onto the diagonal subalgebra $\mathcal{D}=\ell^{\infty}_{n}$ of $\mathrm{M}_{n}$ , where $\mathrm{M}_{n}$ is equipped with its canonical trace. The ¡¡upper triangle subalgebra¿¿ $\mathrm{T}_{n}$ of all upper triangular matrices is a finite subdiagonal algebra of $\mathrm{M}_{n}$ . Suppose that $1<p<\infty$ . In this context, the Hilbert transform $\mathcal{H}\colon S^{p}_{n}\to S^{p}_{n}$ is the ¡¡discrete Schur-Hilbert transform¿¿ defined as being the Schur multiplier with symbol $[-\mathrm{i}\,\operatorname{\mathrm{sign}}(i-j)]$ . The Riesz projection $P\colon S^{p}_{n}\to S^{p}_{n}$ is the triangular projection. We consider the representation $\pi\colon\mathrm{M}_{n}\to\mathrm{B}(S^{p}_{n})$ , $a\mapsto M_{a}$ of the matrix algebra $\mathrm{M}_{n}$ , where $M_{a}\colon S^{p}_{n}\to S^{p}_{n}$ , $y\mapsto ay$ is the multiplication operator by the matrix $a$ . The couple $(S^{p}_{n},F)$ is obviously an odd Banach Fredholm module. The assumptions of Proposition 9.1 are obviously satisfied for the algebra $\mathcal{A}=\mathrm{M}_{n}$ . Since $\mathrm{K}_{1}(\mathrm{M}_{n})=0$ , we obtain the following result.

Proposition 9.2

Suppose that $1<p<\infty$ . For any invertible matrix $a\in\mathrm{M}_{n}$ , we have

(9.7)

\big{\langle}[a],(S^{p}_{n},F)\big{\rangle}_{\mathrm{K}_{1}(\mathrm{M}_{n}),\mathrm{K}^{1}(\mathrm{M}_{n},\mathscr{L}^{p}_{\mathrm{nc}})}=0.

Remark 9.3

It is possible to state in the same spirit more complicated statements for nest algebras.

9.4 Banach Fredholm modules on reduced group $\mathrm{C}^{*}$ -algebras

Orderable groups

An order relation $\preceq$ on a group $G$ is left-invariant (resp. right-invariant) if for all $s,t\in G$ such that $s\preceq t$ , one has $rs\preceq rt$ (resp. $sr\preceq tr$ ) for all $r\in G$ . The relation is bi-invariant if it is simultaneously left-invariant and right-invariant. We will use the term left-ordering (resp. right-ordering, bi-ordering) for referring to a left-invariant (resp. right-invariant, right-invariant and left-invariant) total order on a group.

Following [ClR16, p. 2], we will say that a group $G$ is left-orderable (resp. right-orderable, bi-orderable) if it admits a total order which is invariant by the left (resp. by the right, simultaneously by the left and right). We also refer to [DNR16] for more information on these classes of groups. We will write $r\prec s$ when $r\preceq s$ with $r\neq s$ . Note that by [KaT08, Proposition 7.5 p. 277] or [ClR16, Proposition 1.3 p. 2], any orderable group $G$ is torsion-free.

It is known [DNR16, Proposition 1.1.8 p. 11] [ClR16, Theorem 2.23 p. 28] that a countable group is left-orderable if and only if it is isomorphic to a subgroup of the set of orientation-preserving homeomorphisms of $\mathbb{R}$ .

Example 9.4

Following [ClR16, p. 14], we say that a group is indicable if it has the group of integers $\mathbb{Z}$ as a quotient, and locally indicable if each of its nontrivial finitely-generated sub-groups is indicable. Locally indicable groups are left-orderable by [ClR16, Corollary 1.51 p. 14].

Very large classes of groups are left-orderable as the following list shows.

Example 9.5

Every free group $\mathbb{F}_{n}$ with an arbitrary number $n$ (possibly infinite) of generators is bi-orderable, as shown in [ClR16, Theorem 3.4 p. 35] or [KaT08, Corollary 7.12 p. 280].

Example 9.6

If an orientable, irreducible²²21. A 3-manifold $M$ is irreducible if every smooth 2-sphere in $M$ bounds a 3-dimensional ball in $M$ . 3-manifold $M$ has infinite first homology group, then by [ClR16, Theorem 5.3 p. 66] its fundamental group $\pi_{1}(M)$ is locally indicable, hence left-orderable.

Example 9.7

If $K$ is a knot, then the fundamental group $\pi_{1}(S^{3}\backslash K)$ of its complement $S^{3}\backslash K$ is called the knot group of $K$ . By [ClR16, Theorem 4.9 p. 49], such a group is locally indicable, and consequently left-orderable.

Example 9.8

Braid groups

(9.8)

\mathrm{B}_{n}\overset{\mathrm{def}}{=}\left\langle\sigma_{1},\ldots,\sigma_{n-1}\left|\begin{array}[]{cl}\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},&|i-j|\geqslant 2\\ \sigma_{i}\sigma_{j}\sigma_{i}=\sigma_{j}\sigma_{i}\sigma_{j},&|i-j|=1\end{array}\right.\right\rangle,

where $n\geqslant 1$ is an integer, are left-orderable, according to [KaT08, Theorem 7.15 p. 283], or [ClR16, Theorem 7.23 p. 103]. Note that $\mathrm{B}_{1}=\{0\}$ and $\mathrm{B}_{2}=\mathbb{Z}$ . It is worth noting that the group $\mathrm{B}_{3}$ is isomorphic to the knot group of the trefoil knot by [ClR16, p. 93]. In particular, it is an infinite non-abelian group. For any integer $n\geqslant 3$ , the group $\mathrm{B}_{n}$ is not bi-orderable according to [DDRW08, Proposition 1.2 p. 12].

Example 9.9

The infinite braid group

(9.9)

\mathrm{B}_{\infty}\overset{\mathrm{def}}{=}\left\langle\sigma_{1},\sigma_{2},\ldots\left|\begin{array}[]{cl}\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},&|i-j|\geqslant 2\\ \sigma_{i}\sigma_{j}\sigma_{i}=\sigma_{j}\sigma_{i}\sigma_{j},&|i-j|=1\end{array}\right.\right\rangle,

is left-orderable by [ClR16, Problem 10.11 p. 136].

Example 9.10

According to [ClR16, Theorem 7.10 p. 95], the pure braid group $\mathrm{P}_{n}$ is bi-orderable for any integer $n\geqslant 2$ .

Example 9.11

Torsion-free nilpotent groups are bi-orderable, as shown in [DNR16, p. 15]. By the way, we refer to the paper [Sud17] for some information on the K-theory of the reduced group $\mathrm{C}^{*}$ -algebra of these groups.

Example 9.12

For each integer $n\geqslant 2$ , the Baumslag-Solitar group $\mathrm{BS}(1,n)\overset{\mathrm{def}}{=}\langle a,b\mid a^{-1}ba=b^{n}\rangle$ is bi-orderable by [DNR16, p. 17]. The K-theory groups of its reduced group $\mathrm{C}^{*}$ -algebra are described in [PoV18]. Actually, we have $\mathrm{K}_{0}(\mathrm{C}^{*}(\mathrm{BS}(1,n)))=\mathbb{Z}$ and $\mathrm{K}_{1}(\mathrm{C}^{*}(\mathrm{BS}(1,n)))=\mathbb{Z}\oplus\mathbb{Z}/(n-1)\mathbb{Z}$ .

Example 9.13

The Thompson group $\mathrm{F}$ is bi-orderable. See Example LABEL:Example-Thompson below for more information.

Example 9.14

If $M$ is an orientable closed Seifert fibered manifold with infinite first homology group, then the fundamental group $\pi_{1}(M)$ is left-orderable, according to [ClR16, Theorem 6.12 p. 83].

Example 9.15

The fundamental groups of all closed surfaces, orientable or not, with the exceptions of the projective plane and the Klein bottle are bi-orderable by [RoW01, Theorem 3 p. 314] or [ClR16, Theorem 3.11 p. 36]. If $\mathscr{K}$ is the Klein Bottle then by [ClR16, Problem 1.10 p. 4] its fundamental group $\pi_{1}(\mathscr{K})=\langle x,y\mid xyx^{-1}=y^{-1}\rangle$ is not bi-orderable. However, it is left-orderable by [RoW01, Theorem 3 p. 314]. Finally, the fundamental group $\pi_{1}(\mathrm{P}_{2}(\mathbb{R}))=\mathbb{Z}/2\mathbb{Z}$ of the projective plane $\mathrm{P}_{2}(\mathbb{R})$ is not left-orderable (see also [ClR16, Proposition 6.13 p. 84]). A description of the K-theory groups of the reduced group $\mathrm{C}^{*}$ -algebras of these ¡¡surface groups¿¿ is given in [BBV99].

Example 9.16

Let $\Sigma^{b}_{g,n}$ be an orientable connected surface with genus $g$ , $b$ boundary components, and $n$ punctures³³32. Recall that $P=\{p_{1},\ldots,p_{n}\}$ is a finite subset of distinct points ¡¡removed¿¿ from the surface. $p_{1},\ldots,p_{n}$ . If $b>0$ and if the Euler characteristic of $\Sigma^{b}_{g,n}$ satisfies $\chi(\Sigma^{b}_{g,n})<0$ then by [ClR16, Theorem 7.29 p. 107], the mapping class group $\mathrm{Mod}(\Sigma^{b}_{g,n})$ is left-orderable. This group is the group of isotopy classes of orientation-preserving homeomorphisms that restrict to the identity on the boundary of $\Sigma^{b}_{g,n}$ .

Example 9.17

Let $M$ be an $n$ -dimensional connected piecewise linear manifold (PL manifold). Consider a non-empty closed $(n-1)$ -dimensional PL submanifold $K$ of $M$ . Then by [ClR16, Theorem 8.12 p. 119] the group $\mathrm{PL}_{+}(M,K)$ of orientation-preserving PL homeomorphisms of $M$ , fixed on $K$ , is locally indicable and therefore left-orderable.

Example 9.18

Let $M$ be a connected smooth $n$ -manifold and let $K$ be a non-empty closed $(n-1)$ -submanifold of $M$ . According to [ClR16, Theorem 8.13 p. 119] the group $\mathrm{Diff}^{1}_{+}(M,K)$ of orientation-preserving homeomorphisms of $M$ which are fixed on $K$ and which are continuously differentiable is locally indicable, hence left-orderable.

Example 9.19

The universal cover $\widetilde{\mathrm{PSL}}_{2}(\mathbb{R})$ of the group $\mathrm{PSL}_{2}(\mathbb{R})$ is left-orderable by [ClR16, Example 1.15 p. 5].

Finally, note that both left-orderability and bi-orderability are clearly preserved under taking subgroups and finite products using lexicographic ordering, see [ClR16, Example 1.6 p. 3]

Fourier multipliers

Let $G$ be a discrete group. We denote by $\mathrm{VN}(G)$ its group von Neumann algebra. Suppose that $1\leqslant p<\infty$ . We say that a complex function $\varphi\colon G\to\mathbb{C}$ induces a bounded Fourier multiplier $\mathrm{M}_{\varphi}\colon\mathrm{L}^{p}(\mathrm{VN}(G))\to\mathrm{L}^{p}(\mathrm{VN}(G))$ if the linear map $\mathbb{C}[G]\to\mathbb{C}[G]$ , $\lambda_{s}\mapsto\varphi_{s}\lambda_{s}$ extends to a bounded map $M_{\varphi}$ on the noncommutative $\mathrm{L}^{p}$ -space $\mathrm{L}^{p}(\mathrm{VN}(G))$ , we use the canonical normalized normal finite faithful trace $\tau$ on the algebra $\mathrm{VN}(G)$ for defining the noncommutative $\mathrm{L}^{p}$ -space. Here $\mathbb{C}[G]$ is the group algebra of $G$ . We say that $\varphi$ is the symbol of $\mathrm{M}_{\varphi}$ .

Hilbert transforms

Assume that $G$ is a left-orderable discrete group and endowed with a left-order $\preceq$ . In this context, we can introduce the function $\operatorname{\mathrm{sign}}\colon G\to\mathbb{C}$ defined by

\operatorname{\mathrm{sign}}s\overset{\mathrm{def}}{=}\begin{cases}1&\text{ if }e\prec s\\ 0&\text{ if }s=e\\ -1&\text{ if }s\prec e\end{cases}.

Moreover, we can consider the subalgebra $\mathcal{D}\overset{\mathrm{def}}{=}\mathbb{C}1$ of the group von Neumann algebra $\mathrm{VN}(G)$ . The linear map $\mathrm{E}\colon\mathrm{VN}(G)\to\mathrm{VN}(G)$ , $f\mapsto\tau(f)1$ is a trace preserving normal faithful conditional expectation onto the ¡¡diagonal¿¿ subalgebra $\mathcal{D}$ . If $G$ is left-orderable and endowed with a left-order $\preceq$ then it is essentially⁴⁴43. The proposed definition in $\mathrm{H}^{\infty}(\mathrm{VN}(G))$ is slightly incorrect since the ¡¡noncommutative Fourier series¿¿ of elements in $\mathrm{VN}(G)$ does not converges in general. observed in [GPX22] that

\mathrm{H}^{\infty}(\mathrm{VN}(G))\overset{\mathrm{def}}{=}\overline{\bigg{\{}\text{finite sums }\sum_{e\preceq s}a_{s}\,\lambda_{s}:a_{s}\in\mathbb{C}\bigg{\}}}^{\mathrm{w}*}

is a subdiagonal algebra whose Hilbert transform $\mathcal{H}\colon\mathrm{L}^{p}(\mathrm{VN}(G))\to\mathrm{L}^{p}(\mathrm{VN}(G))$ coincides with the bounded Fourier multiplier with symbol $-\mathrm{i}\operatorname{\mathrm{sign}}$ . Actually the boundedness of the last operator was already known for abelian groups [ABG90].

The map $F\overset{\eqref{def-de-F}}{=}\mathrm{i}\mathcal{H}+\mathrm{E}\colon\mathrm{L}^{p}(\mathrm{VN}(G))\to\mathrm{L}^{p}(\mathrm{VN}(G))$ is the Fourier multiplier defined by the symbol $\varphi\colon G\to\mathbb{C}$ defined by

\varphi_{s}\overset{\mathrm{def}}{=}\begin{cases}1&\text{ if }e\preceq s\\ -1&\text{ if }s\prec e\end{cases}.

Finally, the linear map $P_{+}\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{H}^{p}(\mathcal{M})$ satisfies $P_{+}(\lambda_{s})=\lambda_{s}=0$ for any $s\in G$ satisfying $e\preceq s$ and $P_{+}(\lambda_{s})=0$ otherwise.

The following result says that the assumptions of Proposition 9.1 are satisfied.

Proposition 9.20

Suppose that $1<p<\infty$ . Let $G$ be a left-orderable discrete group endowed with a left-order. The commutator $[\mathcal{H},M_{a}]$ is a compact operator acting on the Banach space $\mathrm{L}^{p}(\mathrm{VN}(G))$ for any element $a$ of the reduced group $\mathrm{C}^{*}$ -algebra $\mathrm{C}^{*}_{\mathrm{red}}(G)$ of the group $G$ . Moreover, $(\mathrm{L}^{p}(\mathrm{VN}(G)),F)$ is an odd Banach Fredholm module.

Proof : For any $s,t\in G$ , we have

	$\displaystyle\left[\mathcal{H},M_{\lambda_{s}}\right](\lambda_{t})=\mathcal{H}M_{\lambda_{s}}(\lambda_{t})-M_{\lambda_{s}}\mathcal{H}(\lambda_{t})=\mathcal{H}(\lambda_{st})+\mathrm{i}\operatorname{\mathrm{sign}}(t)M_{\lambda_{s}}\lambda_{t}$
	$\displaystyle=-\mathrm{i}\operatorname{\mathrm{sign}}(st)\lambda_{st}+\mathrm{i}\operatorname{\mathrm{sign}}(t)\lambda_{st}=\mathrm{i}\big{[}-\operatorname{\mathrm{sign}}(st)+\operatorname{\mathrm{sign}}(t)\big{]}\lambda_{st}.$

If $f=\sum a_{s}\lambda_{s}$ is a finite sum belonging to the group algebra $\mathbb{C}[G]$ then we deduce by linearity that the commutator $\left[\mathcal{H},M_{f}\right]$ is a finite-rank operator. We conclude that for any $a\in\mathrm{C}^{*}_{\mathrm{red}}(G)$ the commutator $\left[\mathcal{H},M_{a}\right]$ is compact, as a limit of finite-rank operators, since the homomorphism $\pi\colon\mathrm{C}^{*}_{\mathrm{red}}(G)\to\mathrm{B}(\mathrm{L}^{p}(\mathrm{VN}(G)))$ , $a\mapsto M_{a}$ is continuous.

Remark 9.21

Suppose that $p=2$ . For any $s\in G$ , it is easy to check that the singular values of the operator $\left[\mathcal{H},M_{\lambda_{s}}\right]$ are given by

s_{n}(\left[\mathcal{H},M_{\lambda_{s}}\right])=|\operatorname{\mathrm{sign}}(t_{n})-\operatorname{\mathrm{sign}}(st_{n})|,

where $(t_{n})$ is an enumeration of the elements of $G$ . So the prevous Fredholm module will rarely be finitely summable.

9.5 The index pairing for the reduced group $\mathrm{C}^{*}$ -algebra of the braid group $\mathrm{B}_{3}$

[KaT08, Theorem 7.15 p. 283], or [ClR16, Theorem 7.23 p. 103] [DDRW08]

Let $n\geqslant 2$ be an integer. The Dehornoy ordering (also known as the $\sigma$ -ordering) of the braid group $\mathrm{B}_{n}$ is a left-ordering introduced in [Deh94] that is defined in terms of representative words of braids as follows. Following [ClR16, p. 95](see also [DDRW08, Definition 1.6 p. 12]), if $1\leqslant i\leqslant n-1$ , a word $w$ in the generators $\sigma_{1},\ldots,\sigma_{n-1}$ is called $i$ -positive if $w$ contains at least one occurrence of $\sigma_{i}$ , no occurrences of $\sigma_{1},\ldots,\sigma_{i-1}$ , and every occurrence of $\sigma_{i}$ has positive exponent. Replacing ¡¡positive¿¿ by ¡¡negative¿¿, we obtain the definition of a $i$ -negative word.

A braid $\beta\in\mathrm{B}_{n}$ is called $i$ -positive (respectively $i$ -negative) if it admits a representative word $w$ in the generators $\sigma_{1},\ldots,\sigma_{n-1}$ that is $i$ -positive (respectively $i$ -negative). The Dehornoy ordering of the braid group $\mathrm{B}_{n}$ is the ordering whose positive elements are the braids $\beta\in\mathrm{B}_{n}$ that are $i$ -positive for some $i$ .

Recall that the non-abelian group $\mathrm{B}_{3}$ is defined by

(9.10)

\mathrm{B}_{3}\overset{\eqref{Braid-groups-presentation}}{=}\left\langle\sigma_{1},\sigma_{2}\mid\sigma_{1}\sigma_{2}\sigma_{1}=\sigma_{2}\sigma_{1}\sigma_{2}\right\rangle.

The 2-positive elements are precisely the elements $\sigma_{2}^{k}$ with $k\geqslant 0$ . Examples of 1-positive element are given by $\sigma_{2}\sigma_{1}^{k}\sigma_{2}^{-2}$ or $\sigma_{1}^{k}$ with $k\geqslant 0$ .

The K-theory groups of the braid group $\mathrm{B}_{3}$ are known. Actually, by [ABARW22, Proposition 6.4 p. 1288], we have isomorphisms

\mathrm{K}_{0}(\mathrm{C}^{*}_{\mathrm{red}}(\mathrm{B}_{3}))=\mathrm{K}_{1}(\mathrm{C}^{*}_{\mathrm{red}}(\mathrm{B}_{3}))=\mathbb{Z}.

Moreover, the group $\mathrm{K}_{0}(\mathrm{C}^{*}_{\mathrm{red}}(\mathrm{B}_{3}))$ is generated by the unit of $\mathrm{C}^{*}_{\mathrm{red}}(\mathrm{B}_{3})$ and the group $\mathrm{K}_{1}(\mathrm{C}^{*}_{\mathrm{red}}(\mathrm{B}_{3}))$ is generated by $[\lambda_{\sigma_{1}}]=[\lambda_{\sigma_{2}}]$ . In the next result, we compute the index pairing of (9.6).

Proposition 9.22

Suppose that $1<p<\infty$ . For any integer $k\in\mathbb{Z}$ , we have

(9.11)

\big{\langle}[\lambda_{\sigma_{1}^{k}}],(\mathrm{L}^{p}(\mathrm{VN}(\mathrm{B}_{3})),F)\big{\rangle}_{\mathrm{K}_{1}(\mathrm{C}^{*}_{\mathrm{red}}(\mathrm{B}_{3})),\mathrm{K}^{1}(\mathrm{C}^{*}_{\mathrm{red}}(\mathrm{B}_{3})),\mathscr{L}^{p}_{\mathrm{nc}})}=-k.

Proof : If $i=1$ or $i=2$ , we have for any $s\in\mathrm{B}_{3}$

T_{\lambda_{\sigma_{i}^{k}}}(\lambda_{s})=P_{+}(\lambda_{\sigma_{i}^{k}}\lambda_{s})=P_{+}(\lambda_{\sigma_{i}^{k}s}).

Suppose that $k\geqslant 0$ . If $s$ is 1-positive then $\lambda_{\sigma_{1}^{k}s}$ is also 1-positive, so $T_{\lambda_{\sigma_{1}^{k}}}(\lambda_{s})=\lambda_{\sigma_{1}^{k}s}$ . If $s$ is 2-positive then $\lambda_{\sigma_{2}^{k}s}$ is also 2-positive. Hence $T_{\lambda_{\sigma_{2}^{k}}}(\lambda_{s})=\lambda_{\sigma_{2}^{k}s}$ . If $s=e$ then $T_{\lambda_{\sigma_{2}^{k}}}(e)=\lambda_{\sigma_{2}^{k}}$ . By linearity, it is easy to check that $\operatorname{Ker}T_{\lambda_{\sigma_{1}^{k}}}=\{0\}$ . Consequently, we have

	$\displaystyle\operatorname{Index}T_{\lambda_{\sigma_{1}^{k}}}\overset{\eqref{Fredholm-Index}}{=}\dim\operatorname{Ker}T_{\lambda_{\sigma_{1}^{k}}}-\dim\mathrm{L}^{p}(\mathrm{VN}(\mathrm{B}_{3}))/\operatorname{Ran}T_{\lambda_{\sigma_{1}^{k}}}$
	$\displaystyle=-\dim\mathrm{L}^{p}(\mathrm{VN}(\mathrm{B}_{3}))/\operatorname{Ran}T_{\lambda_{\sigma_{1}^{k}}}=-k.$

We conclude with (9.6).

Now, suppose that $k<0$ . If $s=\sigma_{2}^{r}$ is $2$ -positive then $\sigma_{2}^{k}s=\sigma_{2}^{k}\sigma_{2}^{r}=\sigma_{2}^{r+k}$ is 2-positive if and only if $r+k\geqslant 0$ . It is obvious that the left multiplication by $\sigma_{2}^{k}$ transforms any 1-positive element into a 1-positive element. So, it is easy to check that $\dim\operatorname{Ker}T_{\lambda_{\sigma_{2}^{k}}}=k$ . We conclude that

\displaystyle\operatorname{Index}T_{\lambda_{\sigma_{1}^{k}}}\overset{\eqref{Fredholm-Index}}{=}\dim\operatorname{Ker}T_{\lambda_{\sigma_{1}^{k}}}-\dim\mathrm{L}^{p}(\mathrm{VN}(\mathrm{B}_{3}))/\operatorname{Ran}T_{\lambda_{\sigma_{1}^{k}}}=\dim\operatorname{Ker}T_{\lambda_{\sigma_{1}^{k}}}=k.

Remark 9.23

In [LOS21, Example 5.10], the $\mathrm{K}$ -theory of the reduced group $\mathrm{C}^{*}$ -algebra of the group $\mathrm{B}_{4}$ is computed to be

\mathrm{K}_{0}(\mathrm{C}^{*}_{\mathrm{red}}(\mathrm{B}_{4}))\simeq\mathbb{Z}\oplus(\mathbb{Z}/2\mathbb{Z}),\qquad\qquad\mathrm{K}_{1}(\mathrm{C}^{*}_{\mathrm{red}}(\mathrm{B}_{4}))\simeq\mathbb{Z}.

The abelian group $\mathrm{K}_{1}(\mathrm{C}^{*}_{\mathrm{red}}(\mathrm{B}_{4}))$ is generated by the element $[\lambda_{\sigma_{1}}]=[\lambda_{\sigma_{2}}]=[\lambda_{\sigma_{3}}]$ . It is not difficult to compute the index pairing.

9.6 Bi-orderings of the Thompson group $\mathrm{F}$

The Thompson group $\mathrm{F}$ consists of orientation-preserving piecewise linear homeomorphisms $f$ of the interval $[0,1]$ such that

1.

there are only finitely many points of non-differentiability of $f$ ,
2.

each point of non-differentiability of $f$ occurs at a dyadic rational number $\frac{a}{2^{n}}$ where $a,n\in\mathbb{N}$ ,
3.

the derivatives on intervals of differentiability are powers of 2.

See Figure 1 for an example of element of this group. We refer to [CFP96] for an introduction to this group.

Figure 1: an element of Thompson group $\mathrm{F}$

A description of all bi-orderings on Thompson group is given in the paper [NaR10]. We describe four of them. Let $f\in\mathrm{F}$ with $f\not=\mathrm{Id}$ . We consider the corresponding lateral derivatives $f^{\prime}_{-}$ and $f^{\prime}_{+}$ . We denote by $x^{-}_{f}$ the leftmost point $x$ for which $f^{\prime}_{+}(x)\neq 1$ and we denote by $x^{+}_{f}$ the rightmost point $x$ for which $f^{\prime}_{-}(x)\neq 1$ . In this context, we can define

1.

the bi-ordering $\preceq_{x^{-}}^{+}$ for which $\mathrm{Id}\prec f$ if and only if $f^{\prime}_{+}(x^{-}_{f})>1$ ,
2.

the bi-ordering $\preceq_{x^{-}}^{-}$ for which $\mathrm{Id}\prec f$ if and only if $f^{\prime}_{+}(x^{-}_{f})<1$ ,
3.

the bi-ordering $\preceq_{x^{+}}^{+}$ for which $\mathrm{Id}\prec f$ if and only if $f^{\prime}_{-}(x^{+}_{f})<1$ ,
4.

the bi-ordering $\preceq_{x^{+}}^{-}$ for which $\mathrm{Id}\prec f$ if and only if $f^{\prime}_{-}(x^{+}_{f})>1$ .

Our results pave the way for a better understanding of the K-theory and K-homology groups of Thompson group, whose properties remain largely unknown.

10 Banach Fredholm module associated to the free Hilbert transform

Let $G=\mathbb{F}_{\infty}$ be the free group with a countable sequence of generators $g_{1},g_{2},\ldots$ . For any integer $n\geqslant 1$ , let $L_{n}^{+}\colon\mathrm{L}^{2}(\mathrm{VN}(\mathbb{F}_{\infty}))\to\mathrm{L}^{2}(\mathrm{VN}(\mathbb{F}_{\infty}))$ and $L_{n}^{-}\colon\mathrm{L}^{2}(\mathrm{VN}(\mathbb{F}_{\infty}))\to\mathrm{L}^{2}(\mathrm{VN}(\mathbb{F}_{\infty}))$ be the orthogonal projections such that

L_{n}^{+}(\lambda_{s})=\begin{cases}\lambda_{s}&s\text{ starts with the letter }g_{n}\\ 0&\text{otherwise}\end{cases},\quad L_{n}^{-}(\lambda_{s})=\begin{cases}\lambda_{s}&s\text{ starts with the letter }g_{n}^{-1}\\ 0&\text{otherwise}\end{cases}.

Let us further consider signs $\varepsilon_{n}^{+},\varepsilon_{n}^{-}\in\{-1,1\}$ for any integer $n\geqslant 1$ . Following [MeR17], we define the free Hilbert transform associated with $\varepsilon=(\varepsilon_{n}^{\pm})$ as

(10.1)

\mathcal{H}_{\varepsilon}\overset{\mathrm{def}}{=}\sum_{n\geqslant 1}\big{(}\varepsilon_{n}^{+}L_{n}^{+}+\varepsilon_{n}^{-}L_{n}^{-}\big{)}.

Clearly, since the ranges of the various projections $L_{n}^{\pm}$ are mutually orthogonal, the linear map $\mathcal{H}_{\varepsilon}$ is bounded on the Hilbert space $\mathrm{L}^{2}(\mathrm{VN}(\mathbb{F}_{\infty}))$ . The far-reaching generalization in [MeR17, Section 4] is that the map $\mathcal{H}_{\varepsilon}$ induces a completely bounded map on the Banach space $\mathrm{L}^{p}(\mathrm{VN}(\mathbb{F}_{\infty}))$ for any $1<p<\infty$ .

In the sequel, we consider the contractive homomorphism $\pi\colon\mathrm{C}^{*}_{\mathrm{red}}(\mathbb{F}_{\infty})\to\mathrm{B}(\mathrm{L}^{p}(\mathrm{VN}(\mathbb{F}_{\infty}))$ , $a\mapsto M_{a}$ , where $M_{a}\colon\mathrm{L}^{p}(\mathrm{VN}(\mathbb{F}_{\infty}))\to\mathrm{L}^{p}(\mathrm{VN}(\mathbb{F}_{\infty}))$ , $y\mapsto ay$ is the multiplication operator by $a$ .

Proposition 10.1

Suppose that $1<p<\infty$ . Consider a family $\varepsilon=(\varepsilon_{n}^{\pm})_{n\geqslant 1}$ of signs. Then $(\mathrm{L}^{p}(\mathrm{VN}(\mathbb{F}_{\infty})),\mathcal{H}_{\varepsilon})$ is a Banach Fredholm module over the reduced group $\mathrm{C}^{*}$ -algebra $\mathrm{C}^{*}_{\mathrm{red}}(\mathbb{F}_{\infty})$ .

Proof : Let $s,t\in\mathbb{F}_{\infty}$ . Assume that $s$ begins with $g_{k}^{\eta_{s}}$ and that $ts$ begins with $g_{l}^{\mu_{ts}}$ , where $\eta_{s},\mu_{ts}\in\{-1,1\}$ (identified with $\{-,+\}$ ). We have

	$\displaystyle\left[\mathcal{H}_{\varepsilon},M_{\lambda_{t}}\right](\lambda_{s})=\mathcal{H}_{\varepsilon}M_{\lambda_{t}}(\lambda_{s})-M_{\lambda_{t}}\mathcal{H}_{\varepsilon}(\lambda_{s})$
	$\displaystyle\overset{\eqref{free-Hilbert}}{=}\bigg{(}\sum_{n\geqslant 1}\varepsilon_{n}^{+}L_{n}^{+}+\varepsilon_{n}^{-}L_{n}^{-}\bigg{)}\lambda_{ts}-M_{\lambda_{t}}\bigg{(}\sum_{n\geqslant 1}\varepsilon_{n}^{+}L_{n}^{+}+\varepsilon_{n}^{-}L_{n}^{-}\bigg{)}(\lambda_{s})$
	$\displaystyle=\varepsilon_{l}^{\mu_{ts}}\lambda_{ts}-M_{\lambda_{t}}(\varepsilon_{k}^{\eta_{s}}\lambda_{s})=\varepsilon_{l}^{\mu_{ts}}\lambda_{ts}-\varepsilon_{k}^{\eta_{s}}\lambda_{ts}.$

The other cases where $s=e$ or $ts=e$ are left to the reader. We deduce that the rank of the commutator $\left[\mathcal{H}_{\varepsilon},M_{\lambda_{t}}\right]$ is zero or one. If $a=\sum_{s\in\mathbb{F}_{\infty}}a_{s}\lambda_{s}$ is a finite sum, where each $a_{s}$ is a complex number, belonging to the group algebra $\mathbb{C}[\mathbb{F}_{\infty}]$ then we see by linearity that the commutator $\left[\mathcal{H},M_{a}\right]$ is a finite-rank operator. We infer that for any $a\in\mathrm{C}^{*}_{\mathrm{red}}(\mathbb{F}_{\infty})$ the commutator $\left[\mathcal{H},M_{a}\right]$ is compact, as a limit of finite-rank operators, since the homomorphism $\pi\colon\mathrm{C}^{*}_{\mathrm{red}}(\mathbb{F}_{\infty})\to\mathrm{B}(\mathrm{L}^{p}(\mathrm{VN}(\mathbb{F}_{\infty})))$ , $a\mapsto M_{a}$ is continuous.

Note that $\mathcal{H}_{\varepsilon}(\lambda_{e})=0$ . Consequently, with (10.1) it is easy to check that $\mathcal{H}_{\varepsilon}^{2}\sim\mathrm{Id}_{\mathrm{L}^{p}(\mathrm{VN}(\mathbb{F}_{\infty})}$ .

Remark 10.2

By restriction, we can obtain a Banach Fredholm module over each reduced group $\mathrm{C}^{*}$ -algebra $\mathrm{C}^{*}_{\mathrm{red}}(\mathbb{F}_{n})$ for any integer $n\geqslant 1$ . By [PiV82, Corollary 3.2 p. 152], we have $\mathrm{K}_{0}(\mathrm{C}^{*}_{\mathrm{red}}(\mathbb{F}_{n}))=\mathbb{Z}$ and the generator of $\mathrm{K}_{0}(\mathrm{C}^{*}_{\mathrm{red}}(\mathbb{F}_{n}))$ is $[1]$ . The same reference gives an isomorphism $\mathrm{K}_{1}(\mathrm{C}^{*}_{\mathrm{red}}(\mathbb{F}_{n}))=\mathbb{Z}^{n}$ and the generators of the group $\mathrm{K}_{1}(\mathrm{C}^{*}_{\lambda}(\mathbb{F}_{n}))$ are $[\lambda_{g_{1}}],\ldots,[\lambda_{g_{n}}]$ . We invite the reader to determine the index pairing.

11 Semigroups, Fredholm modules and vectorial Riesz transforms

Hilbert bimodules

We start by reviewing the concept of Hilbert bimodule which is crucial for defining the derivations that allow the introduction of the Hodge-Dirac operators considered in this paper. Let $\mathcal{M}$ be a von Neumann algebra. A Hilbert $\mathcal{M}$ -bimodule is a Hilbert space $\mathcal{H}$ together with a $*$ -representation $\Phi\colon\mathcal{M}\to\mathrm{B}(\mathcal{H})$ and a $*$ -anti-representation $\Psi\colon\mathcal{M}\to\mathrm{B}(\mathcal{H})$ such that $\Phi(x)\Psi(y)=\Psi(y)\Phi(x)$ for any $x,y\in\mathcal{M}$ . For all $x,y\in\mathcal{M}$ and any $\xi\in\mathcal{H}$ , we let $x\xi y\overset{\mathrm{def}}{=}\Phi(x)\Psi(y)\xi$ . We say that the bimodule is normal if $\Phi$ and $\Psi$ are normal, i.e. weak* continuous. The bimodule is said to be symmetric if there exists an antilinear involution $\mathcal{J}\colon\mathcal{H}\to\mathcal{H}$ such that $\mathcal{J}(x\xi y)=y^{*}\mathcal{J}(\xi)x^{*}$ for any $x,y\in\mathcal{M}$ and any $\xi\in\mathcal{H}$ .

$\mathrm{W}^{*}$ -derivations

Now, we introduce a notion of derivation that can be viewed as an abstract version of a gradient. If $(\mathcal{H},\Phi,\Psi)$ is a Hilbert $\mathcal{M}$ -bimodule, then following [Wea96, p. 267] we define a $\mathrm{W}^{*}$ -derivation to be a weak* closed densely defined unbounded operator $\partial\colon\operatorname{dom}\partial\subset\mathcal{M}\to\mathcal{H}$ such that the domain $\operatorname{dom}\partial$ is a weak* dense unital $*$ -subalgebra of $\mathcal{M}$ and

(11.1)

\partial(fg)=f\partial(g)+\partial(f)g,\quad f,g\in\operatorname{dom}\partial.

We say that a $\mathrm{W}^{*}$ -derivation is symmetric if the bimodule $(\mathcal{H},\Phi,\Psi)$ is symmetric and if we have $\mathcal{J}(\partial(f))=\partial(f^{*})$ for any $x\in\operatorname{dom}\partial$ . In the sequel, we let $\mathcal{B}\overset{\mathrm{def}}{=}\operatorname{dom}\partial$ .

The triple $(\mathrm{L}^{\infty}(\mathcal{M}),\mathrm{L}^{2}(\mathcal{M})\oplus_{2}\mathcal{H},D)$

From a derivation, we will now explain how to introduce a triple in the spirit of noncommutative geometry. Let $\partial\colon\operatorname{dom}\partial\subset\mathcal{M}\to\mathcal{H}$ be a $\mathrm{W}^{*}$ -derivation where the von Neumann algebra $\mathcal{M}$ is equipped with a normal faithful finite trace $\tau$ . Suppose that the operator $\partial\colon\operatorname{dom}\partial\subset\mathrm{L}^{2}(\mathcal{M})\to\mathcal{H}$ is closable. We denote again its closure by $\partial$ . Note that the subspace $\mathcal{B}$ is a core of $\partial$ . Recall that it is folklore and well-known that a weak* dense subalgebra of $\mathcal{M}$ is dense in the space $\mathrm{L}^{2}(\mathcal{M})$ . As the operator $\partial$ is densely defined and closed, by [Kat76, Theorem 5.29 p. 168] the adjoint operator $\partial^{*}$ is densely defined and closed on $\mathrm{L}^{2}(\mathcal{M})$ and $\partial^{**}=\partial$ .

Following essentially [HiT13b] and [Cip16], we introduce the unbounded closed operator $D$ on the Hilbert space $\mathrm{L}^{2}(\mathcal{M})\oplus_{2}\mathcal{H}$ introduced in (11.6) and defined by

(11.2)

D(f,g)\overset{\mathrm{def}}{=}\big{(}\partial^{*}(g),\partial(f)\big{)},\quad f\in\operatorname{dom}\partial,\ g\in\operatorname{dom}\partial^{*}.

We call it the Hodge-Dirac operator associated to $\partial$ and it can be written as in (11.6). It is not difficult to check that this operator is selfadjoint. If $f\in\mathrm{L}^{\infty}(\mathcal{M})$ , we define the bounded operator $\pi_{f}\colon\mathrm{L}^{2}(\mathcal{M})\oplus_{2}\mathcal{H}\to\mathrm{L}^{2}(\mathcal{M})\oplus_{2}\mathcal{H}$ by

(11.3)

\pi_{f}\overset{\mathrm{def}}{=}\begin{bmatrix}\mathrm{M}_{f}&0\\ 0&\Phi_{f}\\ \end{bmatrix},\quad f\in\mathrm{L}^{\infty}(\mathcal{M})

where the linear map $\mathrm{M}_{f}\colon\mathrm{L}^{2}(\mathcal{M})\to\mathrm{L}^{2}(\mathcal{M})$ , $g\mapsto fg$ is the multiplication operator by $f$ and where $\Phi_{f}\colon\mathcal{H}\to\mathcal{H}$ , $h\mapsto fh$ is the left bimodule action.

Note that completely Dirichlet forms give rise to $\mathrm{W}^{*}$ -derivations, see [CiS03], [Cip97], [Cip08], [Cip16] and [Wir22] and references therein. More precisely, if $\mathcal{E}$ is a completely Dirichlet form on a noncommutative $\mathrm{L}^{2}$ -space $\mathrm{L}^{2}(\mathcal{M})$ with associated semigroup $(T_{t})_{t\geqslant 0}$ and associated operator $A_{2}$ , there exist a symmetric Hilbert $\mathcal{M}$ -bimodule $(\mathcal{H},\Phi,\Psi,\mathcal{J})$ and a $\mathrm{W}^{*}$ -derivation $\partial\colon\operatorname{dom}\partial\subset\mathcal{M}\to\mathcal{H}$ such that $\partial\colon\operatorname{dom}\partial\subset\mathrm{L}^{2}(\mathcal{M})\to\mathcal{H}$ is closable with closure also denoted by $\partial$ and satisfying $A_{2}=\partial^{*}\partial$ . In other words, this means that the opposite of the infinitesimal generator $-A_{2}$ of a symmetric sub-Markovian semigroup of operators can always be represented as the composition of a ¡¡divergence¿¿ $\partial^{*}$ with a ¡¡gradient¿¿ $\partial$ . Note that the symmetric Hilbert $\mathcal{M}$ -bimodule $(\mathcal{H},\Phi,\Psi,\mathcal{J})$ is not unique since one can always artificially expand $\mathcal{H}$ . However, if we add an additional condition, namely that ¡¡the bimodule is generated by the derivation $\partial$ ¿¿, we obtain uniqueness, see [Wir22, Theorem 6.9] for a precise statement. In this case, the bimodule (and its associated derivation) is called the tangent bimodule associated with the completely Dirichlet form (or the associated semigroup).

This fundamental result allows anyone to introduce a triple $(\mathrm{L}^{\infty}(\Omega),\mathrm{L}^{2}(\Omega)\oplus_{2}\mathcal{H},D)$ associated to the semigroup in the spirit of noncommutative geometry. Here $D$ is the unbounded selfadjoint operator acting on a dense subspace of the Hilbert space $\mathrm{L}^{2}(\Omega)\oplus_{2}\mathcal{H}$ defined by

(11.4)

D\overset{\mathrm{def}}{=}\begin{bmatrix}0&\partial^{*}\\ \partial&0\end{bmatrix}.

The Hodge-Dirac operator $D$ of (11.6) is related to the operator $A_{2}$ by

(11.5)

D^{2}\overset{\eqref{Hodge-Dirac-I}}{=}\begin{bmatrix}0&\partial^{*}\\ \partial&0\end{bmatrix}^{2}=\begin{bmatrix}\partial^{*}\partial&0\\ 0&\partial\partial^{*}\end{bmatrix}=\begin{bmatrix}A_{2}&0\\ 0&\partial\partial^{*}\end{bmatrix}.

Now, suppose that $1<p<\infty$ . Sometimes, the map $\partial_{2}$ induces a closable unbounded operator $\partial\colon\operatorname{dom}\partial\subset\mathrm{L}^{p}(\mathcal{M})\to\mathcal{X}_{p}$ for some Banach space $\mathcal{X}_{p}$ . Denoting by $\partial_{p}$ its closure, we can consider the $\mathrm{L}^{p}$ -realization of the previous operator

(11.6)

D_{p}\overset{\mathrm{def}}{=}\begin{bmatrix}0&(\partial_{p^{*}})^{*}\\ \partial_{p}&0\end{bmatrix}

as acting on a dense subspace of the Banach space $\mathrm{L}^{p}(\mathcal{M})\oplus_{p}\mathcal{X}_{p}$ . If the operator $D_{p}$ is bisectorial and admits a bounded $\mathrm{H}^{\infty}(\Sigma_{\theta}^{\mathrm{bi}})$ functional calculus for some $\theta\in(0,\frac{\pi}{2})$ , we have

\displaystyle\operatorname{\mathrm{sign}}D_{p}=D_{p}|D_{p}|^{-1}=D_{p}(D_{p}^{2})^{-\frac{1}{2}}=\begin{bmatrix}0&(\partial_{p^{*}})^{*}\\ \partial_{p}&0\end{bmatrix}\left(\begin{bmatrix}A_{p}&0\\ 0&\partial_{p}(\partial_{p^{*}})^{*}\end{bmatrix}\right)^{-\frac{1}{2}}=\begin{bmatrix}0&*\\ \partial_{p}A_{p}^{-\frac{1}{2}}&0\end{bmatrix}.

In particular, the operator $\partial_{p}A_{p}^{-\frac{1}{2}}\colon\mathrm{L}^{p}(\mathcal{M})\to\mathcal{X}_{p}$ is bounded.

Definition 11.1

We say that $\mathcal{R}\overset{\mathrm{def}}{=}\partial_{p}A_{p}^{-\frac{1}{2}}$ is the (abstract) vectorial Riesz transform associated with the symmetric sub-Markovian semigroup $(T_{t})_{t\geqslant 0}$ .

From the perspective of the action of the algebra or the kernel of the Dirac operator, the Banach space $\mathcal{X}_{p}$ is sometimes too large and needs to be replaced by a smaller space. In this case, we consider the closed subspace

(11.7)

\Omega_{p}\overset{\mathrm{def}}{=}\overline{\mathrm{span}\big{\{}g\partial_{p}(f):f\in\operatorname{dom}\partial_{p},g\in\mathrm{L}^{\infty}(\mathcal{M})}\big{\}}

of the Banach space $\mathcal{X}_{p}$ . And we replace the Banach space $\mathcal{X}_{p}$ by the subspace $\Omega_{p}$ . The advantage of this space is the following result.

Proposition 11.2

The Banach space $\Omega_{p}$ is a $\mathrm{L}^{\infty}(\mathcal{M})$ -bimodule.

Proof : For any $f\in\operatorname{dom}\partial$ and any $g,h\in\mathrm{L}^{\infty}(\mathcal{M})$ , note that

g(h\partial(f))=gh\partial(f).

Thus by linearity and density, $\Omega_{p}$ is a right $\mathrm{L}^{\infty}(\mathcal{M})$ -module. Moreover, for any $g\in\mathrm{L}^{\infty}(\mathcal{M})$ and any $f,h\in\operatorname{dom}\partial$ , we have

\displaystyle g\partial(f)h\overset{\eqref{Leibniz}}{=}g\big{[}\partial(fh)-f\partial(h)\big{]}=g\partial(hf)-gf\partial(h).

Thus $g\partial(f)h$ belongs to $\Omega_{p}$ . Since $\operatorname{dom}\partial$ is a core for $\partial_{p}$ , the same holds for $f\in\operatorname{dom}\partial_{p}$ . If $h\in\mathrm{L}^{\infty}(\mathcal{M})$ is a general element, we approximate it in the strong operator topology by a bounded net in $\operatorname{dom}\partial$ and obtain again that the same holds for $h\in\mathrm{L}^{\infty}(\mathcal{M})$ . By linearity and density, we deduce that $\Omega_{p}$ is a left $\mathrm{L}^{\infty}(\mathcal{M})$ -module, so finally a $\mathrm{L}^{\infty}(\mathcal{M})$ -bimodule.

Remark 11.3

It is interesting to observe that if $\mathrm{L}^{\infty}(\mathcal{M})$ is commutative, i.e. the algebra $\mathrm{L}^{\infty}(\Omega)$ of a (localizable) measure space $\Omega$ , the previous bimodule is not necessarily commutative, i.e. we does not have $gkh=hkg$ for any $g,h\in\mathrm{L}^{\infty}(\Omega)$ and $k\in\Omega_{p}$ . In our previous paper, we referred to this phenomenon as ¡¡hidden noncommutative geometry¿¿.

If $f\in\mathrm{L}^{\infty}(\mathcal{M})$ , we define the bounded operator $\pi_{f}\colon\mathrm{L}^{p}(\mathcal{M})\oplus_{p}\Omega_{p}\to\mathrm{L}^{p}(\mathcal{M})\oplus_{p}\Omega_{p}$ by

(11.8)

\pi_{f}\overset{\mathrm{def}}{=}\begin{bmatrix}\mathrm{M}_{f}&0\\ 0&\Phi_{f}\\ \end{bmatrix},\quad f\in\mathrm{L}^{\infty}(\mathcal{M})

where the linear map $\mathrm{M}_{f}\colon\mathrm{L}^{p}(\mathcal{M})\to\mathrm{L}^{p}(\mathcal{M})$ , $g\mapsto fg$ is the multiplication operator by $f$ and where $\Phi_{f}\colon\Omega_{p}\to\Omega_{p}$ , $h\mapsto fh$ is the left bimodule action provided by Proposition 11.2.

If $\bigg{(}\mathrm{L}^{p}(\mathcal{M})\oplus_{p}\Omega_{p},\operatorname{\mathrm{sign}}D_{p},\begin{bmatrix}-\mathrm{Id}&0\\ 0&\mathrm{Id}\\ \end{bmatrix}\bigg{)}$ is a (well-defined) even Banach Fredholm module over a subalgebra $\mathcal{A}$ of $\mathrm{L}^{\infty}(\mathcal{M})$ , the pairing with the K-homology group $\mathrm{K}_{0}(\mathcal{A})$ is given by the following formula. If $e\in\mathrm{M}_{n}(\mathcal{A})$ is an idempotent then

(11.9)

\big{\langle}[e],(\mathrm{L}^{p}(\mathcal{M})\oplus_{p}\Omega_{p},\operatorname{\mathrm{sign}}D)\big{\rangle}_{\mathrm{K}_{0}(\mathcal{A}),\mathrm{K}^{0}(\mathcal{A},\mathscr{L}^{p}_{\mathrm{nc}})}\overset{\eqref{pairing-even-1}}{=}\operatorname{Index}e_{n}(\mathrm{Id}\otimes\mathcal{R})e_{n}.

Remark 11.4

Note that with Proposition 3.13, we can obtain sometimes an odd Banach Fredholm module with $\operatorname{\mathrm{sign}}D_{p}$ , so a pairing with the group $\mathrm{K}_{1}(\mathcal{A})$ .

12 Future directions

We plan to elaborate on the ideas presented in the final section in a future update of this very preliminary preprint. We also plan to expand some sections and explain the analogies with [ScS22]. Finally, we will conduct an exhaustive study of the groups $\mathrm{K}^{0}(\mathcal{A},\mathscr{B})$ and $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$ (and some generalizations) in a forthcoming paper.

Declaration of interest

None.

Competing interests

The author declares that he has no competing interests.

Data availability

No data sets were generated during this study.

Acknowledgment

The author would like to thank Hermann Schulz-Baldes for bringing his recent book [ScS22] to our attention on his own initiative.

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Cédric Arhancet
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(7.4)		$\displaystyle\left[\operatorname{\mathrm{sign}}D,a\right]=D\|D\|^{-1}a-aD\|D\|^{-1}=(Da-aD)\|D\|^{-1}+D(\|D\|^{-1}a-a\|D\|^{-1})$
	$\displaystyle=[D,a]\|D\|^{-1}+D[\|D\|^{-1},a]=[D,a]\|D\|^{-1}-D\|D\|^{-1}[\|D\|,a]\|D\|^{-1}$
	$\displaystyle=\big{(}\underbrace{[D,a]-\operatorname{\mathrm{sign}}(D)[\|D\|,a]\|}_{\overset{\mathrm{def}}{=}R}\big{)}\|D\|^{-1}.$