Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers (Lecture Notes in Mathematics) 3030990109, 9783030990107

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Lecture Notes in Mathematics  2304

Cédric Arhancet Christoph Kriegler

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

Lecture Notes in Mathematics Volume 2304

Editors-in-Chief Jean-Michel Morel, CMLA, ENS, Cachan, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Alessio Figalli, ETH Zurich, Zurich, Switzerland Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Sylvia Serfaty, NYU Courant, New York, NY, USA Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

This series reports on new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes: 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research. Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is sometimes more important than its form, which may be preliminary or tentative. Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews, and zbMATH.

Cédric Arhancet • Christoph Kriegler

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

Cédric Arhancet Albi, France

Christoph Kriegler Laboratoire de Mathematiques Blaise Pascal (LMBP), CNRS Université Clermont Auvergne Clermont-Ferrand, France

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-030-99010-7 ISBN 978-3-030-99011-4 (eBook) https://doi.org/10.1007/978-3-030-99011-4 Funding Information: Agence Nationale de la Recherche anr-18-ce40-0021 anr-17-ce40-0021 (http://dx.doi.org/10.13039/501100001665) Mathematics Subject Classification: 46L51, 46L07, 47D03, 58B34 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

We dedicate this book to our families: Clément, Lise, Nina, Christine, Raphael and Clara.

Preface

The starting point of the writing of this book is an open problem of functional calculus of operators explicitly stated in a remarkable recent paper of Junge, Mei and Parcet. This problem is connected to Riesz transforms defined by semigroups of operators acting on spaces associated to discrete groups. In this book, we give a solution for a large class of groups relying in particular on some approximation properties of groups. Along the way, we also present a significant number of topics which are new and of independent interest: a transference principle between Fourier multipliers and Fourier multipliers on crossed products, Khintchine type equivalences for deformed Gaussians in spaces associated to crossed products, and Hodge decompositions. We think that these tools will be used in the next developments of this theory. Indeed, Riesz transforms and functional calculus are still evolving mathematical fields. It is transparent for us that a lot of things remain to be done in this area. Consequently, we describe an incomplete story. In this book, we also want to popularize the fact that noncommutative spaces appear naturally in the classical and old topics of Riesz transforms and functional calculus. This is an astonishing and really striking fact linked to cohomology. Furthermore, we will show in the last chapter that the language of noncommutative geometry is natural and allows us to illuminate these topics with a new geometric point of view. In some sense, there is some “hidden noncommutative geometry” in some semigroups of operators acting on classical spaces. Moreover, we also describe and define a Banach space generalization of the notion of spectral triple of noncommutative geometry. We believe that this generalization will be used in a large number of settings in the future. We also prove similar results suitable for semigroups of Schur multipliers, and we wanted to show the analogies and the differences between those two settings. It is known that Fourier multipliers and Schur multipliers are connected, and we continue in this book to display interesting features and links. We hope we made the topic accessible. Our guiding principle has been to give all details of computations and approximation arguments. We think that it is easier for a reader to have complete proofs than proofs left to the reader. We equally wanted to vii

viii

Preface

close some gaps in the literature. We have tried to reduce the required prerequisites to a minimum. We also hope that our effort will consolidate the beautiful setting of modern noncommutative harmonic analysis which is a mixture of probabilities, analysis, algebra and geometry. Albi, France Clermont-Ferrand, France February 2022

Cédric Arhancet Christoph Kriegler

Acknowledgements

The authors are supported by the grant of the French National Research Agency ANR-18-CE40-0021 (project HASCON). The first author would like to thank Françoise Lust-Piquard and Cyril Lévy for some discussions. We are grateful to Marek Bo˙zejko for the reference [41], Quanhua Xu for the reference [100] and to Frédéric Latrémolière for his confirmation of Proposition 5.1. Finally, we are indebted to Markus Haase for his help to devise rigorously the end of the proof of Lemma 5.6. The second author acknowledges support by the grant ANR-17-CE400021 (project Front).

ix

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Operators, Functional Calculus and Semigroups.. . . . . . . . . . . . . . . . . . . . 2.2 q-Gaussian Functors, Isonormal Processes and Probability .. . . . . . . . 2.3 Vector-Valued Unbounded Bilinear Forms on Banach Spaces . . . . . . 2.4 Transference of Fourier Multipliers on Crossed Product Von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Hilbertian Valued Noncommutative Lp -Spaces . .. . . . . . . . . . . . . . . . . . . . 2.6 Carré Du Champ  and First Order Differential Calculus for Fourier Multipliers .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7 Carré Du Champ  and First Order Differential Calculus for Schur Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

23 23 34 37 41 46 55 60

3 Riesz Transforms Associated to Semigroups of Markov Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 65 3.1 Khintchine Inequalities for q-Gaussians in Crossed Products .. . . . . . 65 3.2 Lp -Kato’s Square Root Problem for Semigroups of Fourier Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 79 3.3 Extension of the Carré du Champ  for Fourier Multipliers . . . . . . . . 88 3.4 Lp -Kato’s Square Root Problem for Semigroups of Schur Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 93 3.5 Meyer’s Problem for Semigroups of Schur Multipliers.. . . . . . . . . . . . . 123 4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Boundedness of Functional Calculus of Hodge-Dirac Operators for Fourier Multipliers . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Extension to Full Hodge-Dirac Operator and Hodge Decomposition .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Hodge-Dirac Operator on Lp (VN(G)) ⊕ ψ,q,p .. . . . . . . . . . . . . . . . . . . 4.4 Bimodule ψ,q,p,c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

135 135 150 158 163 xi

xii

Contents

4.5 4.6 4.7

Hodge-Dirac Operators Associated to Semigroups of Markov Schur Multipliers .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 165 Extension to Full Hodge-Dirac Operator and Hodge Decomposition .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 171 Independence from H and α . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 173

5 Locally Compact Quantum Metric Spaces and Spectral Triples . . . . . . . 5.1 Background on Quantum Locally Compact Metric Spaces . . . . . . . . . 5.2 Quantum Compact Metric Spaces Associated to Semigroups of Fourier Multipliers . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Gaps and Estimates of Norms of Schur Multipliers.. . . . . . . . . . . . . . . . . 5.4 Seminorms Associated to Semigroups of Schur Multipliers . . . . . . . . 5.5 Quantum Metric Spaces Associated to Semigroups of Schur Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 Gaps of Some Markovian Semigroups of Schur and Fourier Multipliers .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7 Banach Spectral Triples . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.8 Spectral Triples Associated to Semigroups of Fourier Multipliers I . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.9 Spectral Triples Associated to Semigroups of Fourier Multipliers II . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.10 Spectral Triples Associated to Semigroups of Schur Multipliers I .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.11 Spectral Triples Associated to Semigroups of Schur Multipliers II .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.12 Bisectoriality and Functional Calculus of the Dirac Operator II . . . .

181 181 186 195 199 203 210 218 224 232 240 250 257

A Appendix: Lévy Measures and 1-Cohomology . . . . . . .. . . . . . . . . . . . . . . . . . . . 263 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 265 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 275

Chapter 1

Introduction

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

The continuity of the Hilbert transform H on Lp (R) by Riesz [199] is known as one of the greatest discoveries in analysis of the twentieth century. This transformation is defined by the principal value def

(Hf )(x) =

1 p. v. π

 R

f (y) dy, x−y

f ∈ S (R), a.e. x ∈ R.

(1.1)

It is at the heart of many areas: complex analysis, harmonic analysis, Banach space geometry, martingale theory and signal processing. We refer to the thick books [109, 110, 137, 138] and references therein for more information. Directional Riesz transforms Rj are higher-dimensional generalizations of the Hilbert transform defined by the formula 1

Rj = ∂j ◦ (−)− 2 , def

j = 1, . . . , n

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Arhancet, C. Kriegler, Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers, Lecture Notes in Mathematics 2304, https://doi.org/10.1007/978-3-030-99011-4_1

(1.2)

1

2

1 Introduction

where  is the Laplacian on Rn . Generalizing Riesz’s result, Calderón and Zygmund proved in [47] that these operators are bounded on Lp (Rn ) if 1 < p < ∞ by developing a theory of singular integrals. Indeed, each Riesz transform Rj is defined by the following integral (Rj f )(x) =

( n+1 2 ) π

n+1 2

 p. v.

Rn

f (y)

xj − yj dy, |x − y|n+1

f ∈ S (R), a.e. x ∈ Rn .

It is known that the Lp -norms do not depend on the dimension n by Banuelos and Wang [30] and Iwaniec and Martin [112]. More precisely, we have  π   tan 2p Rj  p n = n p L (R )→L (R ) π cot 2p

if 1 < p  2 if 2  p < ∞

(1.3)

.

The case n = 1 is a classical result of Pichorides [186], see [91, 92] for a short proof. In [211] (see also [30, 112]), Stein showed that the vectorial Riesz transform 1 ∇(−)− 2 satisfies  n 1  2     2  ∇(−)−12 f  p n 2 p f Lp (Rn ) , i.e.  |Rj f |   L (R , )

p f Lp (Rn )

Lp (Rn )

j =1

(1.4) def

with free dimensional bound where ∇f = (∂1 f, . . . , ∂n f ) is the gradient of a function f belonging to some suitable subspace of Lp (Rn ). It is an open problem to find the best constants in (1.4), explicitly stated in [28, Problem 6]. Furthermore, by duality we have by e.g. [67, Proposition 2.1], [204, p. 5] an equivalence of the form   (−) 21 f 

  1  2  n  1  2     2 ∇f  ≈ ≈ |∂ f | p (Rn , 2 ) , i.e. (−) f n p p j p p L   L (R ) L n j =1

. Lp (Rn )

(1.5) Note that the case p = 2 is easy since an integration by parts gives 1   (−) 21 f  2 n = −f, f  2 2 n = L (R ) L (R )

 n j =1

  ∂j f 2 2 n L (R )

1 2

= ∇f L2 (Rn , 2n ) .

1

Moreover, we can show that the domain of the operator (−) 2 is the Sobolev space def  W1,2 (Rn ) = f ∈ L2 (Rn ) : ∂i f ∈ L2 (Rn ) for any 1  i  n .

1 Introduction

3

Note that the case p = 2 can be seen as a very particular case of the famous Kato square root problem solved in [21, 23], see also [106, 217]. In this context, the def

Laplacian − is replaced by an operator L = −div(A∇) where A : Rn → Mn (C) is a measurable bounded function satisfying the ellipticity condition λ|ξ |2  Re Aξ, ξ  2n

and | Aξ, ζ  2n |  |ξ ||ζ |,

ξ, ζ ∈ Cn

for some constants 0 < λ  . If A(x) = I for any x ∈ Rn we recover the Laplacian −. Note that the precise definition of L relies on the theory of sesquilinear forms on Hilbert spaces, see [136, Chapter 6]. The square root problem essentially introduced by Kato and refined by some other authors consisted to establish the estimate  1  L 2 f  2 n ≈ ∇f  2 n 2 L (R , n ) L (R ) 1

with constants depending only on n, λ and , and the equality dom L 2 = W1,2 (Rn ). Some authors investigated the Lp -version of this problem. The corresponding Lp equivalence is not necessarily true. We refer to the memoir [20] and references therein. The study of Riesz transforms and variants in many contexts has a long history and was the source of many fundamental developments such as the CalderónZygmund theory [47] or the classical work of Stein [210] on Littlewood-Paley Theory. Nowadays, Riesz transforms associated to various geometric structures is a recurrent theme in analysis and geometry. For example, the singular integral (Bf )(z) = −

1 p. v. π

 C

f (w) dw, (z − w)2

f ∈ Cc∞ , a.e. z ∈ C

where the integration is with respect to the Lebesgue measure on the plane, is known as the complex Hilbert transform or the Beurling-Ahlfors transform. This operator is fundamental in complex analysis, since it intertwines the two partial differential operators def

∂z =

1 (∂x − i∂y ), 2

def

∂z =

1 (∂x + i∂y ) 2

in the sense that B ◦ ∂z = ∂z . For any 1 < p < ∞, this integral induces a bounded operator B : Lp (C) → Lp (C). In the remarkable works of Donaldson and Sullivan [77] and Iwaniec and Martin [111], it has been observed that the knowledge of the exact value or a good estimate of the Lp -norm of this operator will lead to important applications in the study

4

1 Introduction

of quasi-conformal mappings and related nonlinear geometric partial differential equations as well as in the Lp -Hodge decomposition theory. The computation of the exact value is an important open problem known as Iwaniec’s conjecture. In the field of harmonic analysis in the discrete setting, a discrete Hilbert transform was introduced by Hilbert at the beginning of twentieth century. This operator H maps a finite support sequence a = (an )n∈Z of complex numbers to the sequence Ha defined by (Ha)n =

1 π

 an−m , m

n ∈ Z.

m∈Z\{0}

Riesz and Titchmarsh proved that for any 1 < p < ∞, H induces a bounded 1 def  p p p p . Recently, the exact operator H : Z → Z where a p = n∈Z |an | Z computation of the norm of this operator was given in [29], a century after its introduction. The result is identical with the norm (1.3) of the continuous Hilbert transform H defined in (1.1). Riesz transforms associated with Ornstein-Uhlenbeck operators were studied and play a fundamental role in the Malliavin calculus on the Wiener space [161, 178]. Let n be a fixed positive integer and suppose that γn is the standard Gaussian measure on Rn , i.e. dγn (x) =

1 (2π)

n 2

e−

|x| 2

x ∈ Rn .

dx,

Recall that the Ornstein-Uhlenbeck operator L on L2 (Rn , γn ) is a negative operator defined by def

L = −x ·∇ =

 n  2  ∂ ∂ . − x j ∂xj ∂xj2 j =1

This operator generates the Ornstein-Uhlenbeck semigroup in n dimensions. In [171], Meyer introduced Riesz transforms associated with the Ornstein-Uhlenbeck operator L by RL = ∇ ◦ (−L)− 2 . def

1

He proved that for any 1 < p < ∞, the Riesz transform RL induces a bounded operator on the space Lp (Rn , γn ). This result remains true on the infinitedimensional Wiener space, see e.g. [178, Proposition 1.5.2]. In [187], Pisier gave a different proof of Meyer’s result by using the Lp -boundedness of the Hilbert transform H of (1.1).

1 Introduction

5

Arcozzi considered in [6] the Riesz transform RS n = ∇S n ◦ (−Sn )− 2 1

def

def

on the n-dimensional unit sphere S n = {x ∈ Rn+1 : x = 1}. Here, Sn denotes the Laplace-Beltrami operator on S n and ∇S n is the gradient assocated to S n . For any 1 < p < ∞, he showed that  RS n Lp (S n )→Lp (S n ,TS n )  2

1 p−1

if 1 < p  2

p−1

if 2  p < ∞

.

(1.6)

We refer to [7] for other Riesz transforms on spheres. A well-known√fact is that many geometric objects on the (n − 1)-dimensional sphere of radius n pass in the limit to the corresponding objects on the infinitedimensional Wiener space, see e.g. [166]. This is often referred to as Poincaré’s limit or Poincaré’s observation, though the argument can be traced back to the work of Mehler [168]. Combining this observation and the estimates (1.6), Arcozzi obtained for any 1 < p < ∞ that  RL Lp (Rn ,γn )→Lp (Rn ,γn , 2n )  2

1 p−1

if 1 < p  2

p−1

if 2  p < ∞

and the same estimate also for the Riesz transforms acting on the infinitedimensional Wiener space. The exact value of the norm is an open question. See [142] for a study of the growth of the constants when p → 1 and when p → ∞. Generalizing the cases of Rn and the spheres S n , Strichartz raised in [213] the question concerning the structure of a complete Riemannian manifold M which guarantees that the Riesz transform 1

RM = ∇M ◦ (−M )− 2 def

induces a bounded operator on the space Lp (M) for 1 < p < ∞ where M is the Laplace-Beltrami operator of M and ∇M the gradient. A classical result of Bakry [25] is that it is true for any complete Riemannian manifold with nonnegative Ricci curvature and any 1 < p < ∞. Note that there exist some manifolds for which RM does not induce a bounded operator on Lp (M) for some (or all) p > 2, see e.g. [4] or [50]. Is it conjectured in [67, Conjecture 1.1] that for any 1 < p < 2 there exists a constant Cp > 0 such that RM Lp (M)→Lp (M,TM)  Cp

6

1 Introduction

for all complete Riemannian manifolds M. By Coulhon and Duong [66, Theorem 1.1], we know that the Riesz transform RM is bounded on Lp (M) for 1 < p  2 as soon as the manifold M satisfies the doubling property V (x, 2r)  V (x, r),

x ∈ M, r > 0

and a diagonal estimate pt (x, x) 

1 √ , V (x, t)

x ∈ M, t > 0 def

on the kernel pt (x, y) of the heat semigroup (et  )t 0 , where V (x, r) = μ(B(x, r)) is the Riemannian volume of the geodesic ball B(x, r) of center x and radius r > 0. See the survey [65] and the recent paper [115] and references therein for more information. We equally refer to the surveys [63, 64] for a deeper discussion of heat kernels. Some authors investigated some generalizations of (1.2) to Lie groups. Let G be a unimodular connected Lie group equipped with a Haar measure μG and g its Lie algebra which can be identified with the Lie algebra of left invariant vector fields on G where the product is the Lie bracket (X, Y ) → [X, Y ]. Consider a family X = (X1 , . . . , Xm ) of left-invariant vector fields on G satisfying the Hörmander condition, which means that the Lie subalgebra generated by the Xk ’s is g, and such that the vectors X1 (e), . . . , Xm (e) are linearly independent. We can consider the sublaplacian G defined by1 def

G = −

m 

Xj2 .

j =1

Suppose that the Lie group G has polynomial volume growth, i.e. that there exists an integer D  0 such that μG (B(x, r)) ≈ r D ,

r1

where B(x, r) is the open ball centred at x ∈ G and of radius r with respect to the Carnot-Carathéodory metric. Alexopoulos proved in [1, Theorem 2] that in this case the Riesz transforms def

−1

RG,j = Xj G 2 ,

j = 1, . . . , n

induce bounded operators on Lp (G) for any 1 < p < ∞. His approach relies on some ideas inspired by homogenization theory. Recall that a connected nilpotent

1

From now on, the Laplacian type operators incorporate a minus sign and thus are positive definite.

1 Introduction

7

Lie group or a connected compact Lie group have polynomial volume growth (with D = 0 in the compact case). We continue with the case of the hypercube {−1, 1}n where n ∈ N. For any function f : {−1, 1}n → R and any index j ∈ {1, . . . , n} we can consider the hypercube partial derivative ∂j f : {−1, 1}n → R defined by def

(∂j f )(ε) = f (ε) − f (ε1 , . . . , εj −1 , −εj , εj +1 , . . . , εn ),

ε ∈ {−1, 1}n .

(1.7)

We recall the usual Fourier–Walsh expansion of a function f : {−1, 1}n → R. For any subset A of {−1, 1}n consider the corresponding Walsh function wA : {−1, 1}n → R given by def

wA (ε) =

εj ,

ε ∈ {−1, 1}n

j ∈A

def 1 and denote fˆ(A) = n 2



f (ε)wA (ε).

ε∈{−1,1}n

(1.8) Then we have f (ε) =

1 2n



fˆ(A)wA (ε),

ε ∈ {−1, 1}n .

A⊂{−1,1}n

For any function f : {−1, 1}n → R we equally define the discrete Laplacian f : {−1, 1}n → R by 

def

(f )(ε) =

|A| fˆ(A)wA (ε),

ε ∈ {−1, 1}n

(1.9)

A⊂{1,...,n}

where |A| is the cardinal of A. This operator is sometimes called the number operator since (wA ) = |A|wA . Then we can introduce the hypercube Riesz transforms R1 , . . . , Rn by def

Rj =

1 1 ∂j ◦ − 2 , 2

j ∈ {−1, 1}n .

More concretely, we can show that (Rj f )(ε) =



fˆ(A) √ wA (ε), |A| A⊂{1,...,n} j ∈A

ε ∈ {−1, 1}n .

(1.10)

8

1 Introduction

Similarly to (1.4), Lust-Piquard proved2 in [154, Theorem 0.1 (a)] that for any 2  p < ∞ and any function f ∈ {−1, 1}n → R of mean 0,3 we have 1 p3/2

f Lp ({−1,1}n )

 n 1  2   2    (Rj f ) 

 p f Lp ({−1,1}n ) .

Lp ({−1,1}n )

j =1

(1.11) where the Lp norm is taken with respect to the normalized counting measure on the discrete hypercube {−1, 1}n , i.e. def

f Lp ({−1,1}n ) =



1 2n

1



|f (ε)|

p

p

.

ε∈{−1,1}n

We can rewrite these inequalities as   1  2   n 1  2    12 f  p  (∂ f ) j n)   L ({−1,1} 3/2 p

Lp ({−1,1}n )

j =1

 1   p 2 f Lp ({−1,1}n ) (1.12)

which are analogues of (1.5). Note that the obtained constants are dimension independent. In sharp contrast with the case of Rn or the Wiener space, these inequalities are false for 1 < p < 2 as showed in [154], see also [82, Section 5.5]. Indeed, Lust-Piquard shows that we can use a transference trick to the fermion algebra in order to reduce the problem for 2  p < ∞ to some noncommutative estimates. This striking dichotomy between the cases 1 < p < 2 and 2  p < ∞ is very natural in noncommutative analysis due to noncommutative Khintchine inequalities [153, 158] (see Theorem 1.5 for a generalization to qgaussians and crossed products) which imply in general two formulations of the results in noncommutative analysis according to the value of p. Actually, LustPiquard showed a more complicated substitute for (1.11) in the case 1 < p < 2: f p p

    1  1   2 2  n  n 2 2    p f p  inf |gj | + |Tej (hj )|    Rj (f )=gj +hj j =1

p

j =1

p

(1.13)

2 3

The constants proved by Lust-Piquard were worse than the ones of (1.11). That means that 1 2n

 ε∈{−1,1}n

f (ε) = 0.

1 Introduction

9

where the infimum is taken over all decompositions with gj , hj : {−1, 1}n → R and where Tej is the translation by ej = (1, . . . , −1, 1, . . . , 1) where −1 occurs at coordinate j . For the study of Hilbert transforms and Riesz transforms in other contexts, the interested reader can consult the papers [79, 152, 209, 210, 218] (Lie groups), [5] (symmetric spaces), [154] (fermion algebras), [155] (deformed Gaussian algebras), [76, 156] (abelian groups), [157] (generalized Heisenberg groups), [24, 55, 203] (graphs), [19, 73, 103] (Schrödinger operators), [101, 102] (fields of p-adic numbers), [114, 163, 164, 173, 220] (fractal sets and measures), [169] (von Neumann algebras of free groups), [51] (free Araki-Woods factors), [119, 184, 192] (von Neumann algebras), [52] (quantum groups) and [224] (free probability). Indeed, Riesz transforms have become a cornerstone of analysis and the literature is quite huge and it would be impossible to give complete references here. We refer to [28, 204] for nice surveys. An important generalization of (1.5) was given by Meyer [171]. It consists in replacing the Laplacian  by the Lp -realization Ap of the negative infinitesimal generator A of a Markov semigroup (Tt )t 0 of operators acting on the Lp -spaces of a measure space  and to replace the gradient ∇ by the “carré du champ”  introduced by Roth [202] (see also [105]) defined4 by def

(f, g) =

 1 A(f )g + f A(g) − A(f g) . 2

(1.14)

In the case of the Heat semigroup (e−t  )t 0 with generator , we recover the gradient form ∇f, ∇g 2n . Meyer was interested in the equivalence    12 1 Ap (f ) p ≈ (f, f ) 2 Lp () L () p 1

(1.15)

on some suitable subspace (ideally dom Ap2 ) of Lp (). Meyer proved such equivalence for the Ornstein-Uhlenbeck semigroup. Nevertheless, with sharp contrast, if 1 < p < 2 these estimates are surprisingly false for the Poisson semigroup on Lp (Rn ) which is a Markov semigroup of Fourier multipliers, see [131, Appendix D]. Actually, as we said, other examples of semigroups illustrating this phenomenon are already present in the papers of Lust-Piquard [156, Proposition 2.9] and [154, p. 283] relying on an observation of Lamberton. Of course, when something goes wrong with a mathematical problem it is rather natural to change slightly the formulation of the problem in order to obtain a natural positive statement. By introducing some gradients with values in a noncommutative space, Junge, Mei and Parcet obtained in [131] dimension free estimates for Riesz transforms associated with arbitrary Markov semigroups (Tt )t 0 ˆ where G is for example of Fourier multipliers acting on classical Lp -spaces Lp (G) 4

Here, the domain of A must contain a suitable involutive algebra.

10

1 Introduction

ˆ (and more generally on the an abelian discrete group with (compact) dual group G noncommutative Lp -spaces Lp (VN(G)) associated with a nonabelian group G). We denote by ψ : G → C the symbol of the (negative) infinitesimal generator A of the semigroup. In the spirit of (1.5), the previous authors proved estimates of the form    12  Ap (f ) p ˆ ≈p ∂ψ,1,p (f ) p ∞ L (G) L (L ()

α G)

(1.16)

where ∂ψ,1,p is some kind of gradient defined on a dense subspace of the classical ˆ It takes values in a closed subpace ψ,1,p of a noncommutative Lp -space Lp (G). p p L -space L (L∞ () α G) associated with some crossed product L∞ () α G where  is a probability space and where α : G → Aut(L∞ ()) is an action of G on L∞ () determined by the semigroup. Let us explain the simplest case, i.e. the case where α is trivial. In this non-crossed and very particular situation, we have ˆ and an identification of Lp (L∞ () α G) with the classical Lp -space Lp ( ⊗ G) p ˆ the map ∂ψ,1,p is defined on the span of characters s, ·G,Gˆ in L (G) with values ˆ It is defined by in Lp ( ⊗ G).

def ∂ψ,1,p s, ·G,Gˆ = W(bψ (s)) ⊗ s, ·G,Gˆ .

(1.17)

where W : H → L0 () is an H -isonormal Gaussian process5 for some real Hilbert space H and where bψ : G → H is a specific function satisfying  2 ψ(s) = bψ (s)H ,

s ∈ G.

(1.18)

We refer to Sect. 2.6 for the (crossed) general situation where the action α is obtained by second quantization from an orthogonal representation π : G → B(H ) associated to the semigroup, see (2.83). The approach by Junge, Mei and Parcet highlights an intrinsic noncommutativity since ψ,1,p is in general a highly noncommutative object although the group G may be abelian. It is fair to say that this need of noncommutativity was first noticed and explicitly written by Lust-Piquard in [154, 156] in some particular cases under a somewhat different but essentially equivalent form of (1.16), see (1.13). Moreover, it is remarkable that the estimates of [156] were exploited in a decisive way by Naor [172] to understand subtle geometric phenomena. Finally, note that the existence of gradients suitable for arbitrary Markov semigroups of linear operators appears already in the work of Sauvageot and Cipriani, see [57, 205] and the survey [56]. Finally, we refer to [17, 128, 139, 191] and references therein for more information on noncommutative Lp -spaces. In the context of Riesz transforms, the authors of the classical and remarkable paper [23] were the first to introduce suitable Hodge-Dirac operators. The Lp -

5

In particular, for any h ∈ H the random variable W(h) is a centred real Gaussian.

1 Introduction

11

boundedness of the H∞ calculus of this unbounded operator allows everyone to obtain immediately the Lp -boundedness of Riesz transforms. The authors of [131] introduced a similar operator in the context of Markov semigroups (Tt )t 0 of Fourier multipliers acting on classical Lp -spaces and more generally on noncommutative Lp -spaces Lp (VN(G)) associated with group von Neumann algebras VN(G) where 1 < p < ∞ and where G is a discrete group. We refer to the papers [56, Definition 10.4], [58, 107, 108, 160, 165, 176] for Hodge-Dirac operators in related contexts. Recall that if G is a discrete group then the von Neumann algebra VN(G), whose elements are bounded operators acting on the Hilbert space 2G , is generated by the left translation unitaries λs : 2G → 2G , δr → δsr where r, s ∈ G. If G is abelian, ˆ of essentially bounded functions then VN(G) is ∗-isomorphic to the algebra L∞ (G) ˆ ˆ as on the dual group G of G. In this case, we can see the functions of L∞ (G) 2 ˆ multiplication operators on L (G). As basic models of quantum groups, these von Neumann algebras play a fundamental role in operator algebras. Moreover, we can equip VN(G) with a normalized trace (=noncommutative integral) and if 1  p  ∞ we have a canonical identification ˆ Lp (VN(G)) = Lp (G).

(1.19)

A Markov semigroup (Tt )t 0 of Fourier multipliers on VN(G) is characterized by a conditionally negative length ψ : G → C such that the symbol of each operator Tt of the semigroup is e−t ψ . Moreover, the symbol of the (negative) infinitesimal generator Ap on the noncommutative Lp -space Lp (VN(G)) of the semigroup is ψ. Introducing the Banach space Lp (VN(G))⊕p ψ,1,p , the authors of [131] define the Hodge-Dirac operator def

Dψ,1,p =



0

∂ψ,1,p

(∂ψ,1,p )∗ 0

 (1.20)

which is an unbounded operator defined on a dense subspace. In [131, Problem def

C.5], the authors ask for dimension free estimates for the operator sgn Dψ,1,p = Dψ,1,p |Dψ,1,p |−1 . We affirmatively answer this question for a large class of groups including all amenable discrete groups and free groups by showing the following def

result in the spirit of [23] (see also [22]). Here we use the bisector ω± = ω ∪ def  (−ω ) where ω = z ∈ C\{0} : | arg z| < ω for any angle ω ∈ (0, π2 ). Theorem 1.1 (see Theorems 4.3, 4.4 and 4.6 and Remark 4.3) Suppose 1 < p < ∞. Let G be a weakly amenable discrete group such that the crossed product L∞ () α G has QWEP. The Hodge-Dirac operator Dψ,1,p is bisectorial on Lp (VN(G)) ⊕p ψ,1,p and admits a bounded H∞ (ω± ) functional calculus on a

12

1 Introduction

bisector ω± . Moreover, the norm of the functional calculus is bounded by a constant Kω,p which depends neither on G nor on the semigroup.6 We refer to [83, 110] for more information on bisectorial operators. Roughly speaking, our result says that   f (Dψ,1,p )

Lp (VN(G))⊕p ψ,1,p →Lp (VN(G))⊕p ψ,1,p

 Kω,p f H∞ (ω± )

(1.21) def

for any suitable function f of H∞ (ω± ). Using the function sgn defined by sgn(z) = 1ω (z)−1−ω (z), we obtain the estimate for the operator sgn Dψ,1,p . Our result can be seen as a strengthening of the dimension free estimates (1.16) of Riesz transforms of the previous authors since it is almost immediate that the boundedness of the H∞ functional calculus implies the equivalence (1.16), see Remark 4.1. Note that the H∞ functional calculus of bisectorial Hodge-Dirac operators plays an important role in the geometry of Riemannian manifolds, notably for the regularity properties of geometric flows, the Riesz continuity of the Atiyah-Singer Dirac operator or even for boundary value problems of elliptic operators. We refer to the recent survey [27] for more information. We expect that our H∞ functional calculus result from Theorem 4.4 and its dimension free estimate from Remark 4.3 will have similar geometric consequences for the noncommutative manifolds (i.e. spectral triples) investigated in Chap. 5. Our argument relies in part on a new transference argument between Fourier multipliers on crossed products and classical Fourier multipliers (see Proposition 2.8) which is of independent interest and which needs that a crossed product has QWEP, which is an approximation property, see [181]. So we need a QWEP assumption on the von Neumann algebra L∞ ()α G of Theorem 1.1. Note that this assumption is satisfied for amenable groups by Ozawa [181, Proposition 4.1] and for free groups Fn by the same reasoning as used in the proof of [10, Proposition 4.8]. Moreover, we assume that the discrete group G is weakly amenable. Indeed, we need in the proof some form of Lp -summability of noncommutative Fourier series in order to work with elements whose Fourier series have finite support. The weak amenability assumption and our transference result allow us to have uniformly bounded approximations of arbitrary elements of noncommutative Lp spaces associated to crossed products, see the discussion at the end of Sect. 2.4. We also show a q-gaussian version of Theorem 1.1, see Theorem 4.3. This kind of generalization will be useful in Chap. 5. In this introduction, we will explain its interest in the discussion following Theorem 1.4. With the help of an extension of this result (Theorem 4.4), we obtain a Hodge decomposition (see Theorem 4.5). In particular, we are able to deduce functional calculus for an extension of Dψ,1,p on the whole space Lp (VN(G)) ⊕p Lp (L∞ () α G).

6 In particular, it is independent of the dimension of the Hilbert space H associated with the 1cocycle by Proposition 2.3.

1 Introduction

13

Below, we describe concrete semigroups in which Theorem 1.1 applies. Semigroups on Abelian Groups Recall that a particular case of [33, Corollary 18.20] says that a function ψ : G → R on a discrete abelian group G is a conditionally negative length if and only if there exists a quadratic form7 q : G → ˆ − {0} such that R+ and a symmetric positive measure μ on G 

ˆ G−{0}

1 − Re χ(s) dμ(χ) < ∞

for any s ∈ G satisfying  ψ(s) = q(s) +

ˆ G−{0}

1 − Re χ(s) dμ(χ),

s ∈ G.

(1.22)

In this case, μ is the so called Lévy measure of ψ and q is determined by the . This is the Lévy-Khintchine representation of ψ formula q(s) = limn→+∞ ψ(ns) n2 as a continuous sum of elementary conditionally negative lengths.8 (1.19)

(a) If G = Zn , we recover the semigroups on the Lp -spaces Lp (VN(Zn )) = Lp (Tn ) of the torus Tn . For example, taking μ = 0 and ψ(k1 , . . . , kn ) = q(k1, . . . , kn ) = k12 + · · · + kn2 , we obtain the function defining the heat semigroup (e−t  )t 0 . By choosing q = 0 and the right measure μ, we can 1

obtain the Poisson semigroup (e−t  2 )t 0 or more generally the semigroups associated with the fractional Laplacians α with 0  α < 2 [200, 201]. Note that in the particular case of the Poisson semigroup on the Lp -space Lp (T) associated with the torus T, the Lévy measure is given by  dμ(eix ) = Re

−2eix (1 − eix )2

 dμT (eix ),

where μT denotes the normalized Haar measure on the torus. (b) Fix some integer n  1. We consider the group G of Walsh functions wA defined in (1.8) where ε = (ε1 , . . . , εn ) belongs to the discrete abelian group ˆ = {−1, 1}n . For any 1  i  n, we let ei def = (1, . . . , 1, −1, 1, . . . , 1). G  ˆ − {(1, . . . , 1)} and If we consider the atomic measure μ = 12 ni=1 δei on G

That means that 2q(s) + 2q(t) = q(s + t) + q(s − t) for any s, t ∈ G. Recall that a quadratic form q : G → R+ and a function G → R, s → 1 − Re χ(s) are conditionally negative lengths by Berg and Forst [33, Proposition 7.19, Proposition 7.4 (ii), and Corollary 7.7].

7 8

14

1 Introduction

q = 0, we obtain9 ψ(wA ) = |A|. So we recover the discrete Heat semigroup10 of [110, p. 19] whose generator is the discrete Laplacian (1.9). It is also related to [82, 154, 156, 172]. Semigroups on Finitely Generated Groups Let G be a finitely generated group and S be a generating set for G such that S −1 = S and e ∈ S. Any element s admits a decomposition s = s1 s2 · · · sn

(1.23)

where s1 , . . . , sn are elements of S. The word length |s| of s with respect to the generating set S is defined to be the minimal integer n of such a decomposition and is a basic notion in geometric group theory. As a special case, the neutral element e has length zero. (a) Coxeter groups Here, we refer to [45] and references therein for more information. Recall that a group G = W is called a Coxeter group if W admits the following presentation:    W = S  (s1 s2 )m(s1 ,s2 ) = e : s1 , s2 ∈ S, m(s1 , s2 ) = ∞ where m : S × S → {1, 2, 3, . . . , ∞} is a function such that m(s1 , s2 ) = m(s2 , s1 ) for any s1 , s2 ∈ S and m(s1 , s2 ) = 1 if and only if s1 = s2 . The pair (W, S) is called a Coxeter system. In particular, every generator s ∈ S has order two. By Bo˙zejko [41, Theorem 7.3.3], the word length |s| is a conditionally negative length and our results can be used with the semigroup generated by this function. Recall that dihedral groups   Dn = s1 , s2 | s12 = s22 = (s1 s2 )2 = e , product groups Z2 ×· · ·×Z2 , symmetric groups Sn with S = {(n, n+1) : n ∈ N} and the infinite symmetric group S∞ of all finite permutations of the set N with S = {(n, n + 1) : n ∈ N} are examples of Coxeter groups. In the case of symmetric groups, the length |σ | of σ is the number of crossings in the diagram which represents the permutation σ .

(1.8)

For any 1  i  n, note that wA (ei ) = we have

9

(1.22)

ψ(wA ) =



j ∈A (ei )j

is equal to 1 if i ∈ A and to −1 if i ∈ A. So

  n n  1 1 

1 − wA (ei ) = wA (ei ) = |A|. n− 2 2 i=1

i=1

10 By analogy with the case of Rn , the name “discrete Poisson semigroup”

seems more appropriate.

1 Introduction

15

Consider a Coxeter group W . If s ∈ W , note that the sequence s1 , . . . , sn in (1.23) chosen in such a way that n is minimal is not unique in general. However, the set of involved generators is unique, i.e. if s = s1 s2 · · · sn = s1 s2 · · · sn are minimal words of s ∈ W then {s1 , s2 , . . . , sn } = {s1 , s2 , . . . , sn }. This subset {s1 , s2 , . . . , sn } of S is denoted Ss and is called the colour of s, following [45, def

p. 585]. We define the colour-length of s putting s = card Ss . We always have w  |w|. By Bo˙zejko et al. [45, Theorem 4.3 and Corollary 5.4], if 0  α  1 the functions | · |α and · are conditionally negative lengths on S∞ . Finally, see also [129, p. 1971] for other examples for Sn for n < ∞. (b) Free groups Our results can be used with the noncommutative Poisson semigroup [93], [128, Definition 10.1] on free groups Fn (1  n  ∞) whose negative generator is the length | · |. Moreover, we can use the characterization [95, Theorem 1.2] of radial functions ψ : Fn → C with ψ(e) = 0 which are conditionally negative definite. If ϕz : Fn → C denotes the spherical function11 of parameter z ∈ C, these functions can be written  ψ(s) =

1 −1

ψz (s) dν(z),

s ∈ Fn def 1−ϕz (s) 1−z ,

for some finite positive Borel measure ν on [−1, 1] where ψz (s) = def

z (s) z ∈ C \ {1} and ψ1 (s) = limz→1 1−ϕ 1−z . Finally, [130] contains (but without proof) examples of weighted forms of the word length which are conditionally negative. We can write Fn = ∗ni=1 Gi with Gi = Z. Every element s of Fn − {e} has a unique representation s = si1 si2 · · · sim , where sik ∈ Gik are distinct from the corresponding neutral elements and i1 = i2 = · · · = ik . The number m is called the block length of s and denoted s. By Junge and Zeng [126, Example 6.14], this function is a conditionally negative length. (c) Cyclic groups The word length on Zn is given by |k| = min{n, n − k}. It is known that this length is negative definite, see for example [131, p. 553], [130, Appendix B], [126, Section 5.3] and [127, Example 5.9] for more information.

n2 By Junge et al. [132, p. 925], the function ψn defined on Zn by ψn (k) = 2π 2 1− cos( 2πk n ) is another example of conditionally negative length on Zn .

Semigroups on the Discrete Heisenberg Group Let H = Z2n+1 be the discrete Heisenberg group with group operations (a, b, t) · (a  , b  , t  ) = (a + a  , b + b  , t + t  + ab ) (a, b, t)−1 = (−a, −b, −t + ab)

11 In

the case n = ∞, we have ϕz (s) = z|s| for any s ∈ F∞ and any z ∈ C.

and (1.24)

16

1 Introduction

where a, b, a  , b  ∈ Zn and t, t  ∈ Z. By Junge and Zeng [126, Proposition 5.13] and [127, p. 261], the map ψ : H → R, (a, b, t) → |a| + |b| is a conditionally negative length. We also prove in this paper an analogue of the equivalences (1.16) for markovian p def

semigroups (Tt )t 0 of Schur multipliers acting on Schatten spaces SI = S p ( 2I ) for 1 < p < ∞ where I is an index set. In this case, by Arhancet [10, Proposition 5.4], the Schur multiplier symbol [aij ] of the negative generator A of (Tt )t 0 is given by aij = αi − αj 2H for some family α = (αi )i∈I of vectors of a real Hilbert space H . We define a gradient operator ∂α,1,p as the closure of the unbounded linear p operator MI,fin → Lp (, SI ), eij → W(αi − αj ) ⊗ eij where W : H → L0 () is an H -isonormal Gaussian process,  is the associated probability space and where p MI,fin is the subspace of SI of matrices with a finite number of non null entries. Then the result reads as follows. Theorem 1.2 (see Theorem 3.3 and (3.88)) Let I be an index set and A be the negative generator of a markovian semigroup (Tt )t 0 of Schur multipliers on B( 2I ). Suppose 1 < p < ∞. For any x ∈ MI,fin , we have  12    Ap (x) p ≈p ∂α,1,p (x) p p . S L (,S ) I

(1.25)

I

We also obtain an analogue of Theorem 1.1. With this result, we are equally able to obtain a Hodge decomposition, see Theorem 4.10. Theorem 1.3 (see Theorems 4.9 and 4.11) Suppose 1 < p < ∞. The unbounded   0 (∂α,1,p∗ )∗ def p p operator Dα,1,p = on the Banach space SI ⊕ Lp (, SI ) is ∂α,1,p 0 bisectorial and admits a bounded H∞ (ω± ) functional calculus on a bisector ω± . Moreover, the norm of the functional calculus is bounded by a constant Kω,p which depends neither on I nor on the semigroup. In particular it is independent of the dimension of H . Moreover, we also relate the equivalences (1.16) and (1.25) with the ones of Meyer’s formulation (1.15). To achieve this, we define and study in the spirit of (1.14) a carré du champ  (see (2.78) and (2.91)) and its closed extension in the sense of Definition 2.7 and we connect this notion to some approximation properties of groups. It leads us to obtain alternative formulations of (1.16) and (1.25). Note that some carrés du champ were studied in the papers [56], [57, Section 9], [119, 126, 205] mainly in the σ -finite case and for L2 -spaces (see [70] for related things) but unfortunately their approach does not suffice for our work on Lp -spaces. By the way, it is rather surprising that even in the commutative setting, no one has examined the carré du champ on Lp -spaces with p = 2. The following is an example of result that we have achieved, and which can be compared with (1.15).

1 Introduction

17

Theorem 1.4 (see Theorem 3.5) Suppose 2  p < ∞. Let A be the negative generator of a markovian semigroup of Schur multipliers on B( 2I ). For any x ∈ 1

dom Ap2 , we have   21      Ap (x) p ≈p max (x, x) 12  p , (x ∗ , x ∗ ) 12  p . S S S I

I

(1.26)

I

The maximum is natural in noncommutative analysis due to the use of noncommutative Khintchine inequalities. It is remarkable that the point of view of Hodge-Dirac operators fits perfectly into the setting of noncommutative geometry if p = 2. If G is a discrete group, the def

def

Hilbert space H = L2 (VN(G)) ⊕2 ψ,1,2 , the ∗-algebra A = span {λs : s ∈ G} of trigonometric polynomials and the Hodge-Dirac operator Dψ,1,2 on L2 (VN(G)) ⊕2 ψ,1,2 define a triple (A, H, Dψ,1,2 ) in the spirit of noncommutative geometry [60, 90, 222]. Recall that the notion of spectral triple (A, H, D) (= noncommutative manifold) à la Connes covers a huge variety of different geometries such as Riemmannian manifolds, fractals, quantum groups or even non-Hausdorff spaces. We refer to [62] for an extensive list of examples and to [53, 61, 141, 208, 222] for some surveys. From here, it is apparent that we can see Markov semigroups of Fourier multipliers as geometric objects. The same observation is true for Markov semigroups of Schur multipliers. Nevertheless, the Hilbert space setting of the noncommutative geometry is too narrow to encompass our setting on Lp -spaces. So, we develop in Sect. 5.7 a natural Banach space variant (A, X, D) of a spectral triple where the selfadjoint operator D acting on the Hilbert space H is replaced by a bisectorial operator D acting on a (reflexive) Banach space X, allowing us to use (noncommutative) Lp -spaces (1 < p < ∞). It is well-known that Gaussian variables are not bounded, i.e. do not belong to L∞ (). From the perspective of noncommutative geometry, this is problematic under technical aspects as the boundedness of the commutators [D, π(a)] of a spectral triple (A, H, D). Indeed, the noncommutative gradients (1.17) appear naturally in the commutators of our spectral triples and these gradients are defined with Gaussian variables. Fortunately, the noncommutative setting is very flexible and allows us to introduce a continuum of gradients ∂ψ,q,p and ∂α,q,p indexed by a new parameter −1  q  1 replacing Gaussian variables (q = 1) by bounded noncommutative q-deformed Gaussian variables (q < 1) and L∞ () by the von Neumann algebra q (H ) of [43, 44]. Note that −1 (H ) is the fermion algebra and that L∞ () can be identified with the boson algebra 1 (H ). Our main theorems on Hodge-Dirac operators admit extensions in these cases, see Theorems 4.3 and 4.6. We expect some differences of behaviour when q varies. We also prove that

18

1 Introduction

Theorem 1.2 and its counterpart for markovian semigroups of Fourier multipliers are valid under the forms  12    Ap (x) p ≈p ∂α,q,p (x) p S L (q (H )⊗B( 2 )) I

(1.27)

I

   21  Ap (x) p ≈ ∂ψ,q,p (x)Lp (q (H )α G) L (VN(G)) p

(1.28)

with gradients ∂α,q,p and ∂ψ,q,p taking values in Lp spaces over amplifications of the q-deformed Gaussian von Neumann algebra q (H ). A major ingredient for (1.27) and (1.28) are Khintchine inequalities for q-Gaussians for that we give full proofs. They read in the following form. Theorem 1.5 (Khintchine Inequalities, see Theorem 3.1 and Lemma 3.19) Consider −1  q  1 and 1 < p < ∞. 1. Let G be a discrete group. Denote the conditional expectation E : q (H )α G → VN(G), xλs → τq (H ) (x)λs . Then for any finite sum f = s,h fs,h sq (h)λs , we have in case 1 < p < 2 f Lp (q (H )α G) ≈p

inf

f =g+h

 

1   E(g ∗ g) 2 

Lp (VN(G))



1  ,  E(hh∗ ) 2 Lp (VN(G))



and in case 2 < p < ∞   



1  1  f Lp (q (H )α G) ≈p max  E(f ∗ f ) 2 Lp (VN(G)) ,  E(ff ∗ ) 2 Lp (VN(G)) .

2. Let I be an index set. Denote the conditional expectation E : q (H )⊗B( 2I ) →  B( 2I ), x ⊗y → τq (H ) (x)y. Then for any finite sum f = i,j,h fi,j,h sq (h)⊗eij , we have in case 1 < p < 2 f Lp (q (H )⊗B( 2 )) ≈p I

  

1  

1  ∗ ∗ 2  2    p, p E(hh ) E(g g) inf S S

f =g+h

I

I

and in case 2 < p < ∞   

1  

1  f Lp (q (H )⊗B( 2 )) ≈p max  E(f ∗ f ) 2 S p ,  E(ff ∗ ) 2 S p . I

I

I

In Chap. 5, we give a first elementary study of the triples associated to Markov semigroups of multipliers. In particular, we give sufficient conditions for the verification of axioms of noncommutative geometry and more generally the axioms of our Banach spectral triples. One of our main results in this part reads as follows and can be used with arbitrary amenable groups or free groups. Here C∗r (G) denotes the reduced C∗ -algebra of the discrete group G.

1 Introduction

19

Theorem 1.6 (see Theorem 5.4) Suppose 1 < p < ∞ and −1  q < 1. Let G be a weakly amenable discrete group. Let (Tt )t 0 be a markovian semigroup of Fourier multipliers on the group von Neumann algebra VN(G). Consider the associated function bψ : G → H from (1.18). Assume that the Hilbert space H is finite-dimensional, that bψ is injective and that def

Gapψ =

inf

bψ (s)=bψ (t )

  bψ (s) − bψ (t)2 > 0. H

Finally assume that the crossed product von Neumann algebra q (H ) α G has QWEP. Let π : C∗r (G) → B(Lp (VN(G)) ⊕p Lp (q (H ) α G)) def

be the Banach algebra homomorphism such that π(a)(x, y) = (ax, (1  a)y) where x ∈ Lp (VN(G)) and y ∈ Lp (q (H ) α G). Let Dψ,q,p be the Hodge-Dirac operator on the Banach space Lp (VN(G)) ⊕p Lp (q (H ) α G) defined in (5.45). Then (C∗r (G), Lp (VN(G))⊕p Lp (q (H )α G), Dψ,q,p ) is a Banach spectral triple in the sense of Definition 5.10. We are equally interested in the metric aspect [147] of noncommutative geometry, see Sect. 5.1 for background on quantum (locally) compact metric spaces. We introduce new quantum (locally) compact metric spaces in the sense of [143, 147, 197] associated with these spectral triples. It relies on Lp -variants of the seminorms of [119, Section 1.2]. Here, we check carefully the axioms taking into account all problems of domains required by this theory. Note that it is not clear how to do the same analysis at the level p = ∞ considered in [119, Section 1.2] since we cannot hope for the boundedness of the Riesz transform on L∞ which is an important tool in this part. So, on this point, Lp -seminorms seem more natural. We observe significant differences between the case of Fourier multipliers (Theorem 5.1) and the one of Schur multipliers (Theorem 5.3). For example, we need to use C∗ -algebras for semigroups of Fourier multipliers and which produces quantum compact metric spaces contrarily to the case of semigroups of Schur multipliers which requires order-unit spaces and that leads to quantum locally compact metric spaces if I is infinite. Furthermore, our analysis with quantum compact metric spaces relying on noncommutative Lp -spaces makes appear a new phenomenon when the value of the parameter p changes, see Remark 5.3. Note that the combination of our spectral triples and our quantum (locally) compact metric spaces is in the spirit of the papers [32, 150] (see also [59]) but it is more subtle here since the link between the norms of the commutators and the seminorms of our quantum metric spaces is not as direct as the ones of [32, 150]. Finally, we are interested in other Dirac operators related to Riesz transforms. We study the associated noncommutative geometries in Sects. 5.9 and 5.11 and their properties of functional calculus in Sect. 5.12.

20

1 Introduction

The paper is organized as follows. Chapter 2 gives background and preliminary results. In Sect. 2.1, we collect some elementary information on operator theory, functional calculus and semigroup theory. In Sect. 2.2, we recall some important information on isonormal Gaussian processes and more generally q-Gaussian functors. These notions are fundamental for the construction of the noncommutative gradients. In Sect. 2.3, we develop a short theory of vector-valued unbounded bilinear forms on Banach spaces that we need for our carré du champ with values in a noncommutative Lp -space. In Sect. 2.4, we introduce a new transference result (Proposition 2.8) between Fourier multipliers on crossed products and classical Fourier multipliers. In Sect. 2.5, we discuss Hilbertian valued noncommutative Lp -spaces which are fundamental for us. We complete and clarify some technical points of the literature. In Sect. 2.6, we introduce the carré du champ  and the gradients with values in a noncommutative Lp -space for semigroups of Fourier multipliers. In Sect. 2.7, we discuss the carré du champ  and the gradients associated to Markov semigroups of Schur multipliers. In Chap. 3, we investigate dimension free Riesz estimates for semigroups of Schur multipliers and we complement the results of [131] for Riesz transforms associated to semigroups of Markov Fourier multipliers. In Sect. 3.1, we obtain Khintchine inequalities for q-Gaussians in crossed products generalizing some result of [131]. In Sect. 3.2, we extend the equivalences (1.16) of [131] to qdeformed Gaussians variables on a larger (maximal) domain, see (3.31). We also examine the constants. In Sect. 3.3, we are interested in Meyer’s equivalences (1.15) in the context of semigroups of Fourier multipliers in the case p  2, see Theorem 3.2. In Sect. 3.4, we show dimension free estimates for Riesz transforms associated to semigroups of Markov Schur multipliers, see Theorem 3.3 and (3.88). Moreover, we examine carefully the obtained constants in this equivalence. In Sect. 3.5, we turn again to the formulation of these equivalences in the spirit of Meyer’s equivalence (1.15) in connection with the carré du champ. We also obtain concrete equivalences similar to the ones of Lust-Piquard [154, 156] and related to (1.4). Note that for the case 1 < p < 2, the statement becomes more involved, see Corollary 3.1. In the next Chap. 4, we show our main results on functional calculus of HodgeDirac operators. In Sect. 4.1, we start with the case of the Hodge-Dirac operator for semigroups of Fourier multipliers. We show the boundedness of the functional calculus on Lp (VN(G)) ⊕p Ran ∂ψ,q,p , see Theorems 4.2 and 4.3. In Sect. 4.2, we extend this boundedness to the larger space Lp (VN(G)) ⊕ Lp (q (H ) α G). In Sect. 4.3, we obtain the boundedness on Lp (VN(G)) ⊕ ψ,q,p . In Sect. 4.4, we examine the bimodule properties of ψ,q,p . In Sect. 4.5, we change the setting and we consider the Hodge-Dirac operator associated to semigroups of Schur multipliers p and we show the boundedness of the functional calculus on SI ⊕p Ran ∂α,q,p , see p Theorem 4.8. In a second step, we extend in Sect. 4.6 this boundedness to SI ⊕ p 2 L (q (H )⊗B( I )) in Corollary 4.2 and Theorem 4.9. In Sect. 4.7, we show that the constants are independent from H and α.

1 Introduction

21

In Chap. 5, we examine the noncommutative geometries induced by the HodgeDirac operators in both settings and another two Hodge-Dirac operators. In Sect. 5.1, we give background on quantum (locally) compact metric spaces. In Sect. 5.2, we introduce Lp -versions of the quantum compact metric spaces of [119]. In Sect. 5.3, we establish fundamental estimates for the sequel and we introduce the constant Gapα which plays an important role. In Sect. 5.4, we introduce the seminorm which allows us to define the new quantum (locally) compact metric spaces in the next Sect. 5.5. In Sect. 5.6, we compute the gaps of some explicit semigroups and we compare the notions in both of our settings (Fourier multipliers and Schur multipliers), see Proposition 5.8. In Sect. 5.7, we define a Banach space generalization of the notion of spectral triple. In Sect. 5.8, we give sufficient conditions to make sure that the triple induced by the Hodge-Dirac operator for a semigroup of Fourier multipliers gives birth to a Banach spectral triple satisfying axioms of noncommutative geometry and more generally the axioms of Sect. 5.7. In Sect. 5.9, we introduce a second Hodge-Dirac operator for Fourier multipliers and we study the noncommutative geometry induced by this operator. In Sect. 5.10, we give sufficient conditions to ensure that the triple induced by the Hodge-Dirac operator for a semigroup of Schur multipliers gives rise to a spectral triple. In Sect. 5.11, we introduce a second related Hodge-Dirac operator and we study the noncommutative geometry induced by this operator. In Sect. 5.12, in the bosonic case q = 1 we establish the bisectoriality and the boundedness of the functional calculus of Dirac operators introduced in Sects. 5.9 and 5.11. Finally, in the short Appendix A, we discuss how the Lévy measure of a continuous conditionally of negative type function ψ on a locally compact abelian group G induces a 1-cocycle.

Chapter 2

Preliminaries

Abstract In this chapter, we start by recalling the used properties of the notions used in this book: operators, semigroups, H∞ functional calculus, noncommutative Lp -spaces and probabilities. In particular, the construction of our markovian semigroups of Fourier and Schur multipliers is a standing assumption in the rest of the book. We equally investigate vector-valued unbounded bilinear forms on Banach spaces which will be used as a framework for (the domain of) the carré du champ. We also show a transference result between Fourier multipliers on group von Neumann algebras and Fourier multipliers on crossed product von Neumann algebras. Then we give useful results on Hilbertian valued noncommutative Lp spaces for the sequel of the book. Finally, we examine in detail the carré du champ and the first order differential calculus for semigroups of Fourier multipliers and semigroups of Schur multipliers.

2.1 Operators, Functional Calculus and Semigroups In this short section, we describe various notions that play a role in this book at different places. Closed and Weak* Closed Operators An operator T : dom T ⊂ X → Y is closed if for any sequence (xn ) of dom T with xn → x and T (xn ) → y with x ∈ X (2.1) and y ∈ Y we have x ∈ dom T and T (x) = y. By Kato [136, p. 165], an operator T : dom T ⊂ X → Y is closed if and only if its domain dom T is a complete space with respect to the graph norm def

xdom T = xX + T (x)Y .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Arhancet, C. Kriegler, Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers, Lecture Notes in Mathematics 2304, https://doi.org/10.1007/978-3-030-99011-4_2

(2.2)

23

24

2 Preliminaries

A linear subspace C of dom T is a core of T if C is dense in dom T for the graph norm, that is for any x ∈ dom T there is (xn ) of C s. th. xn → x in X and T (xn ) → T (x) in Y. (2.3) If T is closed, a subspace C of dom T is a core for T if and only if T is the closure of its restriction T |C. Recall that if T : dom T ⊂ X → Y is a densely defined unbounded operator then dom T ∗ is equal to  ∗ y ∈ Y ∗ : there exists x ∗ ∈ X∗ s. th. T (x), y ∗ Y,Y ∗ = x, x ∗ X,X ∗ for all x ∈ dom T .

(2.4) If y ∗ ∈ dom T ∗ , the previous x ∗ ∈ X∗ is determined uniquely by y ∗ and we let T ∗ (y ∗ ) = x ∗ . For any x ∈ dom T and any y ∗ ∈ dom T ∗ , we have

T (x), y ∗ Y,Y ∗ = x, T ∗ (y ∗ )X,X∗ .

(2.5)

Recall that the unbounded operator T : dom T ⊂ X → Y is closable [136, p. 165] if and only if xn ∈ dom T , xn → 0 and T (xn ) → y imply y = 0.

(2.6)

If the unbounded operator T : dom T ⊂ X → Y is closable then x ∈ dom T iff there exists (xn ) ⊂ dom T s. th. xn → x and T (xn ) → y for some y. (2.7) In this case T (x) = y. Finally, if T is a densely defined closable operator then ∗ T ∗ = T by Kato [136, Problem 5.24]. If T is densely defined, by Kato [136, Problem 5.27 p. 168], we have Ker T ∗ = (Ran T )⊥ .

(2.8)

If T is a densely defined unbounded operator and if T ⊂ R, by Kato [136, Problem 5.25 p. 168] we have R∗ ⊂ T ∗ .

(2.9)

If T R and T are densely defined, by Kato [136, Problem 5.26 p. 168] we have R ∗ T ∗ ⊂ (T R)∗ .

(2.10)

In Sect. 5, we need the following lemma. To this end, an operator T : dom T ⊂ X → Y between dual spaces X and Y is called weak* closed if (2.1) holds, when the convergences are in the weak* sense.

2.1 Operators, Functional Calculus and Semigroups

25

Lemma 2.1 Let X, Y be Banach spaces and T : dom T ⊂ X∗ → Y ∗ be a weak* closed operator with a weak*-core1 D ⊂ dom T . Then any a ∈ dom T admits a bounded net (aj ) of elements of D such that the net (T (aj )) is also bounded, aj → a and T (aj ) → T (a) both for the weak* topology. Proof We consider the Banach space X∗ ⊕Y ∗ which is canonically the dual space of X ⊕ Y and thus carries a weak* topology. Let N = {(a, T (a)) : a ∈ D} ⊂ X∗ ⊕ Y ∗ and N1 = N ∩ B1 where B1 is the closed unit ball of X∗ ⊕ Y ∗ . w∗ w∗ We claim that CN1 = N . The inclusion “⊂” is clear. For the inclusion “⊃” ∗ w it suffices to show that CN1 contains N and is weak* closed. That it contains N is easy to see. For the weak*-closedness, we will employ the Krein-Smulian Theorem [167, w∗ Theorem 2.7.11]. So it suffices to show that CN1 ∩ Br is weak* closed for any w∗ r > 0. Let (xj ) be a net in CN1 ∩ Br converging to some x ∈ X∗ ⊕ Y ∗ . Since Br is weak* closed (by Alaoglu Theorem), we have x ∈ Br . Moreover, for any j w∗ we can write xj = λj yj with yj ∈ N1 and λj ∈ C. We have a freedom in choice of both factors and can take yj X∗ ⊕Y ∗ = 1. Thus, |λj |  r. By Alaoglu Theorem, w∗

(yj ) admits a weak* convergent subnet (yjk ) with a limit thus in N1 . Moreover, we can suppose that the net (λjk ) is convergent. Thus, xjk = λjk yjk converges for w∗

the weak* topology with limit a fortiori equal to x. Thus x ∈ CN1 . We infer that w∗ w∗ w∗ x ∈ CN1 ∩ Br . Consequently CN1 ∩ Br is weak*-closed and hence CN1 is ∗ ∗ w w weak*-closed. So we have shown CN1 = N . w∗ Since D is a weak*-core of T , the closure N equals the graph of T . Now if w∗ w∗ a ∈ dom T , the elements (a, T (a)) belongs to N = CN1 , so that we can choose a net (aj ) of graph norm less than 1 and λ ∈ C such that aj ∈ D, λaj → a and T (λaj ) → T (a) both for the weak* topology.   The following is [219, Corollary 5.6 p. 144]. Theorem 2.1 Let T be a densely defined closed operator on a Hilbert space H . Then the operator T ∗ T on (Ker T )⊥ is unitarily equivalent to the operator T T ∗ on (Ker T ∗ )⊥ . Semigroups and Sectorial Operators If (Tt )t 0 is a strongly continuous semigroup on a Banach space X with (negative) infinitesimal generator A, we have 1

Tt (x) − x) −−−→ −A(x), t t →0+

x ∈ dom A

(2.11)

by Engel and Nagel [85, Definition 1.2 p. 49] and Ker A = {x ∈ X : Tt (x) = x for all t  0} by Engel and Nagel [85, p. 337]. Moreover, by Engel and Nagel For any a ∈ dom T there exists a net (aj ) in D such that aj → a and T (aj ) → T (a) both for the weak* topology.

1

26

2 Preliminaries

[85, Corollary 5.5 p. 223], for any t  0, we have −n  t x −−−−→ Tt (x), Id + A n→+∞ n

x ∈ X.

(2.12)

Furthermore, by Engel and Nagel [85, (1.5) p. 50], if x ∈ dom A and t  0, then Tt (x) belongs to dom A and Tt A(x) = ATt (x).

(2.13)

For any angle σ ∈ (0, π), we will use the open sector symmetric around the positive real half-axis with opening angle 2σ def  σ+ = σ = z ∈ C\{0} : | arg z| < σ . It will be convenient to set 0+ = (0, ∞). We refer to [83, 98, 110, 128] for background on sectorial and bisectorial operators and their H∞ functional calculus, as well as R-boundedness. A sectorial operator A : dom A ⊂ X → X of type σ on a Banach space X is a closed densely defined operator such that σ (A) ⊂ σ for some σ ∈ [0, π) and such that sup{zR(z, A) : z ∈ σ  } < ∞ for any σ  ∈ (σ, π). We introduce the angle of sectoriality def

ω(A) = inf σ.

(2.14)

An operator A is called R-sectorial of type σ if it is sectorial of type σ and if for any θ ∈ (σ, π) the set  zR(z, A) : z ∈ θ

(2.15)

is R-bounded. If A is the negative generator of a bounded strongly continuous semigroup (Tt )t 0 on a Banach space X then A is sectorial of type π2 by Hytönen [110, Example 10.1.2]. If A is sectorial operator on a reflexive Banach space X, we have by Haase [98, Proposition 2.1.1 (h)] or [110, Proposition 10.1.9] a decomposition X = Ker A ⊕ Ran A.

(2.16)

If in addition A is the negative generator of a bounded strongly continuous semigroup (Tt )t 0 , this means that (Tt )t 0 is mean ergodic [8, Definition 4.3.3], t [85, p. 338]. In particular, the map P : X → X, x → limt →∞ 1t 0 Ts (x) ds is the bounded projection on Ker A associated to (2.16). By Arendt et al. [8, Corollary 4.3.2] (see also [110, (10.5) and p. 367]), it is also given by P (x) = lim λ(λ + A)−1 x, λ→0

x ∈ X.

(2.17)

2.1 Operators, Functional Calculus and Semigroups

27

Let A be the negative generator of a bounded strongly continuous semigroup (Tt )t 0 on X. For any x ∈ X and any z ∈  π2 , we have by e.g. [85, p. 55] the following expression of the resolvent as a Laplace transform (z − A)−1 x = −



∞

ezt Tt (x) dt.

(2.18)

0

The following is [110, Proposition G.2.4 p. 526]. Lemma 2.2 Let (Tt )t 0 be a strongly continuous semigroup of bounded operators on a Banach space X with (negative) generator A. If Y is a subspace of dom A which is dense in X and invariant under each operator Tt , then Y is a core of A. Bisectorial Operators In a similar manner, for σ ∈ [0, π2 ), we consider the open def

bisector σ± = σ ∪ (−σ ) and we say that a closed densely defined operator A is  (R-)bisectorial of type σ if σ (A) ⊂ σ± for some σ ∈ [0, π2 ) and if zR(z, A) : z ∈ σ± is (R-)bounded for any σ  ∈ (σ, π2 ). By Hytönen [110, p. 447] (see also [110, Proposition 10.3.3]), a linear operator is (R-)bisectorial if and only if iR − {0} ⊂ ρ(A) and

sup

t ∈R+ −{0}

tR(it, A) < ∞

(2.19)

(resp. {tR(it, A) : t ∈ R+ − {0}} is R-bounded). Self-adjoint operators are bisectorial of type 0. If A is bisectorial of type σ then by Hytönen [110, Proposition 10.6.2 (2)] the operator A2 is sectorial of type 2σ and we have Ran A2 = Ran A

and

Ker A2 = Ker A.

(2.20)

Functional Calculus of Sectorial and Bisectorial Operators For any 0 < θ < π, let H∞ (θ ) be the algebra of all bounded analytic functions f : θ → C, equipped  def with the supremum norm f H∞ (θ ) = sup |f (z)| : z ∈ θ . Let H∞ 0 (θ ) ⊂ H∞ (θ ) be the subalgebra of bounded analytic functions f : θ → C for which c|z|s there exist s, c > 0 such that |f (z)|  (1+|z|) 2s for any z ∈ θ . Given a sectorial operator A of type 0 < ω < π, a bigger angle θ ∈ (ω, π), and a function f ∈ H∞ 0 (θ ), one may define a bounded operator f (A) by means of a Cauchy integral on θ (see e.g. [94, Section 2.3] or [140, Section 9]). The resulting mapping H∞ 0 (θ ) → B(X), f → f (A) is an algebra homomorphism. By definition, A has a bounded H∞ (θ ) functional calculus provided that this homomorphism is bounded, that is if there exists a positive constant C such that   f (A)  C f H∞ (θ ) for any f ∈ H∞ 0 (θ ). In the case where A has X→X a dense range, the latter boundedness condition allows a natural extension of f → f (A) to the full algebra H∞ (θ ). For a bisectorial operator A, we can make a similar definition by integrating on the boundary of a bisector. In particular, A is said to have a bounded H∞ (θ± ) calcu-

28

2 Preliminaries

  lus, if there is some C > 0 such that f (A)X→X  C f H∞ ( ± ) for any f of the θ ± def  ∞ ( ± ) : ∃c, s > 0 ∀ z ∈  ± : |g(z)|  c|z|s /(1 + |z|)2s . space H∞ ( ) = g ∈ H θ θ θ 0 The following is a particular case of [176, Proposition 2.3], see also [110, Theorem 10.6.7]. Proposition 2.1 Suppose that A is an R-bisectorial operator on a Banach space X of finite cotype. Then A2 is R-sectorial and for each ω ∈ (0, π2 ) the following assertions are equivalent. 1. The operator A admits a bounded H∞ (ω± ) functional calculus. 2. The operator A2 admits a bounded H∞ (2ω ) functional calculus. Fractional Powers We refer to [8, 98, 99, 140, 162] for more information on fractional powers. Let A be a sectorial operator of type σ on a Banach space X. If α ∈ (0, πσ ), then by Haase [98, Proposition 3.1.2] the operator Aα is sectorial of angle ασ . For all α, β with Re α, Re β > 0 we have Aα Aβ = Aα+β . By Haase [98, page 62], [98, Corollary 3.1.11] and [162, p. 142], for any α ∈ C with Re α > 0 we have Ran Aα = Ran A

and

Ker Aα = Ker A.

(2.21)

If A is densely defined and 0 < Re α < 1, then the space dom A is a core of Aα by Haase [98, page 62]. def

For t, s > 0, we let ft (s) = 

t2

√ t e− 4s . 4πs 3

+∞

It is well-known [8, Lemma 1.6.7] that

e−sλ ft (s) ds = e−t

√ λ

.

(2.22)

0

Let A be the negative generator of a bounded semigroup (e−t A )t 0 acting on a Banach space X. By Haase [98, Example 3.4.6], for any x ∈ X we have 1

e−t A 2 x =



+∞

ft (s)e−sA (x) ds.

(2.23)

0

Compact Operators and Complex Interpolation Recall the following result [69, Theorem 9] (see also [135, Theorem 5.5]) which allows to obtain compactness via complex interpolation. Theorem 2.2 Suppose that (X0 , X1 ) and (Y0 , Y1 ) are Banach couples and that X0 is a UMD-space. Let T : X0 +X1 → Y0 +Y1 such that its restrictions T0 : X0 → Y0 resp. T1 : X1 → Y1 are compact resp. bounded. Then for any 0 < θ < 1 the map T : (X0 , X1 )θ → (Y0 , Y1 )θ is compact. Noncommutative Lp -Spaces We give a description of noncommutative Lp -spaces associated with a semifinite von Neumann algebra. We refer the reader to [191] and the references therein for further information on these spaces.

2.1 Operators, Functional Calculus and Semigroups

29

Let M be a semifinite von Neumann algebra equipped with a normal semifinite faithful trace τ . We let M+ denote the positive part of M. Let S+ be the set of all x ∈ M+ whose support projection has a finite trace. Then any x ∈ S+ has a finite trace. Let S ⊂ M be the linear span of S+ , then S is a weak* dense ∗-subalgebra of M. Let 0 < p < ∞. For any x ∈ S, the operator |x|p belongs to S+ and we set 1 def

xLp (M) = τ (|x|p ) p , def

x∈S

1

where |x| = (x ∗ x) 2 denotes the modulus of x. It turns out that ·Lp (M) is a norm on S if p  1, and a p-norm if p < 1. By definition, the noncommutative Lp -space associated with (M, τ ) is the completion Lp (M) of (S,  p ). For convenience, we def

also set L∞ (M) = M equipped with its operator norm. Note that by definition, Lp (M) ∩ M is dense in Lp (M) for any 1  p < ∞. Assume that M acts on some Hilbert space H . We denote by L0 (M) the set of all measurable operators which is a ∗-algebra under suitable operations. For any x p in L0 (M) and any 0 < p < ∞, the operator |x|p = (x ∗ x) 2 belongs to L0 (M). Let L0 (M)+ be the positive part of L0 (M), that is, the set of all selfadjoint positive operators in L0 (M). Then the trace τ extends to a positive tracial functional on L0 (M)+ , still denoted by τ , in such a way that for any 0 < p < ∞, we have an isometric identification  Lp (M) = x ∈ L0 (M) : τ (|x|p ) < ∞ , 1

where the latter space is equipped with xp = (τ (|x|p )) p . Furthermore, τ uniquely extends to a bounded linear functional on L1 (M), still denoted by τ . Indeed we have |τ (x)|  τ (|x|) = x1 for any x ∈ L1 (M). For any 0 < p  ∞ and any x ∈ Lp (M), the adjoint operator x ∗ belongs to p Lp (M) as well, with x ∗ p = xp . Clearly, we also have that x ∗ x ∈ L 2 (M) and |x| ∈ Lp (M), with  |x| p = xp . We let Lp (M)+ = L0 (M)+ ∩ Lp (M) denote the positive part of Lp (M). The space Lp (M) is spanned by Lp (M)+ . We recall the noncommutative Hölder inequality. If 0 < p, q, r  ∞ satisfy 1 1 1 + p q = r , then xyr  xp yq ,

x ∈ Lp (M), y ∈ Lq (M).

(2.24)

Conversely for any z ∈ Lr (M), there exist x ∈ Lp (M) and y ∈ Lq (M) such that z = xy with zr = xp yq . For any x ∈ L2p (M) and any y ∈ Lp (M)+ , we have if α > 0  ∗  x x  = x2 2p p

and

 α y  p = yα . p α

(2.25)

30

2 Preliminaries def

p For any 1  p < ∞, let p∗ = p−1 be the conjugate number of p. Applying ∗ (2.24) with q = p and r = 1, we may define a duality pairing between Lp (M) and ∗ Lp (M) by

x, y = τ (xy),

∗

x ∈ Lp (M), y ∈ Lp (M).

(2.26)

This induces an isometric isomorphism ∗

Lp (M) = (Lp (M))∗ ,

1  p < ∞,

1 1 + = 1. p p∗

(2.27)

In particular, we may identify L1 (M) with the unique predual M∗ of the von Neumann algebra M. By means of the natural embeddings of L∞ (M) = M and L1 (M) = M∗ into 0 L (M), one may regard (L∞ (M), L1 (M)) as a compatible couple of Banach spaces. Then we have (L∞ (M), L1 (M)) 1 = Lp (M), p

1  p  ∞,

(2.28)

where ( , )θ stands for the interpolation space obtained by the complex interpolation method, see [35]. Finally, for any 1 < p < ∞ note that the Banach space Lp (M) is a UMD Banach space and has finite cotype, see [191, Section 7]. The space L2 (M) is a Hilbert space, with inner product given by (x, y) →

x, y ∗  = τ (xy ∗ ). Note that the identity (2.27) provided by (2.26) for p = 2 differs from the canonical (antilinear) identification of a Hilbert space with its dual space. Let T : L2 (M) → L2 (M) be any bounded operator. We will denote by T ∗ the adjoint of T provided by (2.27) and (2.26) defined by



τ T (x)y = τ xT ∗ (y) ,

x, y ∈ L2 (M).

For any 1  p  ∞ and any T : Lp (M) → Lp (M), we can consider the map T ◦ : Lp (M) → Lp (M) defined by T ◦ (x) = T (x ∗ )∗ , def

x ∈ Lp (M).

(2.29)

If p = 2 and if we denote by T † the adjoint of T in the usual sense of Hilbertian operator theory, that is



τ T (x)y ∗ = τ x(T † (y))∗ ,

x, y ∈ L2 (M),

2.1 Operators, Functional Calculus and Semigroups

31

we see that T † = T ∗◦ .

(2.30)

In particular, the selfadjointness of T : L2 (M) → L2 (M) means that T ∗ = T ◦ . Selfadjoint Maps Let T : M → M be a contraction. We say that T is selfadjoint if



τ T (x)y ∗ = τ xT (y)∗ ,

x, y ∈ M ∩ L1 (M).

(2.31)





  In this case, for any x, y in M ∩ L1 (M), we have τ T (x)y  = τ xT (y ∗ )∗   x1 T (y ∗ )∗ ∞  x1 y∞ . Hence the restriction of T to M ∩L1 (M) uniquely extends to a contraction T1 : L1 (M) → L1 (M). Then by (2.28), it also extends to a contraction Tp : Lp (M) → Lp (M) for any 1  p < ∞. We write T∞ = T by convention. If T is in addition weak* continuous, we obtain from (2.31) that (Tp )∗ = (Tp∗ )◦ ,

1  p < ∞,

1 1 + ∗ = 1. p p

By (2.30), this implies that the operator T2 : L2 (M) → L2 (M) is selfadjoint. If T : M → M is in addition positive, then each operator Tp is positive. In particular (Tp )◦ = Tp . Thus in this case, we have Tp∗ = Tp∗ for any 1  p < ∞. Markovian Semigroups of Fourier Multipliers In this paragraph, we recall the basic theory of markovian semigroups of Fourier multipliers. The following definition and properties of a markovian semigroup are fundamental for us. Thus the assumptions and notations which follow these lines are standing for all the book. A companion definition of equal importance is Definition 2.2 together with Proposition 2.3 which follow. Definition 2.1 Let M be a von Neumann algebra equipped with a normal semifinite faithful trace. We say that a weak* continuous semigroup (Tt )t 0 of operators on M is a markovian semigroup if each Tt is a weak* continuous selfadjoint completely positive unital contraction. For any 1  p < ∞, such a semigroup induces a strongly continuous semigroup of operators on each Lp (M) satisfying 1. 2. 3. 4.

each Tt is a contraction on Lp (M), each Tt is selfadjoint on L2 (M), each Tt is completely positive on Lp (M), Tt (1) = 1.

32

2 Preliminaries

By Hellmich [104, Corollary 1.27], if M is finite, then there exists a normal conditional expectation E on the fixed point subalgebra {x ∈ M : Tt (x) = x for all t  0}. A careful reading of the proof shows that E is trace preserving since it is easy to check that each Tt is2 trace preserving. If {x ∈ M : Tt (x) = x for all t  0} = C1, the conditional expectation is p defined by E(x) = τ (x)1. We use the notation L0 (M) for the subspace Ker Ep of p p L (M). We have L0 (M) = Ran Ap if Ap is the Lp -realization of the generator of the semigroup. The following is [119, Theorem 1.1.7]. Note that the proof of this result uses [119, Lemma 1.1.6] which seems false in the light of [135, Problem 5.4]. However, [119, Lemma 1.1.6] can be replaced by Theorem 2.2. Proposition 2.2 Let M be a finite von Neumann algebra equipped with a normal finite faithful trace. Let (Tt )t 0 be a weak* continuous markovian semigroup ∞ on all t  0} = C1 satisfying  L 1(M) with ∞{x ∈ M 1: Tt (x) = x for −1 Tt : L (M) → L (M)  n and such that A is compact on L2 (M). Then 0 0 t2

for all 1  p < q  ∞ such that p q operator A−z : L0 (M) → L0 (M).

2 Re z n

>

1 p

−

1 q

we have a well-defined compact

Let G be a discrete group and consider its left regular representation λ : G → B( 2G ) over the Hilbert space 2G given by left translations λs : δr → δsr where r, s ∈ G. Then the group von Neumann algebra VN(G) is the von Neumann subalgebra of B( 2G ) generated by these left translations. We also recall that C∗r (G) stands for the reduced group C∗ -algebra sitting inside VN(G) and generated by these λs . Let us write def

PG = span {λs : s ∈ G}

(2.32)

for the space of “trigonometric polynomials”. The von Neumann algebra VN(G) is def

equipped with the tracial faithful normal state τ (λs ) = δs=e = δe , λs (δe ) 2 . Now, G we introduce the main class of multipliers in which we are interested. Definition 2.2 Let G be a discrete group. A Fourier multiplier on VN(G) is a weak* continuous linear map T : VN(G) → VN(G) such that there exists a (unique) complex function φ : G → C such that for any s ∈ G we have T (λs ) = φs λs . In this case, we let Mφ = T and we say that φ is the symbol of T . We have the following folklore characterization of markovian semigroups of Fourier multipliers. Proposition 2.3 is central for the remainder of the book and the mappings π and bψ = b will be used throughout tacitly. Recall that a function ψ : G → C is conditionally negative definite if ψ(s) =  ψ(s −1 ) for any s ∈ G, ψ(e)  0 and if the condition ni=1 ci = 0 implies n −1 i,j =1 ci cj ψ(sj si )  0. If H is a real Hilbert space, O(H ) stands for the orthogonal group.

2

If x ∈ M, we have τ (Tt (x)) = τ (Tt (x)1) = τ (xTt (1)) = τ (x).

2.1 Operators, Functional Calculus and Semigroups

33

Proposition 2.3 Let G be a discrete group and (Tt )t 0 be a family of weak* continuous operators on VN(G). Then the following are equivalent. 1. (Tt )t 0 is a markovian semigroup of Fourier multipliers. 2. There exists a (unique) real-valued conditionally negative definite function ψ : G → R satisfying ψ(e) = 0 such that Tt (λs ) = exp(−tψ(s))λs for any t  0 and any s ∈ G. 3. There exists a (unique) function ψ : G → R such that Tt (λs ) = exp(−tψ(s))λs for any t  0 and any s ∈ G with the following property: there exists a real Hilbert space H together with a mapping bψ : G → C and a homomorphism π : G → O(H ) such that the 1-cocycle law holds πs (bψ (t)) = bψ (st) − bψ (s),

i.e.

bψ (st) = bψ (s) + πs (bψ (t))

(2.33)

for any s, t ∈ G and such that ψ(s) = bψ (s)2H ,

s ∈ G.

(2.34)

Under these conditions, we say that ψ is a conditionally negative length. Note also the equality bψ (s −1 ) = −πs −1 bψ (s)

(2.35)

which is immediate from (2.33). We refer to Appendix A for a link between Lévy measures and 1-cohomology strongly related to this result. Truncations of Matrices See [9, p. 488] for the following definition. Definition 2.3 We denote by MI,fin the space of matrices indexed by I × I with a finite number of non null entries. Given a set I , the set Pf (I ) of all finite subsets of I is directed with respect to set inclusion. For J ∈ Pf (I ) and A ∈ MI a matrix indexed by I × I , we write TJ (A) for the matrix obtained from A by setting each entry to zero if its row and column index are not both in J . We call TJ (A) J ∈P (I ) f the net of finite submatrices of A. p

p

If J is a finite subset of I , recall that the truncation TJ : SI → SI is a complete p contraction. It is well-known that TJ (x) → x in SI as J → I where 1  p  ∞. Markovian Semigroups of Schur Multipliers Let I be any non-empty index set. Let A = [aij ]i,j ∈I be a matrix of MI . By definition, the Schur multiplier on B( 2I ) associated with this matrix is the unbounded linear operator MA whose domain dom MA is the space of all B = [bij ]i,j ∈I of B( 2I ) such that [aij bij ]i,j ∈I belongs def

to B( 2I ), and whose action on B = [bij ]i,j ∈I is given by MA (B) = [aij bij ]i,j ∈I . In the literature, it is often written A ∗ B instead of MA (B) for the Schur product [aij bij ]i,j ∈I .

34

2 Preliminaries

The following description [10, Proposition 5.4], [15] of a markovian semigroup consisting of Schur multipliers is central for our paper. See also [11] for a generalization to semigroups of non positive Schur multipliers. Proposition 2.4 Let I be some non-empty index set. Then the following are equivalent. 1. If (Tt )t 0 is a weak* continuous semigroup of selfadjoint unital completely positive Schur multipliers on B( 2I ) then there exists a real Hilbert space H and a family α = (αi )i∈I

(2.36)

of elements of H such that for any t  0, the Schur multiplier Tt : B( 2I ) → B( 2I ) is associated to the matrix 

e−t αi −αj H 2

 i,j ∈I

.

(2.37)

2. If there exists a real Hilbert space H and a family α = (αi )i∈I of elements of H then (2.37) defines a weak* continuous semigroup (Tt )t 0 of selfadjoint unital completely positive Schur multipliers on B( 2I ) In this case, the weak* (negative) infinitesimal generator A of this markovian semigroup of Schur multipliers acts by A(eij ) = αi − αj 2H eij .

2.2 q-Gaussian Functors, Isonormal Processes and Probability The definitions (2.84) and (2.95) of the noncommutative gradients associated with our markovian semigroups of Schur and Fourier multipliers need q-deformed Gaussian variables from [44]. We recall here several facts about the associated von Neumann algebras. We denote by Sn the symmetric group. If σ is a permutation  of Sn we denote by |σ | the number card (i, j ) | 1  i, j  n, σ (i) > σ (j ) of inversions of σ . Let H be a real Hilbert space with complexification HC . If −1  q < 1 the q-Fock space over H is def

Fq (H ) = C ⊕



HC⊗n

n1

where  is a unit vector, called the vacuum and where the scalar product on HC⊗n is given by

h1 ⊗ · · · ⊗ hn , k1 ⊗ · · · ⊗ kn q =

 σ ∈Sn

q |σ | h1 , kσ (1) HC · · · hn , kσ (n) HC .

2.2 q-Gaussian Functors, Isonormal Processes and Probability

35

If q = −1, we must first divide out by the null space, and we obtain the usual antisymmetric Fock space. The creation operator (e) for e ∈ H is given by (e) :

−→ Fq (H ) Fq (H ) h1 ⊗ · · · ⊗ hn −→ e ⊗ h1 ⊗ · · · ⊗ hn .

They satisfy the q-relation (f )∗ (e) − q (e) (f )∗ = f, eH IdFq (H ) . We denote by sq (e) : Fq (H ) → Fq (H ) the selfadjoint operator (e) + (e)∗ . The q-von Neumann algebra q (H ) is the von Neumann algebra over Fq (H ) generated by the operators sq (e) where e ∈ H . It is a finite von Neumann algebra with the trace τ defined by τ (x) = , x()Fq (H ) where x ∈ q (H ). Let H and K be real Hilbert spaces and T : H → K be a contraction with complexification TC : HC → KC . We define the following linear map Fq (T ) :

Fq (H ) −→ Fq (K) h1 ⊗ · · · ⊗ hn −→ TC (h1 ) ⊗ · · · ⊗ TC (hn ).

Then there exists a unique map q (T ) : q (H ) → q (K) such that for every x ∈ q (H ) we have

q (T )(x)  = Fq (T )(x).

(2.38)

This map is weak* continuous, unital, completely positive and trace preserving. If T : H → K is an isometry, q (T ) is an injective ∗-homomorphism. If 1  p < ∞, p it extends to a contraction q (T ) : Lp (q (H )) → Lp (q (K)). Moreover, we need the following Wick formula, (see [42, p. 2] and [80, Corollary 2.1]). In order to state this, we denote, if k  1 is an integer, by P2 (2k) the set of 2-partitions of the set {1, 2, . . . , 2k}. If V ∈ P2 (2k) we let c(V) the number of crossings of V, which is given by the number of pairs of blocks of V which cross (see [80, p. 8630] for a precise definition). Then, if f1 , . . . , f2k ∈ H we have

τ sq (f1 )sq (f2 ) · · · sq (f2k ) =

 V∈P2 (2k)

q c(V)



fi , fj H

(2.39)

(i,j )∈V

and for an odd number of factors of q-Gaussians, the trace vanishes,

τ sq (f1 )sq (f2 ) · · · sq (f2k−1 ) = 0.

(2.40)

In particular, for any e, f ∈ H , we have

τ sq (e)sq (f ) = e, f H .

(2.41)

36

2 Preliminaries

Recall that if e ∈ H has norm 1, then the operator s−1 (e) satisfies s−1 (e)2 = IdF−1 (H ) .

(2.42)

The q-Gaussian functor for q = 1 identifies to an H -isonormal process on a probability space (, μ) [175, Definition 6.5], [178, Definition 1.1.1], that is a linear mapping W : H → L0 () with the following properties: for any h ∈ H the random variable W(h) is a centered real Gaussian,

for any h1 , h2 ∈ H we have E W(h1 )W(h2 ) = h1 , h2 H .

(2.44)

The linear span of the products W(h1 )W(h2 ) · · · W(hm ), with m  0

(2.45)

(2.43)

and h1 , . . . , hm in H, is dense in the real Hilbert space L2R (). Here L0 () denotes the space of measurable functions on  and we make the convention that the empty product, corresponding to m = 0 in (2.45), is the constant function 1. Recall that the span of elements eiW(h) is dense in Lp () by Janson [113, Theorem 2.12] if 1  p < ∞ and weak* dense if p = ∞. If 1  p  ∞ and if T : H → H is a contraction, we denote by p 1 (T ) : Lp () → Lp () the (symmetric) second quantization of T acting on the complex Banach space Lp (). Recall that the map 1∞ (T ) : L∞ () → L∞ () preserves the integral.3 If T : H1 → H2 is an isometry between Hilbert spaces then 1∞ (T ) : L∞ (H1 ) → L∞ (H2 ) is a trace preserving injective weak* continuous unital ∗-homomorphism which is surjective if T is surjective. Moreover, we have if 1p 0, we deduce (x, y)Lp (VN(G))

 1 p 1 p p  1 inf λ (x, x)Lp (VN(G)) + (y, y)Lp (VN(G)) . λ 2 p λ>0 1

Ruling out beforehand the easy cases (x, x) = 0 or (y, y) = 0, we choose )1 ( 2 p p λ = (y, y)Lp (VN(G)) / (x, x)Lp (VN(G)) and obtain 1

(x, y)Lp (VN(G)) 

1

2p

 1 p p p 2 2 2 (x, x)Lp (VN(G)) (y, y)Lp (VN(G)) (2.82) 1 2

1 2

= (x, x)Lp (VN(G)) (y, y)Lp (VN(G)) .   Suppose that (Tt )t 0 is associated to the length ψ : G → R+ with associated cocycle b = bψ : G → H and the orthogonal representation π : G → B(H ), s → πs of G on H from Proposition 2.3. Suppose −1  q  1. For any s ∈ G, we will use the second quantization from (2.38) by letting αs = q∞ (πs ) : q (H ) → q (H ) def

(2.83)

which is trace preserving. We obtain an action α : G → Aut(q (H )). So we can consider the crossed product q (H ) α G as studied in Sect. 2.4, which comes equipped with its canonical normal finite faithful trace τ . Suppose −1  q  1 and 1  p < ∞. We introduce the map ∂ψ,q : PG → Lp (q (H ) α G) defined by ∂ψ,q (λs ) = sq (bψ (s))  λs .

(2.84)

which is a slight generalization of the map of [131, p. 535]. Note that Lp (q (H ) α G) is a VN(G)-bimodule with left and right actions induced by def

λs (z  λt ) = αs (z)  λst

def

and (z  λt )λs = z  λt s ,

z ∈ q (H ), s, t ∈ G. (2.85)

The following is stated in the particular case q = 1 without proof in [131, p. 544]. For the sake of completness, we give a short proof.

2.6 Carré Du Champ  and First Order Differential Calculus for Fourier. . .

59

Lemma 2.16 Suppose −1  q  1. Let G be a discrete group. For any x, y ∈ PG , we have ∂ψ,q (xy) = x∂ψ,q (y) + ∂ψ,q (x)y.

(2.86)

Proof For any s, t ∈ G, we have



(2.84) λs ∂ψ,q (λt ) + ∂ψ,q (λs )λt = λs sq (bψ (t))  λt + sq (bψ (s))  λs λt

(2.85) = αs sq (bψ (t))  λst + sq (bψ (s))  λst

= sq πs (bψ (t))  λst + sq (bψ (s))  λst (2.84)

= sq (πs (bψ (t)) + sq (bψ (s))  λst (2.33)

= sq (bψ (st))  λst = ∂ψ,q (λst ) = ∂ψ,q (λs λt ).   The following is a slight generalization of [131, Remark 1.3]. The proof is not difficult.14 Here E : Lp (q (H ) α G) → Lp (VN(G)) denotes the canonical conditional expectation.

14 On

the one hand, for any s, t ∈ G, we have  1 ψ(s −1 ) + ψ(t) − ψ(s −1 t) λs −1 t 2 *      + (2.34) 1  bψ (s −1 )2 + bψ (t)2 − bψ (s −1 t)2 λ −1 = s t 2 *    (2.33) 1  bψ (s −1 )2 + bψ (t)2 = 2  2  2 +   − bψ (s −1 ) − 2 bψ (s −1 ), πs −1 (bψ (t)) H − bψ (t) λs −1 t

(2.79)

(λs , λt ) =

  = − bψ (s −1 ), πs −1 (bψ (t)) H λs −1 t . On the other hand, we have ∗

+ 

∗  (2.84) *

E ∂ψ,q (λs ) ∂ψ,q (λt ) = E sq (bψ (s))  λs sq (bψ (t))  λt *



+ = E αs −1 sq (bψ (s))  λs −1 sq (bψ (t))  λt

(2.57)

*



+ = E sq (πs −1 (bψ (s)))  λs −1 sq (bψ (t))  λt *



+ = E sq (bψ (e) − bψ (s −1 ))  λs −1 sq (bψ (t))  λt

(2.33)

*



+ = −E sq (bψ (s −1 ))  λs −1 sq (bψ (t))  λt

60

2 Preliminaries

Proposition 2.11 Suppose −1  q  1. For any x, y ∈ PG , we have 

∗  (x, y) = E ∂ψ,q (x) ∂ψ,q (y) .

(2.87)

Suppose 1  p < ∞. The following equalities are in [132, pp. 930-931] for q = 1. For any x, y ∈ PG , we have   (x, y) = ∂ψ,q (x), ∂ψ,q (y) Lp (VN(G),L2(q (H ))c,p ) ,

(2.88)

    (x, x) 12  p = ∂ψ,q (x)Lp (VN(G),L2 (q (H ))c,p ) . L (VN(G))

(2.89)

and

We shall also need the following Riesz transform norm equivalence for markovian semigroup of Fourier multipliers from [131, Theorem A2, Remark 1.3]. Proposition 2.12 Let G be a discrete group and (Tt )t 0 a markovian semigroup of Fourier multipliers with symbol ψ of the negative generator A. Then for 2  p < ∞, we have the norm equivalence       1  (x ∗ , x ∗ ) 12  p A 2 (x) p (x, x) 12  p ≈ max , p L (VN(G)) L (VN(G)) L (VN(G)) (2.90) for any x ∈ PG .

2.7 Carré Du Champ  and First Order Differential Calculus for Schur Multipliers This section on markovian semigroups of Schur multipliers is the analogue of Sect. 2.6 on Fourier multipliers. We suppose here that we are given a markovian semigroup of Schur multipliers as in Proposition 2.4 and we fix the associated family (αi )i∈I from (2.36).

(2.56)

+ *( ) sq (bψ (s −1 ))αs −1 (sq (bψ (t)))  λs −1 t

(2.33)

+ *(



) sq (bψ (s −1 ))sq πs −1 bψ (t)  λs −1 t

= −E = −E

  = − bψ (s −1 ), πs −1 bψ (t) H λs −1 t .

(2.41)

2.7 Carré Du Champ  and First Order Differential Calculus for Schur. . .

61

For any x, y ∈ MI,fin , we define the element def

(x, y) =

 1 A(x ∗ )y + x ∗ A(y) − A(x ∗ y) . 2

(2.91)

of MI,fin . As in (2.78) in the case of a markovian semigroup of Fourier multipliers,  is called the carré du champ, associated with the semigroup (Tt )t 0 . For any i, j, k, l ∈ I , note that (eij , ekl ) =

 1   1 A(ej i )ekl + ej i A(ekl ) − A(ej i ekl ) = δi=k aj i + akl − aj l ej l . 2 2 (2.92)

The first part of the following result shows how to construct the carré du champ directly from the semigroup (Tt )t 0 . The proof is similar to the one of Lemma 2.15 to which we refer. Lemma 2.17 1. Suppose 1  p  ∞. For any x, y ∈ MI,fin , we have (x, y) = lim

t →0+

1

Tt (x ∗ y) − (Tt (x))∗ Tt (y) 2t

(2.93)

p

in SI . 2. For any x ∈ MI,fin , we have (x, x)  0.   (x, x) (x, y) 3. For any x, y ∈ MI,fin , the matrix is positive. (y, x) (y, y) 4. For any x, y ∈ MI,fin , we have 1

1

(x, y)S p  (x, x) 2 p (y, y) 2 p . S S I

I

(2.94)

I

This inequality even holds for 0 < p  ∞. We shall link, more profoundly in Sect. 3, the (square root of the) negative generator Ap , the carré du champ  and some noncommutative gradient from (2.95). Suppose 1  p < ∞ and −1  q  1. Note that the Bochner space Lp (q (H )⊗B( 2I )) is equipped with a canonical structure of SI∞ -bimodule whose def

def

p

operations are defined by (f ⊗x)y = f ⊗xy and y(f ⊗x) = f ⊗yx where x ∈ SI , y ∈ SI∞ and f ∈ Lp (q (H )). If q is the q-Gaussian functor of Sect. 2.2 associated with the real Hilbert space H stemming from Proposition 2.4, we can consider the

62

2 Preliminaries

linear map ∂α,q : MI,fin → q (H ) ⊗ MI,fin (resp. ∂α,1 : MI,fin → L0 () ⊗ MI,fin if q = 1) defined by def

∂α,q (eij ) = sq (αi − αj ) ⊗ eij ,

i, j ∈ I.

(2.95)

We have the following Leibniz rule. Lemma 2.18 Suppose −1  q  1. For any x, y ∈ MI,fin , we have ∂α,q (xy) = x∂α,q (y) + ∂α,q (x)y.

(2.96)

Proof On the one hand, for any i, j, k, l ∈ I , we have (2.95)

∂α,q (eij ekl ) = δj =k ∂α,q (eil ) = δj =k sq (αi − αl ) ⊗ eil . On the other hand, for any i, j, k, l ∈ I , we have eij ∂α,q (ekl ) + ∂α,q (eij )ekl



= eij sq (αk − αl ) ⊗ ekl + sq (αi − αj ) ⊗ eij ekl = sq (αk − αl ) ⊗ eij ekl + sq (αi − αj ) ⊗ eij ekl = sq (αk − αl ) ⊗ δj =k eil + sq (αi − αj ) ⊗ δj =k eil

= δj =k sq (αk − αl ) + sq (αi − αj ) ⊗ eil = δj =k sq (αi − αl ) ⊗ eil .  

The result follows by linearity.

Now, we describe a connection between the carré du champ and the map ∂α,q which is analogous to the equality of [205, Section 1.4] (see also [206]). For that, we introduce the canonical trace preserving normal faithful conditional expectation E : q (H )⊗B( 2I ) → B( 2I ). Proposition 2.13 Suppose −1  q  1 and 1  p < ∞. 1. For any x, y ∈ MI,fin , we have

∗   (x, y) = E ∂α,q (x) ∂α,q (y) = ∂α,q (x), ∂α,q (y) S p (L2 (q (H ))c,p ) .

(2.97)

I

2. For any x ∈ MI,fin , we have     (x, x) 12  p = ∂α,q (x) p 2 , S S (L (q (H ))c,p ) I

I

(2.98)

2.7 Carré Du Champ  and First Order Differential Calculus for Schur. . .

63

Proof 1. On the one hand, for any i, j, k, l ∈ I , we have   1 δi=k aj i + akl − aj l ej l 2 (  2 ) 1 = δi=k αj − αi 2H + αk − αl 2H − αj − αl H ej l 2 ( ) 1 = δi=k 2 αk 2 − 2 αi , αk  − 2 αj , αk  + 2 αi , αj  ej l 2

(2.92)

(eij , ekl ) =

= αk − αi , αk − αj H ej l . On the other hand, we have ∗

+ 

∗  (2.95) *

E ∂α,q (eij ) ∂α,q (ekl ) = E sq (αi − αj ) ⊗ eij sq (αk − αl ) ⊗ ekl + * = E sq (αi − αj )sq (αk − αl ) ⊗ ej i ekl = δi=k τ (sq (αi − αj )sq (αk − αl ))ej l = δi=k αi − αj , αk − αl H ej l The second equality is a consequence of (2.73). 2. If x ∈ MI,fin , we have   ∂α,q (x) p 2 S (L ( I

 1  =  ∂α,q (x), ∂α,q (x) S2 p (L2 (

(2.68) q (H ))c,p )

I

q (H ))c,p

   )

p

SI

 1 = (x, x) 2 S p .

(2.97)

I

 

Chapter 3

Riesz Transforms Associated to Semigroups of Markov Multipliers

Abstract In this chapter, we start by proving Khintchine inequalities for qGaussians in crossed products. As a consequence, we obtain boundedness of Lp -Riesz transforms associated with markovian semigroups of Fourier multipliers and defined over these crossed products with q-Gaussians. We also give dependence in p and independence of the group G and the markovian semigroup, of these Lp inequalities. Then we examine in detail the domains of the operators related to Kato’s square root problem for markovian semigroups of Fourier multipliers. We also show how to extend the carré du champ  associated with such a markovian semigroup to a closed form. Moreover, we solve the Kato square root problem for markovian semigroups of Schur multipliers. In particular, in its course, we again prove Khintchine inequalities for q-Gaussians. We also obtain the constants of the Kato square root problem independently of the markovian semigroup and discuss dependence in p. Finally, we also investigate Meyer’s problem for semigroups of Schur multipliers and study Lp -boundedness of directional Riesz transforms.

3.1 Khintchine Inequalities for q-Gaussians in Crossed Products In this section, we consider a markovian semigroup of Fourier multipliers on VN(G), where G is a discrete group satisfying Proposition 2.3 with cocycle (bψ , π, H ). Moreover, we have seen in (2.83) that by second quantization we have an action α : G → Aut(q (H )). The aim of the section is to prove Theorem 3.1 below which generalizes [131, Theorem 1.1]. In the following, the conditional expectation is E : q (H ) α G → q (H ) α G, x  λs → τq (H ) (x)λs . We let  def Gaussq,p, (Lp (VN(G))) = span sq (h)  x : h ∈ H, x ∈ Lp (VN(G))

(3.1)

where the closure is taken in Lp (q (H ) α G) (for the weak* topology if p = ∞ and −1  q < 1). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Arhancet, C. Kriegler, Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers, Lecture Notes in Mathematics 2304, https://doi.org/10.1007/978-3-030-99011-4_3

65

66

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

Lemma 3.1 Let 1 < p < ∞. Let G be a discrete group. If (ek )k∈K is an orthonormal basis of the Hilbert space H , then Gaussq,p, (Lp (VN(G))) equals the closure of the span of the sq (ek )  λs ’s with k ∈ K and s ∈ G. Proof We write temporarily G for the closure of the span of the sq (ek )  λs . First note that it is trivial that G ⊂ Gaussq,p (Lp (VN(G))). For the reverse inclusion, by linearity and density and (3.1), it suffices to show that sq (h)  λs belongs to G for any h ∈ H and any s ∈ G. Let ε > 0 be fixed. Then   there exists some finite subset F of K and αk ∈ C for k ∈ F such that h − k∈F αk ek H < ε. We have   1 k∈F αk sq (ek )  λs ∈ G and        αk sq (ek )  λs  sq (h)  λs −   k∈F

        = sq h − αk ek    k∈F

Lp (

Lp (q (H )α G)

q (H ))

         = sq h − αk ek  λs    k∈F

        ∼ αk ek  = sq h −   k∈F

Lp

< ε.

L2 (

q (H ))

We conclude Gaussq,p, (Lp (VN(G))) ⊂ G by closedness of G .

 

If 2  p < ∞, recall that we have a contractive injective inclusion Gaussq,p, (Lp (VN(G))) ⊂ Lp (q (H ) α G)

Lemma 2.8

⊂

p

Lcr (E)

(3.2)

and that if 1 < p < 2 we have an inclusion  span sq (h)  x : h ∈ H, x ∈ Lp ⊂ Lp (VN(G), L2 (q (H )c,p ) (2.74)

p

(3.3)

p

= Lc (E) ⊂ Lcr (E). p

p

In Theorem 3.1 below, we shall need several operations on Lc (E), Lr (E) and Lp (q (H ) α G). To this end, we have the following result.

1

Recall that α preserves the trace. If s ∈ G and k ∈ H , we have  

p sq (h)  λs pp = τ (sq (h)  λs )∗ (sq (h)  λs ) 2 L (q (H )α G)

p = τ (αs −1 (sq (h)))  λs ((sq (h)  λs )) 2

(2.57)







p p = τ (αs −1 (sq (h)2 )  λe ) 2 = τ π αs −1 (sq (h)2 ) 2 = τ αs −1 (sq (h)p ) p 

= τ sq (h)p = sq (h)Lp (q (H )) .

(2.56)

3.1 Khintchine Inequalities for q-Gaussians in Crossed Products

67

Lemma 3.2 Let −1  q  1 and 1 < p < ∞. Consider P : L2 (q (H )) → L2 (q (H )) the orthogonal projection onto Gaussq,2 (C) = span{sq (e) : e ∈ H }. Let G be a discrete group. Consider Qp = P  IdLp (VN(G)) initially defined on P,G . Then Qp induces well-defined contractions p

p

p

Qp : Lc (E) → Lc (E)

p

and Qp : Lr (E) → Lr (E).

(3.4)

Consequently, according to Proposition 2.10, Qp equally extends to a contraction p on Lcr (E). Proof By (2.71) for the column case and (2.76), the column part of (3.4) follows immediately. We turn to row part of (3.4). Note that the immediate analogue of (2.71) to the row case holds true, but not for (2.76), according to Remark 2.2. So we need to argue differently. We start by showing that for any h ∈ L2 (q (H )) and any s ∈ G we have



αs (P (h))∗ = P αs (h∗ ) .

(3.5)

Indeed, by anti-linearity and L2 (q (H )) continuity of both sides, it suffices to show (3.5) for h = w(en1 ⊗ · · · ⊗ enN ) the Wick word of en1 ⊗ · · · ⊗ enN , where (ek ) is an orthonormal basis of H . But then we have2 αs (P (h)∗ ) = αs (δN=1 sq (en1 )∗ ) = δN=1 αs (sq (en1 )) (2.83)

= δN=1 sq (πs (en1 )) = P (αs (h∗ )).

We deduce that     

   hk  λsk   P  IdLp (VN(G))   k

(3.6) p Lr (E)

 ∗   

    =  P  IdLp (VN(G)) hk  λsk     ∗       = P (hk )  λsk    k

k

p

Lc (E)

    (3.5) 

 ∗ =  P αs −1 (hk )  λs −1  k k   k

2

p

Lc (E)

    (2.57)   ∗ =  αs −1 (P (hk ) )  λs −1  k k   k

p

Lc (E)

p

Lc (E)

In the last equality, we use that w(en1 ⊗ · · · ⊗ enN ) = w(enN ⊗ · · · ⊗ en1 )∗ [177, p. 21].

68

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

    

   ∗ =  P  IdLp (VN(G)) αs −1 (hk )  λs −1  k k  

p

 & ∗ '    (2.57) 

 =  P  IdLp (VN(G)) hk  λsk   

p

k

k

 ∗   (3.4)       hk  λsk    k

Lc (E)

p Lc (E)

Lc (E)

      = hk  λsk    k

.

p Lr (E)

 

We have proved the row part of (3.4).

Now, the noncommutative Khintchine inequalities can be rewritten under the following form. Theorem 3.1 Consider −1  q  1 and 1 < p < ∞. Let G be a discrete group.   1. Suppose 1 < p < 2. For any element f = s,h fs,h sq (h)  λs of span sq (h) : h ∈ H }  PG , we have  f Gaussq,p, (Lp (VN(G))) ≈p

inf

f =g+h

 



1  1  ∗ 2   E(g ∗ g) 2  p , E(hh ) . L (VN(G)) Lp (3.7)

p

p

Here the infimum runs over all g ∈ Lc (E) and h ∈ Lr (E) such that f = g + h. Further one can restrict the infimum to all g ∈ Ran Qp |Lpc (E) , h ∈ Ran Qp |Lpr (E) , where Qp is the mapping from Lemma 3.2. Finally, in case that Lp (VN(G)) has CCAP and VN(G) has QWEP, the infimum can be taken over all g, h ∈ span sq (e)  : e ∈ H }  PG . 2. Suppose 2  p < ∞. For any element f = s,h fs,h sq (h)  λs of the space Gaussq,p, (Lp (VN(G))) with fs,h ∈ C, we have   

1  

1  max  E(f ∗ f ) 2 Lp ,  E(ff ∗ ) 2 Lp  f Gaussq,p, (Lp (VN(G))) 

  



1  1  √ p max  E(f ∗ f ) 2 Lp (VN(G)),  E(ff ∗ ) 2 Lp (VN(G)) . , -. / f Lp

cr (E)

(3.8)

3.1 Khintchine Inequalities for q-Gaussians in Crossed Products

69

Proof 2. Suppose 2 < p < ∞. We begin by proving the upper estimate of (3.8). Fix m  1. We have3 an isometric embedding Jm : H → 2m (H ) defined by def 1  Jm (h) = √ el ⊗ h. m m

(3.9)

l=1

p

Hence we can consider the operator q (Jm ) : Lp (q (H )) → Lp (q ( 2m (H ))) of second quantization (2.38) which is an isometric completely positive map. We define the maps πsm : G → B( 2m (H )), s → (el ⊗ h → el ⊗ πs (h)) and def

αsm = q (πsm ) : q ( 2m (H )) → q ( 2m (H )). For any h ∈ H , any s ∈ G, note that   m m 1  m 1  (3.9) πsm ◦ Jm (h) = πsm √ el ⊗ h = √ πs (el ⊗ h) m m l=1

l=1

1  (3.9) = √ el ⊗ πs (h) = Jm ◦ πs (h). m m

(3.10)

l=1

We deduce that (3.10)

αsm ◦ q (Jm ) = q (πsm ) ◦ q (Jm ) = q (πsm ◦ Jm ) = q (Jm ◦ πs ) = q (Jm ) ◦ αs .

By Lemma 2.6, we obtain a trace preserving unital normal injective ∗homomorphism q (Jm )  IdVN(G) : q (H ) α G → q ( 2m (H )) α m G. This def

p

p p map induces an isometric map ρ = q (Jm ) α Id L (VN(G)) : L (q (H ) α G) → p 2 L (q ( m (H )) α m G). For any finite sum f = s,h fs,h sq (h)  λs of P,G with fs,h ∈ C, we obtain

f Lp (q (H )α G)

       = f s (h)  λ s,h q s   s,h 

(3.11)

Lp (q (H )α G)

        fs,h sq (h)  λs  =  ρ   s,h

Lp (q ( 2m (H ))αm G)

3

If h ∈ H , we have   m   1    el ⊗ h √   m l=1

2m (H )

1 = √ m

 m      el     l=1

2m

hH = hH .

70

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

       m = f  (J )(s (h))  λ s,h q m q α s    s,h        (2.38)  =  fs,h sq,m (Jm (h))  λs     s,h

Lp (q ( 2m (H ))αm G)

Lp (q ( 2m (H ))αm G)

      m    1 (3.9)   =  fs,h sq,m √ el ⊗ h  λs   m  s,h  l=1    1   = √  fs,h sq,m (el ⊗ h)  λs   m

Lp (q ( 2m (H ))αm G)

.

(3.12)

Lp (q ( 2m (H ))αm G)

s,h,l

def  For any 1  l  m, consider the element fl = s,h fs,h sq,m (el ⊗ h)  λs and the conditional expectation E : Lp (q ( 2m (H )) α m G) → Lp (VN(G)). For any 1  l  m, note that   

E(fl ) = E fs,h sq,m (el ⊗ h)  λs = fs,h E sq,m (el ⊗ h)  λs s,h

=



s,h (2.40)

fs,h τ (sq,m (el ⊗ h))λs = 0.

s,h

We deduce that the random variables fl are mean-zero. Now, we prove in addition that the fl are independent over VN(G), so that we will be able to apply the noncommutative Rosenthal inequality (2.48) to them. The rest of the proof follows page 72.   Lemma 3.3 The fl defined previously are independent over VN(G). More precisely, the von Neumann algebras generated by fl are independent with respect to the von Neumann algebra VN(G). Proof Due to normality of E, it suffices to prove E(xy) = E(x)E(y)

(3.13)

for x (resp. y) belonging to a weak∗ dense subset of W∗ (fl ) (resp. of W∗ ((fk )k=l )). Moreover, by bilinearity in x, y of both sides of (3.13) and selfadjointness of sq,m (el ⊗ h), it suffices to take x = sq,m (el ⊗ h)n  λs for some n ∈ N and  y = Tt=1 sq,m (ekt ⊗ h)nt  λu for some T ∈ N, kt = l and nt ∈ N. Since

3.1 Khintchine Inequalities for q-Gaussians in Crossed Products

71

αsm = q (πsm ) is trace preserving, we have E(x)E(y) = τ (sq,m (el ⊗ h) )λs τ n

 T

 sq,m (ekt ⊗ h) λu nt

t =1

= τ (sq,m (el ⊗ h)n )τ

 T

 sq,m (ekt ⊗ h)nt λsu

t =1

   T = τ (sq,m (el ⊗ h)n )τ αsm sq,m (ekt ⊗ h)nt λsu t =1

= τ (sq,m (el ⊗ h) )τ n

 T

 sq,m (ekt ⊗ πs (h))

nt

λsu

t =1

and   T 

E(xy) = E sq,m (el ⊗ h)n  λs sq,m (ekt ⊗ h)nt  λu t =1 (2.56)

= E



sq,m (el ⊗ h)n αsm



sq,m (el ⊗ h)n αsm

 T

 λsu

sq,m (ekt ⊗ h)nt

t =1

& =τ

 T

sq,m (ekt ⊗ h)





' nt

λsu

t =1

  T

n nt = τ sq,m (el ⊗ h) sq,m (ekt ⊗ πs (h)) λsu . t =1

We shall now apply  the Wick formulae (2.39) and (2.40) to the pevious trace term. Note that if n + Tt=1 nt is odd, then according to the Wick formula (2.40) we  have E(xy) = 0. On the other hand, then either n or Tt=1 nt is odd, so according to the Wick formula (2.40), either E(x) = 0 or E(y) = 0. Thus, (3.13) follows in this case.  def Now suppose that n+ Tt=1 nt = 2k is even. Consider a 2-partition V ∈ P2 (2k). T If both n and t =1 nt are odd, then we must have some  (i, j ) ∈ V such that  one term fi , fj  2m (H ) in the Wick formula (2.39) equals el ⊗ h, ekt ⊗ πs (h) = 0, since kt = l. Thus, E(xy) =0, and since n is odd, also E(x) = 0, and therefore, (3.13) follows. If both n and Tt=1 nt are even, then in the Wick formula (2.39), we only need to consider those 2-partitions V without a mixed term, since otherwise we have as previously fi , fj  = 0. Such a V is clearly the disjoint union V =  V1 ∪ V2 of 2-partitions corresponding to n and to Tt=1 nt . Moreover, we have

72

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

for the number of crossings, c(V) = c(V1 ) + c(V2 ). With (f1 , f2 , . . . , fn+T ) = (el ⊗ h, el ⊗ h, . . . , el ⊗ h, ek1 ⊗ πs (h), . . . , ekT ⊗ πs (h)), we obtain , -. / n

& τ sq,m (el ⊗ h)

n

T

' sq,m (ekt ⊗ πs (h))

nt

t=1



(2.39)

=

V∈P2 (2k)

=



q c(V)

=

q c(V1 )+c(V2 )

V1 ,V2



fi , fj  2m (H )

(i,j )∈V

(i1 ,j1 )∈V1

q

c(V1 )

V1



fi1 , fj1  2m (H )

fi1 , fj1  2m (H )

(i1 ,j1 )∈V1

 

q

c(V2 )

V2

' &T

= τ sq,m (el ⊗ h)n τ sq,m (ekt ⊗ πs (h))nt .

(2.39)

fi2 , fj2  2m (H )

(i2 ,j2 )∈V2



fi2 , fj2  2m (H )

(i2 ,j2 )∈V2



t=1

 

Thus, also in this case (3.13) follows.

Proof (End of the Proof of Theorem 3.1) Now we are able to apply the noncommutative Rosenthal inequality (2.48) which yields f Lp (q (H )α G)

 m   1    = √  fl   m

(3.12)

l=1

Lp (

(3.14) 2 q ( m (H ))α m G)

1   m (2.48) 1 p p fl Lp ( ( 2 (H )) m G)  √ p q α m m l=1   m  m 1  1    2 2 √ √       E(fl∗ fl )  + p E(fl fl∗ )  + p     l=1

l=1

Lp (VN(G))

(3.15)  .

Lp (VN(G))

For any integer 1  l  m, note that E(fl∗ fl ) = E

  s,h

(2.57)

=

 s,h,t,k

(2.56)

=

 s,h,t,k

fs,h sq,m (el ⊗ h)  λs

∗  

 ft,k sq,m (el ⊗ k)  λt

t,k

(



) fs,h ft,k E αsm−1 (sq,m (el ⊗ h))  λs −1 sq,m (el ⊗ k)  λt ( ) fs,h ft,k E αsm−1 (sq,m (el ⊗ h))αsm−1 (sq,m (el ⊗ k))  λs −1 t

3.1 Khintchine Inequalities for q-Gaussians in Crossed Products



=

73

fs,h ft,k τ (αsm−1 (sq,m (el ⊗ h)(sq,m (el ⊗ k)))λs −1 t

s,h,t,k



=

fs,h ft,k τ sq,m (el ⊗ h)sq,m (el ⊗ k) λs −1 t

s,h,t,k



(2.41)

=

fs,h ft,k h, kH λs −1 t .

i,j,h,s

and ∗

E(f f ) = E

  s,h

(2.57)

=

 s,h,t,k

(2.56)

=



fs,h sq (h)  λs

∗  

 ft,k sq (k)  λt

t,k

(



) fs,h ft,k E αs −1 (sq (h))  λs −1 sq (k)  λt ) (

fs,h ft,k E αs −1 (sq (h))αs −1 (sq (k))  λs −1 t

s,h,t,k



=

fs,h ft,k τ αs −1 (sq (h)sq (k)) λs −1 t

s,h,t,k



=

(2.41)  fs,h ft,k τ sq (h)sq (k) λs −1 t = fs,h ft,k h, kH λs −1 t

s,h,t,k

i,j,h,s

and similarly for the row terms. We conclude that E(fl∗ fl ) = E(f ∗ f )

and E(fl fl∗ ) = E(ff ∗ ).

(3.16)

Moreover, for 1  l  m, using the isometric map ψl : H → 2m (H ), h → el ⊗ h, we can introduce the second quantization operator q (ψl ) : q (H ) → q ( 2m (H )). For any h ∈ H , any s ∈ G, note that πsm ◦ ψl (h) = πsm (el ⊗ h) = el ⊗ πs (h) = ψl ◦ πs (h).

(3.17)

We deduce that (3.17)

αsm ◦ q (ψl ) = q (πsm ) ◦ q (ψl ) = q (πsm ◦ ψl ) = q (ψl ◦ πs ) = q (ψl ) ◦ αs .

74

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

By Lemma 2.6, we obtain a trace preserving unital normal injective ∗homomorphism q (ψl )  IdVN(G) : q (H ) α G → q ( 2m (H )) α m G. This p map induces an isometric map q (ψl ) α IdLp (VN(G)) : Lp (q (H ) α G) → p 2 L (q ( m (H )) α m G). We have fl Lp (q ( 2m (H ))αm G)

     = fs,h sq,m (el ⊗ h)  λs  

Lp (q ( 2m (H ))αm G)

s,h

(3.18)

    (2.38)   =  fs,h q (ψl )(sq,m (h))  λs        = f s (h)  λ s,h q s 

Lp (q (H )G)

s,h

(3.19)

Lp (q ( 2m (H ))αm G)

s,h

= f Lp (q (H )α G) .

We infer that f Lp (q (H )α G)

(3.14)(3.18)(3.16)



 m 1  2 √     + p E(f ∗ f )    l=1

Lp (VN(G))

 1 1 = √ pm p f Lp (q (H )α G) m +

  1 m p 1 p f Lp (q (H )α G) √ p m l=1  m 1  2 √     + p E(ff ∗ )    l=1

Lp (VN(G))

1  1  √  √ 

pm E(f ∗ f ) 2 Lp (VN(G)) + pm E(ff ∗ ) 2 Lp (VN(G)) 1

= pm p

− 12

f Lp (q (H )α G) +





1  1  √  √ 

p  E(f ∗ f ) 2 Lp + p  E(ff ∗ ) 2 Lp .

Since p > 2, passing to the limit when m → ∞ and noting that Rosenthal’s inequality comes with an absolute constant not depending on the von Neumann algebras under consideration [125, p. 4303], we finally obtain f Lp (q (H )α G) 

  

1  1  √  p  E(f ∗ f ) 2 Lp (VN(G)) +  E(ff ∗ ) 2 Lp (VN(G)) .

Using the equivalence 12 ≈ ∞ 2 , we obtain the upper estimate of (3.8).

3.1 Khintchine Inequalities for q-Gaussians in Crossed Products

75

The lower estimate of (3.8) holds with constant 1 from the contractivity of the p conditional expectation E on L 2 (q (H ) α G): 





1  1  max  E(f ∗ f ) 2 Lp (VN(G)),  E(ff ∗ ) 2 Lp (VN(G))   1 = max E(f ∗ f ) 2 p

L 2 (VN(G))



 1  max f ∗ f  2 p

 1 , E(ff ∗ ) 2 p

 

L 2 (VN(G))

1  , ff ∗  2 p

L 2 (q (H )α G)

L 2 (q (H )α G)

 = f Lp (q (H )α G) .

1. Now, let us consider the case 1 < p < 2. We will proceed by duality as follows. Consider the Gaussian projection Qp from Lemma 3.2 which is a p contraction on Lcr (E). Note that τ (f ∗ g) = τ (Qp (f )∗ g) = τ (f ∗ Qp∗ (g)) for ∗ any g ∈ Lp (q (H ) α G) and the fixed f from the statement of the theorem. p Indeed, this can be seen from selfadjointness of P and (3.5). Note also that Lcr (E) p∗ and Lcr (E) are dual spaces with respect to each other according to Lemma 2.14, with duality bracket f, g = τ (f ∗ g), i.e. the same as the duality bracket between ∗ Lp (q (H ) α G) and Lp (q (H ) α G). p∗ p∗ Using this in the first two equalities, the contractivity Qp∗ : Lcr (E) → Lcr (E) from Lemma 3.2 in the third equality, and the upper estimate of (3.8) together with the density result of Lemma 2.13 in the last inequality, we obtain for any f ∈ Gaussq,2 (C)  PG f Lpcr (E) = =

g

sup

1 p∗ Lcr (E)

sup g p∗ 1 Lcr (E) g∈Ran Qp∗

  τ (f ∗ g) =

g

  τ (f ∗ g)

 f Lp (q (H )α G)

sup

1 p∗ Lcr (E)

  τ (f ∗ Qp∗ (g))

(3.8)

sup g p∗ 1 Lcr (E) g∈Ran Qp∗

gLp∗ (q (H )α G) p f Lp (q (H )α G) .

p

p

Note that in the definition of the Lcr (E)-norm, the infimum runs over all g ∈ Lc (E) p and h ∈ Lr (E) such that f = g + h. However, we can write f = Qp (f ) = (3.4) (3.4)     Qp (g) + Qp (h) and Qp (g)Lp (E)  gLpc (E) and Qp (h)Lp (E)  hLpr (E) . p

c

r

Thus, in the definition of the Lcr (E)-norm of f , we can restrict to g, h ∈ Ran Qp . Our  next goal is to restrict in the lower estimate of (3.7) to elements g, h of span sq (e) : e ∈ H  PG with f = g + h, in the case where f belongs to span sq (e) : e ∈ H  PG , under the supplementary assumptions that Lp (VN(G)) has CCAP and that VN(G) has QWEP. We consider the approximating net (Mϕj )

76

3 Riesz Transforms Associated to Semigroups of Markov Multipliers p

of Fourier multipliers. Moreover, let f = g + h be a decomposition with g ∈ Lc (E) p and h ∈ Lr (E) such that gLpc (E) + hLpr (E)  2 f Lpcr (E) . In light of this, we can already assume that g, h belong to Ran Qp . Then we newly decompose for each j

f = (Idq (H )  Mϕj )(f ) + f − Idq (H )  Mϕj (f )

= (Idq (H )  Mϕj )(g) + f − (Idq (H )  Mϕj )(f ) + (Idq (H )  Mϕj )(h). -. / , -. / , p

p

∈Lc (E)

∈Lr (E)

Note that (Idq (H )  Mϕj )(g) and (Idq (H )  Mϕj )(h) belong to the subspace  span sq (e) : e ∈ H  PG since ϕj is of finite support. Moreover, also f − Idq (H )  Mϕj (f ) belongs to that space.  Now, we will control the norms in this new decomposition. Recall that f = s,e fs,e sq (e)  λs with finite sums. Since ϕj (s) converges pointwise to 1, there is j such that      f − Id (H )  Mϕ (f ) p  |fs,e | sq (e)  λs − ϕj (s)sq (e)  λs Lp (E) q j L (E) c





c

s,e

  |fs,e | |1 − ϕj (s)| sq (e)  λs Lp (E)  ε  f Lpcr (E) c

s,e

 f Lp (q (H )α G) . We turn to the other two parts. With (2.77), we obtain     (Id (H ) ⊗ Mϕ )(g) p + (Id (H ) ⊗ Mϕ )(h) p q j q j L (E) L (E) c

p g

p Lc (E)

+ h

p Lr (E)

r

 f 

p Lcr (E)

p f Lp (q (H )α G) . Together we have shown that also the infimum in (3.7) restricted to elements of the space span sq (e) : e ∈ H  PG is also controlled by f Gaussq,p, (Lp (VN(G))). Now, we will prove the remaining estimate, that is, f Gaussq,p, (Lp (VN(G))) is controlled by the second expression in (3.7). Since 1 < p < 2, the function R+ → p R+ , t → t 2 is operator concave by [37, p. 112]. Using [100, Corollary 2.2] applied with the trace preserving positive map E, we can write  1  2 f Lp (q (H )α G) = |f |2  p

L 2 (q (H )α G)

 1  2  E(|f |2 ) p

L 2 (q (H )α G)

= f Lpc (E)

and similarly for the row term. Thus, passing to the infimum over all decompositions f = g+h, we obtain f Lp (q (H )α G)  f Lpcr (E) , which can be majorised in turn by the infimum of gLpc (E) + hLpr (E) , where f = g + h and g, h ∈ span{sq (h) : h ∈ H }  PG . Hence, we have the last equivalence in the part 1 of the theorem.

3.1 Khintchine Inequalities for q-Gaussians in Crossed Products

77

The case p = 2 is obvious since we have isometrically L2cr (E) = L2 (VN(G), L2 (q (H ))rad,2 ) = L2 (q (H )) ⊗2 L2 (VN(G))  

by Lemma 2.10 and [128, Remark 2.3 (1)].

The remainder of the section is devoted to extend Theorem 3.1 to the case of f being a generic element of Gaussq,p, (Lp (VN(G))). First we have the following lemma. Lemma 3.4 Let −1  q  1 and 1 < p < ∞. Consider again Qp = P  IdLp (VN(G)) the extension of the Gaussian projection from Lemma 3.2. Then Qp extends to a bounded operator Qp : Lp (q (H ) α G) → Lp (q (H ) α G).

(3.20)

Proof First note that the case G = {e} is contained in [118, Theorem 3.5], putting there d = 1. Note that the closed space spanned by {sq (h) : h ∈ H } coincides in this case with G1p,q there. To see that the projection considered in this source is Qp , we refer to [118, Proof of Theorem 3.1]. We turn to the case of  general discrete group G. Since P is selfadjoint, (Qp )∗ = ∗ Qp . We obtain for f = s∈F fs  λs with F ⊂ G finite and fs ∈ Lp (q (H ))   Qp (f )

  Qp (f ) p Lcr (E)    = sup  Qp (f ), g : gLp∗ (E)  1 cr      = sup  f, Qp∗ (g)Lp (E),Lp∗ (E)  : g cr cr      = sup  f, Qp∗ (g)Lp (q (H )α G),Lp∗  : g     f Lp (q (H )α G) sup Qp∗ (g)Lp∗ (q (H )α G) : g Theorem 3.1

Gaussq,p, (Lp (VN(G)))

Theorem 3.1





   f Lp (q (H )α G) sup Qp∗ (g)Lp∗ (E) : gLp∗ (E)  1 cr

cr

 f Lp (q (H )α G) , p∗

where in the last step we used that Qp∗ is bounded on Lcr (E) according to Lemma 3.2.   Now we obtain the following extension of Theorem 3.1.

78

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

Proposition 3.1 Let −1  q  1 and 1 < p < ∞. Assume that Lp (VN(G)) has CCAP and that q (H ) α G has QWEP. Let f be an element of Ran Qp = Gaussq,p, (Lp (VN(G))).4 1. Suppose 1 < p < 2. Then we have  f Gaussq,p, (Lp (VN(G))) ≈p

inf

f =g+h

 



1  1  ∗ 2   E(g ∗ g) 2  p , E(hh ) . L (VN(G)) Lp (3.21)

p

p

Here the infimum runs over all g ∈ Lc (E) and h ∈ Lr (E) such that f = g + h. Further one can restrict the infimum to all g, h ∈ Ran Qp . 2. Suppose 2  p < ∞. Then we have   

1  

1  max  E(f ∗ f ) 2 Lp ,  E(ff ∗ ) 2 Lp  f Gaussq,p, (Lp (VN(G))) 





(3.22)





1  1  √ p max  E(f ∗ f ) 2 Lp (VN(G)),  E(ff ∗ ) 2 Lp (VN(G)) .

Proof We consider the approximating net (Mϕj ) of finitely supported Fourier multipliers guaranteed by the CCAP assumption. Observe that (Idq (H )  Mϕj ) approximates the identity on P,G . Moreover, according to Proposition 2.8, (Idq (H )  Mϕj )j is a bounded net in B(Lp (q (H ) α G)). We conclude by density of P,G in Lp (q (H ) α G) that (Idq (H )  Mϕj )j converges in the point norm topology of Lp (q (H ) α G) to the identity. Moreover, replacing in this argument Proposition 2.8 by Lemma 2.12, the same argument yields that p (Idq (H )  Mϕj )j converges to the identity in the point norm topology of Lc (E) p and of Lr (E). Then again by the same argument, we infer that f Lpcr (E) =   limj (Idq (H )  Mϕj )(f )Lp (E) . Note that for fixed j , (Idq (H )  Mϕj )(f ) = cr (Idq (H )  Mϕj )(Qp f ) = Qp (Idq (H )  Mϕj )(f ) belongs to span{sq (e) : e ∈ H }  PG . Thus, Theorem 3.1 applies to f replaced by Idq (H )  Mϕj (f ), and therefore,   f Gaussq,p, (Lp (VN(G))) = lim (Idq (H )  Mϕj )(f )Gauss

q,p, (L

j

Theorem 3.1

∼ =

p (VN(G)))

  lim (Idq (H )  Mϕj )(f )Lp (E) = f Lpcr (E) . j

cr

This equality follows from the fact that P,G is dense in Lp (q (H ) α G) according to Lemma 2.3, so Qp (P,G ) ⊂ Gaussq,p, (Lp (VN(G))) is dense in Ran Qp . The other inclusion Gaussq,p, (Lp (VN(G))) ⊂ Ran Qp follows from the fact that the span of the sq (h)  x with x ∈ Lp (VN(G)) is dense in Gaussq,p, (Lp (VN(G))) and obviously lies in Ran Qp which in turn is closed.

4

3.2 Lp -Kato’s Square Root Problem for Semigroups of Fourier Multipliers

79

Finally, the fact that one can restrict the infimum to all g, h ∈ Ran Qp can be proved in the same way as that in Theorem 3.1.  

3.2 Lp -Kato’s Square Root Problem for Semigroups of Fourier Multipliers Throughout this section, we consider a discrete group G, and fix a markovian semigroup of Fourier multipliers (Tt )t 0 acting on VN(G) with negative generator A and representing objects bψ : G → H, π : G → O(H ), α : G → Aut(q (H )), see Proposition 2.3 and (2.83). We also have the noncommutative gradient ∂ψ,q : PG ⊂ Lp (VN(G)) → Lp (q (H )α G) from (2.84). The aim of this section is to compare 1

A 2 (x) and ∂ψ,q (x) in the Lp -norm. We shall also extend ∂ψ,q to a closed operator and identify the domain of its closure. To this end, we need some facts from the general theory of strongly continuous semigroups. The following is a straightforward extension of [25, Lemma 4.2]. Proposition 3.2 Let (Tt )t 0 be a strongly continuous bounded semigroup acting on a Banach space X with (negative) generator A. We have   

1   IdX + A 2 (x) ≈ xX + A 21 (x) , X X

x ∈ X.

(3.23) 1

Proof It is well-known [25, Lemma 2.3] that the function f1 : t → (1 + t) 2 (1 + 1 t 2 )−1 is the Laplace transform L(μ1 ) of some bounded measure μ1 . By [98, Proposition 3.3.2] (note that f1 ∈ H∞ (π−ε ) for any ε ∈ (0, π2 ) and f1 has finite limits limz∈π −ε , z→0 f1 (z) and limz∈π −ε , |z|→∞ f1 (z)) (see also [151, Lemma 2.12]), we have 

∞

( ) 1 1 −1 Tt dμ1 (t) = L(μ1 )(A) = f1 (A) = (IdX + A) 2 IdX + A 2 .

0 1

1

Hence for any x ∈ dom(IdX + A 2 ) = dom A 2 , we have 1 2



(IdX + A) x = 0

∞

1 Tt x + A 2 x dμ1 (t).

(3.24)

80

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

∞ By Haase [98, Remark 3.3.3], we know that 0 Tt dμ1 (t) is a bounded operator on X of norm  μ1 . Using the triangular inequality in the last inequality, we conclude that  ∞      

1 1  (IdX + A) 21 x  (3.24)    2 x dμ1 (t) x + A = T t    μ1  x + A 2 x X X 0

X

(

 1  )  μ1  xX + A 2 x X .

1

1

It is easy to see [25, Lemma 2.3] that the functions f2 : t → (1 + t 2 )(1 + t)− 2 and 1 f3 : t → (1 + t)− 2 are the Laplace transforms L(μ2 ) and L(μ3 ) of some bounded measures μ2 and μ3 . So we have 

∞

) ( 1 1 Tt dμ2 (t) = IdX + A 2 (Id + A)− 2

0



∞

and

Tt dμ3 (t) = (IdX + A)− 2 . 1

0

Following the same argument as previously, we obtain       x + A 21 x   μ2  (IdX + A) 12 x  and xX  μ3  (IdX + A) 21 x  . X X X (3.25) Note that  1      A 2 x  =  − x + x + A 12 x   xX + x + A 12 x  . X X X

(3.26)

We conclude that (3.25)   1  (3.26)  1  1  xX + A 2 x X  2xX + x + A 2 x X  (IdX + A) 2 x X .

  We start to observe that the estimates in (3.27) below come with a constant independent of the group G and the cocycle (α, H ). This is essentially the second proof of the appendix [131, pp. 574-575]. Note that we are unfortunately unable5 to check the original proof given in [131, p. 544]. Here, L∞ () = 1 (H ) is the Gaussian space from Sect. 2.2. More precisely, with the notations of [131] we are unable to check that “H  IdG extends to a bounded operator on Lp ”. A part of the very concise explanation given in [131, p. 544] is “H is G-equivariant”. But it seems to be strange. Indeed we have an action α : G → Aut(L∞ (Rnbohr )),   f → x → αg (f )(x) = f (πg (x)) for some map πg : Rnbohr → Rnbohr where g ∈ G and an induced action α from G on L∞ (Rnbohr × Rn , ν × γ ). Now, note that

5

 (H (αg f ))(x, y) =

 p. v.

R

βt αg f

    dt dt (x, y) = p. v. βt (f ◦ πg ) (x, y) t t R

3.2 Lp -Kato’s Square Root Problem for Semigroups of Fourier Multipliers

81

Lemma 3.5 Suppose 1 < p < ∞. For any x ∈ dom PG , we have  21    1 Ap (x) p  ∂ψ,1,p (x) p ∞ L L (L ()α G) ∗ K max(p, p )  3 1  K max(p, p∗ ) 2 Ap2 (x) p L

(3.27)

with an absolute constant K not depending on G nor the cocycle (α, H ). ∗ : P p We define the densely defined unbounded operator ∂ψ,q ,G ⊂ L (q (H ) α p G) → L (q (H )) by

  ∗ ∂ψ,q (f  λs ) = sq (bψ (s)), f Lp∗ (q (H )),Lp (q (H )) λs ,

s ∈ G, f ∈ Lp (q (H )). (3.28)

The following lemma is left to the reader. ∗ are formal adjoints. Lemma 3.6 The operators ∂ψ,q and ∂ψ,q

The next proposition extends (3.27) to the case of q-Gaussians. Proposition 3.3 Suppose 1 < p < ∞ and −1  q  1. For any x ∈ PG , we have 1 K max(p, p∗ )

3 2

 21    Ap (x) p  ∂ψ,q,p (x) p L L (q (H )α G)  1   K max(p, p∗ )2 Ap2 (x)Lp

(3.29)

with an absolute constant K not depending on G nor the cocycle (α, bψ , H ). Proof Start with the case 2  p < ∞. The case q = 1 is covered  by Lemma 3.5. Consider now the case −1  q < 1. Pick some element x = s xs λs of PG . We recall from Sect. 3.1 that we have a conditional expectation E : q (H ) α G → q (H ) α G, x  λs → τq (H ) (x)λs . In the following calculations, we consider an orthonormal basis (ek ) of H . Therefore, using the orthonormal systems (W(ek ))

 = p. v.

R

f (πg (x + ty))

dt t

and     dt dt αg (H (f ))(x, y) = αg p. v. βt f (x, y) = p. v. f (πg (x) + ty) t t R R which could be different if π is not trivial.

82

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

and (sq (ek )) in L2 () and L2 (q (H )) in the third equality, we obtain        = x

e , b (s)W(e )  λ s k ψ k s  p   p s,k Lc (E) Lc (E)      (2.76)   =  xs ek , bψ (s)λs ⊗ W(ek )   s,k  p 2

      xs W(bψ (s))  λs    s 

L (VN(G),L ()c,p )

       = xs ek , bψ (s)λs ⊗ sq (ek )   s,k 

Lp (VN(G),L2 (q (H ))c,p )

    (2.76)   =  xs sq (bψ (s))  λs    s

. p

Lc (E)

We claim that this equality holds also for the row space instead of the column space. Note that according to Remark 2.2, we have to argue differently. We have with def

xs,k = xs ek , bψ (s)H ,  2     xs sq (bψ (s))  λs     p s

Lr (E)

      

∗   = xs,k xt,l E (sq (ek )  λs )(sq (el )  λt )  p s,k,t,l t,l 2

      

(2.57)  =  xs,k xt,l E (sq (ek )  λs )(αt −1 (sq (el ))  λt −1 )   s,k,t,l p 2

      

(2.56)  =  xs,k xt,l E sq (ek )αst −1 (sq (el ))  λst −1   p s,k,t,l      

 = xs,k xt,l τq sq (ek )sq (πst −1 (el )) λst −1   s,k,t,l p        (2.41)   =  xs,k xt,l ek , πst −1 (el ) H λst −1   . s,k,t,l p

2

2

2

The point is that this last quantity does not depend on q. We infer that      xs W(bψ (s))  λs    s

p

Lcr (E)

     = xs sq (bψ (s))  λs   s

. p

Lcr (E)

(3.30)

3.2 Lp -Kato’s Square Root Problem for Semigroups of Fourier Multipliers

83

Then we have (3.27)    12  Ap (x) p  Kp∂ψ,1,p (x)Lp (L∞ () G) L (VN(G)) α               = Kp  xs W(bψ (s))  λs  = Kp  xs ek , bψ (s)W(ek )  λs     s   p

s,k

    (3.8)  1    K p · p 2  x

e , b (s)W(e )  λ s k ψ k s   s,k  p Lcr (E)         (3.30)  3   e = K p2  x , b (s) s (e )  λ s k ψ q k s   s,k  p Lcr (E)     (3.8)      32  e  Kp  xs k , bψ (s) sq (ek )  λs    s,k  p

p

L (q (H )α G)

 3 = K  p 2 ∂ψ,q,p (x)Lp (q (H )α G) . We pass to the converse inequality. We have (3.27)  12  Ap (x) p  L (VN(G))

 1  ∂ψ,1,p (x) p ∞ 3 L (L ()α G) Kp 2             1  1    = x W(b (s))  λ = x

e , b (s)W(e )  λ s ψ s s k ψ k s 3  3   2   Kp 2  s Kp s,k p p      (3.8)  1    xs ek , bψ (s)W(ek )  λs  3     p K p 2  s,k Lcr (E)       1  (3.30)  = xs ek , bψ (s)sq (ek )  λs  3    p K  p 2  s,k Lcr (E)     (3.8)   1   xs ek , bψ (s)sq (ek )  λs  3 1    p K  p 2 · p 2  s,k L (q (H )α G)

=

 1  ∂ψ,q,p (x) p . L (q (H )α G) K  p2

Altogether we have shown (3.29) in the case 2  p < ∞.

84

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

We turn to the case 1 < p < 2. Note that (3.30) still holds in this case. Indeed, for elements f ∈ Gaussq,p, (Lp (VN(G))), the norm f Lpcr (E) = inf{gLpc (E) + hLpr (E) : f = g + h} remains unchanged if g, h are choosen in Ran Qp |Lpc (E) and p p Ran Qp |Lpr (E) , see Theorem 3.1. But for those elements g, h, the Lc (E) and Lr (E) norms remain unchanged upon replacing classical Gaussian variables W(ek ) by qGaussian variables sq (ek ), see the beginning of the proof. Then the proof in the case 1 < p < 2 can be executed as in the case 2  p < ∞, noting that the additional 1 factor p∗ 2 will appear at another step of the estimate, in accordance with the two parts of Theorem 3.1.   Proposition 3.4 Let G be a discrete group. Suppose 1 < p < ∞ and −1  q  1. 1. The operator ∂ψ,q : PG ⊂ Lp (VN(G)) → Lp (q (H ) α G) is closable as a densely defined operator on Lp (VN(G)) into Lp (q (H ) α G). We denote by ∂ψ,q,p its closure. So PG is a core of ∂ψ,q,p . 1

2. PG is a core of Ap2 . 1

1

3. We have dom ∂ψ,q,p = dom Ap2 . Moreover, for any x ∈ dom Ap2 , we have    12  Ap (x) p ≈ ∂ψ,q,p (x)Lp (q (H )α G) . L (VN(G)) p

(3.31)

1

Finally, for any x ∈ dom Ap2 , there exists a sequence (xn ) of elements of PG such 1

1

that xn → x, Ap2 (xn ) → Ap2 (x) and ∂ψ,q,p (xn ) → ∂ψ,q,p (x). 4. If x ∈ dom ∂ψ,q,p , we have x ∗ ∈ dom ∂ψ,q,p and (∂ψ,q,p (x))∗ = −∂ψ,q,p (x ∗ ).

(3.32)

5. Let Mψ : Lp (VN(G)) → Lp (VN(G)) be a finitely supported bounded Fourier multiplier such that the map Id  Mψ : Lp (q (H ) α G) → Lp (q (H ) α G) is a well-defined bounded operator. For any x ∈ dom ∂ψ,q,p , the element Mψ (x) belongs to dom ∂ψ,q,p and we have ∂ψ,q,p Mψ (x) = (Id  Mψ )∂ψ,q,p (x).

(3.33)

Proof 1. This is a consequence of [136, Theorem 5.28 p. 168] together with Lemma 3.6. 1

2. Since dom Ap is a core of dom Ap2 , this is a consequence of a classical argument [180, p.29]. 1

1

3. Let x ∈ dom Ap2 . By the point 3, PG is dense in dom Ap2 equipped with the graph norm. Hence we can find a sequence (xn ) of PG such that xn → x and

3.2 Lp -Kato’s Square Root Problem for Semigroups of Fourier Multipliers 1

85

1

Ap2 (xn ) → Ap2 (x). For any integers n, m, we obtain   xn − xm Lp (VN(G)) + ∂ψ,q,p (xn ) − ∂ψ,q,p (xm )Lp ( (3.29) p

q (H )α G)

1  1  xn − xm Lp (VN(G)) + Ap2 (xn ) − Ap2 (xm )Lp (VN(G)).

which shows that (xn ) is a Cauchy sequence in dom ∂ψ,q,p . By the closedness of ∂ψ,q,p , we infer that this sequence converges to some x  ∈ dom ∂ψ,q,p equipped with the graph norm. Since dom ∂ψ,q,p is continuously embedded into Lp (VN(G)), we have xn → x  in Lp (VN(G)), and therefore x = x  since xn → x. It follows that x ∈ dom ∂ψ,q,p . This proves the inclusion 1

dom Ap2 ⊂ dom ∂ψ,q,p . Moreover, for any integer n, we have (3.29)  1    ∂ψ,q,p (xn ) p p Ap2 (xn )Lp (VN(G)) . L (q (H )α G) 1

Since xn → x in dom ∂ψ,q,p and in dom Ap2 both equipped with the graph norm, we conclude that   1   ∂ψ,q,p (x) p  Ap2 (x)Lp (VN(G)). L (q (H )α G) p The proof of the reverse inclusion and of the reverse estimate are similar. Indeed, by part 1, PG is a dense subspace of dom ∂ψ,q,p equipped with the graph norm. 4. Recall that bψ (e) = 0. For any s ∈ G, we have

∂ψ,q,p (λs )

∗

(2.84)

=

∗ (2.57)

sq (bψ (s))  λs = αs −1 (sq (bψ (s)))  λs −1

(2.33)

= sq (bψ (e) − bψ (s −1 ))  λs −1 (2.84)

= −sq (bψ (s −1 ))  λs −1 = −∂ψ,q,p (λs −1 ) = −∂ψ,q,p (λ∗s ). Let x ∈ dom ∂ψ,q,p . By the point 1, PG is core of ∂ψ,q,p . Hence there exists a sequence (xn ) of PG such that xn → x and ∂ψ,q,p (xn ) → ∂ψ,q,p (x). We have xn∗ → x ∗ and by the first part of the proof ∂ψ,q,p (xn∗ ) = −(∂ψ,q,p (xn ))∗ → −(∂ψ,q,p (x))∗ . By (2.7), we conclude that x ∗ ∈ dom ∂ψ,q,p and that ∂ψ,q,p (x ∗ ) = −(∂ψ,q,p (x))∗ . 5. If s ∈ G, we have (2.84)

(Id  Mψ )∂ψ,q,p (λs ) = (Id  Mψ )(sq (bψ (s))  λs ) = ψs sq (bψ (s))  λs (2.84)

= ψs ∂ψ,q,p (λs ) = ∂ψ,q,p Mψ (λs ).

86

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

By linearity, (3.33) is true for elements of PG . Let x ∈ dom ∂ψ,q,p . There exists a sequence (xn ) of PG such that xn → x and ∂ψ,q,p (xn ) → ∂ψ,q,p (x). We have Mψ (xn ) → Mψ (x) and ∂ψ,q,p Mψ (xn ) = (Id  Mψ )∂ψ,q,p (xn ) −−−−→ (Id  Mψ )∂ψ,q,p (x). n→+∞

By (2.1), we deduce (3.33).   For the property AP in the next proposition, we refer to the preliminary Sect. 2.4. Proposition 3.5 Assume −1  q < 1. Let G be a discrete group with AP. The operator ∂ψ,q : PG ⊂ VN(G) → q (H ) α G is weak* closable.6 We denote by ∂ψ,q,∞ its weak* closure. Proof Suppose  that (xi ) is a net of PG which converges to 0 for the weak* topology with xi = s∈G xi,s λs such that the net (∂ψ,q (xi )) converges for the weak* topology to some y belonging to q (H ) α G. Let (Mϕj ) be the net of Fourier multipliers approximating the identity from Proposition 2.9. For any j , we have     = ∂ψ,q xi,s λs ∂ψ,q (Mϕj xi ) = ∂ψ,q Mϕj =

 ϕj (s)xi,s λs

s∈supp ϕj

s∈G





ϕj (s)xi,s ∂ψ,q (λs ) = (Id  Mϕj )∂ψ,q



s∈supp ϕj

 xi,s λs

s∈G

→ (Id  Mϕj )(y). = (Id  Mϕj )∂ψ,q (xi ) − i

On the other hand, for all s ∈ supp ϕj we have xi,s → 0 as i → ∞. Hence ⎛ ∂ψ,q (Mϕj xi ) = ∂ψ,q ⎝



⎞ ϕj (s)xi,s λs ⎠

s∈supp(ϕj ) (2.84)

=

 s∈supp(ϕj )

ϕj (s)xi,s sq (bψ (s))  λs − → 0. i

This implies by uniqueness of the limit that (Id  Mϕj )(y) = 0 for any j . By the point 3 of Proposition 2.9, we deduce that y = w*- limj (Idq (H )  Mϕj )(y) = 0.  

That is, if (xn ) is a sequence in PG such that xn → 0 and ∂ψ,q (xn ) → y for some y ∈ q (H ) α G both for the weak* topology, then y = 0. 6

3.2 Lp -Kato’s Square Root Problem for Semigroups of Fourier Multipliers

87

Remark 3.1 We do not know if the assumption “AP” in Proposition 3.5 is really necessary. The following generalizes an observation of [131] (in the case q = 1). 1

Lemma 3.7 Let −1  q  1. Suppose 1 < p < ∞. For any x ∈ dom Ap2 and any 1

y ∈ dom Ap2 ∗ , we have ( 1 1 ∗ )

τq (H )α G ∂ψ,q,p (x)(∂ψ,q,p∗ (y))∗ = τG Ap2 (x) Ap2 ∗ (y) .

(3.34)

1

Proof By the fact that PG is a core for ∂ψ,q,p , ∂ψ,q,p∗ , Ap2 and Ap∗ and the norm  equivalence (3.31), we  can assume x, y ∈ PG . Consider some elements x = s∈G xs λs and y = r∈G yr λr of PG where both sums are finite. On the one hand, we have 



τq (H )α G ∂ψ,q (x)(∂ψ,q (y))∗ = xs yr τq (H )α G ∂ψ,q (λs )(∂ψ,q (λr ))∗ s,r∈G (2.84)

=



xs yr τq (H )α G (sq (bψ (s))  λs )(sq (bψ (r))  λr )∗

s,r∈G (2.57)

=



xs yr τq (H )α G (sq (bψ (s))  λs )(αr −1 (sq (bψ (r)))  λr −1

s,r∈G (2.56)

=



xs yr τq (H )α G (sq (bψ (s))αsr −1 (sq (bψ (r))))  λsr −1

s,r∈G

=



2  (2.41) 

xs ys τq (H ) sq (bψ (s))sq (bψ (s)) = xs ys bψ (s)H .

s∈G

s∈G

On the other hand, we have ( 1 1 ) ( 1 

1 ∗ ) ∗ = τG A 2 (x) A 2 (y) xs yr τG A 2 (λs ) A 2 (λr ) =

 s,r∈G

s,r∈G

   2  

 xs yr bψ (s)H bψ (r)H τG λs λ∗r = xs ys bψ (s)H . s∈G

 

88

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

3.3 Extension of the Carré du Champ  for Fourier Multipliers In this section, we consider again a markovian semigroup (Tt )t 0 of Fourier multipliers acting on the von Neumann algebra VN(G) where G is a discrete group. We shall extend the carré du champ  associated with (Tt )t 0 to a closed form and identify its domain. It will be easier to consider simultaneously x → (x, x) and x → (x ∗ , x ∗ ) and the case 2  p < ∞ throughout the section. We will also link 1

the carré du champ with Ap2 and the gradient ∂ψ,q,p . In most of the results of this section, we need approximation properties of Lp (VN(G)) and Lp (q (H ) α G). Note that it is stated in [121, Theorem 1.2] that if G is a discrete group with AP such that VN(G) has QWEP, then Lp (VN(G)) has the completely contractive approximation property CCAP for any 1 < p < ∞. With the following lemma, one can extend the definition of (x, y) to a larger domain. Lemma 3.8 Suppose 2  p < ∞. Let G be a discrete group. The forms a : PG × p p PG → L 2 (VN(G)) ⊕∞ L 2 (VN(G)), (x, y) → (x, y) ⊕ (x ∗ , y ∗ ) and  : PG × p PG → L 2 (VN(G)), (x, y) → (x, y) are symmetric, positive and satisfy the Cauchy-Schwarz inequality. The first is in addition closable and the domain of the 1

closure a is dom Ap2 . Proof According to the point 3 of Lemma 2.15, we have (x, y)∗ = (y, x), so a is symmetric. Moreover, again according to Lemma 2.15, a is positive. For any x, y ∈ PG , we have   p = (x, y) ⊕ (x ∗ , y ∗ ) p (VN(G)) L 2 (VN(G))⊕∞ L 2 (VN(G))    = max (x, y) p , (x ∗ , y ∗ ) p

a(x, y)

p L2

p (VN(G))⊕∞ L 2

2



2

 1  1   max (x, x) (y, y) , (x ∗ , x ∗ ) 2p (y ∗ , y ∗ ) 2p

(2.81)

1 2 p 2

1 2 p 2

2

2

1  1  1    max (x, x) , (x ∗ , x ∗ ) 2p max (y, y) 2p , (y ∗ , y ∗ ) 2p



1 2 p 2

= a(x, x)

1 2

p

2

p

L 2 (VN(G))⊕∞ L 2 (VN(G))

2

a(y, y)

1 2

p

2

p

L 2 (VN(G))⊕∞ L 2 (VN(G))

.

3.3 Extension of the Carré du Champ  for Fourier Multipliers

89

So a satisfies the Cauchy-Schwarz inequality. The assertions concerning  are a similar. Suppose xn − → 0 that is xn ∈ PG , xn → 0 and a(xn − xm , xn − xm ) → 0. For any integer n, m, we have   12 Ap (xn − xm ) p L (VN(G)) (2.90) p

   1 1 max (xn − xm , x − n − xm ) 2 Lp , ((xn − xm )∗ , (xn − xm )∗ ) 2 Lp

= a(xn − xm , xn − xm )

p

p

L 2 (VN(G))⊕∞ L 2 (VN(G))

→ 0.

1 We infer that Ap2 (xn ) is a Cauchy sequence, hence a convergent sequence. Since 1

1

xn → 0, by the closedness of Ap2 , we deduce that Ap2 (xn ) → 0. Now, we have a(xn , xn )

p

p

L 2 (VN(G))⊕∞ L 2 (VN(G))

   1 1 = max (xn , xn ) 2 Lp (VN(G)) , (xn∗ , xn∗ )) 2 Lp (VN(G)) (2.90) p

 12  Ap (xn ) p → 0. L (VN(G))

By Proposition 2.7, we obtain the closability of the form a. Let x ∈ Lp (VN(G)). By (2.52), we have x ∈ dom a if and only if there exists a a sequence (xn ) of PG satisfying xn − → x, that is satisfying xn → x and (xn − xm , xn − xm ) → 0, ((xn − xm )∗ , (xn − xm )∗ ) → 0 as n, m → ∞. By (2.90), this is equivalent to the existence of a sequence xn ∈ PG , such that xn → x, 1

Ap2 (xn − xm ) → 0 as n, m → ∞. Now recalling that PG is a core of the operator 1

1

Ap2 , we conclude that this is equivalent to x ∈ dom Ap2 .

 

Remark 3.2 The closability and biggest reasonable domain of the form  are unclear. 1

If 2  p < ∞, and x, y ∈ dom Ap2 , then we let (x, y) be the first component of p p a(x, y), where a : PG × PG → L 2 (VN(G)) ⊕∞ L 2 (VN(G)), (u, v) → (u, v) ⊕ (u∗ , v ∗ ) is the form in Lemma 3.8. Lemma 3.9 Suppose 2  p < ∞. Let G be a discrete group and assume that Lp (VN(G)) has CCAP and that VN(G) has QWEP. Let (ϕj ) be a net of functions ϕj : G → C with finite support such that the net (Mϕj ) converges to

90

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

  IdLp (VN(G)) in the point-norm topology with supj Mϕj cb,Lp (VN(G)→Lp (VN(G)  1. 1

If x ∈ dom Ap2 then for any j 

 1  Mϕ (x), Mϕ (x)  2 p j j

L 2 (VN(G))

   ∂ψ,q,p (x)Lp (VN(G),L2(q (H ))c,p ) .

1

Proof Let x ∈ dom Ap2 = dom ∂ψ,q,p . Since PG is a core of ∂ψ,q,p by Proposition 3.4, there exists a sequence (xn ) of elements of PG such that xn → x and ∂ψ,q,p (xn ) → ∂ψ,q,p (x). Note that Mϕj : Lp (VN(G)) → Lp (VN(G)) is a complete contraction. The linear map Mϕj ⊗ IdL2 (q (H )) : Lp (VN(G), L2 (q (H ))c,p ) → Lp (VN(G), L2 (q (H ))c,p ) is also a contraction according to (2.70). We deduce that ∂ψ,q,p Mϕj (xn )  = ϕj (s)xns ∂ψ,q,p (λs ) s∈supp ϕj



= Mϕj ⊗ IdL2 (q (H ))

 

 xns ∂ψ,q,p (λs )

s∈G





= Mϕj ⊗ IdL2 (q (H )) (∂ψ,q,p xn ) −−−−→ Mϕj ⊗ IdL2 (q (H )) (∂ψ,q,p x). n→+∞

Since ∂ψ,q,p Mϕj is bounded by Kato [136, Problem 5.22], we deduce that

∂ψ,q,p Mϕj (x) = Mϕj ⊗ IdL2 () (∂ψ,q,p x). Now, we have 1    (Mϕ x, Mϕ x) 2 p = (Mϕ x, Mϕ x) 21  p j j j j L L2

  = ∂ψ,q,p Mϕj (x)Lp (VN(G),L2(q (H ))c,p )   = (Mϕj ⊗ IdL2 (q (H )) )(∂ψ,q,p x)Lp (VN(G),L2(q (H ))c,p )    ∂ψ,q,p (x)Lp (VN(G),L2 (q (H ))c,p ) . (2.89)

  Now, we give a very concrete way to approximate the carré du champ for a large class of groups. Lemma 3.10 Let 2  p < ∞. Assume that Lp (VN(G)) has CCAP and that VN(G) has QWEP. Let (ϕj ) be a net of functions ϕj : G → C with finite support such that the net (Mϕj ) converges to IdLp (VN(G)) in the point-norm topology with

3.3 Extension of the Carré du Champ  for Fourier Multipliers

91

1   supj Mϕj cb,Lp (VN(G)→Lp (VN(G)  1. For any x, y ∈ dom Ap2 , we have in p

L 2 (VN(G))

(x, y) = lim  Mϕj (x), Mϕj (y) .

(3.35)

j

1

Proof If x ∈ dom Ap2 , we have for any j, k  1 (Mϕ x − Mϕ x, Mϕ x − Mϕ x) 2 p j k j k

L 2 (VN(G)

 1 = (Mϕj x − Mϕk x, Mϕj x − Mϕk x) 2 Lp (VN(G)) (2.90) p

1 1     1 A 2 (Mϕ x − Mϕ x) p = Mϕj Ap2 x − Mϕk Ap2 x Lp (VN(G)). j k L (VN(G))

The same inequality holds with ((Mϕj x − Mϕk x)∗ , (Mϕj x − Mϕk x)∗ ) on the left 1

1

hand side. Note that since Ap2 (x) belongs to Lp (VN(G)), (Mϕj Ap2 (x))p is a Cauchy a

net of Lp (VN(G)). Since Mϕj (x) → x, we infer that Mϕj (x) − → x, where a is the first form from Lemma 3.8. Then (3.35) is a consequence of Lemma 3.9 and Proposition 2.6.   Lemma 3.11 Suppose 2  p < ∞ and −1  q  1. 1

1. For any x, y ∈ dom Ap2 = dom ∂ψ,q,p , we have   (x, y) = ∂ψ,q,p (x), ∂ψ,q,p (y) Lp (VN(G),L2 (q (H ))c,p ) ∗

= E ∂ψ,q,p (x) ∂ψ,q,p (y) .

(3.36)

1

2. For any x ∈ dom Ap2 , we have     (x, x) 12  p = ∂ψ,q,p (x)Lp (VN(G),L2 (q (H ))c,p ) . L (VN(G))

(3.37)

Proof 1

1. Consider some elements x, y of dom Ap2 = dom ∂ψ,q,p . According to the point 2 of Proposition 3.4, PG is a core of dom ∂ψ,q,p . So by (2.3) there exist sequences (xn ) and (yn ) of PG such that xn → x, yn → y, ∂ψ,q,p (xn ) → ∂ψ,q,p (x) and ∂ψ,q,p (yn ) → ∂ψ,q,p (y). By Lemma 2.8, we have, since p  2, a contractive inclusion p

(2.76)

Lp (q (H ) α G) → Lc (E) = Lp (VN(G), L2 (q (H ))c,p ).

92

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

We deduce that ∂ψ,q,p (xn ) → ∂ψ,q,p (x) and ∂ψ,q,p (yn ) → ∂ψ,q,p (y) in the space Lp (VN(G), L2 (q (H ))c,p ). We obtain (xn − xm , xn − xm )

p

L 2 (VN(G))

     =  ∂ψ,q,p (xn − xm ), ∂ψ,q,p (xn − xm ) Lp (VN(G),L2 (q (H )))c,p  p L 2 (VN(G))      ∂ψ,q,p (xn − xm )Lp (L2 (q (H ))c,p ) ∂ψ,q,p (xn − xm )Lp (L2 (q (H ))c,p ) (2.89)

−−−−−−→ 0 n,m→+∞

and similarly for xn∗ , yn and yn∗ . We deduce that (2.53)

(x, y) =

lim (xn , yn )

n→+∞

(2.88)

=

lim

n→+∞





 ∂ψ,q,p (xn ), ∂ψ,q,p (yn ) Lp (VN(G),L2( 

q (H ))c,p )

= ∂ψ,q,p (x), ∂ψ,q,p (y) Lp (VN(G),L2(q (H ))c,p ) . 1

2. If x ∈ dom Ap2 , we have   ∂ψ,q,p (x)

Lp (VN(G),L2 (q (H ))c,p )

 1 (2.66)  =  ∂ψ,q,p (x), ∂ψ,q,p (x) L2 p (VN(G),L2(

q (H ))c,p

   )

Lp (VN(G))

1 (3.36)  = (x, x) 2 Lp (VN(G)).

  Now, we can extend Proposition 2.12. Theorem 3.2 Suppose that Lp (VN(G)) has CCAP and that VN(G) has QWEP. 1

Let 2  p < ∞. For any x ∈ dom Ap2 , we have       12  (x, x) 21  p (x ∗ , x ∗ ) 12  p Ap (x) p . ≈ max , p L (VN(G)) L (VN(G)) L (VN(G)) (3.38) 1

Proof Pick any −1  q  1. For any x ∈ dom Ap2 , first note that (3.32)

(3.36)

= E ∂ψ,q,p (x ∗ )∗ (∂ψ,q,p (x ∗ )) = (x ∗ , x ∗ ). E ∂ψ,q,p (x)(∂ψ,q,p (x)∗ ) (3.39)

3.4 Lp -Kato’s Square Root Problem for Semigroups of Schur Multipliers

93

Using Proposition 3.1, in the second equivalence, we conclude that  12   (3.31)  Ap (x) p ≈p ∂ψ,q,p (x)Lp (q (H )α G) L (VN(G))      

1  

1  ∗ ∗ 2 2    ≈p max  E((∂ψ,q,p (x)) ∂ψ,q,p (x))  ,  E(∂ψ,q,p (x)(∂ψ,q,p (x) ))   p

   1 1 (3.36)(3.39) = max (x, x) 2 Lp (VN(G)), (x ∗ , x ∗ ) 2 Lp (VN(G)) .

p

 

3.4 Lp -Kato’s Square Root Problem for Semigroups of Schur Multipliers In this section, we shall consider the Kato square root problem for markovian semigroups of Schur multipliers. Thus, we fix for the whole section such a p markovian semigroup from Proposition 2.4, with its generator Ap on SI and p also the gradient type operator ∂α,q from (2.95). An L -variant of Kato’s square 1

root problem is then the question whether Ap2 (x) and ∂α,q (x) are comparable for the Lp norms. The main results in this section, answering affirmatively to this question, are then Theorem 3.3 (case of classical Gaussians) and Proposition 3.10 (case of q-deformed Gaussians), together with Lemma 3.19 on a Khintchine type equivalence in some Gaussian subspace of Lp (q (H )⊗B( 2I )). Finally, the problem of exact description of the domain of the closure of the gradients is investigated in Proposition 3.11. Throughout this section, if H is a Hilbert space, we denote by Hdisc the abelian group (H, +) equipped with the discrete topology. We will use the trace preserving normal unital injective ∗-homomorphism map J : VN(Hdisc ) → L∞ ()⊗VN(Hdisc), λh → 1 ⊗ λh

(3.40)

and the unbounded Fourier multiplier 1

(−)− 2 : PHdisc ⊂ Lp (VN(Hdisc)) → Lp (VN(Hdisc)) defined by (−)− 2 (λ0 ) = 0 and for h = 0, 1

(−)− 2 (λh ) = 1

The following is inspired by [15].

def

1 λh . 2π hH

(3.41)

94

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

Proposition 3.6 Let H be a Hilbert space. There exists a unique weak* continuous group (Ut )t ∈R of ∗-automorphisms of L∞ ()⊗VN(Hdisc) such that Ut (f ⊗ λh ) = e

√ 2it W(h)

t ∈ R, f ∈ L∞ (), h ∈ H.

f ⊗ λh ,

(3.42)

Moreover, each Ut is trace preserving. Proof For any t ∈√ R, we consider the (continuous) function ut : Hdisc → U(L∞ ()), h → e− 2it W(h). For any t ∈ R and any h1 , h2 ∈ Hdisc, note that √

ut (h1 + h2 ) = e−

2it W(h1 +h2 )

√

= e−

√ 2it W(h1 ) − 2it W(h2 )

e

= ut (h1 )ut (h2 ).

By Arhancet [13, Proposition 2.3] applied with M = L∞ (), G = Hdisc and by considering the trivial action α, for any t ∈ R, we have a unitary Vt : L2 (Hdisc, L2 ()) → L2 (Hdisc , L2 ()), ξ → (h → ut (−h)(ξ(h))) and a ∗isomorphism Ut : L∞ ()⊗VN(Hdisc) → L∞ ()⊗VN(Hdisc ), x → Vt xVt∗ satisfying (3.42). The uniqueness is clear by density. For any t, t  ∈ R any ξ ∈ L2 (Hdisc, L2 ()), note that almost everywhere √ √ 



Vt Vt  (ξ ) (h) = ut (−h) (ut  (−h)(ξ(h)) = e− 2it W(−h)e− 2it W(−h) ξ(h) √

 = e− 2i(t +t )W(−h)ξ(h) = ut +t  (−h)(ξ(h)) = Vt +t  (ξ ) (h).

We conclude that Vt Vt  = Vt +t  . Moreover, for any ξ, η ∈ L2 (Hdisc , L2 ()) = L2 (Hdisc × ), using dominated convergence theorem, we obtain   Vt (ξ ), η L2 (H



disc ,L

2 ())

 =

Hdisc ×

√

Hdisc ×

Hdisc ×

Vt (ξ )(h)(ω)η(h, ω) dμHdisc (h) dμ(ω)

ut (−h)(ξ(h))(ω)η(h, ω) dμHdisc (h) dμ(ω)

 =

=

e

2it W(−h)(ω)

ξ(h, ω)η(h, ω) dμHdisc (h) dμ(ω).



−−→ t →0

Hdisc ×

ξ(h, ω)η(h, ω) dμHdisc (s) dμ(ω) = ξ, ηL2 (Hdisc ,L2 ()) .

So (Vt )t ∈R is a weakly continuous group of unitaries hence a strongly continuous group by [212, Lemma 13.4] or [216, p. 239]. By [216, p. 238], we conclude that (Ut )t ∈R is a weak* continuous group of ∗-automorphisms.

3.4 Lp -Kato’s Square Root Problem for Semigroups of Schur Multipliers

95

Finally, for any t  0, any f ∈ L∞ () and any h ∈ H , we have  

· ⊗ τVN(Hdisc ) 



  √  2it W(h) 2it W(h) e f dμ τVN(Hdisc ) (λh ) = e f dμ δ0,h √

= 

  ( √ )

Ut (f ⊗ λh ) = · ⊗ τVN(Hdisc ) e 2it W(h)f ⊗ λh



= 



   f dμ τVN(Hdisc ) (λh ) = · ⊗ τVN(Hdisc ) (f ⊗ λh ). 

  We consider the unbounded operator δ : span {λh : h ∈ Hdisc } ⊂ Lp (VN(Hdisc)) → Lp (, Lp (VN(Hdisc))) defined by def

δ(λh ) = 2πiW(h) ⊗ λh .

(3.43)

Recall the classical transference principle [36, Theorem 2.8]. Let G be a locally compact abelian group and G → B(X), t → πt be a strongly continuous representation of G on a Banach space X such that c = sup{πt  : t ∈ G} < ∞. 1 Let  k ∈ L (G) and let Tk : X → X be the operator defined by Tk (x) = G k(t)π−t (x) dμG (t). Then Tk X→X  c2 k ∗ ·Lp (G,X)→Lp (G,X) .

(3.44)

If 0 < ε < R, we will use the function [109, p. 388] kε,R (t) =

1 1ε 2, passing to the limit when m → ∞, we finally obtain f Lp (q (H )⊗B( 2 ))  I

√

 

1  p  E(f ∗ f ) 2 

p

SI



1  +  E(ff ∗ ) 2 

 p

SI

.

Using the equivalence 12 ≈ ∞ 2 , we obtain the upper estimate of (3.73). The lower estimate of (3.73) holds with constant 1 from the contractivity of the p conditional expectation E on L 2 (q (H )⊗B( 2I )):     1  1 

 1  

1  ∗ ∗ 2 ∗ 2 ∗ 2 2      p p , E(ff ) max E(f f ) = max E(f f ) p , E(ff ) p S S I

I

 1   max f ∗ f  2 p

L 2 (q (H )⊗B( 2I ))

= f Lp (q (H )⊗B( 2 )) . I

SI2

 1 , ff ∗  2 p

L 2 (q (H )⊗B( 2I ))



SI2

3.4 Lp -Kato’s Square Root Problem for Semigroups of Schur Multipliers

115

1. Let us now consider the case 1 < p < 2. We will proceed by duality as follows. Recall the Gaussian projection Qp from Lemma 3.18. Using in the second equality that Q∗p = Qp∗ and that Qp∗ extends to a p∗

contraction on SI (L2 (q (H ))rad,p∗ ) according to Lemma 3.18, and using the upper estimate of (3.73) and the density of span{sq (h) : h ∈ H } ⊗ MI,fin in p∗ SI (L2 (q (H ))rad,p∗ ) in the last inequality, we obtain for any f = Qp (f ) ∈ p Gaussq,2 (C) ⊗ SI (2.69)

f S p (L2 (q (H ))rad,p ) = I

=

g

sup

1 p∗ SI (L2 (q (H ))rad,p∗ )

sup g p∗ 1 SI (L2 (q (H ))rad,p∗ )

   f, g 

   f, g 

g∈Ran Qp∗

 f Lp (q (H )⊗B( 2 )) I

(3.72) p

sup g p∗ 1 SI (L2 (q (H )rad,p∗ ) g∈Ran Qp∗

gLp∗ (q (H )⊗B( 2 )) I

f Lp (q (H )⊗B( 2 )) . I

p

Note that in the definition of the norm of SI (L2 (q (H ))rad,p ), the infimum runs p p over g, h belonging to Lc (E) and Lr (E). Our next goal is to restrict to those g, h belonging to span{sq (e) : e ∈ H } ⊗ MI,fin . To this end, consider a decomposition p p f = g + h with g ∈ Lc (E) and h ∈ Lr (E) such that gLpc (E) + hLpr (E)  2 f S p (L2 (q (H ))rad,p ) . I

Then for some large enough J and with as before Qp the Gaussian projection, we have f = Qp (Idq (H ) ⊗ TJ )(f ) = Qp (Idq (H ) ⊗ TJ )(g) + Qp (Idq (H ) ⊗ TJ )(h)  and Qp (Idq (H ) ⊗ TJ )(g), Qp (Idq (H ) ⊗ TJ )(h) belong to span sq (e) : e ∈ H } ⊗ MI,fin . We claim that we have   Qp (Id (H ) ⊗ TJ )(g) p  g p q Lc (E) L (E) c

and   Qp (Id (H ) ⊗ TJ )(h) p  h p . q Lr (E) L (E) r

116

3 Riesz Transforms Associated to Semigroups of Markov Multipliers p

p

p

p

Indeed, first note that Qp : Lc (E) → Lc (E) and Qp : Lr (E) → Lr (E)) are contracp p tive according to (3.71). Moreover, since TJ : SI → SI is a complete contraction, p according to (2.70), the linear map TJ ⊗ IdL2 (q (H )) : SI (L2 (q (H ))c,p ) → p SI (L2 (q (H ))c,p ) is also a contraction. So we have 



  Qp Id (H ) ⊗ TJ (g) p (2.74) = Qp TJ ⊗ IdL2 (q (H )) (g)S p (L2 (q (H ))c,p ) q L (E) c

I

(2.70)

(2.74)

 gS p (L2 (q (H ))c,p ) = gLpc (E) . I

The row estimate is similar. Together we have shown that the third expression in (3.72) is controlled by f Gaussq,p (S p ) . I Now, we will prove the remaining estimate, that is, f Gaussq,p (S p ) is controlled I by the second expression in (3.72). Since 1 < p < 2, the function R+ → R+ , p t → t 2 is operator concave by [37, p. 112]. Using [100, Corollary 2.2] applied with the trace preserving positive map E, we can write  1  2 f Lp (q (H )⊗B( 2 )) = |f |2  p

L 2 (q (H )⊗B( 2I ))

I

 1  2  E(|f |2 ) p

L 2 (q (H )⊗B( 2I ))

= f Lpc (E)

and similarly for the row term. Thus, passing to the infimum over all decompositions f = g + h, we obtain f Lp (q (H )⊗B( 2 ))  f S p (L2 (q (H ))rad,p ) , which can be I I majorised in turn by the infimum of gLpc (E) + hLpr (E) , where f = g + h and g, h ∈ span{sq (e) : e ∈ H } ⊗ MI,fin . Hence, we have the last equivalence in the part 1 of the theorem. The case p = 2 is obvious since we have isometrically L2cr (E) = SI2 (L2 (q (H ))rad,2 ) = L2 (q (H )) ⊗2 SI2  

by Lemma 2.9 and [128, Remark 2.3 (1)].

We can equally extend Lemma 3.19 to the case that f ∈ Lp (q (H )⊗B( 2I )) belongs to Ran Qp , where we recall that Qp = IdS p ⊗ P : Lp (q (H )⊗B( 2I )) → I

Lp (q (H )⊗B( 2I )) is the Gaussian projection from Lemma 3.18. Lemma 3.21 Let −1  q  1. Suppose 1 < p < ∞. Then the linear map Qp : Lp (q (H )⊗B( 2I )) → Lp (q (H )⊗B( 2I )) is completely bounded.

3.4 Lp -Kato’s Square Root Problem for Semigroups of Schur Multipliers

Proof Since P is selfadjoint, we have Q∗p q (H )⊗MI,fin ,   Qp (f )

Lemma 3.19 p

Gaussq,p (SI )



  Qp (f )

117

= Qp∗ . We obtain for f

∈

p

Lcr (E)

   = sup  Qp (f ), g : gLp∗ (E)  1 cr      = sup  f, Qp∗ (g)Lp (E),Lp∗ (E)  : g cr cr      = sup  f, Qp∗ (g)Lp (q (H )⊗B( 2 )),Lp∗  : g I     f Lp (q (H )⊗B( 2 )) sup Qp∗ (g)Lp∗ (q (H )⊗B( 2 )) : g I

Lemma 3.19



I

   f Lp (q (H )⊗B( 2 )) sup Qp∗ (g)Lp∗ (E) : gLp∗ (E)  1 I

cr

cr

 f Lp (q (H )⊗B( 2 )) , I

p∗

where in the last step we used that Qp∗ is contractive on Lcr (E) according to Lemma 3.18. This shows that Qp is bounded. Then the fact that Qp is completely bounded follows from a standard matrix amplification argument since we can replace I by I × {1, . . . , N}.   p

Lemma 3.22 Consider −1  q  1 and let f ∈ Ran Qp = Gaussq,p (SI ). 1. Suppose 1 < p < 2. We have f Gaussq,p (S p ) ≈p f S p (L2 (q (H ))rad,p ) I I   

1  

1  ∗ ∗ 2 2    p, p ≈p inf E(hh ) E(g g) S S f =g+h

I

I

where the infimum can be taken over all g, h ∈ Ran Qp . 2. Suppose 2  p < ∞. We have   

1  

1  max  E(f ∗ f ) 2 S p ,  E(ff ∗ ) 2 S p  f Gaussq,p (S p ) I

I

I

  

1  

1  √  p max  E(f ∗ f ) 2 S p ,  E(ff ∗ ) 2 S p . I

(3.81)

I

Proof Observe that Idq (H ) ⊗ TJ approximates the identity on q (H )⊗MI,fin . Moreover, according to Lemma 5.3, the net (Idq (H ) ⊗ TJ ) is bounded in the Banach space B(Lp (q (H )⊗B( 2I ))). We conclude by density of q (H )⊗MI,fin in Lp (q (H )⊗B( 2I )) that the net (Idq (H ) ⊗ TJ ) converges in the point norm

118

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

topology of Lp (q (H )⊗B( 2I )) to the identity. Moreover, replacing in this argup ment boundedness in B(Lp (q (H )⊗B( 2I ))) by boundedness in B(Lc (E)) (resp. p p B(Lr (E)), B(Lcr (E))) according to (2.70) and (2.74) (resp. Remark 2.2), we obtain p that the net (Idq (H ) ⊗ TJ ) converges in the point norm topology of Lc (E) (resp. p p Lr (E), Lcr (E)) to the identity. Note that for fixed J , (Idq (H ) ⊗ TJ )(f ) = (Idq (H ) ⊗ TJ )(Qp f ) = Qp (Idq (H ) ⊗ TJ )(f ) belongs to span{sq (e) : e ∈ H } ⊗ MI,fin . Thus, Lemma 3.19 applies to f replaced by (Idq (H ) ⊗ TJ )(f ) and therefore,   f Gaussq,p (S p ) = lim (Idq (H ) ⊗ TJ )(f )Gauss I

Lemma 3.19

∼ =

p q,p (SI )

J

  lim (Idq (H ) ⊗ TJ )(f )Lp (E) = f Lpcr (E) . cr

J

Finally, the fact that one can restrict the infimum to all g, h ∈ Ran Qp can be proved in the same way as that in Lemma 3.19.   Now we can state the Kato square root problem for the case of the gradient taking values in a q-deformed algebra. Proposition 3.10 Suppose −1  q  1 and 1 < p < ∞. For any x ∈ MI,fin , we have  21  Ap (x)

p

SI

  ≈p ∂α,q (x)Lp (

2 q (H )⊗B( I ))

(3.82)

.

Proof Note that if q = 1, then this is a consequence of Theorem 3.3.  We show the remaining case −1  q < 1 by reducing it to the case q = 1. Let x = i,j xij eij ∈ MI,fin . Choose an orthonormal basis (ek ) of span{αi }, where we consider only those indices i appearing in the previous double sum describing x. Thus, αi − αj =  k αi − αj , ek H ek , where the sum is finite. Then  12  (3.54)     Ap (x) p ≈p ∂α,1 (x) p p = ∂α,1 (x) p SI L (,SI ) Gauss1,p (SI )       (2.95)   =  xij W(αi − αj ) ⊗ eij     i,j p Gauss1,p (SI )

         = xij W

ek , αi − αj H ek ⊗ eij     i,j k         = x

e , α − α  W(e ) ⊗ e ij k i j H k ij     i,j k

p

Gauss1,p (SI )

p

Gauss1,p (SI )

3.4 Lp -Kato’s Square Root Problem for Semigroups of Schur Multipliers

       = W(ek ) ⊗ xij ek , αi − αj H eij      k i,j [128, (2.21), (2.22) p. 12]

≈

119

p

Gauss1,p (SI )

          xij ek , αi − αj H eij ⊗ ek     k  i,j

      Lemma 3.19   ≈ sq (ek ) ⊗ xij ek , αi − αj H eij     k  i,j        = xij sq (αi − αj ) ⊗ eij     i,j

p

SI (Hrc )

p

Gaussq,p (SI )

  = ∂α,q (x)Gauss

(2.95)

p q,p (SI )

p Gaussq,p (SI )

.

  Remark 3.4 Assume −1  q  1. In Proposition 3.10, we obtain again constants depending on p as in (3.56) of a slightly different form, that is, for some absolute constant K > 0, we have for all x ∈ MI,fin ,   21   3 1     ∂α,q (x) p  2 ))  Ap (x) S p  Kp 2 ∂α,q (x) Lp L ( (H )⊗B( q I I Kp2 (3.83) (2  p < ∞) and    12   1  ∗ 3 ∂α,q (x) p    2 ))  Ap (x) S p  K(p ) 2 ∂α,q (x) Lp L ( (H )⊗B( ∗ 2 q I I K(p ) (3.84) (1 < p  2). Let us prove this and start with the case 2  p < ∞. Note that W(ek ) and sq (ek ) are orthonormal systems in their spaces L2 () and L2 (q (H )) respectively. Thus, we have isometrically, for xk ∈ MI,fin ,       xk ⊗ W(ek )    k

p SI (L2 ()rad,p )

      = xk ⊗ sq (ek )   k

. p SI (L2 (q (H ))rad,p )

(3.85)

120

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

Thus, for the upper estimate in (3.83), we have  21  (3.65)   Ap (x) p  Kp∂α,1 (x) p SI Gauss1,p (SI )      = Kp W(ek ) ⊗ xij ek , αi − αj H eij  k Lemma 3.19



p

Gauss1,p (SI )

i,j

    1  K p · p 2  xij ek , αi − αj H eij ⊗ W(ek ) k

     = Kp  xij ek , αi − αj H eij ⊗ sq (ek )

(3.85)



3 2

k

Lemma 3.19



p

SI (L2 (q (H ))rad,p )

i,j

     Kp  sq (ek ) ⊗ xij ek , αi − αj H eij  

p

SI (L2 ()rad,p )

i,j

3 2

k

i,j

 3 = K  p 2 ∂α,q (x)Gauss

p q,p (SI )

p

Gaussq,p (SI )

.

In the other direction, we have, again for 2  p < ∞,  21  (3.56) Ap (x) p  S

 1  ∂α,1 (x) p 3 Gauss1,p (SI ) I Kp 2   1    = W(e ) ⊗ x

e , α − α  e k ij k i j H ij  p 3  Gauss1,p (SI ) Kp 2 k i,j    Lemma 3.19 1      xij ek , αi − αj H eij ⊗ W(ek ) p 2 3  SI (L ()rad,p ) Kp 2 k i,j     (3.85) 1     = xij ek , αi − αj H eij ⊗ sq (ek ) p 2 3  SI (L (q (H ))rad,p ) Kp 2 k i,j    Lemma 3.19 1    sq (ek ) ⊗ xij ek , αi − αj H eij  p 3 1  Gaussq,p (SI ) K p 2 · p 2 k i,j =

 1  ∂α,q (x) p . Gaussq,p (SI )  2 Kp

We turn to the case 1 < p  2. Note that (3.85) equally holds in this case. Then we 1 argue in the same way as in the case p  2, noting that the additional factor p∗ 2 appears at another step of the estimate, in accordance with Lemma 3.19. In the remainder of this section, we shall extend the gradient ∂α,q to a closed p (weak* closed if p = ∞) operator SI → Lp (q (H )⊗B( 2I )), and in particular

3.4 Lp -Kato’s Square Root Problem for Semigroups of Schur Multipliers

121

identify its domain in terms of the generator of the markovian semigroup. All this will be achieved in Proposition 3.11. We define the densely defined unbounded operator ∗ ∂α,q : Lp (q (H )) ⊗ MI,fin ⊂ Lp (q (H ))⊗B( 2I )) → SI

p

by   ∗ ∂α,q (f ⊗ eij ) = sq (αi − αj ), f Lp∗ ,Lp eij = τ (sq (αi − αj )f )eij

(3.86)

where i, j ∈ I and f ∈ Lp (q (H )). ∗ are formal adjoints (with respect to Lemma 3.23 The operators ∂α,q and ∂α,q ∗ duality brackets x, y = Tr(x y) and x, y = τ ⊗ Tr(x ∗ y)).

Proof For any i, j, k, l ∈ I and any f ∈ Lp (q (H )), we have 

∂α,q (ekl ), f ⊗ eij



(3.87)

∗

Lp (q (H )⊗B( 2I )),Lp (q (H )⊗B( 2I ))

  sq (αk − αl ) ⊗ ekl , f ⊗ eij Lp∗ (q (H )⊗B( 2 )),Lp (q (H )⊗B( 2 )) I I   = δki δlj sq (αi − αj ), f Lp∗ ( (H )),Lp ( (H )) q q     = ekl , sq (αi − αj ), f Lp∗ (q (H )),Lp (q (H )) eij p∗ p (2.95)

=



∗ = ekl , ∂α,q (f ⊗ eij )

SI ,SI

 p∗

p

SI ,SI

.  

Proposition 3.11 Suppose 1 < p < ∞ and −1  q  1. p

1. The operator ∂α,q : MI,fin ⊂ SI → Lp (q (H )⊗B( 2I )) is closable as a densely p defined operator on SI into Lp (q (H )⊗B( 2I )). We denote by ∂α,q,p its closure. So MI,fin is a core of ∂α,q,p . 1

2. MI,fin is a core of dom Ap2 . 1

1

3. We have dom ∂α,q,p = dom Ap2 . Moreover, for any x ∈ dom Ap2 , we have    12  Ap (x) p ≈p ∂α,q,p (x) p . S Gaussq,p (S ) I

(3.88)

I

1

Finally, for any x ∈ dom Ap2 there exists a sequence (xn ) of elements of MI,fin 1

1

such that xn → x, Ap2 (xn ) → Ap2 (x) and ∂α,q,p (xn ) → ∂α,q,p (x).

122

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

4. If x ∈ dom ∂α,q,p , we have x ∗ ∈ dom ∂α,q,p and (∂α,q,p (x))∗ = −∂α,q,p (x ∗ ).

(3.89)

5. Suppose that −1  q < 1. Then the operator ∂α,q : MI,fin ⊂ B( 2I ) → q (H )⊗B( 2I ) is weak* closable. We denote by ∂α,q,∞ its weak* closure. p p 6. Let MB : SI → SI be a finitely supported bounded Schur multiplier such that the map Id ⊗ MB : Lp (q (H )⊗B( 2I )) → Lp (q (H )⊗B( 2I )) is a well-defined bounded operator. For any x ∈ dom ∂α,q,p , the element MB (x) belongs to dom ∂α,q,p and we have ∂α,q,p MB (x) = (Id ⊗ MB )∂α,q,p (x).

(3.90)

Proof The proofs of the first three points and of the last point are identical to p the proof of Proposition 3.4 replacing PG by MI,fin , Lp (VN(G)) by SI and Lp (q (H ) α G) by Lp (q (H ) ⊗ B( 2I )). 4. For any i, j ∈ I , we have

∗ ∗ (2.95)

∂α,q,p (eij ) = sq (αi − αj ) ⊗ eij = sq (αi − αj ) ⊗ ej i (2.95)

= −sq (αj − αi ) ⊗ ej i = −∂α,q,p (ej i ) = −∂α,q,p (eij∗ ). Let x ∈ dom ∂α,q,p . By the point 1, MI,fin is core of ∂α,q,p . Hence there exists a sequence (xn ) of MI,fin such that xn → x and ∂α,q,p (xn ) → ∂α,q,p (x). We have xn∗ → x ∗ and by the first part of the proof ∂α,q,p (xn∗ ) = −(∂α,q,p (xn ))∗ → −(∂α,q,p (x))∗ . By (2.7), we conclude that x ∗ ∈ dom ∂α,q,p and that ∂α,q,p (x ∗ ) = −(∂α,q,p (x))∗ .

5. Note that Idq (H ) ⊗ TJ converges for the point weak*  topology to Idq (H )⊗B( 2 ) . Suppose that (xk ) is a net of MI,fin with xk = i,j ∈I xk,i,j eij I which converges to 0 for the weak* topology such that the net (∂α,q (xk )) converges for the weak* topology to some y belonging to q (H )⊗B( 2I ). For any finite subset J of I , we have    ∂α,q TJ (xk ) = ∂α,q xk,i,j eij = xk,i,j ∂α,q (eij ) i,j ∈J



= Idq (H ) ⊗ TJ ∂α,q

→ Idq (H ) ⊗ TJ (y) − k

i,j ∈J



i,j ∈I

xk,i,j eij





= Idq (H ) ⊗ TJ ∂α,q (xk )

3.5 Meyer’s Problem for Semigroups of Schur Multipliers

123

for the weak* topology. On the other hand, for any i, j ∈ I we have xk,ij → 0 as k → ∞. Hence for any finite subset J of I ⎛ ∂α,q TJ (xk ) = ∂α,q ⎝



⎞ xk,i,j eij ⎠ =

(2.95)

i,j ∈J



xk,i,j sq (αi − αj ) ⊗ eij − → 0. k

i,j ∈J



This implies by uniqueness of the q (H )⊗B( 2I )-limit that Idq (H ) ⊗ TJ (y) = 0.

We deduce that y = w*- limJ Idq (H ) ⊗ TJ (y) = 0.   Remark 3.5 It would be more natural to replace the definition (2.95) by the formula def

∂α,q (eij ) = 2πisq (αi − αj ) ⊗ eij . With this new definition, the derivation is symmetric, i.e. ∂α,q (x ∗ ) = ∂α,q (x)∗ . We refer to [225] for more information on weak* closed derivations on von Neumann algebras. Finally, note that the extension of Lemma 3.15 which can be proved with the point 3 of Proposition 3.11. 1

Lemma 3.24 Suppose 1 < p < ∞ and −1  q  1. For any x ∈ dom Ap2 and 1

any y ∈ dom Ap2 ∗ , we have 1



1 (τq (H ) ⊗TrB( 2 ) ) (∂α,q,p (x))∗ ∂α,q,p∗ (y) = TrB( 2 ) (Ap2 (x))∗ Ap2 ∗ (y) . I

I

(3.91)

3.5 Meyer’s Problem for Semigroups of Schur Multipliers In this section, we again fix a markovian semigroup of Schur multipliers, and thus we have some Hilbert space H and a family (αi )i∈I in H representing the semigroup in the sense of Proposition 2.4. We investigate the so-called Meyer’s problem which consists in expressing the Lp norm of the gradient form (3.88) in terms of the carré du champ . Here our results split into the cases 1 < p < 2 and 2  p < ∞, and are more satisfactory in the second one. The first main result will be Theorem 3.5. In the second part of this section, we shall have a look at Riesz transforms (3.99) associated with the semigroup, carrying a direction vector in the Hilbert space H . Then these p directional Riesz transforms have some sort of square functions expressing the SI norm, see Theorem 3.6.

124

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

In the case of finite sized matrices, we are already in the position to state the essence of Meyer’s problem for semigroups of Schur multipliers. Namely, the following is an immediate consequence of Lemma 3.19, Proposition 3.10 and (2.97). We recall that we have a canonical conditional expectation p E : Lp (q (H )⊗B( 2I )) → SI . Corollary 3.1 1. Suppose 1 < p < 2. Let −1  q  1. For any x ∈ MI,fin we have  12  Ap (x) p ≈p S I

inf

∂α,q,p (x)=y+z

yLpr (E) + zLpc (E)

where the infimum is taken over all y ∈ Gaussq,2 (C) ⊗ MI,fin and all z ∈ Gaussq,2 (C) ⊗ MI,fin . 2. Suppose 2  p < ∞. For any x ∈ MI,fin , we have  21  Ap (x)

p SI

   1 1 ≈p max (x, x) 2 S p , (x ∗ , x ∗ ) 2 S p . I

(3.92)

I

Now, we will extend the definition of (x, y) to a larger domain. p

p

Lemma 3.25 Suppose 2  p < ∞. The forms a : MI,fin × MI,fin → SI2 ⊕∞ SI2 , p

(x, y) → (x, y) ⊕ (x ∗ , y ∗ ) and  : MI,fin × MI,fin → SI2 , (x, y) → (x, y) are symmetric, positive and satisfy the Cauchy-Schwarz inequality. The first is in 1

addition closable and the domain of the closure a is dom Ap2 . Proof According to the point 3 of Lemma 2.17, we have (x, y)∗ = (y, x), so a is symmetric. Moreover, again according to Lemma 2.17, a is positive. For any x, y ∈ MI,fin , we have a(x, y)

p SI2

p ⊕∞ SI2

  = (x, y) ⊕ (x ∗ , y ∗ )

   = max (x, y) p , (x ∗ , y ∗ ) p 2



p

p

SI2 ⊕∞ SI2

2

 1  1   max (x, x) (y, y) , (x ∗ , x ∗ ) 2p (y ∗ , y ∗ ) 2p

(2.94)

1 2 p 2

1 2 p 2

2

2

1  1  1    max (x, x) , (x ∗ , x ∗ ) 2p max (y, y) 2p , (y ∗ , y ∗ ) 2p



1 2 p 2

= a(x, x)

1 2

p

p

SI2 ⊕∞ SI2

2

a(y, y)

1 2

p

2

p

SI2 ⊕∞ SI2

.

2

3.5 Meyer’s Problem for Semigroups of Schur Multipliers

125

So a satisfies the Cauchy-Schwarz inequality. The assertions concerning  are a similar. Suppose xn − → 0 that is xn ∈ MI,fin , xn → 0 and a(xn − xm , xn − xm ) → 0. For any integer n, m, we have  (3.92)  12 Ap (xn − xm ) p p SI    1 1 max (xn − xm , xn − xm ) 2 S p , ((xn − xm )∗ , (xn − xm )∗ ) 2 S p I

= a(xn − xm , xn − xm )

p

I

p

SI2 ⊕∞ SI2

→ 0.



1 We infer that Ap2 (xn ) is a Cauchy sequence, hence a convergent sequence. Since 1

1

xn → 0, by the closedness of Ap2 , we deduce that Ap2 (xn ) → 0. Now, we have a(xn , xn )

p

p

SI2 ⊕∞ SI2

 (3.92)  1    1 1 = max (xn , xn ) 2 p , (xn∗ , xn∗ )) 2 p p Ap2 (xn )p → 0. By Proposition 2.7, we obtain the closability of the form a. p Let x ∈ SI . By (2.52), we have x ∈ dom a if and only if there exists a sequence a

(xn ) of MI,fin satisfying xn − → x, that is satisfying xn → x and (xn − xm , xn − xm ) → 0, ((xn −xm )∗ , (xn −xm )∗ ) → 0 as n, m → ∞. By the equivalence (3.92), this is equivalent to the existence of a sequence xn ∈ MI,fin , such that xn → x, 1

Ap2 (xn − xm ) → 0 as n, m → ∞. Now recalling that MI,fin is a core of the operator 1

1

Ap2 , we conclude that this is equivalent to x ∈ dom Ap2 .

 

Remark 3.6 The closability of  and the biggest reasonable domain of the form  are unclear. 1

If 2  p < ∞, and x, y ∈ dom Ap2 , then we let (x, y) be the first component p

p

of a(x, y), where a : MI,fin × MI,fin → SI2 ⊕∞ SI2 , (u, v) → (u, v) ⊕ (u∗ , v ∗ ) is the form in Lemma 3.25. Lemma 3.26 Suppose −1  q  1 and 2  p < ∞.10 1

1. For any x, y ∈ dom Ap2 = dom ∂α,q,p , we have  

(x, y) = ∂α,q,p (x), ∂α,q,p (y) S p (L2 ()c,p ) = E (∂α,q,p (x))∗ ∂α,q,p (y) . I (3.93)

10 In

p

the proof, we recall that Lp (q (H )⊗B( 2I )) ⊂ SI (L2 (q (H ))c,p ).

126

3 Riesz Transforms Associated to Semigroups of Markov Multipliers 1

2. For any x ∈ dom Ap2 = dom ∂α,q,p , we have     (x, x) 12  p = ∂α,q,p (x) p 2 S S (L ( I

I

q (H ))c,p )

(3.94)

.

1

Proof Consider some elements x, y ∈ dom Ap2 = dom ∂α,q,p . According to Proposition 3.11, MI,fin is a core of dom ∂α,q,p . So by (2.3), there exist sequences (xn ) and (yn ) of MI,fin such that xn → x, yn → y, ∂α,q,p (xn ) → ∂α,q,p (x) and ∂α,q,p (yn ) → ∂α,q,p (y). Since p  2, we have a contractive inclusion Lp (q (H )⊗B( 2I )) → p

(2.74)

p

Lc (E) = SI (L2 (q (H ))c,p ). We deduce that ∂α,q,p (xn ) → ∂α,q,p (x) and p ∂α,q,p (yn ) → ∂α,q,p (y) in the space SI (L2 (q (H ))c,p ). Note that (xn − xm , xn − xm )

p

SI2

     =  ∂α,q,p (xn − xm ), ∂α,q,p (xn − xm ) S p (L2 (q (H ))c,p ) 

(2.97)

p

SI2

I

     ∂α,q,p (xn − xm )S p (L2 (q (H ))c,p ) ∂α,q,p (xn − xm )S p (L2 (q (H ))c,p )

(2.65)

I

I

−−−−−−→ 0 n,m→+∞





and similarly for yn . Hence xn − → x and yn − → y. We deduce that (2.53)

(x, y) =

(2.97)

lim (xn , yn ) =

n→+∞

lim



n→+∞

 ∂α,q,p (xn ), ∂α,q,p (yn ) S p (L2 (q (H ))c,p ) I

  = ∂α,q,p (x), ∂α,q,p (y) S p (L2 (q (H ))c,p ) . I

1

2. If x ∈ dom Ap2 , we have    1 (2.66)  ∂α,q,p (x) p 2 =  ∂α,q,p (x), ∂α,q,p (x) S2 p (L2 ( S (L (q (H ))c,p ) I

I

q (H ))c,p

   )

p

SI

 1 = (x, x) 2 S p .

(3.93)

I

  The following is our first main theorem of this section. 1

Theorem 3.5 Suppose 2  p < ∞. For any x ∈ dom Ap2 , we have       21  Ap (x) p ≈p max (x, x) 12  p , (x ∗ , x ∗ ) 12  p . S S S I

I

I

(3.95)

3.5 Meyer’s Problem for Semigroups of Schur Multipliers

127

1

Proof Pick any −1  q  1. For any x ∈ dom Ap2 , first note that (3.89)

(3.93)

E ∂α,q,p (x)(∂α,q,p (x))∗ = E (∂α,q,p (x ∗ ))∗ ∂α,q,p (x ∗ ) = (x ∗ , x ∗ ). (3.96) We conclude that  12  (3.88)   Ap (x) p ≈p ∂α,q,p (x) p SI Gaussq,p (SI )   

1  (3.81) ∗ 2 , E((∂ ≈p max  (x)) ∂ (x)) α,q,p α,q,p   p

   

1   E(∂α,q,p (x)(∂α,q,p (x)∗ )) 2    p

(3.93)(3.96)

=

   1 1 max (x, x) 2 S p , (x ∗ , x ∗ ) 2 S p . I

I

  In Lemma 3.25, we have shown that the carré du champ of  is a closable form. In fact, one can give a concrete way how to reach all of its domain by approximation with MI,fin matrices. This is the content of the next two lemmas. Recall the truncations TJ from Definition 2.3. Lemma 3.27 Suppose −1  q  1 and 2  p < ∞. Let J be a finite subset of I . 1

If x ∈ dom Ap2 , then   

 1  TJ (x), TJ (x)  2 p  ∂α,q,p (x) p 2 . S (L (q (H ))c,p ) I

SI2

1

Proof Let x ∈ dom Ap2 = dom ∂α,q,p . Since MI,fin is a core of ∂α,q,p by Proposition 3.11, there exists a sequence (xn ) of MI,fin such that xn → x and p p ∂α,q,p (xn ) → ∂α,q,p (x). Note that since TJ : SI → SI is a complete contraction, p 2 p the linear map TJ ⊗ IdL2 (q (H )) : SI (L (q (H ))c,p ) → SI (L2 (q (H ))c,p ) is also a contraction according to (2.70). We deduce that ∂α,q,p TJ (xn ) =

 i,j ∈J

 

 xnij ∂α,q,p (eij ) = IdL2 (q (H )) ⊗ TJ xnij ∂α,q,p (eij ) i,j ∈I

= IdL2 (q (H )) ⊗ TJ (∂α,q,p xn ) −−−−→ IdL2 (q (H )) ⊗ TJ (∂α,q,p x).



n→+∞

128

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

Since ∂α,q,p TJ is bounded by [136, Problem 5.22], we deduce that ∂α,q,p TJ (x) =

IdL2 (q (H )) ⊗ TJ (∂α,q,p x). Now, we have  1     (TJ x, TJ x) 2 p = (TJ x, TJ x) 21  p (2.98) = ∂α,q,p TJ (x)S p (L2 (q (H ))c,p ) S SI2

I

I

  = (IdL2 (q (H )) ⊗ TJ )(∂α,q,p x)S p (L2 ( I

q (H ))c,p )

   ∂α,q,p (x)S p (L2 (q (H ))c,p ) .

(2.70)

I

  Now, we give a very concrete way to approximate the carré du champ. p

1

Lemma 3.28 Let 2  p < ∞. For any x, y ∈ dom Ap2 , we have in SI2

(x, y) = lim  TJ (x), TJ (y) .

(3.97)

J

1

Proof If x ∈ dom Ap2 , we have for any finite subsets J, K of I 1  1 (TJ x − TK x, TJ x − TK x) 2 p = (TJ x − TK x, TJ x − TK x) 2 S p I

SI2

(3.95) p

1 1     12 Ap (TJ x − TK x) p = TJ Ap2 x − TK Ap2 x  p . S S I

1

I

1

p

p

Note that since Ap2 (x) belongs to SI , (TJ Ap2 (x))p is a Cauchy net of SI . Since 

p

TJ (x) → x in SI , we infer that TJ (x) − → x. Then (3.97) is a consequence of Lemma 3.27 and Proposition 2.6.   In the second half of this section, we shall show that directional Riesz transforms p associated with markovian semigroups of Schur multipliers decompose the SI  2 norm. Since Ap (eij ) = αi − αj H eij for any i, j ∈ I , we have  p Ran Ap = x ∈ SI : xij = 0 for all i, j with αi = αj .

(3.98)

Note that (Tt )t 0 is strongly continuous on SI∞ and that it is not difficult to show that (3.98) also holds for Ran A∞ , where A∞ denotes the generator of (Tt )t 0 on SI∞ . If h ∈ H , we define the h-directional Riesz transform Rα,h defined on MI,fin by def

Rα,h (eij ) =

αi − αj , hH eij if i, j satisfy αi = αj αi − αj H

(3.99)

3.5 Meyer’s Problem for Semigroups of Schur Multipliers

129

and Rα,h (eij ) = 0 if it is not the case. If (ek )k∈K is an orthonormal basis of the Hilbert space H , we let def

Rα,k = Rα,ek .

(3.100)

Suppose 1 < p < ∞. Consider the contractive linear map U : L2 (q (H )) → 2K , g →

  sq (ek ), g L2 (q (H )) ek . k def

p

By (2.71), we have a bounded map u = IdS p ⊗ U : SI (L2 (q (H ))c,p ) → p

p

I

p

SI ( 2K,c,p ) ⊂ SI (SK ). For any g ∈ L2 (q (H )) and any i, j ∈ I we have u(eij ⊗ g) = eij ⊗



sq (ek ), g



e L2 (q (H )) k1

k

=



sq (ek ), g



e L2 (q (H )) ij

⊗ ek1 .

k

(3.101) With the linear form sq (ek ), · : L2 (q (H )) → C and (2.71), we can introduce the map def

p

p

uk = IdS p ⊗ sq (ek ), · : SI (L2 (q (H ))c,p ) → SI . I

For any g ∈ L2 (q (H )), any i, j ∈ I and any k ∈ K, we have   uk (eij ⊗ g) = sq (ek ), g L2 (q (H )) eij .

(3.102)

p

For any k ∈ I and any f ∈ L2 (q (H )) ⊗ SI , we have u(f ) =



uk (f ) ⊗ ek1 .

(3.103)

k

In the following proposition, recall again that we have a canonical conditional p expectation E : Lp (q (H )⊗B( 2I )) → SI . Proposition 3.12 Let −1  q  1. p

1. Suppose 2  p < ∞. For any f, h ∈ Gaussq,p (SI ) we have E(f ∗ h) = (u(f ))∗ u(h)

(3.104) p

where we identify (u(f ))∗ u(h) as an element of SI2 (with its unique non-zero entry).

130

3 Riesz Transforms Associated to Semigroups of Markov Multipliers p

2. Suppose 1 < p < 2. For any f, h ∈ Gaussq,2 (C) ⊗ SI we have

f, hLpc (E) = (u(f ))∗ u(h)

(3.105)

p

where we identify (u(f ))∗ u(h) as an element of SI2 (with its unique non-zero entry). 3. Suppose 1 < p < ∞. If x ∈ MI,fin ∩ Ran A, we have11 −1

uk ∂α,q,p Ap 2 (x) = Rα,k (x).

(3.106)

p

4. Suppose 1 < p < ∞. If h ∈ SI (L2 (q (H ))c,p ), for any k ∈ K we have

uk (h∗ ))∗ = uk (h).

(3.107)

Proof 1 and 2. For any i, j, r, s ∈ I and any g, w ∈ Gaussq,2 (C) (this assumption is used in a crucial way in the last equality), we have (u(eij ⊗ g))∗ u(ers ⊗ w)  ∗       (3.101)   sq (ek ), g L2 eij ⊗ ek1 sq (el ), w L2 ers ⊗ el1 = =



k



l

 sq (ek ), g L2 (

q

e∗ ⊗ e1k (H )) ij

 

k

=

 



sq (el ), w



 L2 (q

e ⊗ el1 (H )) rs

l

sq (ek ), g





s (e ), w L2 (q (H ) q l



e∗ e L2 (q (H )) ij rs

 ⊗ e1k el1

k,l

=



sq (ek ), g







s (e ), w L2 ( (H )) eij∗ ers L2 (q (H )) q k q

⊗ e11

k

=



g, wL2 (q (H )) eij∗ ers ⊗ e11 . k

On the other hand, we have (2.67)

eij ⊗ g, ers ⊗ wS p (L2 (q (H ))c,p ) = g, wL2 (q (H )) eij∗ ers . I

We conclude by bilinearity and density.

11 Recall

p

that if 2  p < ∞, we have Lp (q (H )⊗B( 2I )) ⊂ SI (L2 (q (H ))c,p ).

3.5 Meyer’s Problem for Semigroups of Schur Multipliers

131

3. For any i, j ∈ I such that αi = αj , we have 

−1

uk ∂α,q,p Ap 2 (eij ) = uk ∂α,q,p (2.95)

=

(2.41)

=

 =

1 uk ∂α,q,p (eij ) αi − αj H

1 uk eij ⊗ sq (αi − αj ) αi − αj H

(3.102)

=

1 eij αi − αj H

  1 sq (ek ), sq (αi − αj ) L2 (q (H )) eij αi − αj H

ek , αi − αj H eij αi − αj H

(3.99)(3.100)

=

Rα,k (eij ).

4. Since sq (ek ) is selfadjoint, we have for any i, j ∈ I and any g ∈ L2 (q (H )) 

uk ((eij ⊗ g)∗ )

∗

 ∗ = uk (ej i ⊗ g ∗ )

( )∗  sq (ek ), g ∗ L2 (q (H )) ej i

(3.102)

=

    = sq (ek ), g ∗ L2 (q (H )) eij = g ∗ , sq (ek ) L2 (q (H )) eij = τ (gsq (ek ))eij   = τ (sq (ek )g)eij = sq (ek ), g L2 (q (H )) eij

(3.102)

= uk (eij ⊗ g).  

We conclude by linearity and density. p

Lemma 3.29 Suppose 1 < p  2. The restriction of u : SI (L2 (q (H ))c,p ) → p p p SI (SK ) on SI (Gaussq,2 (C)c,p ) induces an isometric map and the range of this p restriction is SI ( 2K,c,p ). p

Proof For any f ∈ SI ⊗ Gaussq,2 (C), we have 1 (2.66)  f Lpc (E) =  f, f Lpc (E)  2 p SI2

 1 = (u(f ))∗ u(f ) 2 p = u(f )S p (S p ) .

(3.105)

I

SI2

K

(3.101)

and u(eij ⊗ sq (el )) = eij ⊗ el for any i, j ∈ I  and any k ∈ K.Alternatively, note that the linear map Gaussq,2 (C) → 2K , g → k∈K sq (ek ), g L2 (q (H )) ek is p

p

a surjective isometry. So the associated map SI (Gaussq,2 (C)c,p ) → SI ( 2K,c,p ) is also a surjective isometry.   Theorem 3.6 Suppose 1 < p < ∞. 1. If 1 < p  2 and if x ∈ MI,fin ∩ Ran A we have x

p SI

 1  2   2   ≈p inf |a | k  Rα,k (x)=ak +bk  k∈K

p

SI

 1    ∗ 2 2   + |bk |  k∈K p

where the infimum is taken over all (ak ), (bk ) ∈ SI ( 2K,c ).

p

SI

(3.108)

132

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

2. If 2  p < ∞ and if x ∈ MI,fin ∩ Ran A, we have xS p I

  1    1  4 2   

∗ 2 2  2  ,  ≈p max  |Rα,k (x)| | Rα,k (x) |  p   p .  SI

k∈K

SI

k∈K

Proof Here we use some fixed −1  q  1 and we drop the index in the notation ∂α,(q,)p . 1. Suppose 1 < p < 2. For any x ∈ MI,fin ∩ Ran A, we have  1 − 1  (3.88)   −1 xS p = Ap2 Ap 2 (x)S p ≈p ∂α,p Ap 2 (x)Lp ( (H )⊗B( 2 )) q I

I

(3.72) ≈p

I

gLpc (E) + hLpr (E)

inf

−1

∂α,p Ap 2 (x)=g+h

where the infimum is taken over all g, h ∈ Gauss2 (C) ⊗ MI,fin . By Proposition 3.12, we infer that 1 2

g, gLp (E)

1 = u(g)∗ u(g) 2

(3.105)

c

(3.103)

=

 

uk (g) ⊗ ek1

∗  

k

=





l

1

∗

uk (g)) ul (g) ⊗ e1k el1

ul (g) ⊗ el1

 12

2

=



k,l



1 2 uk (g)) uk (g) . ∗

k

Similarly, we have 1

h∗ , h∗ L2 p (E) c

(3.105)



(3.103)

 

= =

(u(h∗ ))∗ u(h∗ )

1 2

∗

uk (h ) ⊗ ek1

∗  

k (3.107)

=

 



ul (h ) ⊗ el1

 12

l

uk (h) ⊗ e1k

 

k

=

∗

uk (h)uk (h∗ )

∗

ul (h ) ⊗ el1

 12

l

1 2

.

k p

p

p

By Lemma 3.29, the restriction of the map u : SI (L2 (q (H ))c,p ) → SI (SI ) on − 12

p

SI (Gaussq,2 (C)c,p ) is injective. So we have ∂α,p Ap (x) = g + h if and only −1

if u∂α,p Ap 2 (x) = u(g) + u(h). By (3.106) and (3.103), this is equivalent to

3.5 Meyer’s Problem for Semigroups of Schur Multipliers

133

Rα,k (x) = uk (g) + uk (h) for any k. Hence this computation gives that xS p is I comparable to the norm  1 2  

∗  inf uk (g) uk (g)    Rα,k (x)=uk (g)+uk (h)

p

SI

k∈K

 1  2  

∗  uk (h)uk (h )  +  p. SI

k∈K

(3.109) Therefore, using again Lemma 3.29, this infimum is finally equal to the infimum (3.108). 2. Now, suppose 2  p < ∞. For any x ∈ MI,fin ∩ Ran A, we have  1 − 1  (3.88)   −1 xS p = Ap2 Ap 2 (x)S p ≈p ∂α,p Ap 2 (x)Gauss (S p ) q,p I

I

I

1   2  ∂α,p A− p (x) Lprc (E)  ( ( )) 1  

∗ − 21 − 12 2  , = max  E ∂ A (x) ∂ A (x) α,p p α,p p  

(3.73) ≈p

p

( (   1 

∗ )) 12  − 21 2  E ∂α A−  . p (x) ∂α,p Ap (x)   p

We obtain (

) (3.104) 

∗

− 1 ∗ −1 −1 −1 E ∂α,p Ap 2 x ∂α,p Ap 2 x = u ∂α,p Ap 2 (x) u ∂α,p Ap 2 (x) ∗     (3.106) = Rα,k (x) ⊗ ek1 Rα,l (x) ⊗ el1 k∈K

=



l∈K

∗ Rα,k (x) ⊗ e1k

k∈K

 



Rα,l (x) ⊗ el1

l∈K

 

∗ Rα,k (x) Rα,k (x) = |Rα,k (x)|2 . = k∈K

k∈K

Similarly, using (3.107) in addition, we obtain ) ( ∗



−1 − 1 ∗ (3.104)

−1 −1 E ∂α,p Ap 2 (x) ∂α,p Ap 2 x = u (∂α,p Ap 2 x)∗ u (∂α,p Ap 2 x)∗  ∗   



− 12 ∗ − 12 ∗ (3.103)  = uk (∂α,p Ap x) ⊗ ek1 ul (∂α,p Ap x) ⊗ el1 k

(3.107)

=

 k

l



− 1 ∗ uk ∂α,p Ap 2 x ⊗ ek1

∗   l



−1

ul (∂α,p Ap 2 x)

∗

 ⊗ el1

134

3 Riesz Transforms Associated to Semigroups of Markov Multipliers

=



−1 uk ∂α,p Ap 2 x ⊗ e1k



k (3.106)

=



 



− 1 ∗ ul (∂α,p Ap 2 x)

 ⊗ el1

l

Rα,k (x) ⊗ e1k

k

    

  Rα,k (x) ∗ 2 . (Rα,l (x))∗ ⊗ el1 = l

k∈K

The proof is complete.  

Chapter 4

Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

Abstract In this chapter, we introduce Hodge-Dirac operators associated with markovian semigroups of Fourier multipliers and we show that these operators admit a bounded H∞ functional calculus on a bisector. We also provide Hodge decompositions. We equally show a similar result for Hodge-Dirac operators associated with markovian semigroups of Schur multipliers. Particular attention is paid to the domain of the Hodge-Dirac operators and different choices are discussed in the course of the chapter. We also prove independence of the bounds of the H∞ functional calculus, on the group G or index set I , and on the markovian semigroup of multipliers.

4.1 Boundedness of Functional Calculus of Hodge-Dirac Operators for Fourier Multipliers In this section, we consider a discrete group G and a semigroup (Tt )t 0 of Markov Fourier multipliers satisfying Proposition 2.3. If 1  p < ∞, we denote by Ap the (negative) infinitesimal generator on Lp (VN(G)). By Junge et al. [128, (5.2)], we have (Ap )∗ = Ap∗ if 1 < p < ∞. If −1  q  1, recall that by Proposition 3.4, we have a closed operator ∂ψ,q,p : dom ∂ψ,q,p ⊂ Lp (VN(G)) → Lp (q (H ) α G),

λs → sq (bψ (s))  λs

and a closed operator (∂ψ,q,p∗ )∗ : dom(∂ψ,q,p∗ )∗ ⊂ Lp (q (H )  G) → Lp (VN(G)). Now, we will define a Hodge-Dirac operator Dψ,q,p in (4.16), from ∂ψ,q,p and its adjoint. Then the main topic of this section will be to show that Dψ,q,p is R-bisectorial (Theorem 4.2) and has a bounded H∞ functional calculus on Lp (VN(G)) ⊕ Ran ∂ψ,q,p (Theorem 4.3). By Remark 4.1, this extends the Kato square root equivalence from (3.31). Note that we use Proposition 2.8, so we need some assumption on the group G. First, with exception of Proposition 4.4, we assume that q (H ) α G has QWEP (e.g. G is amenable or is a free group and q = ±1 by an adaptation of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Arhancet, C. Kriegler, Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers, Lecture Notes in Mathematics 2304, https://doi.org/10.1007/978-3-030-99011-4_4

135

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

136

[10, Proposition 4.8]). Second, since we need approximating compactly supported Fourier multipliers, from Lemma 4.4 on, we also assume that G is weakly amenable. We start with the following intuitive formula which show that ∂ψ,q,p can be seen as a “gradient” for Ap in the spirit of the link between the classical Laplacian and the classical gradient. Proposition 4.1 Suppose 1 < p < ∞ and −1  q  1. As unbounded operators, we have Ap = (∂ψ,q,p∗ )∗ ∂ψ,q,p .

(4.1)

Proof By Lemma 3.6 and [136, p. 167], ∂ψ,q,p (PG ) is a subspace of dom(∂ψ,q,p∗ )∗ . For any s ∈ G, we have

(2.84) (∂ψ,q,p∗ )∗ ∂ψ,q,p (λs ) = (∂ψ,q,p∗ )∗ sq (bψ (s))  λs (4.2) 2 (2.41)  (3.28)

= τ sq (bψ (s))sq (bψ (s)) λs = bψ (s)H λs = Ap (λs ). Hence for any x, y ∈ PG , by linearity we have 1     12 Ap (x), Ap2 ∗ (y) Lp (VN(G)),Lp∗ (VN(G)) = Ap (x), y Lp (VN(G)),Lp∗ (VN(G))

(4.2) 

 = (∂ψ,q,p∗ )∗ ∂ψ,q,p (x), y Lp (VN(G)),Lp∗ (VN(G))  (2.5)  = ∂ψ,q,p (x), ∂ψ,q,p∗ (y) Lp ,Lp∗ .

Using the part 3 of Proposition 3.4, it is not difficult to see that this identity extends to elements x ∈ dom Ap . For any x ∈ dom Ap and any y ∈ PG , we obtain 

Ap (x), y



∗

Lp (VN(G)),Lp (VN(G))

  = ∂ψ,q,p (x), ∂ψ,q,p∗ (y) Lp ,Lp∗ .

Recall that PG is a core of ∂ψ,q,p∗ by the part 1 of Proposition 3.4. So using (2.3), it is easy to check that this identity remains true for elements y of dom ∂ψ,q,p∗ . By (2.4), this implies that ∂ψ,q,p (x) ∈ dom(∂ψ,q,p∗ )∗ and that (∂ψ,q,p∗ )∗ ∂ψ,q,p (x) = Ap (x). We conclude that Ap ⊂ (∂ψ,q,p∗ )∗ ∂ψ,q,p . To prove the other inclusion we consider some x ∈ dom ∂ψ,q,p such that ∂ψ,q,p (x) belongs to dom(∂ψ,q,p∗ )∗ . By Kato [136, Theorem 5.29 p. 168], we have ∗ (2.9) (2.10)

(∂ψ,q,p∗ )∗∗ = ∂ψ,q,p∗ . We infer that (∂ψ,q,p )∗ ∂ψ,q,p∗ ⊂ (∂ψ,q,p∗ )∗ ∂ψ,q,p ⊂ A∗p . For any y ∈ PG , using ∂ψ,q,p (x) ∈ dom(∂ψ,q,p∗ )∗ in the last equality, we

4.1 Boundedness of Functional Calculus of Hodge-Dirac Operators for Fourier. . .

137

deduce that 

A∗p (y), x



∗

Lp (VN(G)),Lp (VN(G))

  = (∂ψ,q,p )∗ ∂ψ,q,p∗ (y), x Lp∗ (VN(G)),Lp (VN(G))   = ∂ψ,q,p∗ (y), ∂ψ,q,p (x) Lp∗ ,Lp

(2.5)

  = y, (∂ψ,q,p∗ )∗ ∂ψ,q,p (x) Lp∗ ,Lp .

(2.5)

Since PG is a core for A∗p = Ap∗ by definition, this implies [136, Problem 5.24 p. ∗ 168] that x ∈ dom A∗∗   p = Ap and that Ap (x) = (∂ψ,q,p ∗ ) ∂ψ,q,p (x). Now, we show how the noncommutative gradient ∂ψ,q,p commutes with the semigroup and the resolvents of its generator. Lemma 4.1 Let G be a discrete group such that q (H ) α G has QWEP. Suppose 1 < p < ∞ and −1  q  1. If x ∈ dom ∂ψ,q,p and t  0, then Tt,p (x) belongs to dom ∂ψ,q,p and we have

IdLp (q (H ))  Tt,p ∂ψ,q,p (x) = ∂ψ,q,p Tt,p (x).

(4.3)

Proof For any s ∈ G, we have



(2.84)

IdLp (q (H ))  Tt,p ∂ψ,q,p (λs ) = IdLp (q (H ))  Tt,p sq (bψ (s))  λs = e−t bψ (s) sq (bψ (s))  λs 2

(2.84) −t bψ (s)2

= e

∂ψ,q,p (λs )

2 = ∂ψ,q,p e−t bψ (s) λs = ∂ψ,q,p Tt,p (λs ). So by linearity the equality (4.3) is true for elements of PG . Now consider some x ∈ dom ∂ψ,q,p . By Kato [136, p. 166], since ∂ψ,q,p is the closure of ∂ψ,q : PG ⊂ Lp (VN(G)) → Lp (q (H ) α G), there exists a sequence (xn ) of elements of PG converging to x in Lp (VN(G)) such that the sequence (∂ψ,q,p (xn )) converges to ∂ψ,q,p (x). The complete boundedness of Tt,p : Lp (VN(G)) → Lp (VN(G)) implies by Proposition 2.8 that we have a (completely) bounded linear operator IdLp (q (H ))  Tt,p : Lp (q (H ) α G) → Lp (q (H ) α G). We infer that in Lp (VN(G)) and Lp (q (H ) α G) we have Tt,p (xn ) −−−−→ Tt,p (x) and n→+∞





IdLp (q (H ))  Tt,p ∂ψ,q,p (xn ) −−−−→ IdLp (q (H ))  Tt,p ∂ψ,q,p (x). n→+∞

For any integer n, we have IdLp (q (H ))  Tt,p ∂ψ,q,p (xn ) = ∂ψ,q,p Tt,p (xn ) by the first part of the proof. Since the left-hand side converges, we obtain that the sequence (∂ψ,q,p Tt,p (xn )) converges to (IdLp (q (H ))  Tt,p )∂ψ,q,p (x) in Lp (q (H ) α G).

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

138

Since each Tt,p (xn ) belongs to dom ∂ψ,q,p , the closedness of ∂ψ,q,p and (2.1)

shows that Tt,p (x) belongs to dom ∂ψ,q,p and that ∂ψ,q,p Tt,p (x) = IdLp (q (H ))  Tt,p ∂ψ,q,p (x).   Proposition 4.2 Let G be a discrete group such that q (H ) α G has QWEP. Suppose 1 < p < ∞ and −1  q  1. For any s  0 and any x ∈ dom ∂ψ,q,p , we

−1 have Id + sAp (x) ∈ dom ∂ψ,q,p and



−1 Id  (Id + sAp )−1 ∂ψ,q,p (x) = ∂ψ,q,p Id + sAp (x).

(4.4)

Proof Note that for any s > 0 and any x ∈ Lp (VN(G)) (resp. x ∈ dom ∂ψ,q,p ) −1 the continuous functions R+ → Lp (VN(G)), t → e−s t Tt,p (x) and R+ → −1 Lp (q (H ) α G), t → e−s t (Id  Tt,p )∂ψ,q,p (x) are Bochner integrable. If t > 0 and if x ∈ dom ∂ψ,q,p , taking Laplace transforms on both sides of (4.3) and using [110, Theorem 1.2.4] and the closedness of ∂ψ,q,p in the penultimate equality, we ∞ −1 obtain that 0 e−t s Tt,p (x) dt belongs to dom ∂ψ,q,p and that



−1 Id  (s −1 Id + Ap )−1 ∂ψ,q,p (x) = − − s −1 Id − (Id  Ap ) ∂ψ,q,p (x)  ∞ −1 (2.18) = e−s t (Id  Ts,p )∂ψ,q,p (x) dt (4.3)

=



0 ∞ 0

e−s

−1 t



∞

= ∂ψ,q,p

∂ψ,q,p Tt,p (x) dt e−s

−1 t

 Tt,p (x) dt

0

−1 (2.18) = ∂ψ,q,p s −1 Id + Ap (x). We deduce the desired identity by multiplying by s −1 .

 

Now, we prove a result which gives some R-boundedness. Proposition 4.3 Suppose 1 < p < ∞ and −1  q  1. Let G be a discrete group. The family   t∂ψ,q,p (Id + t 2 Ap )−1 : t > 0

(4.5)

of operators of B(Lp (VN(G)), Lp (q (H ) α G)) is R-bounded. −1

Proof Note that the operator ∂ψ,q,p Ap 2 : Ran Ap → Lp (q (H ) α G) is bounded by (3.31). Suppose t > 0. A standard functional calculus argument gives ( ) 1 −1 t∂ψ,q,p (Id + t 2 Ap )−1 = ∂ψ,q,p Ap 2 (t 2 Ap ) 2 (Id + t 2 Ap )−1 .

(4.6)

4.1 Boundedness of Functional Calculus of Hodge-Dirac Operators for Fourier. . .

139

By Arhancet [13, 14] or [12], note that Ap has a bounded H∞ (θ ) functional calculus for some 0 < θ < π2 . Moreover, the Banach space Lp (VN(G)) is UMD by Pisier and Xu [191, Corollary 7.7], hence has the triangular contraction property () by Hytönen et al. [110, Theorem 7.5.9]. We deduce by Hytönen et al. [110, Theorem 10.3.4 (2)] that the operator Ap is R-sectorial. By Hytönen et al. [110, Example 10.3.5] applied with α = 12 and β = 1, we infer that the set 

1

(t 2 Ap ) 2 (Id + t 2 Ap )−1 : t > 0



of operators of B(Lp (VN(G))) is R-bounded. Recalling that a singleton is Rbounded by Hytönen et al. [110, Example 8.1.7], we obtain by composition [110, Proposition 8.1.19 (3)] that the set ( )   1 −1 ∂ψ,q,p Ap 2 (t 2 Ap ) 2 (Id + t 2 Ap )−1 : t > 0 of operators of B(Lp (VN(G)), Lp (q (H ) α G)) is R-bounded. Hence with (4.6) we conclude that the subset (4.5) is R-bounded.   Our Hodge-Dirac operator in (4.16) below will be constructed out of ∂ψ,q,p and the unbounded operator (∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p . Note that the latter is by definition an unbounded operator on the Banach space Ran ∂ψ,q,p with values in Lp (VN(G)) having domain dom(∂ψ,q,p∗ )∗ ∩ Ran ∂ψ,q,p . Lemma 4.2 Let G be a discrete group. Suppose 1 < p < ∞ and −1  q  1. The operator (∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p is densely defined and is closed. More precisely, the subspace ∂ψ,q,p (PG ) of dom(∂ψ,q,p∗ )∗ is dense in Ran ∂ψ,q,p . Proof Let y ∈ Ran  ∂ψ,q,p . Let ε > 0. There exists x ∈ dom ∂ψ,q,p such that y − ∂ψ,q,p (x) < ε. By Proposition 3.4, there exist xfin ∈ PG such that x − xfin Lp (VN(G)) < ε and   ∂ψ,q,p (x) − ∂ψ,q,p (xfin ) p < ε. L (q (H )α G)   We deduce that y − ∂ψ,q,p (xfin )Lp (q (H )α G < 2ε. By Proposition 4.1, ∂ψ,q,p (PG ) is a subspace of dom(∂ψ,q,p∗ )∗ . So ∂ψ,q,p (xfin ) belongs to dom(∂ψ,q,p∗ )∗ . Since (∂ψ,q,p∗ )∗ is closed, the assertion on the closedness is (really) obvious.   According to Lemma 4.1, Id  Tt leaves Ran ∂ψ,q,p invariant for any t  0, so by continuity of Id  Tt also leaves Ran ∂ψ,q,p invariant. By Engel and Nagel [85, pp. 60-61], we can consider the generator def

Bp = (IdLp (q (H ))  Ap )|Ran ∂ψ,q,p . of the restriction of (Id  Tt )t 0 on Ran ∂ψ,q,p .

(4.7)

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

140

Lemma 4.3 Let G be a discrete group such that q (H ) α G has QWEP. Suppose 1 < p < ∞ and −1  q  1. The operator Bp is injective and sectorial on Ran ∂ψ,q,p . Proof The operator Bp is sectorial of type π2 by e.g. [128, p. 25]. For the injectivity, we note that once Bp is known to be sectorial on a reflexive space, we have the projection (2.17) onto the kernel of Bp essentially given by the strong limit P = lim λ(λ + Id  Ap )−1 |Ran ∂ψ,q,p . λ→0+

It is easy to check that P (y) = 0 for any y ∈ ∂ψ,q,p (PG ). Indeed, by linearity, we (2.84)

can assume that y = ∂ψ,q (λs ) = sq (bψ (s))  λs for some s ∈ G such that in addition bψ (s) = 0. We claim that

(λ+IdAp )−1 |Ran ∂ψ,q,p sq (bψ (s))λs =

1  2 sq (bψ (s))λs . λ + bψ (s)

(4.8)

To see (4.8), it suffices to calculate  (λ + Id  Ap ) =

1



 2 sq (bψ (s))  λs λ + bψ (s)

 2

1  2 λsq (bψ (s))  λs + bψ (s) sq (bψ (s))  λs λ + bψ (s)

= sq (bψ (s))  λs . λ 0 Now P (y) = 0 follows from = 0 when λ → 0+ since 2 → λ+bψ (s) bψ (s)2   bψ (s)2 = 0. Since PG is a core for ∂ψ,q,p , it is not difficult to see that P (y) = 0 also for y ∈ Ran ∂ψ,q,p . Finally by continuity of P , we deduce P = 0. Thus, Bp is injective.  

Recall that if G is a weakly amenable discrete group, then by the considerations at the end of Sect. 2.4, there exists a net (IdLp (q (H ))  Mϕj ) of finitely supported crossed product Fourier multipliers converging strongly to the identity. We also recall from (2.58) that Pp,,G is the span of the x  λs ’s, where x ∈ Lp (VN(G)) and s ∈ G. We will use the following result. Lemma 4.4 Let G be a weakly amenable discrete group such that q (H ) α G has QWEP. If z ∈ Pp,,G ∩ Ran ∂ψ,q,p then z ∈ ∂ψ,q,p (PG ). Proof Let (zn ) be a sequence in dom ∂ψ,q,p such that ∂ψ,q,p (zn ) → z. Let Mϕ be a compactly supported completely bounded Fourier multiplier such that (Id 

4.1 Boundedness of Functional Calculus of Hodge-Dirac Operators for Fourier. . .

141

Mϕ )(z) = z. Then   z − ∂ψ,q,p Mϕ (zn )

Lp

(3.33)  

 = z − (Id  Mϕ )∂ψ,q,p (zn )Lp      z − (Id  Mϕ )(z) p + (Id  Mϕ )(z − ∂ψ,q,p (zn ))

Lp

L

   z − (Id  Mϕ )(z)

    −−−−→ 0. Lp + z − ∂ψ,q,p (zn ) Lp − n→+∞

Therefore, the  sequence (∂ψ,q,p Mϕ (zn )) is convergent to z for n → ∞. Write Mϕ (zn ) = s∈F zn,s λs with some zn,s ∈ C and a finite subset F of G. Since for any s ∈ G, the Fourier multiplier Ms associated with the symbol t → δs (t) is completely bounded, (Id  Ms )∂ψ,q,p Mϕ (zn ) = zn,s sq (bψ (s))  λs is convergent for n → ∞. Thus, either bψ (s) = 0 or (zn,s ) is a convergent scalar sequence with limit, say, zs ∈ C. We infer that ∂ψ,q,p Mϕ (zn ) =



zn,s sq (bψ (s))  λs

s∈F

−−−−→ n→+∞

We conclude that z = ∂ψ,q,p (



(2.84)

zs sq (bψ (s))  λs = ∂ψ,q,p



s∈F

 s∈F

 zs λs .

s∈F

zs λs ) belongs to ∂ψ,q,p (PG ).

 

Proposition 4.4 Let G be a discrete group such that q (H ) α G has QWEP. In the first three points below, assume in addition that G is weakly amenable, so that the approximating Fourier multipliers Mϕj exist. Suppose 1 < p < ∞ and −1  q  1. 1. Pp,,G is a core of (∂ψ,q,p∗ )∗ . Furthermore, if y ∈ dom(∂ψ,q,p∗ )∗ , then (Id  Mϕj )(y) belongs to dom(∂ψ,q,p∗ )∗ for any j and (∂ψ,q,p∗ )∗ (Id  Mϕj )(y) = Mϕj (∂ψ,q,p∗ )∗ (y).

(4.9)

2. ∂ψ,q,p (PG ) is a core of (∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p . 3. ∂ψ,q,p (PG ) is equally a core of ∂ψ,q,p (∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p . 4. ∂ψ,q,p (PG ) is equally a core of Bp . Proof 1. Let y ∈ dom(∂ψ,q,p∗ )∗ . Then (IdLp (q (H ))  Mϕj )(y) belongs to Pp,,G . It remains to show that (IdLp (q (H ))  Mϕj )(y) converges to y in the graph norm. Recall that (IdLp (q (H ))  Mϕj )(y) converges to y in Lp (q (H )α G) according to the assumptions. For any s ∈ G, we have   (∂ψ,q,p∗ )∗ (Id  Mϕj )(y), λs   = (Id  Mϕj )(y), ∂ψ,q,p∗ (λs )

(2.5)

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

142

  (3.33)   = y, (Id  Mϕj )(∂ψ,q,p∗ (λs )) = y, ∂ψ,q,p∗ Mϕj (λs )     = (∂ψ,q,p∗ )∗ (y), Mϕj (λs ) = Mϕj (∂ψ,q,p∗ )∗ (y), λs . By linearity and density, we deduce the relation (4.9), i.e. (∂ψ,q,p∗ )∗ (Id  Mϕj )(y) = Mϕj (∂ψ,q,p∗ )∗ (y) which converges to (∂ψ,q,p∗ )∗ (y) in Lp (VN(G)). 2. Let y ∈ dom(∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p and ε > 0. By the proof of the first point, we def

already know that for some j large enough, there is z = (IdMϕj )(y) belonging to Pp,,G such that y − zLp < ε

and

  (∂ψ,q,p∗ )∗ (y) − (∂ψ,q,p∗ )∗ (z) p < ε. L

for δ > 0, since y ∈ Ran ∂ψ,q,p , We claim that z belongs to Ran ∂ψ,q,p . Indeed,  there is some a ∈ dom ∂ψ,q,p such that y − ∂ψ,q,p (a)Lp < δ. But then we have     z − (Id  Mϕ )∂ψ,q,p (a) p = (Id  Mϕ )(y − ∂ψ,q,p (a)) p j j L L      y − ∂ψ,q,p (a) Lp < δ (3.33)

and (IdMϕj )∂ψ,q,p (a) = ∂ψ,q,p Mϕj (a) belongs again to Ran ∂ψ,q,p . Letting δ → 0, it follows that z belongs to Ran ∂ψ,q,p . By Lemma 4.4, we conclude that z belongs to ∂ψ,q,p (PG ). 3. Take y ∈ dom ∂ψ,q,p (∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p , that is, y ∈ dom(∂ψ,q,p∗ )∗ ∩ Ran ∂ψ,q,p such that (∂ψ,q,p∗ )∗ (y) ∈ dom ∂ψ,q,p . We have (Id  Mϕj )(y) → y and (Id  Mϕj )(y) ∈ ∂ψ,q,p (PG ) by the proof of part 2. Moreover, we have ∂ψ,q,p (∂ψ,q,p∗ )∗ (Id  Mϕj )(y) (4.9)

= ∂ψ,q,p Mϕj (∂ψ,q,p∗ )∗ (y)

(3.33)

= (Id  Mϕj )∂ψ,q,p (∂ψ,q,p∗ )∗ (y) −→ ∂ψ,q,p (∂ψ,q,p∗ )∗ (y). j

4. Note that ∂ψ,q,p (PG ) is a dense subspace of Ran ∂ψ,q,p which is clearly a subspace of dom Bp and invariant under each operator (Id  Tt )|Ran ∂ψ,q,p by Lemma 4.1. By Lemma 2.2, we deduce that ∂ψ,q,p (PG ) is a core of (Id  (4.7)

Ap )|Ran ∂ψ,q,p = Bp .

 

4.1 Boundedness of Functional Calculus of Hodge-Dirac Operators for Fourier. . .

143

Proposition 4.5 Let G be a weakly amenable discrete group such that q (H )α G has QWEP. Suppose 1 < p < ∞ and −1  q  1. −1

(∂ψ,q,p∗ )∗ induces a bounded operator 1. For any s > 0, the operator Id + sAp on Ran ∂ψ,q,p .

2. For any s  0 and any y ∈ Ran ∂ψ,q,p ∩ dom(∂ψ,q,p∗ )∗ , the element Id  (Id + sAp )−1 (y) belongs to dom(∂ψ,q,p∗ )∗ and

Id + sAp

−1

(∂ψ,q,p∗ )∗ (y) = (∂ψ,q,p∗ )∗ Id  (Id + sAp )−1 (y).

(4.10)

3. For any t  0 and any y ∈ Ran ∂ψ,q,p ∩ dom(∂ψ,q,p∗ )∗ , the element (Id  Tt )(y) belongs to dom(∂ψ,q,p∗ )∗ and

Tt (∂ψ,q,p∗ )∗ (y) = (∂ψ,q,p∗ )∗ Id  Tt (y). Proof ∗ (2.10)

1. Note that (Id + sAp )−1 (∂ψ,q,p∗ )∗ ⊂ ∂ψ,q,p∗ (Id + sAp∗ )−1 . By Propo

∗ sition 4.3, the operator ∂ψ,q,p∗ (Id + sAp∗ )−1 is bounded. By Lemma 4.2, ∗ the subspace ∂ψ,q,p (PG ) of dom(∂ψ,q,p∗ ) is dense in Ran ∂ψ,q,p . Now, the conclusion is immediate. 2. By Proposition 4.1, for any x ∈ dom Ap we have x ∈ dom ∂ψ,q,p and ∂ψ,q,p (x) ∈ dom(∂ψ,q,p∗ )∗ . Moreover, for all t > 0 we have (4.1)

(2.13)

(4.1)

Tt (∂ψ,q,p∗ )∗ ∂ψ,q,p (x) = Tt Ap (x) = Ap Tt (x) = (∂ψ,q,p∗ )∗ ∂ψ,q,p Tt (x) (4.3)

= (∂ψ,q,p∗ )∗ (Id  Tt )∂ψ,q,p (x).

By taking Laplace transforms with (2.18) and using the closedness of (∂ψ,q,p∗ )∗ , we deduce that the element (Id+sIdAp )−1 ∂ψ,q,p (x) belongs to dom(∂ψ,q,p∗ )∗ for any s  0 and that (Id + sAp )−1 (∂ψ,q,p∗ )∗ ∂ψ,q,p (x) = (∂ψ,q,p∗ )∗ (Id  (Id + sAp ))−1 ∂ψ,q,p (x). (4.11) Let y ∈ Ran ∂ψ,q,p ∩ dom(∂ψ,q,p∗ )∗ . Then according to Lemma 4.2, there exists a sequence (xn ) of PG such that ∂ψ,q,p (xn ) → y. We have (Id  (Id + sAp ))−1 ∂ψ,q,p (xn ) → (Id(Id+sAp ))−1 (y). Since each xn belongs to dom Ap , using the first point in the passage to the limit we deduce that (4.11)

(∂ψ,q,p∗ )∗ (Id  (Id + sAp ))−1 ∂ψ,q,p (xn ) = (Id + sAp )−1 (∂ψ,q,p∗ )∗ ∂ψ,q,p (xn ) −−−−−→ (Id + sAp )−1 (∂ψ,q,p∗ )∗ (y). n→+∞

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

144

Since (∂ψ,q,p∗ )∗ is closed, we infer by (2.1) that (Id  (Id + sAp ))−1 (y) belongs to dom(∂ψ,q,p∗ )∗ and that (∂ψ,q,p∗ )∗ (Id  (Id + sAp ))−1 (y) = (Id + sAp )−1 (∂ψ,q,p∗ )∗ (y).

(4.12)

Thus (4.10) follows. 3. Let y ∈ Ran ∂ψ,q,p ∩ dom(∂ψ,q,p∗ )∗ . If t  0, note that   −n t (2.12) Id  Id + Ap (y) −−−−→ (Id  Tt )(y). n→+∞ n Repeating the commutation relation (4.12) together with the observation that (Id  (Id + sAp ))−1 maps ∂ψ,q,p (PG ) into itself hence by continuity Ran ∂ψ,q,p into itself yields1 for any integer n  1 and any t  0 (∂

ψ,q,p ∗

   −n −n t t ) Id  Id + Ap (y) = Id + Ap (∂ψ,q,p∗ )∗ (y) n n ∗

(2.12)

−−−−→ Tt (∂ψ,q,p∗ )∗ (y). n→+∞

Then by the closedness of (∂ψ,q,p∗ )∗ , we deduce that (Id  Tt )(y) belongs to dom(∂ψ,q,p∗ )∗ and that (∂ψ,q,p∗ )∗ (Id  Tt )(y) = Tt (∂ψ,q,p∗ )∗ (y). Thus the third point follows.   Proposition 4.4 enables us to identify Bp in terms of ∂ψ,q,p and its adjoint. Proposition 4.6 Let G be a weakly amenable discrete group such that q (H )α G has QWEP. Let 1 < p < ∞ and −1  q  1. As unbounded operators, we have Bp = ∂ψ,q,p (∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p . Proof For any s ∈ G, we have  2 (4.1) ∂ψ,q,p (∂ψ,q,p∗ )∗ ∂ψ,q,p (λs ) = ∂ψ,q,p Ap (λs ) = bψ (s)H ∂ψ,q,p (λs ) 2 (2.84)  = bψ (s)H sq (bψ (s))  λs

1

Note that we replace s by nt .

(4.13)

4.1 Boundedness of Functional Calculus of Hodge-Dirac Operators for Fourier. . .

145

= (Id  Ap ) sq (bψ (s))  λs

= (Id  Ap ) ∂ψ,q,p (λs ) .

(2.84)

We deduce that the operators ∂ψ,q,p (∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p and (IdLp (q (H ))  Ap ) coincide on ∂ψ,q,p (PG ). By Proposition 4.4, ∂ψ,q,p (PG ) is a core for each operator. We conclude that they are equal.   In the proof of Theorem 4.1, we shall use the following folklore lemma, see e.g. [151, Proposition 2.13] relying essentially on [110, Proposition 10.7.2]. Lemma 4.5 Let (Tt )t 0 be a bounded strongly continuous semigroup on a Banach space X. Let −A be its infinitesimal generator. For any π2 < θ < π the following are equivalent 1. A admits a bounded H∞ (θ ) functional calculus. 2. There exists a constant C > 0 such that for any b ∈ L1 (R+ ) whose Laplace transform L (b) belongs to H∞ 0 (θ ), we have    

+∞ 0

  b(t)Tt dt  

 C L (b)H∞ (θ ) .

(4.14)

X→X

Recall that the preceding integral is defined in the strong operator topology sense. Note that the QWEP assumption of the following result is satisfied if G = Fn is a free group and q = −1. Theorem 4.1 Let G be a discrete group. Suppose 1 < p < ∞ and −1  q  1. Let G be a weakly amenable discrete group such that q (H ) α G has QWEP. The operators Ap and Id  Ap have a bounded H∞ (θ ) functional calculus of angle θ for any θ > π| p1 − 12 |. Proof For the QWEP property in case of G being a free group, we refer to [10, Proposition 4.8]. According to Proposition 2.8, Id  Tt extends to a (completely) contractive operator on Lp (q (H ) α G). Moreover, since P,G is dense in Lp (q (H ) α G), the property of (Tt )t 0 being a strongly continuous semigroup carries over to (Id  Tt )t 0 . According to [13, Theorem 4.1], [14] or [86, Corollary 4] (see also [12]), Ap has a (completely) bounded H∞ (θ ) functional calculus on Lp (VN(G)) for any π2 < θ < π. By Lemma 4.5, for any b ∈ L1 (R+ ) such that L (b) belongs to H∞ 0 (θ ) we have    

∞ 0

  b(t)Tt dt  

cb,Lp (VN(G))→Lp (VN(G))

 L (b)H∞ (θ ) .

(4.15)

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

146

Note that for any λs ∈ PG , we have 

∞

  b(t)Tt dt (λs ) =

0

∞



∞

b(t)Tt (λs ) dt =

0

b(t)e−t bψ (s) λs dt 2

0



∞

=

b(t)e

−t bψ (s)

2

 dt (λs ).

0

So the map    

∞ 0

∞

b(t)Tt dt is also a Fourier multiplier. Thus, by Proposition 2.8

0

  b(t)Id  Tt (x) dt  

    = Id      

(2.62)

0

Lp (q (H )α G)→Lp (q (H )α G)

  b(t)Tt (x) dt   0  ∞  b(t)Tt (x) dt   ∞

(4.15)

cb,Lp (VN(G))→Lp (VN(G))

 L (b)H∞ (θ ) .

By Lemma 4.5, Id  Ap admits a bounded H∞ (θ ) functional calculus for any θ > π2 . Now, we reduce the angle and conclude with [128, Proposition 5.8].   Suppose 1 < p < ∞. We introduce the unbounded operator def

Dψ,q,p =



0 ∂ψ,q,p

(∂ψ,q,p∗ )∗ 0

 (4.16)

on the Banach space Lp (VN(G)) ⊕p Ran ∂ψ,q,p defined by def

Dψ,q,p (x, y) = (∂ψ,q,p∗ )∗ (y), ∂ψ,q,p (x)

(4.17)

for x ∈ dom ∂ψ,q,p , y ∈ dom(∂ψ,q,p∗ )∗ ∩ Ran ∂ψ,q,p . We call it the Hodge-Dirac operator of the semigroup. This operator is a closed operator and can be seen as a differential square root of the generator of the semigroup (Tt,p )t 0 since we have Proposition 4.7. Theorem 4.2 Let G be a weakly amenable discrete group such that q (H ) α G has QWEP. Suppose 1 < p < ∞ and −1  q  1. The Hodge-Dirac operator Dψ,q,p is R-bisectorial on Lp (VN(G)) ⊕p Ran ∂ψ,q,p . Proof We will start by showing that the set {it : t ∈ R, t = 0} is contained in the resolvent set of Dψ,q,p . We will do this by showing that Id − itDψ,q,p has a

4.1 Boundedness of Functional Calculus of Hodge-Dirac Operators for Fourier. . .

147

two-sided bounded inverse (Id − itDψ,q,p )−1 given by 

(Id + t 2 Ap )−1 it (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ 2 −1 it∂ψ,q,p (Id + t Ap ) Id  (Id + t 2 Ap )−1

 (4.18)

acting on Lp (VN(G)) ⊕p Ran ∂ψ,q,p . By Proposition 4.3 and since the operators Ap and IdLp  Ap satisfy the property (2.15) of R-sectoriality (see Theorem 4.1), the four entries are bounded. It only remains to check that this matrix defines a twosided inverse of Id − itDψ,q,p . We have the following equalities of operators acting on dom Dψ,q,p . 

 it (IdLp + t 2 Ap )−1 (∂ψ,q,p∗ )∗ (Id + t 2 Ap )−1 (Id − itDψ,q,p ) it∂ψ,q,p (Id + t 2 Ap )−1 Id  (Id + t 2 Ap )−1   (4.16) it (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ (Id + t 2 Ap )−1 = it∂ψ,q,p (Id + t 2 Ap )−1 Id  (Id + t 2 Ap )−1 < ; IdLp −it (∂ψ,q,p∗ )∗ × −it∂ψ,q,p IdRan ∂ψ,q,p 

=

(Id + t 2 Ap )−1 + t 2 (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ ∂ψ,q,p it∂ψ,q,p (Id + t 2 Ap )−1 − it (Id  (Id + t 2 Ap )−1 )∂ψ,q,p

−it (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ + it (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ t 2 ∂ψ,q,p (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ + Id  (Id + t 2 Ap )−1 ; (Id + t 2 Ap )−1 + t 2 (Id + t 2 Ap )−1 Ap (4.1)(4.4)(4.10)

−1 = it∂ψ,q,p (Id + t 2 Ap )−1 − it∂ψ,q,p Id + t 2 Ap  0 (t 2 ∂ψ,q,p (∂ψ,q,p∗ )∗ + Id)(Id  (Id + t 2 Ap )−1 ) ; < 0 (4.13)(4.7) IdLp (VN(G)) = 0 IdRan ∂ψ,q,p and similarly 

it (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ (Id + t 2 Ap )−1 (Id − itDψ,p ) it∂ψ,q,p (Id + t 2 Ap )−1 Id  (Id + t 2 Ap )−1 < ; Id −it (∂ψ,q,p∗ )∗ = −it∂ψ,q,p IdRan ∂ψ,q,p 

×

(Id + t 2 Ap )−1 it (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ 2 −1 it∂ψ,q,p (Id + t Ap ) Id  (Id + t 2 Ap )−1







4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

148



(Id + t 2 Ap )−1 + t 2 (∂ψ,q,p∗ )∗ ∂ψ,q,p (Id + t 2 Ap )−1 −it∂ψ,q,p (Id + t 2 Ap )−1 + it∂ψ,q,p (Id + t 2 Ap )−1

 it (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ − it (∂ψ,q,p∗ )∗ Id  (Id + t 2 Ap )−1 t 2 ∂ψ,q,p (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ + Id  (Id + t 2 Ap )−1 < ; IdLp 0 . = 0 IdRan ∂ψ,q,p

=

It remains to show that the set {it (it − Dψ,q,p )−1 : t = 0} = {(Id − itDψ,q,p )−1 : t = 0} is R-bounded. For this, observe that the diagonal entries of (4.18) are Rbounded by the R-sectoriality of Ap and IdLp  Ap . The R-boundedness of the other entries follows from the R-gradient bounds of Proposition 4.3. Since a set of operator matrices is R-bounded precisely when each entry is R-bounded, we conclude that (2.19) is satisfied, i.e. that Dψ,q,p is R-bisectorial.   Proposition 4.7 Let G be a weakly amenable discrete group such that q (H )α G has QWEP. Suppose 1 < p < ∞ and −1  q  1. As densely defined closed operators on Lp (VN(G)) ⊕p Ran ∂ψ,q,p , we have 2 Dψ,q,p

  0 Ap . = 0 (IdLp (q (H ))  Ap )|Ran ∂ψ,q,p

(4.19)

Proof By Proposition 4.6, we have   Ap 0 0 (IdLp (q (H ))  Ap )|Ran ∂ψ,q,p   (4.1)(4.13) (∂ψ,q,p ∗ )∗ ∂ψ,q,p 0 = 0 ∂ψ,q,p (∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p  2 0 (∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p (4.16) 2 = Dψ,q,p . = ∂ψ,q,p 0

def

Cp =

  Now, we can state the following main result of this section. Theorem 4.3 Suppose 1 < p < ∞ and −1  q  1. Let G be a weakly amenable discrete group such that q (H )α G has QWEP. The Hodge-Dirac operator Dψ,q,p is R-bisectorial on Lp (VN(G)) ⊕p Ran ∂ψ,q,p and admits a bounded H∞ (ω± ) functional calculus on a bisector. 

 Ap 0 has a bounded H∞ 0 Bp functional calculus of some angle 2ω < π2 . Since Dψ,q,p is R-bisectorial by 2 Proof By Theorem 4.1, the operator Dψ,q,p

(4.19)

=

4.1 Boundedness of Functional Calculus of Hodge-Dirac Operators for Fourier. . .

149

Theorem 4.2, we deduce by Proposition 2.1 that the operator Dψ,q,p has a bounded H∞ (ω± ) functional calculus on a bisector.   Remark 4.1 The boundedness of the H∞ functional calculus of the operator Dψ,q,p implies the boundedness of the Riesz transforms and this result may be thought of as a strengthening of the equivalence (3.31). Indeed, consider the function sgn ∈ def

H∞ (ω± ) defined by sgn(z) = 1ω+ (z) − 1ω− (z). By Theorem 4.3, the operator Dψ,q,p has a bounded H∞ (ω± ) functional calculus on Lp (VN(G)) ⊕p Ran ∂ψ,q,p . Hence the operator sgn(Dψ,q,p ) is bounded. This implies that |Dψ,q,p | = sgn(Dψ,q,p )Dψ,q,p

and Dψ,q,p = sgn(Dψ,q,p )|Dψ,q,p |.

(4.20)

For any element ξ of dom Dψ,q,p = dom |Dψ,q,p |, we deduce that   Dψ,q,p (ξ ) p L (VN(G))⊕

pL

p (

q (H )α G)

  =  sgn(Dψ,q,p )|Dψ,q,p |(ξ )Lp (VN(G))⊕p Lp (q (H )α G)   p |Dψ,q,p |(ξ ) p p (4.20)

L (VN(G))⊕p L (q (H )α G)

and   |Dψ,q,p |(ξ ) p L (VN(G))⊕p Lp (q (H )α G)   =  sgn(Dψ,q,p )Dψ,q,p (ξ )Lp (VN(G))⊕p Lp (q (H )α G)   p Dψ,q,p (ξ )Lp (VN(G))⊕p Lp (q (H )α G) . (4.20)

Recall that on Lp (VN(G)) ⊕p Ran ∂ψ,q,p , we have (4.19)

|Dψ,q,p | =

⎡ 1 2 ⎣Ap

⎤ 0

0 Id

Lp (

q (H ))

1 2

 Ap |Ran ∂ψ,q,p

⎦.

(4.21)

1

By restricting to elements of the form (x, 0) with x ∈ dom Ap2 , we obtain the desired result. Remark 4.2 In a similar way to Remark 4.1, we also obtain for any element (x, y) of dom ∂ψ,q,p ⊕ dom(∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p that 1    (4.21)  21  

|Dψ,q,p |(x, y) ∼ Ap (x) +  IdLp ( (H ))  Ap2 (y) = q p p p

   (4.16)    ∼ = ∂ψ,q,p (x)p + (∂ψ,q,p∗ )∗ (y)p ∼ = Dψ,q,p (x, y)p .

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

150

Similarly, we have 

 IdLp (

q (H ))

1     Ap2 (y)p ∼ = (∂ψ,q,p∗ )∗ (y)p , y ∈ dom(∂ψ,q,p∗ )∗ ∩ Ran ∂ψ,q,p . (4.22)

Proposition 4.8 Let G be a weakly amenable discrete group such that q (H ) α G has QWEP. Suppose 1 < p < ∞ and −1  q  1. We have Ran Ap = Ran(∂ψ,q,p∗ )∗ , Ran Bp = Ran ∂ψ,q,p , Ker Ap = Ker ∂ψ,q,p , Ker Bp = Ker(∂ψ,q,p∗ )∗ = {0} and Lp (VN(G)) = Ran(∂ψ,q,p∗ )∗ ⊕ Ker ∂ψ,q,p .

(4.23)

Here, by (∂ψ,q,p∗ )∗ we understand its restriction to Ran ∂ψ,q,p . However, we have Ran(∂ψ,q,p∗ )∗ = Ran(∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p (see Corollary 4.1). 2 2 Proof By (2.20), we have Ran Dψ,q,p = Ran Dψ,q,p and Ker Dψ,q,p = Ker Dψ,q,p . It is not difficult to prove the first four equalities using (4.19) and (4.17). The last one is a consequence of the definition of Ap and of [110, p. 361].  

4.2 Extension to Full Hodge-Dirac Operator and Hodge Decomposition We keep the standing assumptions of the preceding section and thus we have a markovian semigroup (Tt )t 0 of Fourier multipliers with generator Ap , the noncommutative gradient ∂ψ,q,p and its adjoint (∂ψ,q,p∗ )∗ , together with the HodgeDirac operator Dψ,q,p . We shall now extend the operator Dψ,q,p to a densely defined bisectorial operator Dψ,q,p on Lp (VN(G)) ⊕ Lp (q (H ) α G) which will also be bisectorial and will have an H∞ (ω± ) functional calculus on a bisector. The key will be Corollary 4.1 below. We let def

Dψ,q,p =



0

∂ψ,q,p

(∂ψ,q,p∗ )∗ 0

 (4.24)

along the decomposition Lp (VN(G)) ⊕ Lp (q (H ) α G), with natural domains for ∂ψ,q,p and (∂ψ,q,p∗ )∗ . Again, except in Lemma 4.6, we need in this section approximation properties of G, see before Lemma 4.7. 1

Consider the sectorial operator Ap2 on Lp (VN(G)). According to (2.16), we 1

1

have the topological direct sum decomposition Lp (VN(G)) = Ran Ap2 ⊕ Ker Ap2 . def

−1

1

We define the operator Rp = ∂ψ,q,p Ap 2 : Ran Ap2 → Lp (q (H ) α G).

4.2 Extension to Full Hodge-Dirac Operator and Hodge Decomposition

151 1

According to the point 3 of Proposition 3.4, Rp is bounded on Ran Ap2 , so extends 1

(2.21)

to a bounded operator on Ran Ap2 = Ran Ap . We extend it to a bounded operator Rp : Lp (VN(G)) → Lp (q (H ) α G), called Riesz transform, by putting 1

Rp | Ker Ap2 = 0 along the previous decomposition of Lp (VN(G)). We equally let def

Rp∗ ∗ = (Rp∗ )∗ . Lemma 4.6 Let −1  q  1 and 1 < p < ∞. Then we have the decomposition Lp (q (H ) α G) = Ran ∂ψ,q,p + Ker(∂ψ,q,p∗ )∗ .

(4.25)

Proof Let y ∈ Lp (q (H ) α G) be arbitrary. We claim that y = Rp Rp∗ ∗ (y) + (Id − 1

Rp Rp∗ ∗ )(y) is the needed decomposition for (4.25). Note that Rp maps Ran Ap2 into 1

Ran ∂ψ,q,p , so by boundedness, Rp maps Ran Ap2 to Ran ∂ψ,q,p . Thus, we indeed have Rp Rp∗ ∗ (y) ∈ Ran ∂ψ,q,p . Next we claim that for any z ∈ Lp (VN(G)) and any x ∈ dom ∂ψ,q,p∗ , we have 

1    Rp (z), ∂ψ,q,p∗ (x) Lp (q (H )α G),Lp∗ (q (H )α G) = z, Ap2 ∗ (x) Lp ,Lp∗ .

1

(4.26)

1

According to the decomposition Lp (VN(G)) = Ran Ap2 ⊕ Ker Ap2 , we can write 1

1

1

z = limn→+∞ Ap2 (zn ) + z0 with zn ∈ dom Ap2 and z0 ∈ Ker Ap2 . Then using Lemma 3.7 in the third equality, we have 

 1   Rp (z), ∂ψ,q,p∗ (x) = lim Rp Ap2 (zn ) + z0 , ∂ψ,q,p∗ (x) n→+∞

= lim

n→+∞



1   1   ∂ψ,q,p (zn ), ∂ψ,q,p∗ (x) = lim Ap2 (zn ), Ap2 ∗ (x)

n→+∞



1 1 1       = z − z0 , Ap∗ (x) = z, Ap2 ∗ (x) − z0 , Ap2 ∗ (x) = z, Ap2 ∗ (x) . 1 2

Thus, (4.26) is proved. Now, for any x ∈ dom ∂ψ,q,p∗ , we have 

     (Id − Rp Rp∗ ∗ )(y), ∂ψ,q,p∗ (x) = y, ∂ψ,q,p∗ (x) − Rp Rp∗ ∗ (y), ∂ψ,q,p∗ (x) (4.26)

=



1    y, ∂ψ,q,p∗ (x) − Rp∗ ∗ (y), Ap2 ∗ (x)

1     = y, ∂ψ,q,p∗ (x) − y, Rp∗ Ap2 ∗ (x)

    −1 1 = y, ∂ψ,q,p∗ (x) − y, ∂ψ,q,p∗ Ap∗2 Ap2 ∗ (x) = 0.

By (2.8), we conclude that Id − Rp Rp∗ ∗ (y) belongs to Ker(∂ψ,q,p∗ )∗ .

 

152

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

From now on, we suppose the discrete group G to be weakly amenable such that q (H ) α G has QWEP (e.g. if G is amenable). In the proof of Proposition 4.9 below, we shall need some information on the Wiener-Ito chaos decomposition for q-Gaussians. This is collected in the following lemma. Lemma 4.7 Let −1  q  1 and 1 < p < ∞. Let G be a weakly amenable discrete group such that q (H ) α G has QWEP. Consider an approximating net (Mϕj ) of finitely supported Fourier multipliers. 1. There exists a completely bounded projection P : Lp (q (H )) → Lp (q (H )) onto the closed space spanned by {sq (h) : h ∈ H }. Moreover, the projections are compatible for different values of p. The mapping P  IdLp (VN(G)) extends to a bounded operator on Lp (q (H ) α G). 2. For any j  and any y ∈ Lp (q (H ) α G), the element (P  Mϕj )(y) can be written as s∈supp ϕj sq (hs )  λs for some hs ∈ H . 3. Denoting temporarily by Pp and Mϕj ,p the operator P and Mϕj on the p-level, the identity mapping on Gaussq,p (C)  span{λs : s ∈ supp ϕj } extends to an isomorphism Jp,2,j : Ran(Pp  Mϕj ,p ) → Ran(P2  Mϕj ,2 )

(4.27)

where Ran(Pp  Mϕj ,p ) ⊂ Lp (q (H ) α G). Proof 1. This is contained in [118, Theorem 3.5], putting there d = 1. Note that the closed space spanned by {sq (h), h ∈ H } coincides in this case with G1p,q there. For the fact that the projections are compatible for different values of p, we refer to [118, Proof of Theorem 3.1]. See also the mapping Qp from Lemma 3.4. 2. Once we know that P  Mϕj = (P  Id) ◦ (Id  Mϕj ) is bounded according to point 1. and Proposition 2.8, this is easy and left to the reader. 3. We have           sq (hs ) p  sq (hs )  λs     L (q (H ))  p s∈supp ϕj s∈supp ϕj L (q (H )G)

q,p



hs H

s∈supp ϕj

(the same estimate on the L2 -level). Note that for any s0 ∈ supp ϕj fixed, we have a completely contractive projection of Lp (q (H ) α G) onto span{x  λs0 : x ∈ Lp (q (H ))}. So we have         sq (hs )  λs     s∈supp ϕj

Lp (q (H )α G)

     sq (hs0 )Lp (q (H )) q,p hs0 H .

4.2 Extension to Full Hodge-Dirac Operator and Hodge Decomposition

153

It follows that        ∼ ∼  hs H =  sq (hs )  λs  =  . s∈supp ϕj  2 s∈supp ϕj Lp (q (H )α G) L

        sq (hs )  λs    s∈supp ϕj 



Thus, (4.27) follows.   Proposition 4.9 Let −1  q  1 and 1 < p < ∞. Let G be a weakly amenable discrete group such that q (H )α G is QWEP. Then the subspaces from Lemma 4.6 have trivial intersection, i.e. Ran ∂ψ,q,p ∩ Ker(∂ψ,q,p∗ )∗ = {0}. Proof We begin with the case p = 2. According to Theorem 5.4, the unbounded operator Dψ,q,2 is selfadjoint on L2 (VN(G)) ⊕ L2 (q (H ) α G). We thus have the orthogonal sum Ran Dψ,q,2 ⊕ Ker Dψ,q,2 = L2 (VN(G)) ⊕ L2 (q (H ) α G). Considering vectors in the second component, that is, in L2 (q (H ) α G), we deduce that Ran ∂ψ,q,2 and Ker(∂ψ,q,2 )∗ are orthogonal, hence have trivial intersection. We turn to the case 1 < p < ∞. Consider an approximating net (Mϕj ). According to Lemma 4.7 point 1. and Proposition 2.8, we have for any j a completely bounded mapping PMϕj = (PId)◦(IdMϕj ) : Lp (q (H )α G) → Lp (q (H ) α G). We claim that the subspace Ran ∂ψ,q,p is invariant under Id  Mϕj

(4.28)

the subspace Ker(∂ψ,q,p∗ )∗ is invariant under Id  Mϕj

(4.29)

the restriction of P  Id on Ran ∂ψ,q,p is the identity mapping.

(4.30)

For (4.28), for any s ∈ G, note that

(2.84) (Id  Mϕj )∂ψ,q,p (λs ) = (Id  Mϕj ) sq (bψ (s))  λs = ϕj (s)sq (bψ (s))  λs . This element belongs to Ran ∂ψ,q,p . By linearity and since PG is a core for ∂ψ,q,p according to Proposition 3.4, we deduce that Id  Mϕj maps Ran ∂ψ,q,p into Ran ∂ψ,q,p . Now (4.28) follows from the continuity of Id  Mϕj . For (4.29), note that if x ∈ dom ∂ψ,q,p∗ and f ∈ Ker(∂ψ,q,p∗ )∗ , then 

   (Id  Mϕj )(f ), ∂ψ,q,p∗ (x) = f, (Id  Mϕj )∂ψ,q,p∗ (x)     = f, ∂ψ,q,p∗ Mϕj (x) = (∂ψ,q,p∗ )∗ (f ), Mϕj (x) = 0.

By (2.8), we conclude that (Id  Mϕj )(f ) belongs to Ker(∂ψ,q,p∗ )∗ and (4.29) follows. For (4.30), for any s ∈ G we have

(2.84) (PId)(∂ψ,q,p (λs )) = (PId) sq (bψ (s))λs = sq (bψ (s))λs ∈ Ran ∂ψ,q,p .

154

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

Now use in a similar manner as before linearity, the fact that PG is a core of ∂ψ,q,p and the continuity of P  Id. Now, let z ∈ Ran ∂ψ,q,p ∩ Ker(∂ψ,q,p∗ )∗ . Then according to (4.28)–(4.30), we infer that (P  Mϕj )(z) belongs2 again to Ran ∂ψ,q,p ∩ Ker(∂ψ,q,p∗ )∗ . We claim that Jp,2,j (P  Mϕj )(z) belongs to Ran ∂ψ,q,2 ∩ Ker(∂ψ,q,2 )∗ , where the mapping Jp,2,j was defined in Lemma 4.7. First (P  Mϕj )(z) belongs to Ran ∂ψ,q,p , so that there exists a sequence (xn ) in dom ∂ψ,q,p such that (P  Mϕj )(z) = lim ∂ψ,q,p (xn ).

(4.31)

n→∞

Then we have in Lp (q (H ) α G), with ψj = 1supp ϕj , which is of finite support and thus induces a completely bounded multiplier Mψj ,3 (4.31)

(P  Mϕj )(z) = (Id  Mψj ) · (P  Mϕj )(z) = = lim ∂ψ,q,p Mψj (xn ) = lim n→∞

lim (Id  Mψj )∂ψ,q,p (xn )

n→∞



n→∞

xn,s ∂ψ,q,p (λs ).

s∈supp ϕj

2 Since Jp,2,j is an isomorphism, this limit also  holds in L (q (H ) α G) and the element Jp,2,j (P  Mϕj )(z) = limn→∞ s∈supp ϕj xn,s ∂ψ,q,2 (λs ) belongs to

Ran ∂ψ,q,2 . Furthermore, for some family (hs ) of elements of H , we have (P  Mϕj )(z) =



sq (hs )  λs .

(4.32)

s∈supp ϕj

Then using that (P  Mϕj )(z) belongs again to Ker(∂ψ,q,p∗ )∗ in the last equality, we obtain    (4.32) ∗ ∗ (∂ψ,q,2 ) Jp,2,j (P  Mϕj )(z) = (∂ψ,q,2 ) sq (hs )  λs = (∂ψ,q,p∗ )∗



s∈supp ϕj





sq (hs )  λs

s∈supp ϕj

= (∂ψ,q,p∗ )∗ (P  Mϕj )(z) = 0. We have shown that Jp,2,j (P  Mϕj )(z) belongs to Ran ∂ψ,q,2 ∩ Ker(∂ψ,q,2 )∗ .

2 3

Note that (P  Id)(z) = z. If x ∈ dom ∂ψ,q,p , it is really easy to check that (Id  Mψj )∂ψ,q,p (x) = ∂ψ,q,p Mψj (x).

4.2 Extension to Full Hodge-Dirac Operator and Hodge Decomposition

155

According to the beginning of the proof, the last intersection is trivial. It follows that Jp,2,j (P  Mϕj )(z) = 0. Since Jp,2,j is an isomorphism, we infer that (P  Mϕj )(z) = 0 for any j . Since G is weakly amenable and q (H ) α G has QWEP, the net (Id  Mϕj ) converges to IdLp (q (H )α G) for the point norm topology of Lp (q (H ) α G). We deduce that (P  Id)(z) = 0. But we had seen in (4.30) that (P  Id)(z) = z, so that z = 0 and we are done.   Combining Lemma 4.6 and Proposition 4.9, we can now deduce the following corollary. The QWEP assumption is satisfied if G is amenable or if G is a free group and q = ±1. Corollary 4.1 Let −1  q  1 and 1 < p < ∞. Let G be a weakly amenable discrete group such that q (H ) α G is QWEP. Then we have a topological direct sum decomposition Lp (q (H ) α G) = Ran ∂ψ,q,p ⊕ Ker(∂ψ,q,p∗ )∗ .

(4.33)

where the associated first bounded projection is Rp Rp∗ ∗ . In particular, we have Ran(∂ψ,q,p∗ )∗ = Ran(∂ψ,q,p∗ )∗ |Ran ∂ψ,q,p . Proof According to Lemma 4.6, the preceding subspaces add up to Lp (q (H ) α G), and according to Proposition 4.9, the sum is direct. By Kadison and Ringrose [134, Theorem 1.8.7], we conclude that the decomposition is topological. In the course of the proof of Lemma 4.6, we have seen that for any y ∈ Lp (q (H ) α G), we have the suitable decomposition y = Rp Rp∗ ∗ (y) + (Id − Rp Rp∗ ∗ )(y). So the associated first bounded projection is Rp Rp∗ ∗ .   Theorem 4.4 Let −1  q  1 and 1 < p < ∞. Let G be a weakly amenable discrete group such that q (H ) α G is QWEP. Consider the operator Dψ,q,p from (4.24). Then Dψ,q,p is bisectorial and has a bounded H∞ (ω± ) functional calculus. Proof According to Corollary 4.1, the space Lp (VN(G))⊕Lp (q (H )α G) admits (4.33)

the topological direct sum decomposition Lp (VN(G)) ⊕ Lp (q (H ) α G) = Lp (VN(G)) ⊕ Ran ∂ψ,q,p ⊕ Ker(∂ψ,q,p∗ )∗ into a sum of three subspaces. Along this decomposition, we can write4 ⎡

0

Dψ,q,p = ⎣∂ψ,q,p 0 (4.24)

4

⎤   (∂ψ,q,p∗ )∗ 0 (4.16) Dψ,q,p 0 . 0 0⎦ = 0 0 0 0

Here the notation (∂ψ,q,p∗ )∗ is used for the restriction of (∂ψ,q,p∗ )∗ on the subspace Ran ∂ψ,q,p .

156

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

According to Theorem 4.3, the operator Dψ,q,p is bisectorial and does have a bounded H∞ (ω± ) functional calculus. So we conclude the same thing for Dψ,q,p . See also Proposition 4.10 below.   Theorem 4.5 (Hodge Decomposition) Suppose 1 < p < ∞ and −1  q  1. Let G be a weakly amenable discrete group such that q (H ) α G is QWEP. If we identify Ran ∂ψ,q,p and Ran(∂ψ,q,p∗ )∗ as the closed subspaces {0} ⊕ Ran ∂ψ,q,p and Ran(∂ψ,q,p∗ )∗ ⊕ {0} of Lp (VN(G)) ⊕ Lp (q (H ) α G), we have Lp (VN(G)) ⊕ Lp (q (H ) α G) = Ran ∂ψ,q,p ⊕ Ran(∂ψ,q,p∗ )∗ ⊕ Ker Dψ,q,p . (4.34) Proof From the definition (4.24), it is obvious that Ker Dψ,q,p = (Ker ∂ψ,q,p ⊕ {0}) ⊕ ({0} ⊕ Ker(∂ψ,q,p∗ )∗ ). We deduce that Lp (VN(G)) ⊕ Lp (q (H ) α G)

(4.23)(4.33)

= Ran(∂ψ,q,p∗ )∗ ⊕ Ker ∂ψ,q,p ⊕ Ker(∂ψ,q,p∗ )∗ ⊕ Ran ∂ψ,q,p = ({0} ⊕ Ran ∂ψ,q,p ) ⊕ (Ran(∂ψ,q,p∗ )∗ ⊕ {0}) ⊕ (Ker ∂ψ,q,p ⊕ {0}) ⊕ ({0} ⊕ Ker(∂ψ,q,p∗ )∗ ) = ({0} ⊕ Ran ∂ψ,q,p ) ⊕ (Ran(∂ψ,q,p∗ )∗ ⊕ {0}) ⊕ Ker Dψ,q,p .   Remark 4.3 An inspection in all the steps of the proof of Theorem 4.4 shows that the angle of the H∞ (ω± ) calculus can be chosen ω > π2 | p1 − 12 | and that the norm of the calculus is bounded by a constant Kω not depending on G nor the cocycle (bψ , H ), in particular it is independent of the dimension of H . Proof First note that since Ap has a (sectorial) completely bounded H∞ (2ω ) calculus with angle 2ω > π| p1 − 12 | by e.g. [13, Theorem 4.1], by the representation (4.19) together with the fact that the spectral multipliers of Ap are Fourier multipliers and with Proposition 2.8, D2ψ,q,p also has a sectorial H∞ (2ω ) calculus. According to [110, Proof of Theorem 10.6.7, Theorem 10.4.4 (1) and (3), Proof of Theorem 10.4.9], the operator Dψ,q,p has then an H∞ (ω± ) bisectorial calculus to the angle ω > π2 | p1 − 12 | with a norm control    f (Dψ,q,p )  Kω M ∞ 2 2ω,D

ψ,q,p

2 ( )2 R f ∞,ω , Mω,D ψ,q,p

(4.35)

4.2 Extension to Full Hodge-Dirac Operator and Hodge Decomposition

157

where Kω is a constant only depending on ω (and not on G nor the cocycle (bψ , H )). Here M ∞ 2 is the H∞ (2ω ) calculus norm of D2ψ,q,p and 2ω,Dψ,q,p

R Mω,D =R ψ,q,p

 ) ( π  π  λ(λ − Dψ,q,p )−1 : λ ∈ C\{0},  | arg(λ)| −  < − ω . 2 2 (4.36)

Thus it remains to show that both M ∞

2ω,D2ψ,q,p

R and Mω,D can be chosen indepenψ,q,p

∞ dently of the Hilbert space H and the cocycle bψ . Let us start with M2ω,D 2

. It

ψ,q,p

∞ M2ω,Id , S p ⊗Ap

is controlled according to the previous reasoning and (4.19), by that is, the completely bounded H∞ calculus norm of Ap . Moreover, an application of [128, Proposition 5.8] shows that it suffices to consider only 2ω > π2 . According to [13, Theorem 4.1], we have a certain decomposition of the semigroup (Tt,p )t 0 generated by Ap , given by IdS p ⊗ Tt,p = (IdS p ⊗ Ep )(IdS p ⊗ Ut,p )(IdS p ⊗ Jp ). Here, IdS p ⊗ Ep and IdS p ⊗ Jp are contractions and IdS p ⊗ Ut,p is a group of isometries. An inspection of the proof of Lemma 4.5 shows that the constant in the second condition there is a bound of the H∞ calculus in the first condition there, so that it suffices to show that the generator of IdS p ⊗ Ut,p has a bounded H∞ (2ω ) calculus with a norm controlled by a constant depending only on 2ω > π2 . By Hytönen et al. [110, Proof of Theorem 10.7.10], the norm of the calculus of Ut,p 2 h is controlled by c2ω βp,X p,X , c2ω denoting a constant depending only on 2ω, βp,X denoting the UMD constant of X = S p (Lp (M)) and hp,X denoting the Hilbert transform norm, i.e. on the space Lp (R, S p (Lp (M))). These are controlled by a constant depending only on p but not on M. Indeed, using [109, Corollary 5.2.11], it suffices to control the UMD constant of S p (Lp (M)). According to [193, Corollary 4.5, Theorem 4.3], this constant is controlled by some universal bound times p2 /(p−1). So we have the desired control ∞ ∞ of M2ω,Id , and thus of M2ω,D . 2 p ⊗Ap S

ψ,q,p

R We turn to the control of Mω,D from (4.36). According to (4.18) extended ψ,q,p to complex times z belonging to some bisector σ± with σ = π2 − ω, it suffices to control the following R-bounds

) ( (Id + z2 Ap )−1 : z ∈ σ± ( ) R z(Id + z2 Ap )−1 (∂ψ,q,p∗ )∗ : z ∈ σ± ) ( R z∂ψ,q,p (Id + z2 Ap )−1 : z ∈ σ± ( ) R Id  (Id + z2 Ap )−1 : z ∈ σ± . R

(4.37) (4.38) (4.39) (4.40)

158

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

Indeed, an operator matrix family is R-bounded if and only if all operator entries in the matrix are R-bounded. According to [110, Theorem 10.3.4 (1)], (4.37) is Rbounded since we can write (Id+z2 Ap )−1 = Id−f (z2 Ap ) with f (λ) = λ(1+λ)−1 . ∞ Moreover, by the same reference, its R-bound is controlled by M2ω−ε,A , which in p turn, by the preceding argument of dilation can be controlled independently of the Hilbert space H and the cocycle bψ . The same argument shows that also (4.40) is R-bounded. Since the operator family in (4.38) consists of the family of the adjoints in (4.39) and R-boundedness is preserved under adjoints, it suffices to prove that (4.39) is R-bounded. To this end, we decompose for z ∈ ω the positive part of the bisector (similarly if z belongs to the negative part of the bisector) z∂ψ,q,p (Id + z Ap ) 2

−1

 =  =

−1 ∂ψ,q,p Ap 2 −1 ∂ψ,q,p Ap 2

( )1 2 z2 Ap (Id + z2 Ap )−1  f (z2 Ap ).

√ with f (λ) = λ(1 + λ)−1 . Again [110, Theorem 10.3.4 (1)] shows that the term f (z2 Ap ) is R-bounded with R-bound controlled by some constant independent of the cocycle. Finally, we are left to show that the Riesz transform is bounded by a constant independent of the cocycle, that is,   ∂ψ,q,p (x)

Lp (

q (H )α G)

 1   C Ap2 (x)Lp (VN(G)).

(4.41)  

For this in turn we refer to Proposition 3.3.

4.3 Hodge-Dirac Operator on Lp (VN(G)) ⊕ ψ,q,p We keep the standing assumptions of the two preceding sections and thus have a markovian semigroup (Tt )t 0 of Fourier multipliers with generator Ap , the noncommutative gradient ∂ψ,q,p and its adjoint (∂ψ,q,p∗ )∗ . We also fix the parameters 1 < p < ∞ and −1  q  1, and assume that the discrete group G is weakly amenable such that q (H ) α G has QWEP, so that the main results from the preceding section are valid. For the rest of this section we consider the Hodge-Dirac operator ⎡

0

def Dψ,q,p = ⎣∂ψ,q,p 0

⎤ (∂ψ,q,p∗ )∗ 0 0 0⎦ 0 0

4.3 Hodge-Dirac Operator on Lp (VN(G)) ⊕ ψ,q,p

159

on the bigger space Lp (VN(G)) ⊕ Ran ∂ψ,q,p ⊕ Ker(∂ψ,q,p∗ )∗ , with domain

def dom Dψ,q,p = dom ∂ψ,q,p ⊕ dom(∂ψ,q,p∗ )∗ ∩ Ran ∂ψ,q,p ⊕ Ker(∂ψ,q,p∗ )∗ . In the following, we consider the bounded operator ⎤ it (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ 0 (Id + t 2 Ap )−1 def ⎢ ⎥ T = ⎣it∂ψ,q,p (Id + t 2 Ap )−1 Id  (Id + t 2 Ap )−1 0 ⎦ 0 0 IdKer(∂ψ,q,p∗ )∗ (4.42) ⎡

on the space Lp (VN(G)) ⊕ Ran ∂ψ,q,p ⊕ Ker(∂ψ,q,p∗ )∗ . Here, we interpret the operators it∂ψ,q,p (Id + t 2 Ap )−1 and it (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ as the bounded extensions Lp (VN(G)) → Ran ∂ψ,q,p ⊂ Lp (q (H ) α G) resp. Ran ∂ψ,q,p ⊂ Lp (q (H ) α G) → Lp (VN(G)) guaranteed by Proposition 4.3. Then this proposition and Theorem 4.5 yield that T is a bounded operator on Lp (VN(G)) ⊕ Lp (q (H ) α G). Proposition 4.10 Let 1 < p < ∞, −1  q  1 and G be a weakly amenable discrete group such that q (H ) α G has QWEP. We have T (Id − itDψ,q,p ) = Iddom Dψ,q,p

(4.43)

(Id − itDψ,q,p )T = IdLp (VN(G))⊕Lp (q (H )α G) .

(4.44)

and

def

Proof If R = (Id + t 2 Ap )−1 , a straightforward calculation shows that T (Id − itDψ,q,p ) ⎡ ⎤ R(Id + t 2 (∂ψ,q,p∗ )∗ ∂ψ,q,p ) 0 0 ⎢ ⎥ = ⎣it∂ψ,q,p R − itId  R∂ψ,q,p t 2 ∂ψ,q,p R(∂ψ,q,p∗ )∗ + Id  R 0 ⎦ 0 0 IdKer(∂ψ,q,p∗ )∗ ⎤ ⎡ (I ) 0 0 def ⎢ ⎥ = ⎣(I I ) (I I I ) 0 ⎦. 0 0 IdKer(∂ψ,q,p∗ )∗ We check the expressions (I ), (I I ), (I I I ). For (I ), note that according to Proposition 4.1, we have (I ) = (Id + t 2 Ap )−1 + t 2 (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ ∂ψ,q,p = Iddom ∂ψ,q,p , since we recall that (Id + t 2 Ap )−1 (∂ψ,q,p∗ )∗ is interpreted as a bounded operator Ran ∂ψ,q,p∗ → Lp (VN(G)). Then (I I ) = 0 on dom ∂ψ,q,p according

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

160

to Proposition 4.2. Finally, we note that on the one hand, it is easy to check with Proposition 4.2 that (I I I ) = Id on P,G . On the other hand, P,G is a core of (∂ψ,q,p∗ )∗ according to Proposition 4.4 and ∂ψ,q,p (Id + t 2 Ap )−1 is bounded, the two of which imply easily that (I I I ) = IdRan ∂ψ,q,p ∩dom(∂ψ,q,p∗ )∗ . Altogether, we have shown (4.43). We turn to (4.44). Again a straightforward calculation shows that (Id − itDψ,q,p )T ⎤ ⎡ (Id + t 2 (∂ψ,q,p∗ )∗ ∂ψ,q,p )R itR(∂ψ,q,p∗ )∗ − it (∂ψ,q,p∗ )∗ Id  R 0 ⎥ ⎢ =⎣ 0 0 t 2 ∂ψ,q,p R(∂ψ,q,p∗ )∗ + Id  R ⎦ 0 0 IdKer(∂ψ,q,p∗ )∗ ⎤ (I ) (I I ) 0 def ⎢ ⎥ = ⎣ 0 (I I I ) 0 ⎦. ∗ 0 0 IdKer(∂ψ,q,p∗ ) ⎡

As for (4.43), one shows that (I ) = IdLp (VN(G)) . With Proposition 4.5, one shows that (I I ) = 0 on dom(∂ψ,q,p∗ )∗ . But (I I ) is closed, since the product AB of a closed operator A and a bounded operator B is closed, so since dom(∂ψ,q,p∗ )∗ ∩ Ran ∂ψ,q,p is dense in Ran ∂ψ,q,p , (I I ) = 0 on Ran ∂ψ,q,p . Finally, we have already shown that (I I I ) = Id on dom(∂ψ,q,p∗ )∗ , and as before, (I I I ) is closed. We infer that (I I I ) = IdRan ∂ψ,q,p .   We recall from Theorem 4.4 that Dψ,q,p has a bounded H∞ (ω± ) functional calculus on a bisector on the space Lp (VN(G)) ⊕ Lp (q (H ) α G). Next we show that an appropriate restriction of Dψ,q,p to the space Lp (VN(G)) ⊕ ψ,q,p , where def

ψ,q,p = span L

p



sq (ξ )  λs : ξ ∈ Hψ , s ∈ G



def

and Hψ = span {bψ (s) : s ∈ G}, is still bisectorial and admits an H∞ (ω± ) functional calculus on a bisector. Lemma 4.8 Assume −1  q  1 and 1 < p < ∞. Let G be a weakly amenable discrete group such that q (H ) α G has QWEP. There is a bounded projection W : Lp (VN(G))⊕Lp (q (H )α G) → Lp (VN(G))⊕Lp (q (H )α G) on ψ,q,p such that Lp (VN(G)) ⊂ Ker W . Proof We remind the reader that the bounded projection P : Lp (q (H )) → Lp (q (H )) from the part 1 of Lemma 4.7 is given in the following way. If (ek )k1 denotes an orthonormal basis of H and for a multi-index i = (i1 , i2 , . . . , in ) of def

length |i| = n ∈ N we let ei = ei1 ⊗ ei2 ⊗ · · · ⊗ ein ∈ Fq (H ), then the Wick word w(ei ) ∈ q (H ) is determined by w(ei ) = ei , where  denotes as usual the vacuum vector. Then a careful  inspection  of [118], in particular Theorem 3.5 there, shows that we have P( i αi ei ) = |i|=1 αi w(ei ) for any finite sum and

4.3 Hodge-Dirac Operator on Lp (VN(G)) ⊕ ψ,q,p

161

αi ∈ C. Here, w(ei ) = sq (ei ) in case |i| = 1. According to Lemma 3.4 or 4.7, P  IdLp (VN(G)) : Lp (q (H ) α G) → Lp (q (H ) α G) is a bounded mapping. Now note that if Q : H → H denotes the orthogonal projection onto the closed subspace Hψ of H , then by Bo˙zejko et al. [44, Theorem 2.11] there exists a trace preserving conditional expectation E : Lp (q (H )) → Lp (q (H )) such that E(w(ei )) = w(Fq (Q)ei ). Note that by (2.33), for any s ∈ G, each πs induces an operator πs : Hψ → Hψ , and thus by orthogonality of πs −1 also πs : Hψ⊥ → Hψ⊥ . (2.83)

Thus each πs and Q commute, whence αs = q (πs ) and E = q (Q) commute. We can use Lemma 2.6 and deduce that E extends to a normal complete contraction E  IdVN(G) : q (H ) α G → q (H ) α G. Again by Lemma 2.6, we infer that E  IdVN(G) is a conditional expectation which is trace preserving, so extends to a contraction on Lp (q (H ) α G) for 1  p  ∞. We claim that PE  IdLp (VN(G)) is a projection. To this end, it suffices5 to check that P and E commute. We assume that the orthonormal basis (ek )k∈N is chosen in such a way that (ek )k∈Nψ is an orthonormal basis of Hψ for some Nψ ⊂ N. From the foregoing, for a Wick word w(ei ) we have PEw(ei ) = Pw(Fq (Q)ei ) = δ|i|=1 w(Fq (Q)ei ) = δ|i|=1 δi∈Nψ w(ei ) and EPw(ei ) = Eδ|i|=1 w(ei ) = δ|i|=1 w(Fq (Q)ei ) = δ|i|=1 δi∈Nψ w(ei ). Thus P and E commute on a total set, so commute on all of Lp (q (H )). It suffices def

to consider the projection W  = PE  IdLp (VN(G)) and to finally extend it to def

W : Lp (VN(G)) ⊕ Lp (q (H ) α G) → ψ by setting W (x, y) = 0 ⊕ W  (y) and observe by a standard density and continuity argument that Ran W = ψ,q,p .   The proof of the following elementary lemma is left to the reader. Lemma 4.9 Let X be a Banach space and let Q1 , Q2 , Q3 , Q4 : X → X be bounded projections. Assume that Q1 + Q2 + Q3 + Q4 = IdX and that Qi Qj = 0 for i < j . Then we have a direct sum decomposition X = Ran(Q1 ) ⊕ Ran(Q2 ) ⊕ Ran(Q3 ) ⊕ Ran(Q4 ). Lemma 4.10 Let 1 < p < ∞ and −1  q  1. Assume that the discrete group G is weakly amenable and that q (H ) α G has QWEP. The subspace Lp (VN(G)) ⊕ ψ,q,p is invariant under the resolvents T from (4.42) of the Hodge-Dirac operator Dψ,q,p .

5

Recall that the product of two commuting projections on a Banach space is a projection.

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

162

Proof With respect to the decomposition projections from (4.33), we can decompose the identity IdLp (VN(G))⊕Lp (q (H )α G) as a sum IdLp (VN(G))⊕Lp (q (H )α G) = P1

⊕P2 ⊕ P3

Lp (VN(G)) ⊕ Lp (q (H ) α G) = Lp (VN(G)) ⊕Ran ∂ψ,q,p ⊕ Ran(∂ψ,q,p∗ )∗ . Then we claim that IdLp (VN(G))⊕Lp (q (H )α G) = P1 Lp (VN(G)) ⊕ Lp (q (H ) α G) = Lp (VN(G))

⊕P2 W ⊕ P3 W ⊕ V ⊕Ran ∂ψ,q,p ⊕ Xψ ⊕ Yψ , -. / =ψ,q,p

def

for some subspaces Xψ and Yψ , coming with projections P1 , P2 W , P3 W and V = (P2 + P3 )(Id − W ). To this end, we apply the auxiliary Lemma 4.9. Note first that P1 +P2 W +P3 W +V = P1 +(P2 +P3 )W +(P2 +P3 )(Id−W ) = P1 +(P2 +P3 ) = Id. Then P2 W , P3 W and V are projections. Indeed, P2 W P2 W = P2 P2 W , since Ran P2 W ⊂ Ran ∂ψ,q,p ⊂ ψ,q,p = Ran W , and P2 P2 W = P2 W , since P2 is a projection. Moreover, P3 W P3 W = P3 W (Id − P1 − P2 )W = P3 W − P3 W P1 W − P3 W P2 W = P3 W −P3 ·0·W −P3 P2 W = P3 W −0−0·W = P3 W . Thus, P2 W and P3 W are projections. Moreover, V = (P2 +P3 )(Id−W ) = (P1 +P2 +P3 )(Id−W )− P1 = Id − P1 − W , and V 2 = (Id − P1 − W )2 = Id + P1 + W − 2P1 − 2W + P1 W + W P1 = Id − P1 − W + 0 + 0 = V . Thus, also V is a projection. Now we check that some products of the four projections vanish as needed to apply Lemma 4.9. We choose the order (Q1 , Q2 , Q3 , Q4 ) = (P1 , P3 W, P2 W, V ). First note that this is clear if one of the factors is P1 . Then P3 W P2 W = 0 since Ran(W P2 W ) = Ran(P2 W ) ⊂ Ran ∂ψ,q,p ⊂ Ker P3 . Moreover, P3 W V = P3 W (P2 +P3 )(Id−W ) = P3 W (Id − W ) = 0 and also P2 W V = P2 W (P2 + P3 )(Id − W ) = 0. We have shown the claim and thus have a direct sum decomposition of the space into four closed subspaces. Now write the resolvent ⎤ ⎡ AB 0 ⎥ ⎢ 0 (4.45) T = ⎣C D ⎦ ∗ 0 0 IdKer(∂ψ,q,p∗ ) along the Hodge decomposition Id = P1 + P2 + P3 . If x ∈ Lp (VN(G)) ⊕ Lp (q (H ) α G), then x belongs to Lp (VN(G)) ⊕ ψ,q,p if and only if V (x) = 0. For such an x, we have T (x) = T (P1 x + P2 W x + P3 W x) = P1 AP1 x + P2 W CP1 x + P3 W CP1 x + V CP1 x + P1 BP2 W x + P2 W DP2 W x + P3 W DP2 W x + V DP2 W x + P3 W x.

4.4 Bimodule ψ,q,p,c

163

The summands starting with P1 and P2 lie in Lp (VN(G)) and Ran ∂ψ,q,p ⊂ ψ,q,p . The remaining summands are P3 W CP1 x = P3 CP1 x = 0 since CP1 x ∈ Ran ∂ψ,q,p ; V CP1 x = 0 since CP1 x ∈ Ran ∂ψ,q,p ⊂ ψ,q,p ; P3 W DP2 W x = P3 DP2 W x = 0; V DP2 W x = 0 since DP2 W x ∈ Ran ∂ψ,q,p ⊂ ψ,q,p . Finally, P3 W x ∈ Xψ ⊂ ψ,q,p . We conclude that T (x) ∈ Lp (VN(G)) ⊕ ψ,q,p .   Theorem 4.6 Let 1 < p < ∞ and −1  q  1. Assume that the discrete group G is weakly amenable and that q (H ) α G has QWEP. Consider the part Dψ,q,p of the Hodge-Dirac operator Dψ,q,p : dom Dψ,q,p ⊂ Lp (VN(G)) ⊕ Lp (q (H ) α G) → Lp (VN(G)) ⊕ Lp (q (H ) α G) on the closed subspace Lp (VN(G)) ⊕ ψ,q,p . Then Dψ,q,p is bisectorial and has a bounded H∞ (ω± ) functional calculus on a bisector. Proof We want to apply [83, Proposition 3.2.15]. To this end, note that we have proved previously that Dψ,q,p is bisectorial on X = Lp (VN(G)) ⊕ Lp (q (H ) α G). Moreover, for any t ∈ R, (Id − itDψ,q,p )−1 leaves invariant Y = Lp (VN(G)) ⊕ ψ,q,p according to Lemma 4.10. Note that then the same holds for t belonging to any bisector  ± to which Dψ,q,p is bisectorial. Indeed, z → (Id − izDψ,q,p )−1 is an analytic function on the subset of C where it is defined. Then also Pi (Id − izDψ,q,p )−1 Pj is analytic for i = 1, 2, 3. Therefore by the uniqueness theorem of analytic functions, (Id − izDψ,q,p )−1 has the same form as (4.45) at least for z belonging to such a bisector  ± . But then the proof of Lemma 4.10 goes through for such z in place of t. Now the theorem follows from an application of [83, Proposition 3.2.15] together with Proposition 4.10.  

4.4 Bimodule ψ,q,p,c In this short section, we continue to consider a markovian semigroup (Tt )t 0 of Fourier multipliers as in Proposition 2.3. This time, we do not need approximation properties on the discrete group G. We shall clarify and generalize some results of [131, pp. 585-586]. We need the following notion of bimodule which is different from the notion of [123, Definition 5.4] and is inspired by the well-known theory of Hilbert bimodules. Recall that the notion of right Lp -M-module is defined in Sect. 2.5. Definition 4.1 Let M and N be von Neumann algebras. Suppose 1  p < ∞. An Lp -N-M-bimodule is a right Lp -M-module X equipped with a structure of left-Np module such that the associated L 2 (M)-valued inner product ·, ·X satisfies

a ∗ x, yX = x, ayX ,

x, y ∈ X, a ∈ N.

An Lp -M-bimodule is an Lp -M-M-bimodule.

(4.46)

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

164

Suppose −1  q  1 and 2  p < ∞ (the case p < 2 is entirely left to the reader). If E : Lp (q (H ) α G) → Lp (VN(G)) is the canonical conditional expectation, it is obvious that the formula

ω, η = E(ω∗ η) def

(4.47)

p

defines an L 2 (VN(G))-valued inner product on Lp (q (H ) α G). We can consider p p the associated right Lp -VN(G)-module Lc (E). It is easy to see that Lc (E) is an p L -VN(G)-bimodule. We consider the closed subspace  def ψ,q,p,c = span ∂ψ,q,p (x)a : x ∈ dom ∂ψ,q,p , a ∈ VN(G)

(4.48)

of Lp (q (H ) α G). For any a, b ∈ VN(G) and any x ∈ dom ∂ψ,q,p , note that (∂ψ,q,p (x)a)b = ∂ψ,q,p (x)ab. Thus by linearity and density, ψ,q,p,c is a right VN(G)-module. Moreover, for any a ∈ VN(G) and any x, b ∈ PG , we have (2.86)

b∂ψ,q,p (x)a =



 ∂ψ,q,p (bx) − ∂ψ,q,p (b)x a = ∂ψ,q,p (bx)a − ∂ψ,q,p (b)xa.

Thus b∂ψ,q,p (x)a belongs to ψ,q,p,c . Since PG is a core for ∂ψ,q,p according to Proposition 3.4, the same holds for x ∈ dom ∂ψ,q,p . If b ∈ VN(G) is a general element, we approximate it in the strong operator topology by a bounded net in PG and obtain again that the same holds for b ∈ VN(G). By linearity and density, we deduce that ψ,q,p,c is a left VN(G)-module, so finally a VN(G)-bimodule. It is p

obvious that the restriction of the bracket (4.47) defines an L 2 (VN(G))-valued inner product on this subspace. We can consider the associated right Lp -VN(G)-module ψ,q,p,c which is also an Lp -VN(G)-bimodule and which identifies canonically to (2.76)

p

a closed subspace of Lc (E) = Lp (VN(G), L2 (q (H ))c,p ). Finally, we recall that Hψ is the real Hilbert space generated by the bψ (s)’s where s ∈ G. Lemma 4.11 Let G be a discrete group. 1. If ξ ∈ Hψ and if s ∈ G then sq (ξ )  λs belongs to ψ,q,p,c . 2. Moreover, we have  ψ,q,p,c = spanψ,q,p,c sq (ξ )  λs : ξ ∈ Hψ , s ∈ G .

(4.49)

Proof 1. For any s ∈ G, we have (2.56)

sq (bψ (s))  1 =



(2.84) sq (bψ (s))  λs (1  λs −1 ) = ∂ψ,q,p (λs )(1  λs −1 ).

4.5 Hodge-Dirac Operators Associated to Semigroups of Markov Schur. . .

165

Hence sq (bψ (s))  1 belongs to (4.48). If ξ belongs to the span of the bψ (s)’s where s ∈ G, we deduce by linearity that sq (ξ )  1 belongs to ψ,q,p,c . Now, for ξ ∈ Hψ , there exists a sequence (ξn ) of elements of the previous span such that ξn → ξ in H . By (2.41), we infer that sq (ξn ) → sq (ξ ) in L2 (q (H )). Hence sq (ξn )1 → sq (ξ )1 in Lp (VN(G), L2 (q (H ))c,p ). We conclude that sq (ξ )1 belongs to ψ,q,p,c . Since ψ,q,p,c is a right VN(G)-module, we conclude that sq (ξ )  λs = (sq (ξ )  1)λs belongs to ψ,q,p,c . 2. For any s, t ∈ G, we have (2.84)

(2.56)

∂ψ,q,p (λs )(1  λt ) = (sq (bψ (s))  λs )(1  λt ) = sq (bψ (s))  λst which belongs to  spanψ,q,p,c sq (ξs )  λs : ξs ∈ Hψ , s ∈ G . Since the closed span of the ∂ψ,q,p (λs )(1  λt )’s is ψ,q,p,c ,6 the proof is complete.  

4.5 Hodge-Dirac Operators Associated to Semigroups of Markov Schur Multipliers In this section, we consider some markovian semigroup (Tt )t 0 of Schur multipliers acting on B( 2I ) that we defined in Proposition 2.4. If 1  p < ∞, we denote by p Ap the (negative) infinitesimal generator on SI which is defined as the closure of p the unbounded operator A : MI,fin → SI , eij → αi − αj 2H eij . So MI,fin is a core of Ap . By Junge et al. [128, (5.2)], we have (Ap )∗ = Ap∗ if 1 < p < ∞. If −1  q  1, recall that by Proposition 3.11, we have a closed operator p

∂α,q,p : dom ∂α,q,p ⊂ SI → Lp (q (H )⊗B( 2I )),

eij → sq (αi − αj ) ⊗ eij .

It is clear that elements of the form ∂ψ,q,p (λs )(1  λt ) belong to ψ,q,p,c ⊂ ψ,q,p,c . On the other hand, for the density, since ψ,q,p,c is by definition dense in ψ,q,p,c , it suffices to approximate ∂ψ,q,p (x)(1 a) by elements of the span of the ∂ψ,q,p (λs )(1 λt ) for x ∈ dom ∂ψ,q,p p and a ∈ VN(G). The approximation needs to be in Lc (E) norm, but according to Lemma 2.8, it suffices to approximate in Lp (q (H ) α G) norm. By definition of VN(G) and Kaplansky’s density theorem, there exists a bounded net (aα ) in PG converging in the strong operator topology to a. Then it is not difficult to see that the net (1  aα ) converges in the strong operator topology to 1  a. Thus we can approximate ∂ψ,q,p (x)(1  a) in Lp norm by elements in the span of the ∂ψ,q,p (x)(1  λt ). Since PG is a core of ∂ψ,q,p , we can then approximate in turn by elements in the span of the ∂ψ,q,p (λs )(1  λt ). 6

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

166

Note that the adjoint operator (∂α,q,p∗ )∗ : dom(∂α,q,p∗ )∗ ⊂ Lp (q (H )⊗B( 2I )) → p SI is closed by Kato [136, p. 168]. We will now define a Hodge-Dirac operator Dα,q,p in (4.59), relying on ∂α,q,p and its adjoint. Then the main topic of this section will be to show that Dα,q,p is R-bisectorial (Theorem 4.7) and has a bounded H∞ p functional calculus on SI ⊕ Ran ∂α,q,p (Theorem 4.8). By Remark 4.4, this extends the Kato square root equivalence from Proposition 3.10. Many arguments in this section are parallel to Sect. 4.1 in the Fourier multiplier case, though easier since there is no need any more for approximation properties as p weak amenability, and MI,fin is always an appropriate dense subspace of SI together with completely contractive projections TJ from Definition 2.3. Proposition 4.11 Suppose 1 < p < ∞ and −1  q  1. As unbounded operators, we have Ap = (∂α,q,p∗ )∗ ∂α,q,p .

(4.50)

Proof For any i, j ∈ I , we have

(2.95) (∂α,q,p∗ )∗ ∂α,q,p (eij ) = (∂α,q,p∗ )∗ sq (αi − αj ) ⊗ eij

(4.51)

= τ sq (αi − αj )sq (αi − αj ) eij

(3.86)

(2.41)

= αi − αj 2H eij = Ap (eij ).

Argue as in Proposition 4.1, replacing Lemma 3.6 and Proposition 3.4 by Lemma 3.23 and Proposition 3.11.   Now, we show that the noncommutative gradient ∂α,q,p commutes with the semigroup and the resolvents of its generator. Lemma 4.12 Suppose 1 < p < ∞ and −1  q  1. If x ∈ dom ∂α,q,p and t  0, then Tt,p (x) belongs to dom ∂α,q,p and we have

IdLp (q (H )) ⊗ Tt,p ∂α,q,p (x) = ∂α,q,p Tt,p (x).

(4.52)

Proof For any i, j ∈ I , we have



(2.95)

IdLp (q (H )) ⊗ Tt,p ∂α,q,p (eij ) = IdLp (q (H )) ⊗ Tt,p sq (αi − αj ) ⊗ eij = e−t αi −αj  sq (αi − αj ) ⊗ eij 2

(2.95) −t αi −αj 2

= e

∂α,q,p (eij )

−t α −α 2 i j eij = ∂α,q,p Tt,p (eij ). = ∂α,q,p e

Argue as in Lemma 4.1, replacing Proposition 2.8 by Junge [117, p. 984].

 

4.5 Hodge-Dirac Operators Associated to Semigroups of Markov Schur. . .

167

Proposition 4.12 Let 1 < p < ∞ and −1  q  1. For any s  0 and any −1

x ∈ dom ∂α,q,p , we have IdS p + sAp x ∈ dom ∂α,q,p and I





−1 Id ⊗ (Id + sAp )−1 ∂α,q,p (x) = ∂α,q,p Id + sAp (x).

(4.53)  

Proof Argue as in Proposition 4.2, replacing (4.3) by (4.52).

We have the following analogue of Proposition 4.3 which can be proved in a similar manner. Proposition 4.13 Suppose 1 < p < ∞ and −1  q  1. The family   t∂α,q,p (Id + t 2 Ap )−1 : t > 0

(4.54)

p

of operators of B(SI , Lp (q (H )⊗B( 2I ))) is R-bounded. Proof Argue as in Proposition 4.3, replacing (3.31) by (3.88). Moreover, the   argument yielding bounded H∞ calculus of Ap is now [15] in place of [13]. Our Hodge-Dirac operator in (4.59) below will be constructed with ∂α,q,p and the unbounded operator (∂α,q,p∗ )∗ |Ran ∂α,q,p . Note that the latter is by definition an unbounded operator on the Banach space Ran ∂α,q,p having domain dom(∂α,q,p∗ )∗ ∩ Ran ∂α,q,p . Lemma 4.13 Let 1 < p < ∞ and −1  q  1. The operator (∂α,q,p∗ )∗ |Ran ∂α,q,p is densely defined and is closed. More precisely, the subspace ∂α,q,p (MI,fin ) of dom(∂α,q,p∗ )∗ is dense in Ran ∂α,q,p . Proof Argue as in Lemma 4.2, replacing Proposition 3.4 resp. Proposition 4.1 by Proposition 3.11 resp. Proposition 4.11.   According to Lemma 4.12, Id ⊗ Tt leaves Ran ∂α,q,p invariant for any t  0, so by continuity of Id ⊗ Tt also leaves Ran ∂α,q,p invariant. By Engel and Nagel [85, pp. 60-61], we can consider the generator def

Bp = (IdLp (q (H )) ⊗ Ap )|Ran ∂α,q,p .

(4.55)

of the restriction of (Id ⊗ Tt )t 0 on Ran ∂α,q,p . Lemma 4.14 Let 1 < p < ∞ and −1  q  1. The operator Bp is injective and sectorial on Ran ∂α,q,p . Proof Argue as in Lemma 4.3.

 

Lemma 4.15 If z belongs to Lp (q (H )) ⊗ MI,fin ∩ Ran ∂α,q,p then z ∂α,q,p (MI,fin ).

∈

168

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

Proof Argue as in Lemma 4.4. But now in place of a compactly supported cb. Fourier multiplier, apply the truncations TJ onto J ×J matrices to a given sequence (zn )n in dom ∂α,q,p .   Proposition 4.14 Let 1 < p < ∞ and −1  q  1. 1. Lp (q (H )) ⊗ MI,fin is a core of (∂α,q,p∗ )∗ . Furthermore, if y ∈ dom(∂α,q,p∗ )∗ and if J is a subset of I , we have (∂α,q,p∗ )∗ (Id ⊗ TJ )(y) = TJ (∂α,q,p∗ )∗ (y).

(4.56)

2. ∂α,q,p (MI,fin ) is a core of (∂α,q,p∗ )∗ |Ran ∂α,q,p . 3. ∂α,q,p (MI,fin ) is equally a core of ∂α,q,p (∂α,q,p∗ )∗ |Ran ∂α,q,p . 4. ∂α,q,p (MI,fin ) is equally a core of Bp . Proof Argue as in Proposition 4.4, replacing (3.33), (4.9), Lemma 4.1 resp. Lemma 4.4 by (3.90), (4.56), Lemma 4.12 resp. Lemma 4.15.   Proposition 4.15 Let 1 < p < ∞ and −1  q  1.

−1 1. For any s > 0, the operator Id + sAp (∂α,q,p∗ )∗ induces a bounded operator on Ran ∂α,q,p .

2. For any s  0 and any y ∈ Ran ∂α,q,p ∩ dom(∂α,q,p∗ )∗ , the element Id ⊗ (Id + sAp )−1 (y) belongs to dom(∂α,q,p∗ )∗ and

−1

Id + sAp (∂α,q,p∗ )∗ (y) = (∂α,q,p∗ )∗ Id ⊗ (Id + sAp )−1 (y).

(4.57)

3. For any t  0 and any y ∈ Ran ∂α,q,p ∩ dom(∂α,q,p∗ )∗ , the element (Id ⊗ Tt )(y) belongs to dom(∂α,q,p∗ )∗ and Tt (∂α,q,p∗ )∗ (y) = (∂α,q,p∗ )∗ (Id ⊗ Tt )(y). Proof Argue as in Proposition 4.5, replacing Proposition 4.3, Lemma 4.2, Proposition 4.1, (4.1), (4.3) resp. (4.10) by Proposition 4.13, Lemma 4.13, Proposition 4.11, (4.50), (4.52) resp. (4.57).   Proposition 4.14 enables us to identify Bp in terms of ∂α,q,p and its adjoint. Proposition 4.16 Let 1 < p < ∞ and −1  q  1. As unbounded operators, we have Bp = ∂α,q,p (∂α,q,p∗ )∗ |Ran ∂α,q,p .

(4.58)

4.5 Hodge-Dirac Operators Associated to Semigroups of Markov Schur. . .

169

Proof For any i, j ∈ I , we have (4.50)

∂α,q,p (∂α,q,p∗ )∗ ∂α,q,p (eij ) = ∂α,q,p Ap (eij ) = αi − αj 2H ∂α,q,p (eij ) = αi − αj 2H (sq (αi − αj ) ⊗ eij )

= (IdLp (q (H )) ⊗ Ap ) sq (αi − αj ) ⊗ eij

= (IdLp (q (H )) ⊗ Ap ) ∂α,q,p (eij ) .  

Argue as in Proposition 4.6, replacing Proposition 4.4 by Proposition 4.14. Note that the results of [15] gives the following result.

Proposition 4.17 Suppose 1 < p < ∞. The operators Ap and Bp have a bounded H∞ (ω ) functional calculus of angle ω for some ω < π2 . Suppose 1 < p < ∞ and −1  q  1. We introduce the unbounded operator def

Dα,q,p =



0

∂α,q,p

(∂α,q,p∗ )∗ |Ran ∂α,q,p 0

 (4.59)

p

on the Banach space SI ⊕p Ran ∂α,q,p defined by def

Dα,q,p (x, y) = (∂α,q,p∗ )∗ (y), ∂α,q,p (x)

(4.60)

for x ∈ dom ∂α,q,p , y ∈ dom(∂α,q,p∗ )∗ ∩ Ran ∂α,q,p . By Lemma 4.13 and Proposition 3.11, this operator is a closed operator and can be seen as a differential square root of the generator of the semigroup (Tt,p )t 0 . We call it the Hodge-Dirac operator of the semigroup since we have Proposition 4.18 below. Theorem 4.7 Suppose 1 < p < ∞ and −1  q  1. The Hodge-Dirac operator p Dα,q,p is R-bisectorial on SI ⊕p Ran ∂α,q,p . Proof Argue as in Theorem 4.2, replacing Proposition 4.3, Theorem 4.1, (4.16), (4.1), (4.4), (4.10), (4.7) resp. (4.13) by Propositions 4.13, 4.17, (4.59), (4.50), (4.53), (4.57), (4.55) resp. (4.58).   Proposition 4.18 Suppose 1 < p < ∞ and −1  q  1. We have 2 Dα,q,p =

  Ap 0 . 0 (IdLp (q (H )) ⊗ Ap )|Ran ∂α,q,p

(4.61)

Proof Argue as in Proposition 4.7, replacing Proposition 4.6, (4.1), (4.13) resp. (4.16) by Proposition 4.16, (4.50), (4.58) resp. (4.59).  

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

170

Now, we can state the following main result of this section. Theorem 4.8 Suppose 1 < p < ∞. The Hodge-Dirac operator Dα,q,p has a p bounded H∞ (ω± ) functional calculus on a bisector, on the Banach space SI ⊕p Ran ∂α,q,p .   Ap 0 has a bounded H∞ 0 Bp functional calculus of angle 2ω < π2 . Since Dα,q,p is R-bisectorial by Theorem 4.7, we deduce by Proposition 2.1 that the operator Dα,q,p has a bounded H∞ (ω± ) functional calculus on a bisector.   (4.61)

2 Proof By Proposition 4.17, the operator Dα,q,p =

Remark 4.4 Similarly to Remark 4.1, the boundedness of the H∞ functional calculus of the operator Dα,q,p implies the boundedness of the Riesz transforms and this result may be thought of as a strengthening of the equivalence (3.88). Remark 4.5 In a similar way to Remark 4.4, we also obtain for (x, y) ∈ dom ∂α,q,p ⊕ dom(∂α,q,p∗ )∗ |Ran ∂α,q,p that  2 (D

α,q,p )

1 2

1  1     (x, y)p ∼ = Ap2 (x)p + (IdLp (q (H )) ⊗ Ap2 )(y)p       ∼ = ∂α,q,p (x)p + (∂α,q,p∗ )∗ (y)p ∼ = Dα,q,p (x, y)p .

Moreover, we have 1     (IdLp ( (H )) ⊗ Ap2 )(y) ∼ (∂α,q,p∗ )∗ (y) , y ∈ dom(∂α,q,p∗ )∗ ∩ Ran ∂α,q,p . q p = p (4.62)

Proposition 4.19 We have Ran Ap = Ran(∂α,q,p∗ )∗ , Ran Bp = Ran ∂α,q,p , Ker Ap = Ker ∂α,q,p , Ker Bp = Ker(∂α,q,p∗ )∗ = {0} and p

SI = Ran(∂α,q,p∗ )∗ ⊕ Ker ∂α,q,p .

(4.63)

Here, by (∂α,q,p∗ )∗ we understand its restriction to Ran ∂α,q,p . However, we shall see in Corollary 4.2 that Ran(∂α,q,p∗ )∗ = Ran(∂α,q,p∗ )∗ |Ran ∂α,q,p . 2 2 = Ran Dα,q,p and Ker Dα,q,p = Ker Dα,q,p . Proof By (2.20), we have Ran Dα,q,p It is not difficult to prove the first four equalities using (4.61) and (4.60). The last one is a consequence of the definition of Ap and of [110, p. 361].  

4.6 Extension to Full Hodge-Dirac Operator and Hodge Decomposition

171

4.6 Extension to Full Hodge-Dirac Operator and Hodge Decomposition We keep the standing assumptions of the previous section and thus we have a markovian semigroup (Tt )t 0 of Schur multipliers with generator Ap , the noncommutative gradient ∂α,q,p and its adjoint (∂α,q,p∗ )∗ , together with the HodgeDirac operator Dα,q,p . We shall now extend the operator Dα,q,p to a densely p defined bisectorial operator Dα,q,p on SI ⊕ Lp (q (H )⊗B( 2I )) which will also be bisectorial and will have an H∞ (ω± ) functional calculus on a bisector. The key will be Corollary 4.2 below. We let def



Dα,q,p =

0

∂α,q,p

(∂α,q,p∗ )∗ 0

 (4.64)

p

along the decomposition SI ⊕ Lp (q (H )⊗B( 2I )), with natural domains for ∂α,q,p and (∂α,q,p∗ )∗ . 1

p

Consider the sectorial operator Ap2 on SI . According to (2.16), we have the p

1

1

topological direct sum decomposition SI = Ran Ap2 ⊕ Ker Ap2 . We define the −1

def

1

operator Rp = ∂α,q,p Ap 2 : Ran Ap2 → Lp (q (H )⊗B( 2I )). According to the 1

point 3 of Proposition 3.11, Rp is bounded on Ran Ap2 , so extends to a bounded 1

operator on Ran Ap2

(2.21)

p

= Ran Ap . We extend it to a bounded operator Rp : SI → 1

Lp (q (H )⊗B( 2I )), called Riesz transform, by putting Rp | Ker Ap2 = 0 along the def

preceding decomposition of SI . We equally let Rp∗ ∗ = (Rp∗ )∗ . p

Lemma 4.16 Let −1  q  1 and 1 < p < ∞. Then we have the subspace sum Lp (q (H )⊗B( 2I )) = Ran ∂α,q,p + Ker(∂α,q,p∗ )∗ . Proof Argue as in Lemma 4.6.

(4.65)  

In the proof of Proposition 4.20 below, we shall need some information on the Wiener-Ito chaos decomposition for q-Gaussians. This is collected in the following lemma. Lemma 4.17 Let −1  q  1 and 1 < p < ∞. 1. There exists a completely bounded projection P : Lp (q (H )) → Lp (q (H )) onto the closed space spanned by {sq (h) : h ∈ H }. Moreover, the projections are compatible for different values of p. 2 p 2. For any finite  subset J of I and y ∈ L (q (H )⊗B( I )), (P ⊗ TJ )(y) can be written as i,j ∈J sq (hij ) ⊗ eij for some hij ∈ H .

172

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

3. Let J be a finite subset of I . Denoting temporarily Pp and TJ,p the operators P p and TJ on the p-level, the identity mapping on Gaussq,p (C) ⊗ SJ extends to an isomorphism Jp,2,J : Ran(Pp ⊗ TJ,p ) → Ran(P2 ⊗ TJ,2 )

(4.66)

where Ran(Pp ⊗ TJ,p ) ⊂ Lp (q (H )⊗B( 2I )). Proof 1. Follows from Lemma 4.7. 2. This is easy and left to the reader. 3. Argue similarly to Lemma 4.7, replacing formally λs for s ∈ G by ⊗eij for i, j ∈ I .   Proposition 4.20 Let −1  q  1 and 1 < p < ∞. Then the subspaces from Lemma 4.16 have trivial intersection, Ran ∂α,q,p ∩ Ker(∂α,q,p∗ )∗ = {0}. Proof Argue as in Proposition 4.9, replacing Theorem 5.4, Lemma 4.7 resp. Proposition 3.4 by Proposition 5.14, Lemma 4.17 resp. Proposition 3.11. Also replace the approximating net of Fourmer multipliers (Mϕj )j by the approximating net of Schur multipliers (TJ )J .   Combining Lemma 4.16 and Proposition 4.20, we can now deduce the following corollary. Corollary 4.2 Let −1  q  1 and 1 < p < ∞. Then we have a topological direct sum decomposition Lp (q (H )⊗B( 2I )) = Ran ∂α,q,p ⊕ Ker(∂α,q,p∗ )∗

(4.67)

where the associated first bounded projection is Rp Rp∗ ∗ . In particular, we have Ran(∂α,q,p∗ )∗ = Ran(∂α,q,p∗ )∗ |Ran ∂α,q,p . Proof Argue as in Corollary 4.1, replacing Lemma 4.6 resp. Proposition 4.9 by Lemma 4.16 resp. Proposition 4.20.   Theorem 4.9 Let −1  q  1 and 1 < p < ∞. Consider the operator Dα,q,p from (4.64). Then Dα,q,p is bisectorial and has a bounded H∞ (ω± ) calculus. Proof Argue as in Theorem 4.4, replacing Corollary 4.1, (4.33), (4.24) resp. (4.16) by Corollary 4.2, (4.67), (4.64) resp. (4.59).   Theorem 4.10 (Hodge Decomposition) Suppose 1 < p < ∞ and −1  q  1. If we identify Ran ∂α,q,p and Ran(∂α,q,p∗ )∗ as the closed subspaces {0} ⊕ Ran ∂α,q,p

4.7 Independence from H and α

173 p

and Ran(∂α,q,p∗ )∗ ⊕ {0} of SI ⊕ Lp (q (H )⊗B( 2I )), we have p

SI ⊕ Lp (q (H )⊗B( 2I )) = Ran ∂α,q,p ⊕ Ran(∂α,q,p∗ )∗ ⊕ Ker Dα,q,p .

(4.68)

Proof Argue as in Theorem 4.5, replacing (4.24), (4.23) resp. (4.33) by (4.64), (4.63) resp. (4.67).  

4.7 Independence from H and α In this short section, we again keep the standing assumptions from the two preceding sections and have a markovian semigroup of Schur multipliers. The main topic will be the proof of Theorem 4.11 showing that the bound of the bisectorial H∞ (ω± ) functional calculus in Theorems 4.8 and 4.9 comes with a constant only depending on ω, q, p, but not on the markovian semigroup nor the associated Hilbert space H nor α : I → H . We show several intermediate lemmas before. Lemma 4.18 Let −1  q  1. Let H be a Hilbert space and i1 : H1 ⊂ H an embedding of a sub Hilbert space H1 . Moreover, let I be an index set and I1 ⊂ I a subset. Consider the mapping J1 :

 q (H1 )⊗B( 2I1 ) a⊗x

→ q (H )⊗B( 2I )

→ q (i1 )(a) ⊗ j1 (x)

,

where j1 : B( 2I1 ) → B( 2I ), x → P1∗ xP1 and P1 : 2I → 2I1 , (ξi )i∈I → (ξi δi∈I1 )i∈I is the canonical orthogonal projection. Then J1 is a normal faithful trace preserving ∗-homomorphism and thus extends to a complete isometry on the Lp level, 1  p  ∞. Proof According to [44, Theorem 2.11], since i1 is an isometric embedding, q (i1 ) is a faithful ∗-homomorphism which preserves the traces. According to [128, p. 97], q (i1 ) is normal. Moreover, it is easy to check that j1 is also a normal faithful ∗homomorphism. Thus also J1 is a normal faithful ∗-homomorphism, see also [189, p. 32]. Since q (i1 ) and j1 preserve the traces, also J1 preserves the trace. Now J1 is a (complete) Lp isometry according to [128, p. 92].   In the following, we let (Hn )n∈N be a sequence 2 of mutually orthogonal C sub Hilbert spaces of some big Hilbert space H = n∈N Hn and let I = n∈N In be a partition of a big index set I into smaller pieces In . Lemma 4.19 Let 1 < p < ∞ and −1  q  1. Consider the mappings p p n : SIn → SI , x → Pn∗ xPn , where Pn : 2I → 2In is the canonical orthogonal

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

174

projection as previously, :

p 

p

p

SIn → SI , (xn ) →



n (xn )

n

n∈N

and J:

p 

Lp (q (Hn )⊗B( 2In )) → Lp (q (H )⊗B( 2I )), (yn ) →



Jn (yn ).

n

n∈N

1. On the L∞ level, Ran(Jn ) · Ran(Jm ) = {0} for n = m. 2. If the previous domains of  and J that are exterior Banach space sums, are as indicated equipped with the p -norms, then  and J are isometries. Proof 1. By a density and normality argument, it suffices to pick a ⊗x ∈ q (Hn )⊗B( 2In ) and b ⊗ y ∈ q (Hm ) ⊗ B( 2Im ) and calculate the product Jn (a ⊗ x)Jm (b ⊗ y). We have Jn (a ⊗ x)Jm (b ⊗ y) = q (in )(a)q (im )(b) ⊗ Pn∗ xPn Pm∗ yPm = 0, since Pn Pm∗ : 2Im → 2In equals 0. 2. According to the first point, we have for any xn ∈ q (Hn )⊗B( 2In ), &

N 

'∗ & Jn (xn )

n=1

N 

' Jm (xm ) =

&N 

m=1

'& Jn (xn∗ )

n=1

'

N 

Jm (xm ) =

m=1

N 

Jn (xn∗ xn ).

n=1

 2   2 In the same way, by functional calculus of  N n=1 Jn (xn ) and Jn (|xn | ), we obtain p N N      Jn (xn ) = |Jn (xn )|p ,    n=1

n=1

and thus, taking traces, we obtain  N &N &N ' p1 ' p1       p p Jn (xn )p xn p Jn (xn ) = = .    n=1

p

n=1

n=1

By a density argument, we infer that J is an isometry. The proof for  is easier and left to the reader.  

4.7 Independence from H and α

175

In the following, we consider in addition the mappings αn : In → Hn and C def associate with it α : I → H given by α(i) = αn (i) if i ∈ In ⊂ k∈N Ik . We thus have noncommutative gradients ∂αn ,q,p and ∂α,q,p . Lemma 4.20 Let 1 < p < ∞ and −1  q  1. Recall the mappings Jn and n from Lemma 4.19. 1. For any n ∈ N and xn ∈ MIn ,fin , we have n (xn ) ∈ MI,fin and ∂α,q,p n (xn ) = Jn ∂αn ,q,p (xn ). 2. For any n ∈ N, yn ∈ q (Hn ) ⊗ MIn ,fin , we have Jn (yn ) ∈ dom(∂α,q,p∗ )∗ and (∂α,q,p∗ )∗ Jn (yn ) = n (∂αn ,q,p∗ )∗ (yn ). Proof 1. We calculate with xn =



∂α,q,p n (xn ) = =

i,j ∈In

xij eij and explicit embedding kn : In → I ,

∂α,q,p (Pn∗ xn Pn ) 

   ∗ = ∂α,q,p Pn xij eij Pn i,j

xij ∂α,q,p (ekn (i)kn (j ) )

i,j

=



xij sq (α(kn (i)) − α(kn (j ))) ⊗ ekn (i)kn (j ) .

i,j

On the other hand, we have  Jn ∂αn ,q,p (xn ) = xij Jn (sq (αn (i) − αn (j )) ⊗ eij ) i,j

=



xij q (in )(sq (αn (i) − αn (j ))) ⊗ Pn∗ eij Pn

i,j

=



xij sq (α(kn (i)) − α(kn (j ))) ⊗ ekn (i)kn (j ) .

i,j

2. First we note that Jn (q (Hn )⊗MIn ,fin ) ⊂ dom(∂α,q,p∗ )∗ and that (∂α,q,p∗ )∗ maps p Jn (q (Hn ) ⊗ MIn ,fin ) into n (SIn ). Indeed for sq (hn ) ⊗ eij ∈ q (Hn ) ⊗ MIn ,fin

4 Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators

176

and x =



p∗

k,l

xkl ekl ∈ SI , we have

x → τq (H )⊗B( 2 ) (sq (in (hn )) ⊗ ekn (i)kn (j ) )∗ · ∂α,q,p∗ (x) I 

xkl τq (H )⊗B( 2 ) sq (in (hn )) ⊗ ekn (j )kn (i) · sq (α(k) − α(l)) ⊗ ekl = I

k,l

= τq (H ) sq (in (hn ))sq (α(kn (j )) − α(kn (i))) xkn (i)kn (j ) . p∗

This defines clearly a linear form on SI , so indeed Jn (q (Hn ) ⊗ MIn ,fin ) ⊂ dom(∂α,q,p∗ )∗ . Moreover, we see that if x = ekl and k, l not both in kn (In ), then

(∂α,q,p∗ )∗ (Jn (yn )), ekl  = 0. Therefore, (∂α,q,p∗ )∗ maps Jn (q (Hn ) ⊗ MIn ,fin ) p into n (SIn ). Now we check the claimed equality in the statement of the lemma p∗

by applying a dual element xn to both sides. By density of MI,fin in SI , we can assume xn ∈ MI,fin . Moreover, we can assume xn ∈ n (MIn ,fin ), so that xn = n n∗ (xn ). Then we have according to the first point of the lemma TrI (n (∂αn ,q,p∗ )∗ (yn )xn∗ ) = TrIn ((∂αn ,q,p∗ )∗ (yn )(n∗ (xn ))∗ ) = τq (Hn )⊗B( 2 ) (yn (∂αn ,q,p∗ n∗ (xn ))∗ ) In

=

τq (H )⊗B( 2 ) (Jn (yn (∂αn ,q,p∗ n∗ (xn ))∗ )) I

= τq (H )⊗B( 2 ) (Jn (yn )(Jn ∂αn ,q,p∗ n∗ (xn ))∗ ) I

= τq (H )⊗B( 2 ) (Jn (yn )(∂α,q,p∗ n n∗ (xn ))∗ ) I

= TrI ((∂α,q,p∗ )∗ Jn (yn )(n n∗ (xn ))∗ ) = TrI ((∂α,q,p∗ )∗ Jn (yn )xn∗ ). Here we have also used that Jn is trace preserving and multiplicative. The lemma is proved.   Lemma 4.21 Let 1 < p < ∞ and −1  q  1. Recall the mappings  and J from Lemma 4.19. p

1. Then for any x = (xn ) with xn ∈ MIn ,fin ⊂ SIn and any y = (yn ) with yn ∈ q (Hn ) ⊗ MIn ,fin ∩ Ran(∂αn ,q,p ) such that only finitely many xn and yn are nonzero, we have ⎡ 0 Dα1 ,q,p      0  0 ⎢ 0 D α Dα,q,p ◦ (x, y) = ◦⎣ 2 ,q,p 0 J 0 J .. . 0

⎤ ... 0 . . .⎥ ⎦ (x, y) .. .

(4.69)

4.7 Independence from H and α

177

2. Let ω ∈ (0, π2 ) such that Dα,q,p and all Dαn ,q,p have an H∞ (ω± ) calculus, e.g. according to Theorem 4.8, ω = π4 . Then for any m ∈ H∞ (ω± ), we have ⎤ ⎡ 0 ... m(Dα1 ,q,p )    0  0 ⎢ 0 m(Dα2 ,q,p ) 0 ⎥ m(Dα,q,p ) ◦ = ◦⎣ ⎦ 0 J 0 J .. .. . . 0 



as bounded operators

2p n∈N

(4.70)

( ) p p SIn ⊕p Ran(∂αn ,q,p ) → SI ⊕p Ran(∂α,q,p ).

Proof 1. According to Lemma 4.20, we have Dα,q,p ((x), J (y)) = Dα,q,p



n (xn ),

n



 Jn (yn )

n

    ∗ = (∂α,q,p∗ ) Jn (yn ), ∂α,q,p n (xn ) =



n

n (∂αn ,q,p∗ )∗ yn ,

n

n





Jn ∂αn ,q,p xn

n

= diag((∂αn ,q,p∗ )∗ : n ∈ N)y, J diag(∂αn ,q,p : n ∈ N)x . This shows (4.69). D 2. According to (4.69), for any λ ∈ ρ(Dα,q,p ) ∩ n∈N ρ(Dαn ,q,p ), we have for (x, y) as in the first part of the lemma, <  ;  0 (λ − Dα1 ,q,p ) 0 (λ − Dα,q,p ) . . (x, y) 0 J . 0      0  0 −1 = (λ − Dα,q,p ) (λ − Dα,q,p ) (x, y) = (x, y) 0 J 0 J 0 : −λ1A  a  λ1A }. The element 1A is called the distinguished order unit. The definition of an order-unit space is due to Kadison [133]. Important examples of order-unit spaces are given by real linear subspaces of selfadjoint elements containing the unit element in a unital C∗ -algebra. The following is a slight extension of [147, Definition 2.3] replacing selfadjoint parts of unital C∗ -algebras by order-unit spaces. Definition 5.1 A unital Lipschitz pair (A, ·) is a pair where A is an order-unit space and where · is a seminorm defined on a dense subspace dom · of A such that 

a ∈ dom · : a = 0 = R1A .

(5.1)

Remark 5.1 Note that if a seminorm · is defined on some subspace dom ·  of a unital C∗ -algebra A such that Asa ∩ dom · is dense in Asa and such that a ∈ dom · : a = 0 = C1A , then its restriction on Asa ∩ dom · defines a unital Lipschitz pair. In this case, we also say that (A, ·) is a unital Lipschitz pair (or a compact quantum metric space if Definition 5.2 is satisfied).

1

Recall that D is an unbounded operator acting on the Hilbert space of L2 -spinors and that the functions of C(M) act on the same Hilbert space by multiplication operators.

5.1 Background on Quantum Locally Compact Metric Spaces

183

If (X, dist) is a compact metric space, a fundamental example [147, Example 2.6], [148, Example 2.9] is given by (C(X)sa , Lip) where C(X) is the commutative unital C∗ -algebra of continuous functions and where Lip is the Lipschitz seminorm, defined for any Lipschitz function f : X → C by 

def

Lip(f ) = sup

 |f (x) − f (y)| : x, y ∈ X, x = y . dist(x, y)

(5.2)

It is immediate that a function f has zero Lipschitz constant if and only if it is constant on X. Moreover, the algebra Lip(X) of real Lipschitz functions is normdense in C(X)sa by the Stone-Weierstrass theorem.2 Now, following essentially [198, Definition 2.2] (see also [145, Definition 2.6], [146, Definition 1.2] or [147, Definition 2.42] for the case where A is the selfadjoint part of a unital C∗ -algebra), we introduce a notion of quantum compact metric space. Recall that a linear functional ϕ on an order-unit space A is a state [2, p. 72] if ϕ = ϕ(1A ) = 1. Definition 5.2 A quantum compact metric space (A, ·) is a unital Lipschitz pair whose associated Monge-Kantorovich metric  def distmk (ϕ, ψ) = sup |ϕ(a) − ψ(a)| : a ∈ A, a  1 ,

ϕ, ψ ∈ S(A)

(5.3)

metrizes the weak* topology restricted to the state space S(A) of A. When a Lipschitz pair (A, ·) is a quantum compact metric space, the seminorm · is referred to as a Lip-norm. In the case of (C(X)sa , Lip), the formula (5.3) gives by Villani [223, Remark 6.5] or Dudley [78, Theorem 11.8.2] the dual formulation of the classical KantorovichRubinstein metric3 between Borel probability measures μ and ν on X         f dν  : f ∈ C(X)sa , Lip(f )  1 , distmk (μ, ν) = sup  f dμ − def

X

X

(5.4) which is a basic concept in optimal transport theory [223]. Considering the Dirac measures δx and δy at points x, y ∈ X instead of μ and ν, we recover the distance dist(x, y) with the formula (5.4).

2 Indeed, Lip(X) contains the constant functions. Moreover, Lip(X) separates points in X. If x0 , y0 ∈ X with x0 = y0 , we can use the lipschitz function f : X → R, x → dist(x, y0 ) since we have f (x0 ) > 0 = f (y0 ). 3 The original formulation [78, p. 329], [223, Definition 6.1] of the distance dist (μ, ν) between mk  two Borel probability measures μ and ν on X is given by the infimum of X×X dist(x, y) dπ(x, y) over all Borel probability measures π on X × X whose marginals are given by μ and ν.

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5 Locally Compact Quantum Metric Spaces and Spectral Triples

The compatibility of (5.4) with the weak* topology is well-known, see e.g. [39, Theorem 8.3.2]. This example is at the root of Definition 5.2. The theory of quantum compact metric spaces is the study of noncommutative generalizations of algebras of Lipschitz functions over compact metric spaces. The compatibility of the Monge-Kantorovich metric with the weak* topology is hard to check directly in general. Fortunately, there exists a condition which is more practical. This condition is inspired by the fact that Arzéla-Ascoli’s theorem shows that for any x ∈ X the set 

f ∈ C(X)sa : Lip(f )  1, f (x) = 0



is norm relatively compact and it is known that this property implies that (5.4) metrizes the weak* topology on the space of Borel probability measures on X. The noncommutative generalization relies on the following result which is a slight generalization of [182, Proposition 1.3] with a similar proof left to the reader. Proposition 5.1 Let (A, ·) be a unital Lipschitz pair. Let μ ∈ S(A) be a state of A. If the set {a ∈ dom · : a  1, μ(a) = 0} is norm relatively compact in A then (A, ·) is a quantum compact metric space. Note that in the case of selfadjoint parts of C∗ -algebras, this condition is also necessary, see [147, Theorem 2.43]. The Lipschitz seminorm Lip associated to a compact metric space (X, dist) enjoys a natural property with respect to the multiplication of functions in C(X), called the Leibniz property [78, page 306], [226, Proposition 1.30]. Indeed, for any Lipschitz functions f, g : X → C we have Lip(fg)  f C(X) Lip(g) + Lip(f ) gC(X) .

(5.5)

Moreover, the Lipschitz seminorm is lower-semicontinuous with respect to the C∗ -norm of C(X), i.e. the uniform convergence norm on X. These two additional properties were not assumed in the previous Definition 5.2, yet they are quite natural. However, as research in noncommutative metric geometry progressed, the need for a noncommutative analogue of these properties for some developments became evident. So, some additional conditions are often added to Definition 5.2 which brings us to the following definition which is a slight generalization of [147, Definition 2.21] for order-unit spaces embedded in unital C∗ -algebras, see also [149, Definition 1.3], [144, Definition 2.2.2], [145, Definition 2.19], [147, Definition 2.45] def def and [148, Definition 2.19]. Here a ◦ b = 12 (ab + ba) and {a, b} = 2i1 (ab − ba). Definition 5.3 1. A unital Leibniz pair (A, ·) is a unital Lipschitz pair where A is a real linear subspace of selfadjoint elements containing the unit element in a unital C∗ algebra A such that: a. the domain of · is a Jordan-Lie subalgebra of Asa ,

5.1 Background on Quantum Locally Compact Metric Spaces

185

b. for any a, b ∈ dom ·, we have: a ◦ b  aA b + a bA

and

{a, b}  aA b + a bA .

2. A unital Leibniz pair (A, ·) is a Leibniz quantum compact metric space when · is lower semicontinuous. Note that neither A nor A ∩ dom · are in general a Jordan or Lie or a usual algebra. We continue with a useful observation [147, p. 18], [148, Proposition 2.17] in the spirit of Remark 5.1. Proposition 5.2 Let A be a unital C∗ -algebra and · be a seminorm defined on a dense C-subspace dom · of A, such that dom · is closed under the adjoint operation, satisfying {a ∈ dom · : a = 0} = C1A and such that ab  aA b + a bA ,

a, b ∈ dom · .

If ·sa is the restriction of · to Asa ∩ dom ·, then (Asa , ·sa ) is a unital Leibniz pair. We also need a notion of quantum locally compact metric spaces. The paper [143] gives such a definition in the case of a Lipschitz pair (A, ·) where A is a C∗ -algebra. However, for semigroups of Schur multipliers, we need a version for order-unit spaces unfortunately not covered by Latrémolière [143]. Consequently, in the sequel, we try to generalize some notions of [143]. The following is a variant of [143, Definition 2.3] and [143, Definition 2.27]. Here uA is the unitization of the algebra A. Definition 5.4 A Lipschitz pair (A, ·) is a closed subspace A of selfadjoint elements of a non-unital C∗ -algebra A and a seminorm · defined on a dense subspace of A ⊕ R1uA such that {x ∈ A ⊕ R1uA : x = 0} = R1uA . A Lipschitz triple (A, · , M) consists of a Lipschitz pair (A, ·) and an abelian C∗ -algebra M of A such that M contains an approximate unit of A. The following is [143, Definition 2.23]. Definition 5.5 Let A be a non-unital C∗ -algebra and M be an abelian C∗ subalgebra of A containing an approximate unit of A. Let μ : A → C be a state of A. We call μ a local state (of (A, M)) provided that there exists a projection e in M of compact support4 in the Gelfand spectrum of M such that μ(e) = 1.

4

It is not clear if the support must be in addition open in [143] since the indicator 1A of a subset A is continuous if and only if A is both open and closed.

186

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Inspired by Latrémolière [143, Theorem 3.10], Latrémolière [147, Theorem 2.73], we introduce the following definition. Of course, we recognize that this definition is a bit artificial. A better choice would be to generalize the results and definitions of [143] to a larger context. Here μ is extended on the unitization as in [143, Notation 2.2]. Definition 5.6 We say that a Lipschitz triple (A, · , M) is a quantum locally compact metric space if for a local state μ of (A, M) and any compactly supported a, b ∈ M, the set  a x ∈ A ⊕ R1uA : x  1, μ(x) = 0 b is totally bounded (i.e. relatively compact) in the norm topology of uA. Now, we give a variant of Definition 5.3. Definition 5.7 Let (A, · , M) be a quantum locally compact metric space. We call it a Leibniz quantum locally compact metric space if (A ⊕ R1uA , ·) is a unital Leibniz pair in the sense of Definition 5.3 and if the seminorm · is lower semicontinuous with respect to the C∗ -norm ·uA . Finally, following [143, Condition 4.3] (see also [32, page 4] for a related discussion), we introduce the following definition. Definition 5.8 We say that a quantum locally compact metric space (A, · , M) in the 5.6 satisfies the boundedness condition if the Lipschitz ball  sense of Definition a ∈ A : a  1 is norm bounded.

5.2 Quantum Compact Metric Spaces Associated to Semigroups of Fourier Multipliers In this section, we consider a markovian semigroup (Tt )t 0 of Fourier multipliers on a group von Neumann algebra VN(G) where G is a discrete group, as in Proposition 2.3. We introduce new compact quantum metric spaces in the spirit of the ones of [119]. We also add to the picture the lower semicontinuity and a careful examination of the domains. By Lemma 3.8, the following definition is correct. It is far from clear if it is possible to do the same analysis at the level p = ∞ considered in [119, Section 1.2]. Definition 5.9 Suppose 2  p < ∞. Let G be a discrete group. Let Ap denote the 1

Lp realization of the (negative) generator of (Tt )t 0 . For any x ∈ dom Ap2 we let    1 1 def x,p = max (x, x) 2 Lp (VN(G)) , (x ∗ , x ∗ ) 2 Lp (VN(G)) . We start with an elementary fact.

(5.6)

5.2 Quantum Compact Metric Spaces Associated to Semigroups of Fourier. . .

187

Proposition 5.3 Suppose 2  p < ∞. Let G be a discrete group. Then ·,p is a 1

seminorm on the subspace dom Ap2 of Lp (VN(G)). 1

Proof 1. If x belongs to dom Ap2 and if k ∈ C, we have    1 1 (5.6) kx,p = max (kx, kx) 2 Lp (VN(G)) , ((kx)∗ , (kx)∗ ) 2 Lp (VN(G))    1 1 = max (kx, kx) 2 Lp (VN(G)), (kx ∗ , kx ∗ ) 2 Lp (VN(G))     1 1 = max |k|(x, x) 2 Lp (VN(G)) , |k|(x ∗ , x ∗ ) 2 Lp (VN(G))  (5.6)   1 1 = |k| max (x, x) 2 Lp (VN(G)) , (x ∗ , x ∗ ) 2 Lp (VN(G)) = |k| x,p . Let us turn to the triangular inequality. Assume that x, y ∈ PG . Note that p L 2 (VN(G)) is a normed space since p  2. According to the part 4 of Lemma 2.15 p and (2.50) applied with Z = L 2 (VN(G)), we have 1

(x + y, x + y) 2 p

L 2 (VN(G))

(2.50)

1

1

 (x, x) 2 p

L 2 (VN(G))

+ (y, y) 2 p

.

L 2 (VN(G))

Using (2.25), we can rewrite this inequality under the form     1 1 (x + y, x + y) 21  p  (x, x) 2 Lp (VN(G)) + (y, y) 2 Lp (VN(G)) . L (VN(G)) (5.7) 1

Then, for the general case where x, y ∈ dom Ap2 , consider some sequences (xn ) a a → x and yn − → y with Lemma 3.8 and (2.52). By and (yn ) of PG such that xn − a Remark 2.1, we have xn + yn − → x + y and thus 1   (2.25)  (x + y, x + y) 2 p (x + y, x + y) 21  p = L (VN(G))

(5.8)

L 2 (VN(G))

 1 = lim (xn + yn , xn + yn ) 2 p

(2.53)

n

(5.9)

L 2 (VN(G))

 1 = lim (xn + yn , xn + yn ) 2 Lp (VN(G))

(2.25)

n

  1 1  lim (xn , xn ) 2 Lp (VN(G)) + lim (yn , yn ) 2 Lp (VN(G))

(5.7)

n

n

 1 1 (2.53)  = (x, x) 2 Lp (VN(G)) + (y, y) 2 Lp (VN(G)).

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5 Locally Compact Quantum Metric Spaces and Spectral Triples

A similar reasoning for x, y replaced by x ∗ , y ∗ gives   (x ∗ + y ∗ , x ∗ + y ∗ ) 12 

Lp (VN(G))

 1  (x ∗ , x ∗ ) 2 Lp (VN(G))  1 + (y ∗ , y ∗ ) 2 Lp (VN(G)) .

(5.10)

Finally, we obtain    1 1 (5.6) x + y,p = max (x + y, x + y) 2 Lp , (x ∗ + y ∗ , x ∗ + y ∗ ) 2 Lp   1 1 max (x, x) 2 Lp (VN(G)) + (y, y) 2 Lp (VN(G)) ,     1 (x ∗ , x ∗ ) 12  p + (y ∗ , y ∗ ) 2  p

(5.8)(5.10)



L (VN(G))

L (VN(G))

   1 1  max (x, x) 2 Lp (VN(G)), (x ∗ , x ∗ ) 2 Lp (VN(G))    1 1 + max (y, y) 2 Lp (VN(G)), (y ∗ , y ∗ ) 2 Lp (VN(G)) (5.6)

= x,p + y,p .  

Now, we prove the Leibniz property of these seminorms. We recall that C∗r (G) is the reduced group C∗ -algebra of the discrete group G containing PG as a dense subspace. Proposition 5.4 Suppose 2  p < ∞. Let G be a discrete group. For any x, y ∈ PG , we have xy,p  xC∗r (G) y,p + x,p yC∗r (G) .

(5.11)

Proof Let x, y ∈ PG . Using the structure of bimodule of Lp (VN(G), L2 ()c,p ) in the second inequality, we see that   (xy, xy) 21 

Lp (VN(G))

   (2.86)  = ∂ψ (xy)Lp (VN(G),L2()c,p ) = x∂ψ (y) + ∂ψ (x)y Lp (VN(G),L2()c,p )      x∂ψ (y)Lp (VN(G),L2 () ) + ∂ψ (x)y Lp (VN(G),L2 () ) c,p c,p          xC∗r (G) ∂ψ (y) Lp (VN(G),L2()c,p ) + ∂ψ (x) Lp (VN(G),L2()c,p ) yC∗r (G) (3.37)

  1 1 = xC∗r (G) (y, y) 2 Lp (VN(G)) + (x, x) 2 Lp (VN(G)) yC∗r (G) .

(3.37)

5.2 Quantum Compact Metric Spaces Associated to Semigroups of Fourier. . .

189

Similarly, we have   (y ∗ x ∗ , y ∗ x ∗ ) 12  p L (VN(G))  ∗  1  x  ∗ (y ∗ , y ∗ ) 2 

Lp (VN(G))

Cr (G)

   1 + (x ∗ , x ∗ ) 2 Lp (VN(G)) y ∗ C∗ (G) r

  1 1 = xC∗r (G) (y ∗ , y ∗ ) 2 Lp (VN(G)) + (x ∗ , x ∗ ) 2 Lp (VN(G)) yC∗r (G) . Therefore, ·,p satisfies the Leibniz property.

 

For the proof of Theorem 5.1, we shall need the following lemma. Note that p dom ∂ψ,q,p was defined in Proposition 3.4. See Theorem 3.1 for the notation Lcr (E). Lemma 5.1 Let 2  p < ∞ and −1  q  1. Let G be a discrete group. Suppose that Lp (VN(G)) has CCAP and that VN(G) has QWEP. The map p ∂ψ,q,p : dom ∂ψ,q,p ⊂ Lp (VN(G)) → Lcr (E) is closed. Proof Let (xn ) be a sequence in dom ∂ψ,q,p such that xn → x where x is p some element of Lp (VN(G)) and ∂ψ,q,p (xn ) → z where z ∈ Lcr (E). We have    (3.38)  1 (3.36)(3.32)(5.6) ∂ψ,q,p (y) p y,p ≈p Ap2 (y)Lp (VN(G)) . Thus the sequence = L (E) 1

cr

1

1

(Ap2 (xn )) converges in Lp (VN(G)). Since Ap2 is closed, x belongs to dom Ap2 = 1

1

dom ∂ψ,q,p and Ap2 (x) = limn Ap2 (xn ). By the previous equivalence, we deduce p that ∂ψ,q,p (xn ) → ∂ψ,q,p (x) in Lcr (E). We infer that z = ∂ψ,q,p (x). We conclude with (2.1).   In the next theorem, recall that if A is the generator of a markovian semigroup of Fourier multipliers, then there exists a real Hilbert space H together with a mapping  2 bψ : G → H such that the symbol ψ : G → C of A satisfies ψ(s) = bψ (s)H . Following [129, p. 1962], we define def

Gapψ =

inf

bψ (s)=bψ (t )

  bψ (s) − bψ (t)2 . H

(5.12)

By [31, Proposition 2.10.2], note that Gapψ is independent of bψ , that is, if bψ : I → Hb and cψ : I → Hc are two cocycles such that ψ(s) = bψ (s)2Hb = cψ (s)2Hc for all s ∈ G, then Gapψ takes the same value in (5.12), whether it is defined via bψ or cψ . Recall that C∗r (G) is the reduced group C∗ -algebra associated with the discrete group G. We also write C∗r (G)0 for the range of the bounded projection C∗r (G) → C∗r (G), λs → (1 − δs=e )λs , that is the space of elements of C∗r (G) with vanishing p trace. Note that L0 (VN(G)) was defined before Proposition 2.2. In the sequel, we 1

denote by ·,p the restriction of (5.6) on dom Ap2 ∩ C∗r (G). In the following theorem, note that G satisfies the assumptions of the second point if G is a free group [46, Corollary 12.3.5] or an amenable group.

190

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Theorem 5.1 Let 2  p < ∞. Let G be a discrete group. Let bψ : G → H be an injective cocycle with values in a finite-dimensional Hilbert space of dimension n < p. Assume that Gapψ > 0. 1. If Lp (VN(G)) has CCAP and VN(G) has QWEP then (C∗r (G), ·,p ) is a quantum compact metric space. 2. If G is in addition weakly  amenable with the approximating net ϕj : G → C satisfying supj Mϕj C ∗ (G)→C ∗ (G)  1 then (C∗r (G), ·,p ) is a Leibniz r r quantum compact metric space. Proof 1

1. Since dom ·,p = dom Ap2 ∩C∗r (G) contains the dense subspace PG of C∗r (G), the C-subspace dom ·,p is dense in C∗r (G). By the first part of Lemma 3.11 and Proposition 3.4, dom ·,p is closed under the adjoint operation. According to Proposition 5.3, ·,p is a seminorm. We will show that  x ∈ dom ·,p : x,p = 0 = C1.

(5.13)

(2.78) 1  ∗ ∗ 2 A(1 )1 + 1 A(1) −

Indeed, we have A(1) = ψ(e)1 = 0, so that (1, 1) =  A(1∗ 1) = 0 and consequently

  1 1 (5.6) 1,p = max (1, 1) 2 Lp (VN(G)) , (1∗ , 1∗ ) 2 Lp (VN(G)) = 0. 1

In the other direction, if x,p = 0, then according to (3.38), we have Ap2 (x) = 0. By (2.21), we deduce that x belongs to Ker Ap . For any s ∈ G, we infer that



0 = τG Ap (x)λ∗s = τG xAp∗ (λs −1 ) = ψ(s −1 )τG (xλs −1 ). Note that by (2.34), the injectivity of bψ and ψ(e) = 0 we have ψ(s −1 ) = 0 for s = e. We deduce that τG (xλ∗s ) = 0 for these s. Finally, we obtain x ∈ C1 by approximation using CCAP. We conclude that (5.13) is true. Since τG : C∗r (G) → C is a state, with Proposition 5.1 it suffices to show that 

x ∈ dom ·,p : x,p  1, τG (x) = 0 is relatively compact in C∗r (G). (5.14)

2 2 Note that A−1 2 : L0 (VN(G)) → L0 (VN(G)) is compact. Indeed, (λs )s∈G\{e} 2 is an orthonormal basis of L0 (VN(G)) consisting of eigenvectors of  −2 A−1 . Moreover, the corresponding eigenvalues are bψ (s) and this 2

H

5.2 Quantum Compact Metric Spaces Associated to Semigroups of Fourier. . .

191

family vanish at infinity.5 For the latter, note that the condition Gapψ = 2  infbψ (s)=bψ (t ) bψ (s) − bψ (t)H > 0 together with the injectivity of bψ imply that any compact subset of H meets the bψ (s) only for a finite number of s ∈ G. As H is finite-dimensional by assumption, the closed ball B(0, √1ε ) for ε > 0 of H is compact, so contains a finite number of bψ (s). We deduce that   bψ (s)−2 → 0. We have shown that A−1 is compact. Now, according to [129, 2 H Lemma 5.8], we have Tt L1 (VN(G))→L∞(VN(G))  0

By Proposition 2.2 applied with z = − 12

1 2

1 n

t2

,

t > 0.

and q = ∞, we have a well-defined

p − 12 compact operator A : L0 (VN(G)) → L∞ 0 (VN(G)). So the image I by A 1 p of the closed unit ball of L0 (VN(G)) = Ran Ap is compact. Note that Ran Ap2 ⊂ Ran Ap by (2.21). Hence the subset6



 1  1  x ∈ C∗r (G)0 ∩ dom Ap2 : Ap2 (x)Lp (VN(G))  1

of I is relatively compact in C∗r (G). 1

Note that for any x ∈ dom Ap2 we have   (3.38)   21  1 1 Ap (x) p p max (x, x) 2 Lp (VN(G)), (x ∗ , x ∗ ) 2 Lp (VN(G)) L (VN(G)) (5.6)

= x,p .

We deduce from this inequality that 1  x ∈ C∗r (G)0 ∩ dom Ap2 : x,p  1

is relatively compact in C∗r (G). Now, we have 1  x ∈ C∗r (G)0 ∩ dom Ap2 : x,p  1 1  = x ∈ C∗r (G) ∩ dom Ap2 : x,p  1, τG (x) = 0 .

We deduce (5.14). 5 Recall that a family (x ) s s∈I vanishes at infinity means that for any ε > 0, there exists a finite subset J of I such that for any s ∈ I − J we have |xs | < ε. 6

1

Write x = A− 2 Ap2 x. 1

192

5 Locally Compact Quantum Metric Spaces and Spectral Triples 1

2. Let x ∈ C∗r (G) and (xj ) be a net of elements of dom Ap2 ∩ C∗r (G) such that xj − xC∗r (G) → 0 and xj ,p  1 for any n. By (3.36), (3.32) and (5.6), this 1   means that ∂ψ,1,p (xj )Lp (E)  1. We have to show that x belongs to dom Ap2 cr  and that x,p  1, i.e. ∂ψ,1,p (x)Lp (E)  1. Since ·Lp (VN(G))  ·C∗r (G) , cr note that xj − xLp (VN(G)) → 0.

Then xj , ∂ψ,1,p (xj ) is a net of elements of the graph of ∂ψ,1,p , which is bounded for the graph norm. Note that this graph is closed by Lemma 5.1 and convex, hence weakly closed by [167, Theorem 2.5.16]. Since bounded sets are compact by [167, Theorem 2.8.2], there exists a subnet

weakly relatively

xji , ∂ψ,1,p (xji ) which converges weakly to an element z, ∂ψ,1,p (z) of the graph of ∂ψ,1,p

. In particular, the net (xji ) converges weakly to z and the net ∂ψ,1,p (xji ) converges weakly to ∂ψ,1,p (z). Then necessarily x = z. By (2.1), 1

we conclude that x belongs to dom ∂ψ,1,p = dom Ap2 . Moreover, with [167, Theorem 2.5.21], we have by weak convergence     ∂ψ,1,p (x) p  lim inf ∂ψ,1,p (xji )Lp (VN(G))  1. L (VN(G)) i

Now, suppose that G is weakly amenable and consider an approximating net (Mϕj ) satisfying the assumptions of the second part and the properties following Lemma 2.5. We will extend the Leibniz property (5.11) to elements x, y of 1

dom Ap2 ∩ C∗r (G). In particular, the net (Mϕj )converges to IdLp (VN(G)) and to    IdC∗r (G) in the point-norm topologies with supj Mϕj cb,Lp (VN(G))→Lp (VN(G))  1. We have x = limj Mϕj (x) and y = limj Mϕj (y) where the convergence holds in C∗r (G) and in Lp (VN(G)). Thus also xy = limj Mϕj (x)Mϕj (y) in C∗r (G) since C∗r (G) is a Banach algebra. Using (3.33) and (3.32) in last inequality, we obtain   Mϕ (x)Mϕ (y) j j ,p          Mϕj (x)C∗ (G) Mϕj (y),p + Mϕj (y)C∗ (G) Mϕj (x),p

(5.11)

r

r

   1 1  xC∗r (G) max (Mϕj (y), Mϕj (y)) 2 Lp , (Mϕj (y)∗ , Mϕj (y)∗ ) 2 Lp    1 1 + yC∗r (G) max (Mϕj (x), Mϕj (x)) 2 Lp , (Mϕj (x)∗ , Mϕj (x)∗ ) 2 Lp (5.6)

(3.36)(2.72)(2.77)



xC∗r (G) y,p + yC∗r (G) x,p .

5.2 Quantum Compact Metric Spaces Associated to Semigroups of Fourier. . .

193 1

Using the lower semicontinuity of ·,p , we conclude that xy ∈ dom Ap2 and that (3.35)

xy,p  xC∗r (G) y,p + yC∗r (G) x,p . The theorem is proved.   1 2

Remark 5.2 In the part 2, the argument proves that dom Ap ∩ C∗r (G) is a ∗subalgebra of C∗r (G). Remark 5.3 Note that in Theorem 5.1, in contrast to its counterpart for Schur multipliers, Theorem 5.3 below, we have a restriction of the exponent p > n, where n denotes the dimension of the representing Hilbert space. This restriction is in fact necessary in general as the following example shows. Consider the abelian discrete group G = Z2 , the canonical identification bψ : Z2 → R2 of Z2 into R2 and the trivial homomorphism α : Z2 → O(R2 ), (n, m) → IdR2 . The associated length function is ψ : Z2 → R+ , (n, m) →  2 ψ(n, m) = bψ (n, m)R2 = n2 + m2 . Now pick the critical exponent p = 2 = dim R2 . Note that the operator space L2 (VN(Z2 )) = L2 (T2 ) is an operator Hilbert space by [188, Proposition 2.1 (iii)] and consequently has CCAP.7 We also calculate (5.12)

Gapψ = =

inf

bψ (n,m)=bψ

inf 

(n,m)=(n

,m )

(n ,m )

  bψ (n, m) − bψ (n , m )2 2 R

  (n, m) − (n , m )2 2 = 1 > 0. R

However, (C∗r (Z2 ), ·,2 ) is not a quantum compact metric space Proof We will use [147, Theorem 2.4.3]. Indeed, the set 1  x ∈ C∗r (Z2 )0 ∩ dom A22 : x,2  1

is not bounded, so in particular, not relatively compact. Indeed, consider the double sequence (αn,m )n,m∈Z defined by 1

def

αn,m =

(1 + n2

+

m2 ) log(2 + n2

+ m2 )

δ(n,m)=(0,0).

(5.15)

More generally, if 1 < p < ∞ the operator space Lp (T2 ) has CCAP by Junge and Ruan [121, Proposition 3.5] since an abelian group is weakly amenable with Cowling-Haagerup constant equal to 1.

7

194

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Fix a number N ∈ N and consider the selfadjoint element 

def

xN =

αn,m e2πi (n,m),·

n2 +m2 N

of C∗r (Z2 ) = C(T2 ). We claim that (xN ) is an unbounded sequence in C∗r (Z2 )0 = C(T2 )0 but bounded in ·,2 -seminorm. Indeed, observe that xN is a trigonometric polynomial without constant term,so belonging indeed to C∗r (Z2 )0 = C(T2 )0 . Moreover, xN C(T2 )  |xN (0)| = n2 +m2 N αn,m . But the series (n,m)∈Z2 αn,m diverges since using an integral test [89, Proposition 7.57] and a change of variables to polar coordinates we have 

αn,m ∼ =

(n,m)∈Z2



1 R2

(1 + x 2



∞

=C 0

+ y 2 ) log(2 + x 2

+ y 2)

dx dy

1 r dr = ∞. (1 + r 2 ) log(2 + r 2 )

On the other hand, using Plancherel theorem in the second equality and again an integral test and a change of variables to polar coordinates, we obtain xN 2,2

2 ( )  2   1 1 ∼     2 2 = A (xN ) L2 (T2 ) =  δn2 +m2 N ψ(n, m) αn,m (n,m)∈Z2  2

(3.38)





=

(5.15)

2 (n2 + m2 )αn,m 

n2 +m2 N

∼ =



R2



 (n,m)∈Z2

Z2

1  2 (1 + n2 + m2 ) log(2 + n2 + m2 )

1  2 dx dy (1 + x 2 + y 2 ) log(2 + x 2 + y 2 ) ∞

=C 0

(1 + r 2 )

1  2 r dr < ∞. log(2 + r 2 )  

We refer to the survey [54] and to [119, pp. 628-629] for more information. Recall that a finitely generated discrete group G has rapid decay of order r  0 if we have an estimate xVN(G)  k r xL2 (VN(G))  for any x = |s|k as λs . Similarly using [119, Lemma 1.3.1], we can obtain the following result.

5.3 Gaps and Estimates of Norms of Schur Multipliers

195

Theorem 5.2 Let G be a finitely generated discrete group with rapid r-decay. Suppose that there exists β > 0 such that 2 2r+1 β  p < ∞ and that inf|s|=k ψ(s)  kβ . 1. If Lp (VN(G)) has CCAP and VN(G) has QWEP then (C∗r (G), ·,p ) is a quantum compact metric space. 2. If G is in addition net ϕj : G → C sat weakly amenable with the approximating   isfying supj Mϕj C ∗ (G)→C ∗(G) < ∞ and supj Mϕj cb,Lp (VN(G))→Lp (VN(G)) = r r 1, then (C∗r (G), ·,p ) is a Leibniz quantum compact metric space. def

Proof Using the notation n = 2 2r+1 β , we have by [119, Lemma 1.3.1] the estimate Tt L2 (VN(G))→VN(G)  0

1 t

2r+1 2β

=

1 n

t4

,

t > 0.

By Junge and Mei [119, Lemma 1.1.2], we deduce that Tt L1 (VN(G))→VN(G)  0

1 n t2

.

Moreover, by the argument of [119, p. 628] (using [74, (15.12.8.1)]), the operator A−1 is compact on L20 (VN(G)). Since 2 2r+1 β  p, we can use Proposition 2.2 with 1

z = 12 and q = ∞. We deduce that A− 2 : L0 (VN(G)) → VN(G)0 is compact. The end of the proof is similar to the proof of Theorem 5.1.   p

5.3 Gaps and Estimates of Norms of Schur Multipliers Consider a semigroup (Tt )t 0 of selfadjoint unital completely positive Schur multipliers on B( 2I ) as in (2.37). In order to find quantum compact metric spaces associated with such a semigroup, we need some supplementary information on the semigroup. One important of them is the notion of the gap, which we study in this section. In all the section, we suppose dim H < ∞. We define the gap of α by def

Gapα =

inf

αi −αj =αk −αl

  (αi − αj ) − (αk − αl )2 . H

(5.16)

Note that by the proof of [10, Proposition 5.4] and [31, Theorem C.2.3], Gapα is then independent of α, that is, if α : I → Hα , β : I → Hβ are two families such that (2.36) and (2.37) hold for both, then Gapα = Gapβ . Lemma 5.2 If dim H = n and Gapα > 0, for any integer k  1, we have   card αi − αj : k 2 Gapα  αi − αj 2H  (k + 1)2 Gapα  (5n − 1)k n−1 .

196

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Proof If Bn denotes the open Euclidean unit ball in H and if ξ1 , ξ2 are distinct and can be written ξ1 = αi − αj and ξ2 = αk − αl for some i, j, k, l ∈ I , we have8 (

3

3

Gapα Bn = ∅. 2 √ Gapα Counting the maximum number of disjoint balls of radius in the annulus 2 √ √ ( ( ) ) 3 3 Gapα Gapα (k +1) Gapα + 2 Bn − k Gapα − 2 Bn combined with the binomial theorem, we obtain   card αi − αj : k 2 Gapα  αi − αj 2H  (k + 1)2 Gapα √ √ (( ) ) (( 3 ) ) 3 Gapα Gap vol (k + 1) Gapα + 2 Bn − vol k Gapα − 2 α Bn  ( √Gap ) α vol Bn 2 ξ1 +

Gapα

Bn ∩ ξ2 + 2

= (2k + 3)n − (2k − 1)n =

n   

n (2k)n−j 3j − (−1)j j j =0

n    n n−j j n−1 3 − (−1)j k 2 j j =1

k

n−1

n   n   n n−j j n−1 2n−j (−1)j = (5n − 1)k n−1 . 2 3 −k j j =0

j =0

  We need the following lemma which says that Schur multipliers with 0−1 entries of diagonal block rectangular shape are completely contractive. Lemma 5.3 Let I be a non-empty index set. Let {I1 , . . . , IN } and {J1 , . . . , JN } be two subpartitions of I , i.e. Ik ⊂ I and Ik ∩ Il = ∅ for k = l (and similarly for J1 , . . . , JN ). The matrix B = [bij ], where bij =

 1 0

if i ∈ Ik , j ∈ Jk for the same k otherwise

,

induces a completely contractive Schur multiplier MB : SI∞ → SI∞ and is also p completely contractive on SI for any 1  p  ∞. √ √ √ √



Gap Gap Gapα Gapα If η ∈ ξ1 + 2 α Bn ∩ ξ2 + 2 α Bn then |ξ1 −ξ2 |  |ξ1 −η|+|η−ξ2 | < + = 2 2 3 Gapα which is impossible.

8

5.3 Gaps and Estimates of Norms of Schur Multipliers

197

Proof For any i ∈ {1, . . . , N} consider the orthogonal projections Pi : 2I → 2I , (ξk )k∈I → (ξk 1k∈Ii )k∈I and Qi : 2I → 2I , (ξk )k∈I → (ξk 1k∈Ji )k∈I . If i = j , the ranges Pi ( 2I ) and Pj ( 2I ) (resp. Qi ( 2I ) and Qj ( 2I )) are orthogonal. Moreover, if  2 C ∈ SI∞ it is obvious that MB (C) = N i=1 Pi CQi . Then for any ξ, η ∈ I , using the Cauchy-Schwarz Inequality in the second inequality, we obtain  N       MB (C)ξ, η 2  =  Pi CQi ξ, η I  i=1

    2  I

N  N         CQi ξ, Pi η 2  =  CQi ξ, Pi η 2   I I  i=1



&N  i=1

i=1

CQi ξ 2 2 I &

 CSI∞

N  i=1

' 12 & N  i=1

' 12 Pi η2 2 I

' 12 & Qi ξ 2 2 I

N      = CSI∞  Qi ξ    i=1

' 12

N  i=1

Pi η2 2 I

N      Pi η    2

I

i=1

 CSI∞ ξ  2 η 2 . I

2I

I

We infer that MB (C)SI∞  CSI∞ . Thus MB : SI∞ → SI∞ is a contraction, and since it is a Schur multiplier, it is even a complete contraction [185, Corollary 8.8]. Moreover, since B has only real entries, MB is symmetric, so also (completely) p contractive on SI1 , and by interpolation also on SI for 1  p  ∞.   Recall that Ran A∞ satisfies (3.98). E : H → C satisfies Lemma 5.4 If dim H = n, Gapα > 0 and if the function m   −(n+ε) m E(ξ )  cn ξ  H

for some ε > 0,

∞ ∞ then the Schur multiplier M[E m(αi −αj )] : SI → SI is completely bounded. Moreover, for any t > 0, we have

Tt cb,Ran A∞ →Ran A∞ 

c(n) n

(Gapα t) 2

.

 Proof Let x = i,j ∈I  xij ⊗ eij ∈ S ∞ (SI∞ ) ⊂ S ∞ (SI∞ ), where I  ⊂ I is finite and is an integer. Let {I1 , . . . , IN } be the partition of I  corresponding to the def ∼ j ⇐⇒ αi = αj , and let J1 = I1 , . . . , JN = IN . Let B0 be equivalence relation i = the Schur multiplier symbol from Lemma 5.3 associated with these partitions.

198

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Moreover for ξ ∈ α(I  ) − α(I  ), we let {I1 , . . . , IM } and {J1 , . . . , JM } be the ξ ξ subsets of {I1 , . . . , IN } such that αi − αj = ξ ⇔ i ∈ Ik and j ∈ Jk for the same k. We let Bξ be the Schur multiplier symbol from Lemma 5.3 associated with these subpartitions. Then, using Lemma 5.2 and our growth assumption on m E in the third inequality, we obtain ξ

       m E(αi − αj )xij ⊗ eij     i,j ∈I 

i,j ∈I  αi =αj

k1

          |E m(0)|  xij ⊗ eij    i,j ∈I   α =α 

+

ξ

S ∞ (SI∞ )

    = m E(αi − αj )xij ⊗ eij + 

i

ξ

 ξ ∈α(I  )−α(I  ) k 2 Gapα ξ 2 0. −1

1. The operator Ap 2 : Ran Ap → Ran Ap ⊂ Ran A∞ is bounded. 2. Suppose −1  q  1. Let Bp = (IdLp (q (H )) ⊗ Ap )|Ran ∂α,q,p : dom Bp ⊂ − 12

Ran ∂α,q,p → Ran ∂α,q,p . Then the operator Bp

is bounded.

Proof −1

1. We begin by showing that A2 2 : Ran A2 → Ran A2 is bounded. For any i, j ∈ I such that αi − αj = 0, we have by the gap condition αi − αj H = αi − αj − −1

0  Gapα . Since aij = αi − αj 2H , we deduce that the diagonal operator A2 2 is bounded on the Hilbert space Ran A2 . 1 Next we show that A− 2 : Ran A∞ → Ran A∞ is bounded. Indeed, note that (Tt )t 0 extends to a bounded strongly continuous semigroup on Ran A∞ . Moreover, Lemma 5.4 (in the case where H is one-dimensional, we inject H beforehand into a two-dimensional Hilbert space so that this lemma yields an n estimate Tt ∞→∞  Ct − 2 with n2 > 12 ) together with Lemma 5.6 yield that

A− 2 is bounded on Ran A∞ . 1 Now, if 2 < p < ∞, it suffices to interpolate the operator A− 2 between levels 2 and ∞ and to note that we have Ran Ap = (Ran A2 , Ran A∞ )θ for the right θ ∈ [0, 1). Indeed, the interpolation identity follows from the fact that the p Ran Ap are complemented subspaces of SI by the complementary projections of the projections of the one of (2.17) (note that (Tt )t 0 is a bounded semigroup p on SI for 2  p  ∞), and that these spectral projections are compatible for different values of 2  p  ∞. Then if 1 < p < 2, we use that Ran Ap is the dual space of Ran Ap∗ via the usual duality bracket using (2.8) and [167, page 1

−1

−1

94], and that Ap 2 on the first space is the adjoint of Ap∗2 on the second space. 2. Note that by (4.55), Bp is the generator of the (strongly continuous) semigroup ((Id ⊗ Tt )t 0 |Ran ∂α,q,p )t 0 . Now, we use Lemma 5.4 to obtain the bound

206

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Id ⊗ Tt ∞→∞  t1d for some d > 12 . With Proposition 4.19, we have Ran Bp = Ran ∂α,q,p . By Lemma 5.6 and an argument as in the first part of −1

the proof, we conclude that Bp 2 : Ran ∂α,q,p → Ran ∂α,q,p is bounded.

 

In the following theorem, we consider the abelian C∗ -subalgebra M of SI∞ consisting of its diagonal operators. We recall that we restrict to p  2 to have a seminorm ·,α,p in the usual sense. Theorem 5.3 Assume that the Hilbert space H is of finite dimension and that Gapα > 0.

1. Suppose 2  p  ∞ and that I is finite. Then Ran A∞ )sa ⊕ RId 2 , ·,α,p I is a Leibniz quantum compact metric space. 2. Suppose 2  p < ∞ and that I is infinite. We consider

the abelian Ran A∞ sa ⊕ C∗ -subalgebra M of diagonal operators of SI∞ . Then RId 2 , ·,α,p , M is a Leibniz quantum locally compact metric space I satisfying the boundedness condition. Proof 1. Note that Ran A∞ is a subspace of the space of matrices of SI∞ with null diagonal and that we have a contractive projection from SI∞ onto the diagonal of SI∞ . So it is easy to check that Tr ⊕IdR : Ran A∞ ⊕ RId 2 → R, x + λ → λ is a state of I the order-unit space (Ran A∞ )sa ⊕ RId 2 . By Proposition 5.1, it suffices to show I that 

x ∈ dom ·,α,p : x,α,p  1, (Tr ⊕IdC )(x) = 0 is rel. compact in SI∞ ⊕ CId 2 .

(5.29)

I

1

Case 2  p < ∞. The operator A− 2 : Ran Ap → Ran A∞ is bounded by finite-dimensionality. Applying this operator to the closed unit ball of Ran Ap = 1

Ran Ap2 equipped with the norm ·S p , we obtain that the set9 I

   1  x ∈ Ran A∞ : Ap2 (x)S p  1 I

9

1

1

1

Since I is finite, we have dom Ap2 = SI . Write x = A− 2 Ap2 x. p

5.5 Quantum Metric Spaces Associated to Semigroups of Schur Multipliers

207

is bounded in SI∞ . Note that for any x ∈ SI we have p

  1  (3.95)     A 2 (x) p p max (x, x) 12  p , (x ∗ , x ∗ ) 12  p (5.18) = x,α,p . S S S I

I

I

(5.30) We deduce from this inequality that  x ∈ Ran A∞ : x,α,p  1

(5.31)

is bounded, hence by finite dimensionality, relatively compact in SI∞ . We have  x ∈ Ran A∞ : x,α,p  1  = x ∈ dom ·,α,p : x,α,p  1, (Tr ⊕IdC )(x) = 0 . We deduce (5.29). Case p = ∞. Fix some 2 < p0 < ∞. By finite-dimensionality, the operator 1 A− 2 : Ran Ap0 → Ran A∞ is bounded. This implies that the set10 

 1  x ∈ Ran A∞ : Ap2 0 (x)S p0  1 I

is bounded in SI∞ . The inequality (5.27) says that the set 

x ∈ Ran A∞ ⊕ CId 2 : x,α,∞  1 and (Tr ⊕IdC )(x) = 0 I  = x ∈ Ran A∞ : x,α,∞  1



is bounded, i.e. relatively compact in SI∞ . The desired result follows from Proposition 5.1. For the Leibniz rule, see the end of the proof. 2. Suppose that J is a finite subset of the infinite set I . We denote by μJ : SI∞ → C the normalized “partial trace” where μJ (x) is 1/|J | times the sum of the diagonal entries of index belonging to J . This state is local since μJ (diag(0, . . . , 0, 1, . . . , 1, 0, . . . , 0)) = 1 and since diag(0, . . . , 0, 1, . . . , 1, 0, . . . , 0) belongs to M. This state extends to the def

unitization by μ = μJ ⊕ IdC . We claim that if a, b ∈ M are compactly supported then the set  a x ∈ Ran A∞ ⊕ CId 2 : x,α,p  1, μJ (x) = 0 b I

10 Since

1

p

I is finite, we have dom Ap2 0 = SI 0 .

(5.32)

208

5 Locally Compact Quantum Metric Spaces and Spectral Triples

is relatively compact. Recall that the Gelfand spectrum of M is I since we can identify M with c0 (I ). Thus, compactly supported elementsare those lying in span{eii : i ∈ I }. So we can write a = i∈Ja ai eii and b = j ∈Jb bj ejj where Ja and Jb are finite subsets of I . Now, we see that the previous set is contained in span{eij : i ∈ Ja , j ∈ Jb }, which is a finite-dimensional space since Ja and Jb are finite. Thus it suffices to show boundedness of the previous set (5.32). Using the fact that matrices in Ran A∞ have null diagonal, Lemma 5.7 and Leibniz’ rule (5.20) it is easy to check that (5.32) is equal to  a x ∈ Ran A∞ ∩ MJa ,Jb : x,α,p  1 b

(5.33)

Essentially by mimicking the proof where MJa ,Jb is the space of Ja ×Jb matrices.  of (5.31) with Lemma 5.7, the subset x ∈ Ran A∞ ∩ MJa ,Jb : x,α,p  1 is bounded. Hence the subset (5.33) is also bounded. Note we show that this quantum locally compact metric space satisfies Defi−1

nition 5.8. According to Lemma 5.7, the operator Ap 2 : Ran Ap → Ran Ap ⊂ Ran A∞ is bounded. So the set   1  1  x ∈ Ran A∞ ∩ dom Ap2 : Ap2 (x)S p  1 I

is bounded in SI∞ . By (5.30), we deduce that 1  x ∈ Ran A∞ ∩ dom Ap2 : x,α,p  1

is bounded in SI∞ . So Definition 5.8 is satisfied. We check that the seminorm ·,α,p equally satisfies the Leibniz inequality xy,α,p  xB( 2 ) y,α,p + x,α,p yB( 2 ) , I

I

1

x, y ∈ dom Ap2 0 ⊕ CId 2 . I

(5.34) Indeed, if we write x = x0 + λId 2 and y = y0 + μId 2 then we have I

I

xy = x0 y0 + λy0 + μx0 + λμId 2 . I

p

Thus, we have with limits in SI norm, observing that TJ (Id 2 )TJ (z) = TJ (z)TJ (Id 2 ) = TJ (z) I

I

(5.35)

5.5 Quantum Metric Spaces Associated to Semigroups of Schur Multipliers p

p

209

p

and that the product SI × SI → SI is continuous [38, p. 225], xy − λμId 2 = x0 y0 + λy0 + μx0 = lim [TJ (x0 )TJ (y0 ) + λTJ (y0 ) + μTJ (x0 )] J

I

*

= lim TJ (x0 + λId 2 )TJ (y0 ) + μTJ (x0 ) J

+

I

*

+ = lim TJ (x0 + λId 2 )TJ (y0 + μId 2 ) − λμTJ (Id 2 )TJ (Id 2 ) . J

I

I

I

I

p

Therefore, we have xy − λμId 2 = limJ TJ (x)TJ (y) − TJ (λμId 2 ), limit in SI . I I Moreover, similarly to (5.21) we have     TJ (x)TJ (y) − TJ (λμId 2 )  TJ (x)TJ (y),p + TJ (λμId 2 ),p ,p I

(5.20)(2.92)



I

TJ (x)SI∞ TJ (y),p + TJ (y)S ∞ TJ (x),p I

(2.92)

 xB( 2 ) TJ (y0 ),p + yB( 2 ) TJ (x0 ),p I

I

 xB( 2 ) y0 ,p + yB( 2 ) x0 ,p I

I

(5.25)

= xB( 2 ) y,α,p + yB( 2 ) x,α,p . I

I

Then by the lower semicontinuity of the seminorm ·,p from Proposition 5.5, we 1

deduce that xy − λμId 2 belongs to dom Ap2 and that I

  xy  ,α,p

(5.25)(5.35)

=

  xy − λμId 2  ,p  xB( 2 ) y,α,p + yB( 2 ) x,α,p . I

I

I

Finally, we check both properties of the point 1.(b) of Definition 5.3 as in the proof of [148, Proposition 2.17]. Thus, (Ran A∞ ⊕ RId 2 , ·) is a unital Leibniz pair. I We check the semicontinuity property. To this end, let (xn ) be a sequence in 1

dom Ap2 0 ⊕ CId 2 ⊂ B( 2I ) such that xn → x in B( 2I ) with xn ,α,p  1 for I 1

all n. We can write xn = xn,0 + λn Id 2 with xn,0 ∈ dom Ap2 0 and λn ∈ C. Since I

Ran A∞ ⊕ CId 2 is a closed subspace of B( 2I ), we have x = x0 + λId 2 for some I

I

x0 ∈ Ran A∞ and λ ∈ C. Since the map Ran A∞ ⊕ CId 2 → C, x0 + λId 2 → λ I I is continuous, we have λn → λ, and therefore also xn,0 → x0 in SI∞ . Appealing to p Proposition 5.5, it suffices to show that xn,0 converges weakly to x0 in SI . Since the ∞ convergence already holds in SI norm, it suffices to show that xn,0 is bounded in  1  (5.25) p SI . But this follows from Ap2 (xn,0 )S p  xn,0 ,p = xn ,α,p  1 and the −1

I

fact that Ap 2 : Ran Ap → Ran Ap is bounded according to Lemma 5.7.

210

5 Locally Compact Quantum Metric Spaces and Spectral Triples

1. (Leibniz rule) Note that if I is finite, then the

previous proof of Leibniz quantum locally compact metric space shows that Ran A∞ sa ⊕ RId 2 , ·,α,p is a I Leibniz quantum compact metric space, in case 2  p < ∞. We indicate how the same proof also works for p = ∞. Note first that when I is finite, we have SI∞ = MI,fin , so that the domain of ·,α,∞ is the full space Ran A∞ ⊕ CId 2 : I as in the proof of Proposition 5.5, one can show that for any x, y ∈ MI,fin , one has xy,∞  xSI∞ y,α,∞ + yS ∞ x,α,∞ . I

Thus, in particular the same holds if x, y ∈ dom ·,α,∞ = Ran A∞ ⊕ CId 2 . I For the lower semicontinuity, if suffices to note that the seminorm satisfies x,α,∞  xSI∞ , and by the reversed triangle inequality, we also have | x,α,∞ − y,α,∞ |  x − y,α,∞ .  

5.6 Gaps of Some Markovian Semigroups of Schur and Fourier Multipliers In this section, we study some typical examples of markovian semigroups of Schur and Fourier multipliers. We equally calculate their gaps (5.12) and (5.16) and examine the injectivity of their Hilbert space representation. This information is important for applications to compact quantum metric spaces in Sects. 5.2 and 5.5. Heat Schur Semigroup and Poisson Schur Semigroup In the following, I is equal to {1, . . . , n}, N or Z. We consider the heat Schur semigroup (Tt )t 0 acting on B( 2I ) defined by   2 Tt : [xij ] → e−|i−j | t xij .

(5.36)

Moreover, we also consider the Poisson Schur semigroup (Tt )t 0 acting on B( 2I ) defined by   Tt : [xij ] → e−|i−j |t xij .

(5.37)

These two semigroups are examples of noncommutative diffusion semigroups consisting of Schur multipliers. Indeed, for the first one, we can take the real Hilbert space H = R and put αi = i for α ∈ I . So we have αi − αj 2R = |i − j |2 . For the second one, we can consider  the real Hilbert space H = 2I and put αi = ik=0 ek for i ∈ I with i  0 and |i| αi = k=0 e−k for i ∈ I with i < 0 (if I contains negative elements). Then for

5.6 Gaps of Some Markovian Semigroups of Schur and Fourier Multipliers

211

i > j  0, using the orthogonality of the el ’s in the third equality, we have αi − αj 2 2 I

2   2    j     i  i    = e − e = e k k k    k=0 2I k=0  2 k=j +1 I

=

i  k=j +1

ek 2 2 = I

i 

1 = i − j = |i − j |.

k=j +1

In a similar way, we obtain for i, j in general position that αi − αj 22 = |i − j |. I

Note that both mappings α are injective.

Lemma 5.8 Consider the previous heat Schur semigroup from (5.36). We have Gapα = 1. Proof We have (5.16)

Gapα =

inf

αi −αj =αk −αl

  (αi − αj ) − (αk − αl )2 = R

inf

i−j =k−l

(i − j ) − (k − l)2R  

which is clearly equal to 1.

We will now calculate Gapα for the previous Poisson Schur semigroup. First we have Lemma 5.9 Consider the previous Poisson Schur semigroup from (5.37) with I = Z. Then for any i, j, k, l ∈ Z, i  j  k  l, we have

αi − αj , αk − αl H = 0. Proof Indeed, we have

αi − αj , αk − αl H =

 i r=0

er −

j  r=0

er ,

k  r=0

er −

l  r=0

=

er H

  i r=j +1

er ,

k  r=l+1

=0

er H

  Lemma 5.10 Consider again the previous Poisson Schur semigroup from (5.37). We have Gapα = 1. Proof It is clear that it suffices to examine the case I = Z. For the inequality Gapα  1, it suffices to take i = 1, j = k = l = 0, in which case we have   (αi − αj ) − (αk − αl )2 = αi − αj 2 = |i − j | = 1. H H

212

5 Locally Compact Quantum Metric Spaces and Spectral Triples

For the reverse inequality Gapα  1, consider any i, j, k, l such that αi − αj =  2 αk − αl . We want to estimate (αi − αj ) − (αk − αl ) from below. Using that x = −x and exchanging names of indices, we can assume without loss of generality that max(i, j, k, l) = i. First case: We have l = min(i, j, k, l). Then exchanging the names of indices j and k if necessary, we have i  j  k  l. Thus, according to Lemma 5.9, we have

αi − αj , αk − αl  = 0. Consequently,   (αi − αj ) − (αk − αl )2 = αi − αj 2 + αk − αl 2 − 2 αi − αj , αk − αl  = |i − j | + |k − l| − 0. Clearly, if this expression is 0, then i = j and k = l, which is excluded by αi −αj = αk − αl , or αi − αk = αj − αl in case that we had exchanged names of indices. In any other case, this expression is  1 since i, j, k, l take entire values. Second case: We have min(i, j, k, l) ∈ {j, k} and l is the second smallest value among i, j, k, l. Then exchanging the names of indices j and k if necessary, we can suppose min(i, j, k, l) = k. So we have i  j  l  k, and thus by Lemma 5.9

αi − αj , αl − αk  = 0. We calculate   (αi − αj ) − (αk − αl )2 = αi − αj 2 + αk − αl 2 + 2 αi − αj , αl − αk  = |i − j | + |k − l| + 0. We argue as before to see that this quantity is  1. Third case: We have min(i, j, k, l) ∈ {j, k} and l is the second biggest value among i, j, k, l. Then exchanging the names of indices j and k if necessary, we can suppose min(i, j, k, l) = k. So we have i  l  j  k, and thus by Lemma 5.9

αi − αj , αk − αl  = αi − αl , αk − αl  + αl − αj , αk − αl  = −0 + αl − αj , αk − αj  + αl − αj , αj − αl  = −0 − 0 − αl − αj 2 = −|l − j |. Then we calculate   (αi − αj ) − (αk − αl )2 = αi − αj 2 + αk − αl 2 − 2 αi − αj , αk − αl  = |i − j | + |k − l| + 2|l − j | = i − j + l − k + 2l − 2j = i − k + 3(l − j ).

Again we argue as before to see that this quantity is  1.

 

Markovian Semigroups of Herz-Schur Multipliers vs. Markovian Semigroups of Fourier Multipliers Let G be a discrete group and ψ : G → R be a function. Suppose that ψ(e) = 0. Recall that by [31, Corollary C.4.19], the function ψ is conditionally negative definite if and only if for any t  0, the function e−t ψ is

5.6 Gaps of Some Markovian Semigroups of Schur and Fourier Multipliers

213

of positive type. On the one hand, by [71, Proposition 4.2] that exactly means that def

ψ induces a completely positive Fourier multiplier Tt = Mexp(−t ψ) : VN(G) → VN(G) for any t  0. On the other hand, by [31, Definition C.4.1] and [46, Theorem D.3] that is equivalent to say that ψ induces a completely positive Herz-Schur multiplier TtHS : B( 2G ) → B( 2G ) for any t  0 (whose symbol is   def exp(−tφ(s, r)) s,r∈G where φ(s, r) = ψ(s −1 r) for any s, r ∈ G). In this case, by Proposition 2.3, we obtain a markovian semigroup (Tt )t 0 of Fourier multipliers and it is easy to check and well-known that we obtain a markovian semigroup (TtHS )t 0 of Herz-Schur multipliers. Hence there is a bijective correspondence between markovian semigroups of Fourier multipliers on VN(G) and markovian semigroups of Herz-Schur multipliers on B( 2G ). So any triple (b, π, H ) associated to a markovian semigroup (Tt )t 0 of Fourier multipliers by Proposition 2.3 gives a couple (α, H ) associated to (TtHS )t 0 and conversely. More precisely, if ψ(s) =   bψ (s)2 then for any s, r ∈ G H  2   φ(s, r) = ψ(s −1 r) = bψ (s −1 r)

H

 2   = bψ (s −1 ) + πs −1 (bψ (r))

(2.33)

 2  2 = −πs −1 (bψ (s)) + πs −1 (bψ (r))H = bψ (r) − bψ (s)H .

H

(2.35)

So we can consider the couple (bψ , H ) for the semigroup (TtHS )t 0 . Next we compare the gaps of Herz-Schur and Fourier markovian semigroups as we encountered them in Sects. 5.5 and 5.2. We will see in Proposition 5.10 that a strict inequality may occur in the following result. Proposition 5.8 Let (Tt )t 0 and (TtHS )t 0 be markovian semigroups of Fourier multipliers and Herz-Schur multipliers as previously. Consider a triple (b, π, H ) for def the first semigroup and the couple (α, H ) such that α = b for the second semigroup. We have Gapα  Gapψ

(5.38)

where Gapα and Gapψ are defined in (5.16) and (5.12). (2.33)

Proof For any i, j, l, k ∈ G, we have αi − αj = b(i) − b(j ) = πj (b(j −1 i)) = πj (αj −1 i ) and αk − αl = πl (αl −1 k ). Thus, we have (5.16)

Gapα = =

inf

αi −αj =αk −αl

inf

αi −αj =αk −αl

  (αi − αj ) − (αk − αl )2 H

  πj (α −1 ) − πl (α −1 )2 j i l k H

 2  2 = inf πl −1 j (αj −1 i ) − αl −1 k H = inf πj (αs ) − αr H , i,j,k,l

j,s,r

214

5 Locally Compact Quantum Metric Spaces and Spectral Triples

where the infimum is taken over those j, s, r ∈ G such that the considered norm is = 0. On the other hand, we have, since b(s) = αs and by considering j = e and πe = IdH in the inequality (5.12)

Gapψ =

inf

b(s)=b(t )

b(s) − b(t)2H = inf αs − αr 2H αs =αr

 2  inf πj (αs ) − αr H = Gapα , j,s,r

 

(again infimum over non-zero quantities).

Finite-Dimensional Hilbert Spaces and Coboundary Cocycles over Infinite Groups: A Dichotomy Between Non-injective Cocycles and the Condition Gapψ = 0 Now, we consider particular markovian semigroups of Fourier multipliers and their corresponding semigroups of Herz-Schur multipliers and ask whether the representations b = α are injective and calculate the gaps Gapψ and Gapα defined in (5.16) and (5.12). We start with a general observation. Consider an infinite discrete group G and a finite-dimensional orthogonal representation π : G → O(H ). Consider a 1-cocycle b : G → H with respect to π. So we have a markovian semigroup of Fourier multipliers on VN(G) and a semigroup of HerzSchur multipliers on B( 2G ). For 1-coboundaries, the situation is not as nice as in Sect. 5.2. Indeed, we have the following proposition. Proposition 5.9 Let G be an infinite discrete group and b : G → H be a 1-coboundary with respect to some finite-dimensional orthogonal representation π : G → O(H ). If b is injective, then Gapψ = Gapα = 0. So if Gapψ > 0 or Gapα > 0 then b is non-injective. Proof By definition [31, Definition 2.2.3], there exists ξ ∈ H such that b(s) = πs (ξ ) − ξ for any s ∈ G. Clearly, the formula b(s) = πs (ξ ) − ξ implies that ξ = 0 in the case where b is injective. For simplicity, we assume that ξ H = 1. Since b is injective, O = {πs (ξ ) : s ∈ G} is an infinite subset of the sphere S of H . Since H is finite-dimensional, S is compact. So there exists some accumulation point η in the sphere of the orbit O. We have thus a convergent sequence (πsn (ξ )) consisting by injectivity of b of different points. We infer that (5.12)

0  Gapψ 

2  lim b(sn ) − b(sn+1 )2 = lim πsn+1 (ξ ) − πsn (ξ ) = 0.

n→∞

(5.38)

Thus 0  Gapα  Gapψ = 0.

n→∞

 

Remark 5.6 In the paper [129, p. 1967], the authors were able to find the original paper which contains the famous Bieberbach Theorem. Unfortunately, this theorem is badly written in [129, p. 1967] (since a crucial assumption is missing; to compare

5.6 Gaps of Some Markovian Semigroups of Schur and Fourier Multipliers

215

with a textbook, e.g. [194, Th 7.2.4 p. 306]). So the proof of [129, Theorem 6.4] is doubtful since this “more general version” of Bieberbach Theorem is used in the proof. Nevertheless, we think that an additional “proper” assumption could lead to a correct statement of this interesting idea. Recall that a topological group has property (FH) if every affine isometric action of G on a real Hilbert space has a fixed point, see [31, Definition 2.1.4]. Note that by [31, Theorem 2.12.4], if a discrete group G has (T) then G has (FH) and the converse is true if G is countable. A finite group has (FH). The groups Zk and free groups Fk do not have (FH) if k  1. Using [31, Proposition 2.2.10], we deduce the following result. In [129, p. 1968], it is written that “infinite groups satisfying Kazhdan property (T) do not admit finite-dimensional standard cocycles” (i.e. injective, finite-dimensional with Gapψ > 0). But from our point of view, the proof is missing. Proposition 5.9 allows us to give a proof and to obtain a slightly more general version. Corollary 5.1 Let G be an infinite discrete group with property (FH) and b : G → H be a 1-cocycle with respect to some finite-dimensional orthogonal representation π : G → O(H ). If b is injective, then Gapψ = Gapα = 0. Semigroups on Finite Groups Let G be a finite group. By [129, pp. 1970-1971], there always exists an orthogonal representation π : G → H on some finitedimensional real Hilbert space H and an injective 1-cocycle b : G → H with respect to π. In this case, since G is finite, b only takes a finite number of values. (5.12)

This implies that Gapψ = infb(s)=b(t ) b(s) − b(t)2H > 0. For example, we can consider the left regular representation π : G → B( 2G ) defined by πs (et ) = est for any s, t ∈ G and the cocycle b : G → 2G , s → πs (ξ ) − ξ where ξ is some vector of 2G satisfying πs (ξ ) = ξ for any element s of G − {eG }. We refer to [129, a) and b) p. 1971] for other interesting examples of 1-cocycles. Note in addition that in the context of Schur multipliers, we also have Gapα > 0 if α = b. Heat Semigroup on Tn Here G = Zn . We consider the Heat semigroup (Tt )t 0 2 on L∞ (Tn ) = VN(Zn ) defined by Tt : L∞ (Tn ) → L∞ (Tn ), eik· → e−t |k| eik· . The associated finite-dimensional injective cocycle is given by the canonical inclusion b : Zn → Rn equipped with the trivial action πk = IdRn for all k ∈ Zn . Lemma 5.11 Consider the previous heat semigroup. We have Gapψ = 1. Proof Indeed, we have (5.12)

Gapψ =

inf

bψ (k)=bψ (l)

  bψ (k) − bψ (l)2 n = R

inf

k=l,k,l∈Zn

|k − l|2 = 1.  

The Donut Type Markovian Semigroup of Fourier and Herz-Schur Multipliers Consider now the example of donut type Fourier multipliers in the spirit of [129,

216

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Section 5.3]. That is, we consider the group G = Z with cocycle def

b(n) = e2πiαn , e2πiβn − (1, 1) ∈ C2 = R4 ,

(5.39)

where α, β ∈ R and we consider C2 as the real Hilbert space R4 . The associated cocycle orthogonal representation is11 def

πn (x, y) = e2πiαn x, e2πiβn y

n ∈ Z, x, y ∈ C.

(5.40)

Proposition 5.10 Consider the case that both α and β take rational values. Then b : Z → C2 is not injective, but Gapψ , Gapα > 0. Moreover, the strict inequality Gapα < Gapψ may happen. Proof Consider p, q ∈ Z and N ∈ N∗ such that α =

p N

and β =

q N.

We have



b(n + N) = e2πi(p+αn) , e2πi(q+βn) − (1, 1) = e2πiαn , e2πiβn − (1, 1) = b(n), so that b is N-periodic, and hence only takes a finite number of values. In particular, the function b : Z → C2 is not injective and the set {b(n) − b(m) : n, m ∈ Z} is finite, which readily implies that Gapψ = infb(n)=b(m) b(n) − b(m)2R4 > 0. In the same manner, we obtain Gapα = infb(i)−b(j )=b(k)−b(l) b(i) − b(j ) − (b(k) − b(l))2R4 > 0. We turn to the statement of strict inequality. To this end, we take α = β = 18 . Then  2   Gapψ = 2 inf (e2πiαn − 1) − (e2πiαm − 1)  2 = 2 inf e2πiα(n−m) − 1C  2  2 1 + i   2πi 81   = 2 e − 1 = 2 ·  √ − 1 2

11 For

C

any n, m ∈ Z, we have the cocycle law 



b(n) + πn (bm ) = e2πiαn , e2πiβn − (1, 1) + πn e2πiαm , e2πiβm − (1, 1)





= e2πiαn , e2πiβn − (1, 1) + e2πiα(n+m) , e2πiβ(n+m) − e2πiαn , e2πiβn

= e2πiα(n+m) , e2πiβ(n+m) − (1, 1) = b(n + m).

5.6 Gaps of Some Markovian Semigroups of Schur and Fourier Multipliers

&

1 =2· √ −1 2   1 =4 1− √ . 2

2

1 + √ ( 2)2

217

'

On the other hand, according to the proof of Proposition 5.8, we have  2   Gapα = 2 inf e2πiαn (e2πiαr − 1) − (e2πiαs − 1)  2    2 e2πiα·1(e2πiα·2 − 1) − (e2πiα·4 − 1)

C

C

2 2     1 + i  1    = 2  √ (i − 1) − (−1 − 1) = 2  √ (i − 1 − 1 − i) + 2 2 2 C   2  1  = 2 2 1 − √  2 2    1 1 =8 1− √ 0. Finally assume that the von Neumann crossed product q (H ) α G has QWEP. −1

1. The operator Ap 2 : Ran Ap → Ran Ap is compact. − 12

2. Suppose −1  q  1. Then the operator Bp (4.7).

is compact where Bp is defined in

Proof −1

1. As in the proof of Theorem 5.1 the operator A2 2 : Ran A2 → Ran A2 is compact. −1

We show that A∞2 : Ran A∞ → Ran A∞ is bounded. Note that (Tt,∞ )t 0 is a bounded weak* continuous semigroup with weak* generator A∞ . Using a mild adaptation of Lemma 5.6 to weak* Markov semigroups, it suffices to establish the bound Tt Ran A∞ →Ran A∞  t1d for some d > 12 (if dim H = 1 we need to embed H in a larger Hilbert space before). We conclude with [129, Lemma 5.8] −1

and the contractive inclusion VN(G) ⊂ L1 (VN(G)). Thus, A∞2 : Ran A∞ → Ran A∞ is bounded. Now, assume that p > 2. We will use complex interpolation. Since the resolvents of the operators Ap are compatible for different values of p, the complementary projections associated to the ones of (2.17) onto the spaces Ran Ap are compatible. Hence, the Ran Ap ’s form an interpolation scale. Observe that Ran A2 is a Hilbert space, hence a UMD space. Then we obtain the compactness −1

of Ap 2 : Ran Ap → Ran Ap by means of complex interpolation between a compact and a bounded operator with Theorem 2.2. If p < 2, we conclude by duality and Schauder’s Theorem [167, Theorem 3.4.15], since Ran Ap is the dual space of Ran Ap∗ (use (2.8) and [167, page 94]) −1

−1

and Ap 2 defined on the first space is the adjoint of Ap∗2 defined on the second space. 2. Since A2 = (∂ψ,q,2 )∗ ∂ψ,q,2 and B2 = ∂ψ,q,2 (∂ψ,q,2 )∗ |Ran ∂ψ,q,2 , by Theorem 2.1 and Proposition 4.8 together with a functional calculus argument, −1

− 12

we obtain that the operators A2 2 |Ker(∂ψ,q,2 )⊥ and B2 Since − 12

−1 A2 2

are unitarily equivalent.

: Ran A2 → Ran A2 is compact and Ker(∂ψ,q,2 )⊥ = Ran A2 , −1

A2 |Ker(∂ψ,q,2 )⊥ and finally B2 2 : Ran ∂ψ,q,2 → Ran ∂ψ,q,2 is compact. Recall that by (4.7), Bp is the generator of the (strongly continuous) semigroup (IdLp (q (H ))  Tt |Ran ∂ψ,q,p )t 0 . Now, we use [129, Lemma 5.8] and Proposition 2.8 to obtain the bound Id  Tt ∞→∞  t1d for some d > 12 . According to Proposition 4.8, we have Ran Bp = Ran ∂ψ,q,p . By Lemma 5.6, we −1

obtain that Bp 2 : Ran Bp → Ran Bp is bounded.

226

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Now, it suffices to interpolate with Theorem 2.2 the compactness −1

of the operator B2 2 : Ran ∂ψ,q,2 of

−1 Bp02

: Ran ∂ψ,q,p0

→

→

Ran ∂ψ,q,2 and the boundedness

Ran ∂ψ,q,p0 to obtain the compactness of

−1 Bp 2

: Ran ∂ψ,q,p → Ran ∂ψ,q,p for 2  p < p0 < ∞ and the fact that on the L2 -level, we have a Hilbert space hence a UMD space. Note that the spaces Ran ∂ψ,q,p ⊂ Lp (q (H )⊗B( 2I )) interpolate by the complex interpolation method since we have bounded projections Pp : Lp (q (H )  G) → Ran ∂ψ,q,p ⊂ Lp (q (H )  G) which are compatible for different values of p according to (2.17). We use duality if p < 2.   In the following theorem, recall the weak* closed operator ∂ψ,q,∞ : dom ∂ψ,q,∞ ⊂ q (H ) α G → q (H ) α G from Proposition 3.5. The latter is valid if G has AP and q = 1. Finally, the QWEP assumption of the following result is satisfied if G is amenable or G is a free group and q = −1. Theorem 5.4 Suppose 1 < p < ∞ and −1  q < 1. The triple (C∗r (G), Lp (VN(G)) ⊕p Lp (q (H ) α G), Dψ,q,p ) is a Banach spectral triple in the case where G is weakly amenable, bψ : G → H is injective, Gapψ > 0, H is finite-dimensional and q (H ) α G has QWEP. Moreover, we have the following properties. 1. We have (Dψ,q,p )∗ = Dψ,q,p∗ . In particular, the operator Dψ,q,2 is selfadjoint. 2. We have PG ⊂ LipDψ,q,p (VN(G)).

(5.47)

3. For any a ∈ PG , we have    Dψ,q,p , π(a) 

Lp (VN(G))⊕p Lp →Lp (VN(G))⊕p Lp

   ∂ψ,q (a)

q (H )α G

.

(5.48) 4. Suppose that G has AP. We have dom ∂ψ,q,∞ ⊂ LipDψ,q,p (VN(G)).

(5.49)

5.8 Spectral Triples Associated to Semigroups of Fourier Multipliers I

227

5. Suppose that G has AP. For any a ∈ dom ∂ψ,q,∞ , we have    Dψ,q,p , π(a) 

Lp ⊕p Lp (q (H )α G)→Lp ⊕p Lp (q (H )α G)

(5.50)

   ∂ψ,q,∞ (a)q (H )α G .

6. Assume that G is weakly amenable and that q (H ) α G has QWEP. If bψ : G → H is injective, Gapψ > 0 and if H is finite-dimensional, then the operator |Dψ,q,p |−1 : Ran(∂ψ,q,p∗ )∗ ⊕ Ran ∂ψ,q,p → Ran(∂ψ,q,p∗ )∗ ⊕ Ran ∂ψ,q,p is compact. Proof ∗

∗

1. An element (z, t) of Lp (VN(G)) ⊕p∗ Lp (q (H ) α G) belongs to ∗ dom(Dψ,q,p )∗ if and only if there exists (a, b) ∈ Lp (VN(G)) ⊕p∗ ∗ Lp (q (H ) α G) such that for any (x, y) ∈ dom ∂ψ,q,p ⊕ dom(∂ψ,q,p∗ )∗ we have          0 (∂ψ,q,p∗ )∗ x z x a , = , , ∂ψ,q,p 0 y t y b that is 

   (∂ψ,q,p∗ )∗ (y), z + ∂ψ,q,p (x), t = y, b + x, a.

(5.51)

If z ∈ dom ∂ψ,q,p∗ and if t ∈ dom(∂ψ,q,p )∗ the latter holds with b = ∂ψ,q,p∗ (z) and a = (∂ψ,q,p )∗ (t). This proves that dom ∂ψ,q,p∗ ⊕ dom(∂ψ,q,p )∗ ⊂ dom(Dψ,q,p )∗ and that

(Dψ,q,p )∗ (z, t) = (∂ψ,q,p )∗ (t), ∂ψ,q,p∗ (z)    0 (∂ψ,q,p )∗ z (5.45) = = Dψ,q,p∗ (z, t). ∂ψ,q,p∗ 0 t Conversely, if (z, t) ∈ dom(Dψ,q,p )∗ , choosing y = 0 in (5.51) we obtain t ∈ dom(∂ψ,q,p )∗ and taking x = 0 we obtain z ∈ dom ∂ψ,q,p∗ . 2. Recall that the subspaces PG and P,G are dense in Lp (VN(G)) and Lp (q (H ) α G) and are contained in the domains of ∂ψ,q,p and (∂ψ,q,p∗ )∗ . So PG ⊕P,G is contained in dom Dψ,q,p . For any a ∈ PG , we have La (PG ) ⊂ PG and L˜ a (P,G ) ⊂ P,G . We infer that π(a) · (PG ⊕ P,G ) ⊂ dom Dψ,q,p . Note also that π(a)∗ · (PG ⊕ P,G ) ⊂ dom Dψ,q,p∗ = dom(Dψ,q,p )∗ (condition 2 of Proposition 5.12).

228

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Let a ∈ PG . A simple computation shows that 

Dψ,q,p , π(a)

 

     (∂ψ,q,p∗ )∗ La 0 La 0 0 (∂ψ,q,p∗ )∗ − ∂ψ,q,p 0 0 0 L˜ a 0 L˜ a ∂ψ,q,p     0 (∂ψ,q,p∗ )∗ L˜ a 0 La (∂ψ,q,p∗ )∗ = − ˜ La ∂ψ,q,p ∂ψ,q,p La 0 0   0 (∂ψ,q,p∗ )∗ L˜ a − La (∂ψ,q,p∗ )∗ . = ∂ψ,q,p La − L˜ a ∂ψ,q,p 0

(5.45)(5.46)

=

0

We calculate the two non-zero components of the commutator. For the lower left corner, if x ∈ PG and if we consider the canonical map J : Lp (VN(G)) → Lp (q (H ) α G), x → 1  x, we have14 (∂ψ,q,p La − L˜ a ∂ψ,q,p )(x) = ∂ψ,q,p La (x) − L˜ a ∂ψ,q,p (x)

(5.52)

(2.86)

= ∂ψ,q,p (ax) − a∂ψ,q,p (x) = ∂ψ,q (a)x = L∂ψ,q (a) J (x). For the upper right corner, if E is the conditional expectation associated to J , note that for any y ∈ P,G and any x ∈ PG , (we recall that we have the duality brackets f, g antilinear in the first variable)  

(∂ψ,q,p∗ )∗ L˜ a − La (∂ψ,q,p∗ )∗ (y), x     = (∂ψ,q,p∗ )∗ L˜ a (y), x − La (∂ψ,q,p∗ )∗ (y), x     = L˜ a (y), ∂ψ,q,p∗ (x) − (∂ψ,q,p∗ )∗ (y), La ∗ (x)     = y, L˜ a ∗ ∂ψ,q,p∗ (x) − y, ∂ψ,q,p∗ La ∗ (x)     = y, L˜ a ∗ ∂ψ,q,p∗ (x) − ∂ψ,q,p∗ La ∗ (x) = y, a ∗ ∂ψ,q,p∗ (x) − ∂ψ,q,p∗ (a ∗ x)       = y, −∂ψ,q (a ∗ )x = y, −L∂ψ,q (a ∗ ) (1  x) = y, L(∂ψ,q (a))∗ (1  x)     = L∂ψ,q (a) (y), 1  x = EL∂ψ,q (a)(y), x Lp (VN(G)),Lp∗ (VN(G)).

(2.86)

We conclude that

(∂ψ,q,p∗ )∗ L˜ a − La (∂ψ,q,p∗ )∗ (y) = EL∂ψ,q (a) (y).

14 Recall

that the term ∂ψ,q,p (a)x is by definition equal to ∂ψ,q,p (a)(1  x).

(5.53)

5.8 Spectral Triples Associated to Semigroups of Fourier Multipliers I

229

The two non-zero components of the commutator are  bounded linear operators on PG and on P,G . We deduce that Dψ,q,p , π(a) is bounded on PG ⊕ P,G . By Proposition 5.12, it extends to a bounded operator on Lp (VN(G)) ⊕p Lp (q (H ) α G). Hence PG is a subset of LipDψ,q,p (VN(G)). 3. If (x, y) ∈ dom Dψ,q,p and a ∈ PG , we have     Dψ,q,p , π(a) (x, y) p 



   =  (∂ψ,q,p∗ )∗ L˜ a − La (∂ψ,q,p∗ )∗ y, ∂ψ,q,p La − L˜ a ∂ψ,q,p x 

(5.54) p



   

p 

p 1 =  (∂ψ,q,p∗ )∗ L˜ a − La (∂ψ,q,p∗ )∗ y  +  ∂ψ,q,p La − L˜ a ∂ψ,q,p x  p p

p

  p 1

 EL∂ (a) (y)pp = + ∂ψ,q (a)J (x)Lp (q (H )α G) p ψ,q L (VN(G))    ∂ψ,q (a)q (H )α G (x, y)p . (5.53)(5.52)

So we obtain (5.48). 4. Let a ∈ dom ∂ψ,q,∞ . Let (aj ) be a net in PG such that aj → a and ∂ψ,q,∞ (aj ) → ∂ψ,q,∞ (a) both for the weak* topology. The existence of such a net is guaranteed by Proposition 3.5. By Lemma 2.1, we can suppose that the nets (aj ) and (∂ψ,q,∞ (aj )) are bounded. By the point 4 of Proposition 5.11, we deduce that a ∈ LipDψ,q,p (VN(G)). By continuity of π, note that π(aj ) → π(a) for the weak operator topology. For any ξ ∈ dom Dψ,q,p and any ζ ∈ dom(Dψ,q,p )∗ , we have 

 [Dψ,q,p , π(aj )]ξ, ζ Lp (VN(G))⊕p Lp (q (H )α G),Lp∗ (VN(G))⊕ ∗ Lp∗ (q (H )α G) p   = (Dψ,q,p π(aj ) − π(aj )Dψ,q,p )ξ, ζ     = Dψ,q,p π(aj )ξ, ζ − π(aj )Dψ,q,p ξ, ζ     = π(aj )ξ, (Dψ,q,p )∗ ζ − π(aj )Dψ,q,p ξ, ζ     → π(a)ξ, (Dψ,q,p )∗ ζ − π(a)Dψ,q,p ξ, ζ − j

      = Dψ,q,p π(a)ξ, ζ − π(a)Dψ,q,p ξ, ζ = [Dψ,q,p , π(a)]ξ, ζ . Since the net ([Dψ,q,p , π(aj )]) is bounded by (5.48), we deduce that the net ([Dψ,q,p , π(aj )]) converges to [Dψ,q,p , π(a)] for the weak operator topology by a “net version” of [136, Lemma 3.6 p. 151]. Furthermore, it is (really) easy to check that L∂ψ,q,p (aj ) J → L∂ψ,q,p (a)J and −EL∂ψ,q (aj ) → −EL∂ψ,q (a) both for the weak operator topology. By uniqueness of the limit, we deduce that the commutator is given by the same formula as that in the case of elements of PG . 5. We obtain (5.50) as in (5.54).

230

5 Locally Compact Quantum Metric Spaces and Spectral Triples

6. Recall that Ran Ap = Ran(∂ψ,q,p∗ )∗ by Proposition 4.8 and that on the space Lp (VN(G)) ⊕p Ran ∂ψ,q,p , (4.19) D2ψ,q,p =

    Ap 0 (4.7) Ap 0 = . 0 (IdLp (q (H ))  Ap )|Ran ∂ψ,q,p 0 Bp

So on Ran Dψ,q,p , |Dψ,q,p |−1

⎡ 1 ⎤ − 12  −2 Ap |Ran Ap 0 A |Ran A 0 p ⎦. = =⎣ p −1 0 Bp 0 Bp 2

With Proposition 5.13, we conclude that |Dψ,q,p |−1 is compact.

 

Remark 5.8 Suppose that G has AP. We do not know if LipDψ,q,p (VN(G)) = dom ∂ψ,q,∞ . We will examine this question in a future work. Remark 5.9 Consider the case that the assumptions of Theorem 5.4 are satisfied. Note that the (Banach) spectral triple (C∗r (G), Lp (VN(G)) ⊕p Lp (q (H ) α G), Dψ,q,p ) is even. Indeed, the Hodge-Dirac operator Dψ,q,p anti-commutes with the involution   0 def −IdLp (VN(G)) γp = : 0 IdLp (q (H )α G) Lp (VN(G)) ⊕p Lp (q (H ) α G) → Lp (VN(G)) ⊕p Lp (q (H ) α G). (which is selfadjoint if p = 2), since Dψ,q,p γp + γp Dψ,q,p       (5.45) 0 (∂ψ,q,p∗ )∗ −Id 0 −Id 0 0 (∂ψ,q,p∗ )∗ = + ∂ψ,q,p 0 0 0 Id 0 Id ∂ψ,q,p     0 −(∂ψ,q,p∗ )∗ 0 (∂ψ,q,p∗ )∗ + = 0. = −∂ψ,q,p 0 ∂ψ,q,p 0 Moreover, for any a ∈ VN(G), we have         −La 0 La 0 −Id 0 (5.46) −Id 0 La 0 γp π(a) = = = = π(a)γp . 0 L˜ a 0 L˜ a 0 L˜ a 0 Id 0 Id (5.46)

5.8 Spectral Triples Associated to Semigroups of Fourier Multipliers I

231

Remark 5.10 The estimate (5.48) is in general not optimal. Indeed, already in the case p = 2 and a = λs ∈ PG for some s ∈ G, we have according to (5.52) and (5.53),   [Dψ,q,2 , π(a)] 2 L (VN(G))⊕2 L2 (q (H )α G)→L2 (q (H ))⊕2 L2 (q (H )α G)    max L∂ψ,q (a)J L2 (VN(G))→L2( (H ) G) , q α    EL∂ (a) 2 . ψ,q L ( (H ) G)→L2 (VN(G)) q

(5.55)

α

Note that we have the Hilbert space adjoints (L∂ψ,q (a) J )∗ = J ∗ L∗∂ψ,q (a) = EL(∂ψ,q (a))∗ = −EL∂ψ,q (a ∗ ) . Thus, in the maximum of (5.55), it suffices to consider  the second term. For any element x = t xt  λt of L2 (q (H ) α G), we have EL∂ψ,q (λs ) (x) = E



(2.56)

= E

sq (bψ (s))  λs

 

 xt  λt

t





sq (bψ (s))αs (xt )  λst

=



t

τ (sq (bψ (s))αs (xt ))λst .

t

Thus we have  EL∂

ψ,q (λs )

2    τ (sq (bψ (s))αs (xt ))2 (x)2 = t

  2 αs (xt )2L2 (  sq (bψ (s))L2 (q (H ))

q (H ))

t

  2 xt 2L2 ( = sq (bψ (s))L2 (q (H )) t

 2 = bψ (s)H x2L2 (

(2.41)

2 q (H )⊗B( I ))

q (H ))

.

We infer that   [Dψ,q,2 , π(λs )] 2 L ⊕

2L

2 →L2 ⊕ L2 2

     EL∂ψ,q (λs ) 2→2  bψ (s)H .

(5.56)

In the case where −1 < q < 1 and bψ (s) = 0, this quantity is strictly less than   ∂ψ,q (λs )

q (H )α G

    = sq (bψ (s))  λs q (H )α G = sq (bψ (s))q (H )

(2.84)

[44, Th. 1.10]

=

  2 bψ (s) . √ H 1−q

232

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Remark 5.11 In [131, p. 588], the authors obtain a similar estimate to (5.48) on the commutator for the case q = 1 and p = 2. The differences are that the space on which the commutator is regarded, is the smaller L2 (VN(G)) ⊕2 ψ (G)  L2 (VN(G)) ⊕2 L2 (1 (H ) α G) [131, p. 587], where ψ (G) = span {s1 (h)  λs : h ∈ Hψ , s ∈ G} and Hψ is the Hilbert space spanned by the bψ (s), s ∈ G [131, Lemma C.1]. On the other hand, [131, p. 588] obtains an exact estimate (without proof)   [Dψ,q=1,p=2 , π(a)]

L2 ⊕2 ψ (G)→L2 ⊕2 ψ (G)

(5.57)

1   12  2 . = max (a, a)VN(G) , (a ∗ , a ∗ )VN(G)

Note that when q = 1 and a ∈ PG , then the  right hand  side of (5.57) is finite, whereas the right hand side of (5.48), that is, ∂ψ,1,∞ (a) (H )α G is infinite. We 1 do not know whether (5.57) holds on our full space L2 (VN(G))⊕2 L2 (q (H )α G) or if some intermediate estimate out of (5.48) and (5.56) holds in that case. But we plan to examine this question.

5.9 Spectral Triples Associated to Semigroups of Fourier Multipliers II We consider in this section a discrete group G and a cocycle bψ : G → H . Now, we define another “Hodge-Dirac operator” by generalizing the construction of [131, p. 588] which corresponds to the case q = 0 and p = 2 below. Suppose 1 < p < ∞ and −1  q < 1. We let def

Dψ,q (x  λs ) = sq (bψ (s))x  λs ,

x ∈ q (H ), s ∈ G.

(5.58)

We can see Dψ,q as an unbounded operator acting on the subspace P,G of Lp (q (H ) α G). Finally, note that in [131, p. 588], the authors refer to a real structure but we warn the reader that the antilinear isometry J : L2 (q (H ) α G) → L2 (q (H ) α G), x → x ∗ used in [131] does not15 commute or anticommute with the Dirac operator Dψ,q .

15 For

any x ∈ q (H ) and any s ∈ G, we have

∗ (5.58)

J Dψ,q (x  λs ) = J sq (bψ (s))x  λs = sq (bψ (s))x  λs (2.57)

= αs −1 (x ∗ sq (bψ (s)))  λs −1

5.9 Spectral Triples Associated to Semigroups of Fourier Multipliers II

233

Lemma 5.13 Suppose 1 < p < ∞ and −1  q < 1. 1. For any a, b ∈ P,G we have 

   Dψ,q (a), b = a, Dψ,q (b)

(5.59)

where we use as usual the duality bracket x, y = τ (x ∗ y). 2. The operator Dψ,q : P,G ⊂ Lp (q (H )α G) → Lp (q (H )α G) is closable. Proof 1. For any s, t ∈ G and any x, y ∈ q (H ), we have 

 (5.58)   Dψ,q (x  λs ), y  λt = sq (bψ (s))x  λs , y  λt

= τ (sq (bψ (s))x  λs )∗ (y  λt )

= τ (αs −1 (x ∗ sq (bψ (s)))  λs −1 )(y  λt )



(2.56) = τ αs −1 (x ∗ sq (bψ (s)))αs −1 (y)  λs −1 t = τ αs −1 (x ∗ sq (bψ (s))y) δs=t (2.57)

and    (5.58)  x  λs , Dψ,q (y  λt ) = x  λs , sq (bψ (t))y  λt

= τ (x  λs )∗ (sq (bψ (t))y  λt )

(2.57) = τ (αs −1 (x ∗ )  λs −1 )(sq (bψ (t))y  λt )

(2.56) = τ αs −1 (x ∗ )αs −1 (sq (bψ (t))y)  λs −1 t

= τ αs −1 (x ∗ sq (bψ (s))y) δs=t . Thus, (5.59) follows by linearity. ∗ 2. Since P,G is dense in Lp (q (H ) α G), this is a consequence of [136, Theorem 5.28 p. 168].  

and

Dψ,q J (x  λs ) = Dψ,q αs −1 (x ∗ )  λs −1



(5.58)

= sq (bψ (s −1 ))αs −1 (x ∗ )  λs −1 = −sq (πs −1 (bψ (s)))αs −1 (x ∗ )  λs −1

= −αs −1 (sq (bψ (s)))αs −1 (x ∗ )  λs −1 = −αs −1 (sq (bψ (s))x ∗ )  λs −1 . .

234

5 Locally Compact Quantum Metric Spaces and Spectral Triples

We denote by Dψ,q,p : dom Dψ,q,p ⊂ Lp (q (H )α G) → Lp (q (H )α G) its closure. So P,G is a core of Dψ,q,p . We define the homomorphism π : VN(G) → B(Lp (q (H ) α G)) by def

π(a) = Idq (H )  La ,

a ∈ VN(G)

(5.60)

where La is the left multiplication operator by a. Note that π(a) is equal to the map L˜ a of Sect. 5.8. It is not difficult to see that π : VN(G) → B(Lp (q (H ) α G)) is continuous when VN(G) is equipped with the weak* topology of VN(G) and when B(Lp (q (H ) α G)) is equipped with the weak operator topology. We will also use the restriction of π to C∗r (G). Theorem 5.5 Suppose 1 < p < ∞ and −1  q < 1. 1. The operator Dψ,q,2 is selfadjoint. ∗ 2. Assume that Lp (VN(G)) has CCAP and that q (H ) α G has QWEP. For 1 < p < ∞, the unbounded operators Dψ,q,p and Dψ,q,p∗ are adjoint to each other (with respect to the duality bracket x, y = τ (x ∗ y)). 3. We have PG ⊂ LipDψ,q,p (VN(G)).

(5.61)

[Dψ,q,p , π(a)] = L∂ψ,q (a)

(5.62)

4. For any a ∈ PG , we have

and   [Dψ,q,p , π(a)] p L (

q (H )α G)→L

p (

q (H )α G)

  = ∂ψ,q (a)

q (H )α G

.

(5.63) 5. Suppose that G has AP. We have dom ∂ψ,q,∞ ⊂ LipDψ,q,p (VN(G)).

(5.64)

6. Suppose that G is weakly amenable. We have LipDψ,q,p (VN(G)) = dom ∂ψ,q,∞ .

(5.65)

7. Suppose that G has AP. For any a ∈ dom ∂ψ,q,∞ , we have [Dψ,q,p , π(a)] = L∂ψ,q,∞ (a).

(5.66)

5.9 Spectral Triples Associated to Semigroups of Fourier Multipliers II

235

8. Suppose that G has AP. For any a ∈ dom ∂ψ,q,∞ , we have     [Dψ,q,p , π(a)] p = ∂ψ,q,∞ (a)q (H )α G . L (q (H )α G)→Lp (q (H )α G) (5.67) 9. Suppose p = 2 and q = −1. We have (Dψ,−1,2 )2 = Id  A2 . 10. If −1 (H ) α G has QWEP, bψ : G → H is injective, Gapψ > 0 and if H is finite-dimensional then the operator |Dψ,−1,2 |−1 : Ran Dψ,−1,2 → Ran Dψ,−1,2 is compact. Proof 1. It suffices to show that Dψ,q : P,G ⊂ L2 (q (H )α G) → L2 (q (H )α G) is essentially selfadjoint. By (5.59), we infer that Dψ,q is symmetric. By [195, Corollary p. 257], it suffices to prove that Ran(Dψ,q ± iId) is dense in L2 (q (H ) α G).  2 Let z = s zs  λs ∈ L (q (H ) α G) be a vector which is orthogonal to Ran(Dψ,q + iId) in L2 (q (H ) α G). For any s ∈ G and any x ∈ L2 (q (H )), we have 

 sq (bψ (s))x  λs , z L2 (

q (H )α G)

− iτ (x ∗ zs )

  = Dψ,q (x  λs ), z L2 (q (H )α G) − i x  λs , zL2 (q (H )α G)   = (Dψ,q − iId)(x  λs ), z L2 (q (H )α G) = 0. (5.58)

We infer that 

x, sq (bψ (s))zs

 L2 (q (H ))

= τ (x ∗ sq (bψ (s))zs ) = iτ (x ∗ zs ) = x, izs L2 (q (H )) .

We deduce that sq (bψ (s))zs = izs . Recall that by [128, p. 96], the map q (H ) → Fq (H ), y → y() extends to an isometry  : L2 (q (H )) → Fq (H ). For any x ∈ q (H ) and any y ∈ L2 (q (H )), it is easy to check that (xy) = x(y). We infer that sq (bψ (s))(zs ) = (sq (bψ (s))zs ) = (izs ) = i(zs ). Thus the vector (zs ) of Fq (H ) is zero or an eigenvector of sq (bψ (s)). Since sq (bψ (s)) is selfadjoint, i is not an eigenvalue. So (zs ) = 0. Since  is injective, we infer that zs = 0. It follows that z = 0. The case with −i instead of i is similar. 2. By (5.59) and [136, Problem 5.24 p. 168], Dψ,q,p and Dψ,q,p∗ are formal adjoints. Hence Dψ,q,p∗ ⊂ (Dψ,q,p )∗ by [136, p. 167]. Now, we will show the reverse inclusion. Let z ∈ dom(Dψ,q,p )∗ . For any y ∈ dom Dψ,q,p , we have     Dψ,q,p (y), z = y, (Dψ,q,p )∗ (z) .

(5.68)

236

5 Locally Compact Quantum Metric Spaces and Spectral Triples ∗

Note that (Id  Mϕj )(z) → z in Lp (q (H ) α G), where (Mϕj )j is the approximating net of Fourier multipliers granted by the CCAP assumption. Now, for any y ∈ P,G , we have 

   y, Dψ,q,p∗ (Id  Mϕj )(z) = Dψ,q,p (y), (Id  Mϕj )(z)     = (Id  Mϕj )Dψ,q,p (y), z = Dψ,q,p (Id  Mϕj )(y), z (5.68)

=



   (Id  Mϕj )(y), (Dψ,q,p )∗ (z) = y, (Id  Mϕj )(Dψ,q,p )∗ (z) .

Hence → (Dψ,q,p )∗ (z) Dψ,q,p∗ (Id  Mϕj )(z) = (Id  Mϕj )(Dψ,q,p )∗ (z) − j

By (2.1), we deduce that z ∈ dom Dψ,q,p∗ and that Dψ,q,p∗ (z) = (Dψ,q,p )∗ (z). 3. and 4. For any s, t ∈ G and any x ∈ q (H ), using the relation αs = q∞ (πs ) in the sixth equality, we have 

 Dψ,q,p , π(λs ) (x  λt ) = Dψ,q,p π(λs )(x  λt ) − π(λs )Dψ,q,p (x  λt ) (5.60)(5.58)

=

Dψ,q,p (αs (x)  λst ) − π(λs )(sq (bψ (t))x  λt )

(5.60)

= Dψ,q,p (αs (x)  λst ) − αs (sq (bψ (t))x)  λst

(5.58)

= sq (bψ (st))αs (x)  λst − αs (sq (bψ (t))x)  λst

(2.33)

= sq (bψ (s))αs (x)  λst + sq (πs (bψ (t)))αs (x)  λst − αs (sq (bψ (t))x)  λst (2.56)

(2.84) = sq (bψ (s))αs (x)  λst = sq (bψ (s))  λs (x  λt ) = ∂ψ,q (λs )(x  λt ) = L∂ψ,q (λs ) (x  λt ).   By linearity and density, we obtain Dψ,q,p , π(λs ) = L∂ψ,q (λs ) . It suffices to use Proposition 5.12. Finally by linearity, for any a ∈ PG we obtain (5.62). In addition, we have   [Dψ,q,p , π(a)] p L (q (H )α G)→Lp (q (H )α G)

(5.69)

    = L∂ψ,q (a) Lp (q (H )α G)→Lp (q (H )α G) = ∂ψ,q (a)q (H )α G .

(5.62)

5.9 Spectral Triples Associated to Semigroups of Fourier Multipliers II

237

5. We next claim that dom ∂ψ,q,∞ ⊂ LipDψ,q,p (VN(G)).

(5.70)

Let a ∈ dom ∂ψ,q,∞ and consider with Proposition 3.5 a net (aj ) of elements of PG such that aj → a and ∂ψ,q (aj ) → ∂ψ,q,∞ (a) both for the weak* topology. By Lemma 2.1, we can suppose that the nets (aj ) and (∂ψ,q (aj )) are bounded. So by (5.63), the net ( Dψ,q,p , π(aj ) ) is also bounded. Now by the part 4 of Proposition 5.11, a belongs to LipDψ,q,p (VN(G)). We have shown (5.70). 6. Next we pass to the reverse inclusion of (5.70) and claim LipDψ,q,p (VN(G)) ⊂ def

dom ∂ψ,q,∞ . To this end, let a ∈ LipDψ,q,p (VN(G)) and denote aˆ r = a, λr  = τ (aλr −1 ) its Fourier coefficients for r ∈ G. We define a linear form Ta : P,G → C by def

Ta (x  λs ) =



τ (aˆ r sq (bψ (r))  λr ) · (x  λs ) .

r∈G

Since the trace vanishes for r = s −1 , the sum over r is finite. We will show that it extends to a bounded linear form on Lp (q (H ) α G). To this end consider def

the bounded net (aj ) in P,G defined by aj = Mϕj (a), where (ϕj ) is the approximating net guaranteed by the fact that G has AP. By Proposition 2.9, we have aj → a in the weak* topology. Then for any ξ, η ∈ P,G , using the point 2 in the first equality, we obtain 

     [Dψ,q,p , π(a)]ξ, η = π(a)ξ, Dψ,q,p∗ η − Dψ,q,p ξ, π(a)∗ η       = lim π(aj )ξ, Dψ,q,p∗ η − Dψ,q,p ξ, π(aj )∗ η = lim [Dψ,q,p , π(aj )]ξ, η j

j

  = lim ∂ψ,q (aj )ξ, η = τ ((∂ψ,q (aj )ξ )∗ η) = lim τ (ξ ∗ ∂ψ,q (aj )∗ η)

(5.62)

j

j

= lim τ (η∗ ∂ψ,q (aj )ξ ) = lim τ (∂ψ,q (aj )ξ η∗ ) = Ta (ξ η∗ ), j

j

where in the last equality, we used ϕj (s) → 1 for each fixed s ∈ G and the definition of Ta . Since a ∈ LipDψ,q,p (VN(G)), for any ξ, η ∈ P , we deduce the estimate   |Ta (ξ η∗ )|  [Dψ,q,p , π(a)]p→p ξ p ηp∗ . Choosing the element η = 1 of P,G , we get |Ta (ξ )|  ξ p , so that Ta induces an element of (Lp (q (H ) α G))∗ . We infer that the Fourier coefficients sequence (aˆ r sq (bψ (r)))r∈G belongs to ∗ an element b ∈ Lp (q (H ) α G). But then the previous calculation shows that for any ξ, η ∈ P,G 

 [Dψ,q,p , π(a)]ξ, η = lim ∂ψ,q (aj )ξ, η = bξ, η. j

(5.71)

238

5 Locally Compact Quantum Metric Spaces and Spectral Triples

It follows from a ∈ LipDψ,q,p (VN(G)) that   | bξ, η|  [Dψ,q,p , π(a)]p→p ξ p ηp∗ ,   ∗ so that bξ p  [Dψ,q,p , π(a)]p→p ξ p . Thus, the element b ∈ Lp (q (H )α G) is a pointwise multiplier Lp → Lp . It follows16 that b ∈ q (H ) α G. It remains to show that a ∈ dom ∂ψ,q,∞ . To this end, thanks to the AP property def

of G, we consider again the bounded net (aj ) defined by aj = Mϕj (a). By Proposition 2.9, we have aj → a in the weak* topology. Moreover, we have                ∂ψ,q (aj ) = ∂ψ,q aˆ r ϕj (r)λr  =  aˆ r ϕj (r)sq (bψ (r))  λr  q (H )α G     r r         = aˆ r (Id  Mϕj )(sq (bψ (r))  λr ) = (Id  Mϕj )(b) (H ) G q α  r     Id  Mϕj  (H ) G→ (H ) G bq (H )α G . q

α

q

α

Using Proposition 2.9 in the last estimate (since G is weakly amenable), we deduce that (∂ψ,q (aj )) is a bounded net. Now, using Banach-Alaoglu theorem, we can suppose that ∂ψ,q (aj ) → y for the weak* topology for some y. Since the graph of the unbounded operator ∂ψ,q,∞ is weak* closed, we conclude that a belongs to the subspace dom ∂ψ,q,∞ . 7. Let a ∈ dom ∂ψ,q,∞ . Consider with Proposition 3.5 a net (aj ) of elements of PG such that aj → a and ∂ψ,q (aj ) → ∂ψ,q,∞ (a) both for the weak* topology. By Lemma 2.1, we can suppose that the nets (aj ) and (∂ψ,q (aj )) are bounded. By continuity of π, π(aj ) → π(a) for the weak operator topology of B(Lp (q (H ) α G)). For any ξ, ζ ∈ P,G , we have   [Dψ,q,p , π(aj )]ξ, ζ Lp (q (H )α G),Lp∗ (q (H )α G)   = (Dψ,q,p π(aj ) − π(aj )Dψ,q,p )ξ, ζ     = Dψ,q,p π(aj )ξ, ζ − π(aj )Dψ,q,p ξ, ζ     = π(aj )ξ, Dψ,q,p∗ ζ − π(aj )Dψ,q,p ξ, ζ       → π(a)ξ, Dψ,q,p∗ ζ − π(a)Dψ,q,p ξ, ζ = [Dψ,q,p , π(a)]ξ, ζ . − j

16 For t > 0, let μ (b) denote the generalized singular number of |b|, so that τ (φ(|b|)) = t  ∞ 0 φ(μt (b))dt for any continuous function R+ → R of bounded variation [227, p. 30]. If b were unbounded, then μt (b) → ∞ as t → 0. Taking ξ = φ(|b|) = 0 with φ a smoothed indicator

function of an interval [x, y] with large x, it is not difficult to see that bξ p  x ξ p , which is the desired contradiction.

5.9 Spectral Triples Associated to Semigroups of Fourier Multipliers II

239

Since the net ([Dψ,q,p , π(aj )]) is bounded by (5.63), we deduce that the net ([Dψ,q,p , π(aj )]) converges to [Dψ,q,p , π(a)] for the weak operator topology by a “net version” of [136, Lemma 3.6 p. 151]. Furthermore, it is (really) easy to check that L∂ψ,q,∞ (aj ) → L∂ψ,q,∞ (a) for the weak operator topology of B(Lp (q (H ) α G)). By uniqueness of the limit, we deduce that the commutator is given by the same formula as that in the case of elements of PG . 8. We obtain (5.67) as in (5.69). 9. If x ∈ L2 (−1 (H )) and s ∈ G, note that

(5.58) (5.58) (Dψ,−1,2 )2 (x  λs ) = Dψ,−1,2 s−1 (bψ (s))x  λs = s−1 (bψ (s))2 x  λs  2 (2.42) = x  bψ (s)H λs = x  ψ(s)λs = (IdL2 (−1 (H ))  A2 )(x  λs ). By Hytönen et al. [110, Proposition G.2.4], note that P,G is a core of IdL2 (−1(H ))  2 is again selfadjoint, thus closed, we have IdL2 (−1(H ))  A2 . Then since Dψ,−1,2 def

2 . Now it follows from the fact that both T1 = IdL2 (−1 (H ))  A2 and A2 ⊂ Dψ,−1,2 def

2 T2 = Dψ,−1,2 are selfadjoint that both operators are in fact equal. Indeed, consider some λ ∈ ρ(T1 ) ∩ ρ(T2 ) (e.g. λ = i). Then T1 − λ ⊂ T2 − λ, the operator T1 − λ is surjective and T2 − λ is injective. Now, it suffices to use the result [207, p. 5] to conclude that T1 − λ = T2 − λ and thus T1 = T2 . −1

10. By Proposition 5.13, the operator A2 2 : Ran A2 → Ran A2 is compact. So there exists a sequence (Tn ) of finite rank bounded operators Tn : Ran A2 → Ran A2 −1

which approximate A2 2 in norm. Similarly to the proof of [121, Theorem 4.4], we may assume, without loss of generality, that the range of each Tn is contained in PG . Composing on the right with the orthogonal projection from L2 (VN(G)) − 12

onto Ran A2 , we can see A2

and each Tn as operators on L2 (VN(G)) which is − 12

an operator Hilbert space by Pisier [189, p. 139]. Moreover, A2 multiplier (with symbol ψ(s)

− 12

is then a Fourier

δs=e ). Since the operator space OH is homogeneous −1

by Pisier [189, Proposition 7.2 (iii)], each Tn and A2 2 are completely bounded. −1/2 Proposition 2.8 says that Id  A2 is bounded. Using the projection PG of [17, Corollary 4.6], we obtain for any n a completely bounded Fourier multiplier PG (Tn ) : L2 (VN(G)) → L2 (VN(G)) which has finite rank by the proof of [121, Theorem 4.4]. Moreover, by the contractivity of PG we have for any n  −1/2  A − PG (Tn ) 2

cb,L2 (VN(G))→L2 (VN(G))

  −1/2 = PG (A2 − Tn )cb,L2 →L2

  −1/2  A2 − Tn cb,L2 (VN(G))→L2 (VN(G))   −1/2 = A − Tn  2 −−−−→ 0. 2 2

L (VN(G))→L (VN(G)) n→+∞

240

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Now using the transference of Proposition 2.8, we obtain   Id  A−1/2 − Id  PG (Tn ) 2 2 L (q (H )α G)→L2 (q (H )α G)



−1/2  = Id  A2 − PG (Tn ) L2 (q (H )α G)→L2 (q (H )α G)  −1/2   A2 − PG (Tn )cb,L2 (VN(G))→L2 (VN(G)) −−−−→ 0. n→+∞

Since H is finite-dimensional, note that the fermionic space F−1 (H ) is finitedimensional. It is clear that Id  PG (Tn ) is again a finite rank operator. −1/2 Hence Id  A2 is a norm-limit of finite rank bounded operators, hence −1

compact. Thus, Id  A2 2 : Ran(Id  A2 ) → Ran(IdL2 (−1 (H ))  A2 ) is also compact. Finally, we note that Ran(Id  A2 ) = Ran(Dψ,−1,2 )2 = Ran Dψ,−1,2 , where the last equality follows from the fact that Dψ,−1,2 is selfadjoint, see the first point.  

5.10 Spectral Triples Associated to Semigroups of Schur Multipliers I In this section, we consider a markovian semigroup of Schur multipliers on B( 2I ) with associated gradient ∂α,q,p . Suppose 1 < p < ∞ and −1  q < 1. Recall that the (full) Hodge-Dirac operator Dα,q,p with domain dom Dα,q,p = dom ∂α,q,p ⊕ dom(∂α,q,p∗ )∗ is defined in (4.64) by the formula def

Dα,q,p =



0

∂α,q,p

 (∂α,q,p∗ )∗ . 0

(5.72)

We will see in the main results of this section (Propositions 5.14 and 5.16) how this Hodge-Dirac operator gives rise to a Banach spectral triple. Note that the compactness criterion needs particular attention and supplementary assumptions. The Banach spectral triple of this section will be locally compact. Let us turn to the description of the homomorphism π. For any a ∈ B( 2I ), we def denote by L˜ a = IdLp (q (H ) ⊗ La : Lp (q (H )⊗B( 2 )) → Lp (q (H )⊗B( 2 )), f ⊗ I

I

eij → f ⊗ aeij the left action. If a ∈ B( 2I )), we define the bounded operator p p π(a) : SI ⊕p Lp (q (H )⊗B( 2I )) → SI ⊕p Lp (q (H )⊗B( 2I )) by def

π(a) =

  La 0 , 0 L˜ a

a ∈ B( 2I )

(5.73)

5.10 Spectral Triples Associated to Semigroups of Schur Multipliers I p

241

p

where La : SI → SI , x → ax is the left multiplication operator. Moreover, it is easy to check that π(a)∗ = π(a ∗ ) in the case where p = 2. Finally, we will also use the restriction of π on SI∞ . Lemma 5.14 Let 1 < p < ∞ and −1  q  1. The map π is weak* continuous. p

Proof Indeed, we show that the map π˜ : B( 2I ) → B(SI ⊕p Lp (q (H )⊗B( 2I ))),   La 0 a → is weak* continuous. Let (aj ) be a bounded net of B( 2I ) converging 0 L˜ a in the weak* topology to a. It is obvious that the nets (Laj ) and (L˜ aj ) are bounded. If p∗

x ∈ SI and if y ∈ SI , we have Laj (x), yS p ,S p∗ = Tr((aj x)∗ y) = Tr(yx ∗ aj∗ ) − → p

Tr(yx ∗ a ∗ )

= La (x), y

p p∗ SI ,SI

since

yx ∗

I

∈

j

I

SI1 .

So (Laj ) converges to La in the

p SI

weak operator topology. Since is reflexive, the weak operator topology and p the weak* topology of B(SI ) coincide on bounded sets. We  conclude that (Laj ) converges to (La ) in the weak* topology. If k zk ⊗tk and if l cl ⊗bl are elements of q (H ) ⊗ MI,fin we have 

L˜ aj



  zk ⊗ tk , cl ⊗ b l

k

=

 k,l



= L˜ a

l

∗

Lp (q (H )⊗B( 2I )),Lp (q (H )⊗B( 2I ))

τ (zk∗ cl ) Tr((aj tk )∗ bl ) − → j

 k

 k,l

  zk ⊗ tk , cl ⊗ b l l

τ (zk∗ cl ) Tr((atk )∗ bl )

∗

.

Lp (q (H )⊗B( 2I )),Lp (q (H )⊗B( 2I ))

By a “net version” of [136, Lemma 3.6 p. 151], the bounded net (L˜ aj ) converges to L˜ a in the weak operator topology. Once again, since Lp (q (H )⊗B( 2I )) is reflexive, the weak operator topology and the weak* topology of B(Lp (q (H )⊗B( 2I ))) coincide on bounded sets. We infer that (L˜ aj ) converges to (L˜ a ) in the weak* topology. By Blecher and Le Merdy [38, Theorem A.2.5 (2)], we conclude that π˜ is weak* continuous.   Recall in the following proposition the weak* closed operator ∂α,q,∞ : dom(∂α,q,∞ ) ⊂ q (H )⊗B( 2I ) → q (H )⊗B( 2I ) from the point 5 of Proposition 3.11. Note that the latter proposition is applicable since we suppose that q = 1 in the following.

242

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Proposition 5.14 Let 1 < p < ∞ and −1  q < 1. 1. We have (Dα,q,p )∗ = Dα,q,p∗ . In particular, the operator Dα,q,2 is selfadjoint. 2. We have dom ∂α,q,∞ ⊂ LipDα,q,p (B( 2I )).

(5.74)

3. For any a ∈ dom ∂α,q,∞ , we have    Dα,q,p , π(a) 

p

p

SI ⊕p Lp (q (H )⊗B( 2I ))→SI ⊕p Lp (q (H )⊗B( 2I ))

   ∂α,q,∞ (a)

2 q (H )⊗B( I )

(5.75)

.

Proof 1. The proof is identical to the one of the first point of Theorem 5.4. 2. and 3. By Proposition 3.11 and Proposition 4.14, MI,fin and Lp (q (H )) ⊗ MI,fin are cores of ∂α,q,p and (∂α,q,p∗ )∗ . So MI,fin ⊕ (Lp (q (H )) ⊗ MI,fin ) is a core of Dα,q,p . For any a ∈ MI,fin , we have La (MI,fin ) ⊂ MI,fin and L˜ a (Lp (q (H )) ⊗ MI,fin ) ⊂ Lp (q (H )) ⊗ MI,fin . We infer that π(a) · (MI,fin ⊕ (Lp (q (H )) ⊗ MI,fin )) ⊂ dom Dα,q,p . So the condition (a) of the first point of Proposition 5.12 is satisfied. Note also that π(a)∗ · (MI,fin ⊕ (Lp (q (H )) ⊗ MI,fin )) ⊂ dom Dα,q,p∗ = dom(Dα,q,p )∗ . Let a ∈ MI,fin . A standard calculation shows that   Dα,q,p , π(a)

     La 0 0 (∂α,q,p∗ )∗ (∂α,q,p∗ )∗ La 0 − ∂α,q,p 0 0 0 L˜ a 0 L˜ a ∂α,q,p     0 La (∂α,q,p∗ )∗ 0 (∂α,q,p∗ )∗ L˜ a − ˜ = La ∂α,q,p ∂α,q,p La 0 0   0 (∂α,q,p∗ )∗ L˜ a − La (∂α,q,p∗ )∗ = . ∂α,q,p La − L˜ a ∂α,q,p 0

(5.72)(5.73)

=



0

We calculate the two non-zero components of the commutator. For the lower left corner, if x ∈ MI,fin we have17 (∂α,q,p La − L˜ a ∂α,q,p )(x) = ∂α,q,p La (x) − L˜ a ∂α,q,p (x) = ∂α,q,p (ax) − a∂α,q,p (x) (2.96)

= ∂α,q (a)x = L∂α,q (a) J (x)

17 Recall

that the term ∂α,q,p (a)x is by definition equal to ∂α,q (a)(1 ⊗ x).

(5.76)

5.10 Spectral Triples Associated to Semigroups of Schur Multipliers I

243

p

where J : SI → Lp (q (H )⊗B( 2I )), x → 1 ⊗ x. For the upper right corner, note that for any y ∈ Lp (q (H )) ⊗ MI,fin and any x ∈ MI,fin , (we recall that we have the duality brackets f, g antilinear in the first variable) 

 (∂α,q,p∗ )∗ L˜ a − La (∂α,q,p∗ )∗ (y), x     = (∂α,q,p∗ )∗ L˜ a (y), x − La (∂α,q,p∗ )∗ (y), x     = L˜ a (y), ∂α,q,p∗ (x) − (∂α,q,p∗ )∗ (y), La ∗ (x)     = y, L˜ a ∗ ∂α,q,p∗ (x) − y, ∂α,q,p∗ La ∗ (x)     = y, L˜ a ∗ ∂α,q,p∗ (x) − ∂α,q,p∗ La ∗ (x) = y, a ∗ ∂α,q,p∗ (x) − ∂α,q,p∗ (a ∗ x)       = − y, ∂α,q (a ∗ )x = y, −L∂α,q (a ∗ ) (1 ⊗ x) = y, L(∂α,q (a))∗ (1 ⊗ x)     = L∂α,q (a) (y), 1 ⊗ x = EL∂α,q (a) (y), x S p ,S p∗ .

(2.96)

I

I

p

Here, E : Lp (q (H )⊗B( 2I )) → SI , t ⊗ z → τ (t)z denotes the canonical conditional expectation. We conclude that

(∂α,q,p∗ )∗ L˜ a − La (∂α,q,p∗ )∗ (y) = EL∂α,q (a) (y).

(5.77)

The two non-zero components of the commutator are bounded linear  operators on MI,fin and on Lp (q (H )) ⊗ MI,fin . We deduce that Dα,q,p , π(a) is bounded on the core MI,fin ⊕(Lp (q (H ))⊗MI,fin ) of Dα,q,p . By Proposition 5.12, this operator p extends to a bounded operator on SI ⊕p Lp (q (H )⊗B( 2I )). Hence MI,fin is a subset of LipDα,q,p (B( 2I )). If (x, y) ∈ dom Dα,q,p and a ∈ MI,fin , we have in addition     Dα,q,p , π(a) (x, y) p 



   =  (∂α,q,p∗ )∗ L˜ a − La (∂α,q,p∗ )∗ y, ∂α,q,p La − L˜ a ∂α,q,p x 

(5.78) p



 

 p p 1 

 =  (∂α,q,p∗ )∗ L˜ a − La (∂α,q,p∗ )∗ y  p +  ∂α,q,p La − L˜ a ∂α,q,p x  p SI

(5.77)(5.76)  EL∂ =

p

p  p    α,q (a) (y) S p + ∂α,q (a)J (x) Lp ( I

   ∂α,q (a)q (H )⊗B( 2 ) (x, y)p .

1 2 q (H )⊗B( I ))

p

I

We conclude that    Dα,q,p , π(a) 

p p SI ⊕p Lp (q (H )⊗B( 2I ))→SI ⊕p Lp (q (H )⊗B( 2 I ))

   ∂α,q (a)q (H )⊗B( 2 ) . I

(5.79)

244

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Let a ∈ dom ∂α,q,∞ . Let (aj ) be a net in MI,fin such that aj → a and ∂α,q,∞ (aj ) → ∂α,q,∞ (a) both for the weak* topology. By Krein-Smulian Theorem (see Lemma 2.1), we can suppose that the nets (aj ) and (∂α,q,∞ (aj )) are bounded. By the point 4 of Proposition 5.11, we deduce that a ∈ LipDα,q,p (B( 2I )). By continuity of π, note that π(aj ) → π(a) for the weak operator topology. For any ξ ∈ dom Dα,q,p and any ζ ∈ dom(Dα,q,p )∗ , we have 

[Dα,q,p , π(aj )]ξ, ζ



p∗

p

SI ⊕p Lp (q (H )⊗B( 2I )),SI ⊕p∗ Lp

∗

  = (Dα,q,p π(aj ) − π(aj )Dα,q,p )ξ, ζ     = Dα,q,p π(aj )ξ, ζ − π(aj )Dα,q,p ξ, ζ     = π(aj )ξ, (Dα,q,p )∗ ζ − π(aj )Dα,q,p ξ, ζ     − → π(a)ξ, (Dα,q,p )∗ ζ − π(a)Dα,q,p ξ, ζ j

  = [Dα,q,p , π(a)]ξ, ζ . Since the net ([Dα,q,p , π(aj )]) is bounded by (5.79), we deduce that the net ([Dα,q,p , π(aj )]) converges to [Dα,q,p , π(a)] for the weak operator topology by a “net version” of [136, Lemma 3.6 p. 151]. Furthermore, it is (really) easy to check that L∂α,q,∞ (aj ) J → L∂α,q,∞ (a) J and −EL∂α,q,∞ (aj ) → −EL∂α,q,∞ (a) both for the weak operator topology. By uniqueness of the limit, we deduce that the commutator is given by the same formula as in the case of elements of MI,fin . From here, we obtain (5.75) as in (5.78).   Remark 5.12 We do not know if LipDα,q,p (B( 2I )) = dom ∂α,q,∞ . We will investigate this question in subsequent work. Remark 5.13 The estimate (5.75) is in general not optimal. Indeed, already in the case p = 2 and a = eij ∈ MI,fin ⊂ dom ∂α,q,∞ for some i, j ∈ I , we have according to (5.76) and (5.77),   [Dα,q,2 , π(a)]

SI2 ⊕2 L2 (q (H )⊗B( 2I ))→SI2 ⊕2 L2 (q (H )⊗B( 2I ))

   max L∂α,q (a)J S 2 →L2 ( I

2 q (H )⊗B( I ))

  , EL∂α,q (a)L2 (

q

(5.80)  . 2 2 (H )⊗B( ))→S I

I

Note that we have the Hilbert space adjoints (L∂α,q (a)J )∗ = J ∗ L∗∂α,q (a) = EL(∂α,q (a))∗ = −EL∂α,q (a ∗ ) . Thus, in the maximum of (5.80), it suffices to consider  the second term. We have for x = k,l xkl ⊗ ekl ∈ L2 (q (H )⊗B( 2I ))    

EL∂α,q (eij ) (x) = E sq (αi − αj ) ⊗ eij xkl ⊗ ekl =E

 l

k,l

sq (αi − αj )xj l ⊗ eil

 =

 l

τ (sq (αi − αj )xj l )eil .

5.10 Spectral Triples Associated to Semigroups of Schur Multipliers I

245

Consequently     τ (sq (αi − αj )xj l )2 EL∂ (e ) (x)2 = α,q ij 2 l

 2  sq (αi − αj )L2 (

 q (H ))

xj l 2L2 (

q (H ))

l

  2 xkl 2L2 (  sq (αi − αj )L2 (q (H ))

q (H ))

k,l

= αi − αj 2H x2L2 (

2 q (H )⊗B( I ))

.

We infer that   [Dα,q,2 , π(a)] 2 2 SI ⊕2 L (q (H )⊗B( 2I ))→SI2 ⊕2 L2 (q (H )⊗B( 2I ))   = EL∂α,q (eij ) 2→2  αi − αj H . In the case where −1 < q < 1 and αi − αj = 0, this quantity is strictly less than    (2.95)  ∂α,q (eij ) = sq (αi − αj ) ⊗ eij q (H )⊗B( 2 ) q (H )⊗B( 2I ) I   = sq (αi − αj )q (H ) eij B( 2 ) I

  = sq (αi − αj )

[44, Th. 1.10] q (H )

=

2 αi − αj H . √ 1−q

Under additional assumptions on the family (αi )i∈I of our semigroup, we are able to prove that the triple from Proposition 5.14 is a locally compact spectral triple (see Definition 5.11). In the following, we shall need the restriction ARp = Ap |Rp of Ap to a row def

Rp = span{e0j : j ∈ I },

(5.81)

where 0 is some fixed element of I . Note that Ap leaves clearly Rp invariant. Proposition 5.15 Let 1 < p < ∞. Assume that α : I → H is injective, where H is a Hilbert space of dimension n ∈ N and satisfies Gapα > 0 where Gapα is defined in (5.16). −1

1. The operator ARp2 : Ran ARp → Ran ARp is compact. ∗ 2. Suppose −1  q  1. Let BRp = ∂α,q,p ∂α,q,p : dom BRp ⊂ ∗ |Ran ∂ α,q,p |Rp Ran ∂α,q,p |Rp → Ran ∂α,q,p |Rp . Then BRp is sectorial and injective, and the −1

operator BRp2 is compact.

246

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Proof −1

1. We begin by showing that AR22 : Ran AR2 → Ran AR2 is compact. Note that it is obvious that AR2 is selfadjoint and that the e0i ’s where i ∈ I , form an orthonormal basis of R2 consisting of eigenvectors of AR2 . Thus the e0i ’s where i ∈ I \{0} form an orthonormal basis of Ran AR2 consisting of eigenvectors −1

of AR22 : Ran AR2 → Ran AR2 associated to the eigenvalues αi − α0 −1 H . It suffices to observe that the condition Gapα > 0, the injectivity of α, together with the finite-dimensionality of H imply that any bounded subset of H meets the αi − α0 H only for a finite number of i ∈ I . Hence theses eigenvalues 18 αi − α0 −1 H vanish at infinity. We have proved the compactness. −1

Next we show that AR∞2 : Ran AR∞ → Ran AR∞ is bounded where R∞ ⊂ ∞ SI . Since (Tt |R∞ )t 0 is a semigroup  with generator AR∞ , it suffices by Lemma 5.6 to establish the bound Tt |R∞ Ran A →R∞  t1d for some d > 12 . R∞

−1

We conclude with Lemma 5.4 and by restriction. Thus, AR∞2 : Ran AR∞ → Ran AR∞ is bounded. Now, assume that p > 2. We will use complex interpolation. Since the resolvents of the operators ARp are compatible for different values of p (in fact, they are equal since Rp0 = Rp1 for 1  p0 , p1  ∞), the complementary projections of (2.17) onto the spaces Ran ARp are compatible. Hence, the Ran ARp ’s form an interpolation scale. Observe that Ran AR2 is a Hilbert space, −1

hence a UMD space. Then we obtain the compactness of ARp2 : Ran ARp → Ran ARp by means of complex interpolation between a compact and a bounded operator with Theorem 2.2. If p < 2, we conclude by duality and Schauder’s Theorem [167, Theorem −1

3.4.15], since Ran ARp is the dual space of Ran ARp∗ and ARp2 defined on the −1

first space is the adjoint of ARp2∗ defined on the second space. def

2. We will use the shorthand notation ∂p = ∂α,q,p |Rp and p

∂p∗ = (∂α,q,p∗ )∗ |Lp (q (H )) ⊗ Rp . def

Note that ∂p and ∂p∗ are again closed and densely defined. We begin with the case p = 2. The operators ∂2 and ∂2∗ are indeed adjoints to each other (with the chosen domain). According to Theorem 2.1, the unbounded operators ∂2∗ ∂2 |(Ker ∂2 )⊥ and ∂2 ∂2∗ |(Ker ∂2∗ )⊥ are unitarily equivalent.

18 Recall

that a family (xi )i∈I vanishes at infinity means that for any ε > 0, there exists a finite subset J of I such that for any i ∈ I − J we have |xi |  ε.

5.10 Spectral Triples Associated to Semigroups of Schur Multipliers I

247

Note that we have AR2 = A2 |R2 = ∂2∗ ∂2 . Moreover, according to (2.8), (Ker ∂2 )⊥ = Ran ∂2∗ , which in turn equals Ran(∂α,q,2 )∗ ∩ R2

Prop. 4.19

=

Ran A2 ∩

−1 AR22

R2 = Ran AR2 . By the first part of the proof, : Ran AR2 → Ran AR2 is compact, hence ∂2∗ ∂2 |(Ker ∂2 )⊥ is invertible, and by functional calculus, the −1

previous unitary equivalence also holds between AR22 on Ran AR2 and (∂2 ∂2∗ )− 2 1

−1

(2.3)

on (Ker ∂2∗ )⊥ = Ran ∂2 . We infer that BR22 is compact.

−1

We turn to the case of general p and start by showing that BRp2 is bounded. Proposition 4.6

∗ Note that BRp = ∂α,q,p ∂α,q,p = Bp |Ran ∂p . Since resolvents ∗ |Ran ∂p of Bp leave Ran ∂p invariant and Bp is sectorial by Lemma 4.14, by [83, Proposition 3.2.15], BRp is also sectorial. Since Bp is injective according to Lemma 4.14, also BRp is injective by restriction. Then again by [83, Proposition −1

−1

−1

3.2.15], BRp2 = Bp 2 |Ran ∂p . By Lemma 5.7 and restriction, we infer that BRp2 is bounded. To improve boundedness to compactness, we pick some 2 < p < ∞ and fix some auxiliary p < p0 < ∞. Finally, it suffices interpolate compactness −1

−1

−1

of BR22 and boundedness of BRp2 to conclude with Theorem 2.2 that BRp2 is 0 compact. Details are left to the reader. For the case p < 2, we use duality.   Proposition 5.16 Let 1 < p < ∞ and −1  q < 1. Assume that H is finitep dimensional, that α : I → H is injective and that Gapα > 0. Then (MI,fin , SI ⊕ Lp (q (H )⊗B( 2I )), Dα,q,p ) is a locally compact Banach spectral triple. In other words, we have the following properties. 1. Dα,q,p is densely defined and has a bounded H∞ functional calculus on a bisector. 2. For any a ∈ dom ∂α,q,∞ , we have a ∈ LipDα,q,p (B( 2I )). 3. For any a ∈ SI∞ , π(a)(iId + Dα,q,p )−1 and π(a)|Dα,q,p |−1 are compact p operators between the spaces Ran Dα,q,p → SI ⊕ Lp (q (H )⊗B( 2I )). Proof The first and second points are already contained in Theorem 4.9 and Proposition 5.14. We turn to the third point. Note that by bisectorial H∞ -calculus, we have a bounded operator f (Dα,q,p ) : Ran Dα,q,p → Ran Dα,q,p , where f (λ) = √ λ2 (i + λ)−1 belongs to H∞ (ω± ). Thus, recalling that |Dα,q,p |−1 is the functional calculus of the function n(λ) = √1 2 , we obtain λ

π(a)(iId + Dα,q,p )−1 = π(a)|Dα,q,p |−1 f (Dα,q,p ) p

as operators Ran Dα,q,p → SI ⊕ Lp (q (H )⊗B( 2I )). By composition, if the operator π(a)|Dα,q,p |−1 is compact on Ran Dα,q,p , then so is π(a)(iId+Dα,q,p )−1 . It thus suffices to consider the former in the sequel.

248

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Recall that Ran Ap = Ran(∂α,q,p∗ )∗ by Proposition 4.19 and that as operators on Ran Dα,q,p , (4.61) D2α,q,p =

  Ap |Ran Ap 0 . 0 (IdLp (q (H )) ⊗ Ap )|Ran ∂α,q,p

So as operators on Ran Dα,q,p , |Dα,q,p |−1



0 Ap |Ran Ap = 0 (IdLp (q (H )) ⊗ Ap )|Ran ∂α,q,p

− 12

=

⎡ 1 −2 ⎣Ap 0

⎤ 0 ⎦ . −1

Bp

2

Hence we have π(a)|Dα,q,p |

⎤ ⎡ ⎤ ⎡ 1   − −1 La 0 ⎣ A p 2 0 ⎦ ⎣ La A p 2 0 ⎦ = = . −1 −1 0 L˜ a 0 Bp 2 0 L˜ a Bp 2

−1 (5.73)

− − p Now, it suffices to show that La Ap 2 : Ran Ap → SI and L˜ a Bp 2 : Ran ∂α,q,p → Lp (q (H )⊗B( 2I )) are compact. We start with the first operator. Recall that p p Ap (MI,fin ) is a dense subspace of Ran Ap . If Pj : SI → SI is the Schur multiplier projecting onto the j th line associated with the matrix [δj =k ]kl ,19 if we choose first a = eij , and if ekl ∈ Ap (MI,fin ) we have 1

−1

1

−1

La Ap 2 (ekl ) = Leij Ap 2 (ekl ) = αk − αl −1 eij ekl −1

−1

= δj =k αk − αl −1 eil = δj =k Leij Ap 2 (ekl ) = La Ap 2 Pj (ekl ). −1

Then by span density of such ekl in Ran Ap , we infer that La Ap 2 −1 La Ap 2 Pj

=

p SI .

as bounded operators Ran Ap → From Lemma 5.3, we infer that Pj is a completely contractive projection. It now suffices to show that −1

p

La Ap 2 Pj : Pj (Ran Ap ) → SI is compact. We use the space Rp from (5.81), −1

−1

where we choose 0 = j there. Since Ap 2 Pj = Ap 2 |Ran ARp , we infer by −1

Proposition 5.15 that Ap 2 Pj is indeed compact on Pj (Ran Ap ) = Ran ARp . We −1

−1

infer by composition that La Ap 2 Pj = La Ap 2 is compact as an operator on Ran Ap , −1

in case a = eij . By linearity in a, La Ap 2 is also compact for a ∈ MI,fin . Finally, if a ∈ SI∞ is a generic element, then a = limn→∞ an in SI∞ for some sequence 19 The

entries are 1 on the j -row and zero anywhere else.

5.10 Spectral Triples Associated to Semigroups of Schur Multipliers I

249

−1

p

−1

an ∈ MI,fin , whence La = limn→∞ Lan in B(SI ) and La Ap 2 = limn→∞ Lan Ap 2 −1

p

in B(Ran Ap , SI ). Thus, La Ap 2 is the operator norm limit of compact operators, and hence itself compact. −1 We turn to the second operator L˜ a Bp 2 . Let x ⊗ ekl ∈ ∂α,q,p (MI,fin ) (with x = sq (αk − αl )), which is a dense subspace of Ran ∂α,q,p according to Proposition 3.11. We first fix a = eij and turn to general a ∈ SI∞ at the end. − L˜ a Bp 2 (x ⊗ ekl ) = αk − αl −1 (1 ⊗ eij )(x ⊗ ekl ) = δj =k αk − αl −1 x ⊗ eil 1

1

1

− − = δj =k (1 ⊗ eij )Bp 2 (x ⊗ ekl ) = L˜ a Bp 2 (IdLp ⊗ Pj )(x ⊗ ekl ). −1

−1

By the same arguments as previously, we infer that L˜ a Bp 2 = L˜ a Bp 2 (IdLp ⊗ Pj ) as bounded operators Ran ∂α,q,p → Lp (q (H )⊗B( 2I )). We infer by Proposi− − tion 5.15 that Bp 2 (IdLp ⊗ Pj ) is compact, thus by composition, also L˜ a Bp 2 = 1

1

− L˜ a Bp 2 (IdLp ⊗Pj ) is compact on Ran ∂α,q,p . Then by linearity (resp. operator norm 1

−1

limit), L˜ a Bp 2 is also compact for any a ∈ MI,fin (resp. for any a ∈ SI∞ ). We finally deduce that π(a)|Dα,q,p |−1 is compact on Ran Dα,q,p .   Remark 5.14 Note that the locally compact (Banach) spectral triple p

(MI,fin , SI ⊕p Lp (q (H )⊗B( 2I )), Dα,q,p ) is even.;Indeed, the Hodge-Dirac operator Dα,q,p anti-commutes with the involution < def −IdS p 0 p p I : SI ⊕p Lp (q (H )⊗B( 2I )) → SI ⊕p Lp (q (H )⊗B( 2I )) γp = 0 IdLp (which is selfadjoint if p = 2) since Dα,q,p γp + γp Dα,q,p < ; 0.   Bisectoriality of the Dirac Operator II for Schur Multipliers on the Banach Space Lp (L∞ ()⊗B( 2I )) In the following, we consider the case q = 1 which corresponds to the classical Gaussian space. Consider the “Dirac operator” Dα,1 defined on L∞ () ⊗ MI,fin with values in L0 () ⊗ MI,fin by def

Dα,1 (f ⊗ eij ) = W(αi − αj )f ⊗ eij ,

x ∈ L∞ (), i, j ∈ I. def

(5.98)

Moreover, we consider the subspace dom∞ (Dα,1 ) = span{f ⊗ eij : i, j ∈ I, f ∈ L∞ () : W(αi − αj )f ∈ L∞ ()} of L∞ () ⊗ MI,fin such that Dα,1 (dom∞ (Dα,1 )) ⊂ L∞ () ⊗ MI,fin .

260

5 Locally Compact Quantum Metric Spaces and Spectral Triples

Proposition 5.18 1. There is a weak* continuous group (Ut )t ∈R of trace preserving ∗-automorphisms Ut : L∞ ()⊗B( 2I ) → L∞ ()⊗B( 2I ), such that its weak* generator iDα,1,∞ is an extension of the restriction iDα,1 |dom∞ (Dα,1 ) . p 2. Let 1  p < ∞. The operator iDα,1 : L∞ () ⊗ MI,fin ⊂ Lp (, SI ) → p Lp (, SI ) is closable and its closure iDα,1,p generates the strongly continuous p group (Ut,p )t ∈R of isometries on Lp (, SI ). def  it W(αk ) ⊗ Proof For t ∈ R, we define the block diagonal operator Vt = k∈I e 2 ∞ ekk ∈ L ()⊗B( I ). Note that (Vt )t ∈R is a group of unitaries. By the proof of [15, Theorem 5.2], Ut : L∞ ()⊗B( 2I ) → L∞ ()⊗B( 2I ), t → Vt xVt∗ defines a weak* continuous group (Ut )t ∈R of ∗-automorphisms. For any t ∈ R, it is easy to check that Ut is trace preserving. So it induces for 1  p < ∞ a (uniquely) strongly p continuous group (Ut,p )t ∈R of complete isometries on Lp (, SI ). Thus according to [174, Propositions 1.1.1 and 1.1.2 and Theorem 1.2.3], we have a weak* closed and weak* densely defined generator iDα,1,∞ of (Ut )t ∈R and a generator iDα,1,p for (Ut,p )t ∈R . Now, let us show that iDα,1,∞ contains iDα,1 |dom∞ (Dα,1 ) . To this end, let i, j ∈ I , f ⊗ eij ∈ dom∞ (Dα,1 ) and y be an element of L1 (, SI1 ). Using differentiation under the integral sign by domination (note that W(αi − αj )f ∈ L∞ ()), we obtain

)  1 1 ( y, (Ut − Id)(f ⊗ eij ) L1 (,S 1 ),L∞ ()⊗B( 2 ) = τ y ∗ ((eit W(αi −αj ) − 1)f ⊗ eij ) I I t t 

it W(α −α )(ω) 1 i j e − 1 f (ω)yij (ω) dμ(ω) = t     iW(αi − αj )(ω)f (ω)yij (ω) dμ(ω) = y, iW(αi − αj )f ⊗ eij . −−−→ t →0



(5.99) We deduce that f ⊗eij belongs to dom iDα,1,∞ and that iDα,1,∞ (f ⊗eij ) = iW(αi − αj )f ⊗ eij . Next we show that the generator iDα,1,p is the closure of iDα,1 : L∞ () ⊗ MI,fin ⊂ Lp (, SI ) → Lp (, SI ). p

p

For any f ∈ L∞ () and i, j ∈ I , using [85, Proposition 4.11 p. 32], we have 1 1

(Ut,p − Id)(f ⊗ eij ) = eit W(αi −αj ) − 1 f ⊗ eij t t −−→ iW(αi − αj )f ⊗ eij . t →0

5.12 Bisectoriality and Functional Calculus of the Dirac Operator II

261

We infer that f ⊗ eij belongs to dom iDα,1,p and iDα,1,p (f ⊗ eij ) = iW(αi − αj )f ⊗ eij . So it suffices to prove that L∞ () ⊗ MI,fin is a core for iDα,1,p . Note that IdLp () ⊗ TJ commutes with Ut,p . This is easy to check on elements of the p form f ⊗ eij and extends by a density argument to all of Lp (, SI ). Thus, for x ∈ dom Dα,1,p , we have 1 1 (Ut,p − Id)(IdLp () ⊗ TJ )(x) = (IdLp () ⊗ TJ ) (Ut,p − Id)(x) t t −−→ i(IdLp () ⊗ TJ )Dα,1,p (x). t →0

We infer that (IdLp () ⊗ TJ )(x) belongs to dom Dα,1,p and Dα,1,p (IdLp () ⊗ TJ )(x) = (IdLp () ⊗ TJ )Dα,1,p (x) −−−→ Dα,1,p (x). Moreover, (IdLp () ⊗ J →I

TJ )(x) → x as J → I . Therefore, to show the core property, it suffices to approximate an element f ⊗ eij where f ∈ Lp () in the graph norm of Dα,1,p by elements in L∞ () ⊗ MI,fin . To this end, it suffices to take 1|f |n f ⊗ eij and to argue similarly as beforehand, noting that



1|f |n ⊗ 1B( 2 ) Ut,p (f ⊗ eij ) = Ut,p 1|f |n f ⊗ eij . I

  Similarly to Corollary 5.2, we obtain the following result. Corollary 5.3 Let 1 < p < ∞. Then the operator Dα,1,p is bisectorial and has a bounded H∞ (ω± ) functional calculus to any angle ω > 0. Remark 5.17 Now we study the properties of the operator Dα,q from (5.82) in the case where I is finite and −1  q < 1. For any i, j ∈ I , we will use the maps Jij : Lp (q (H )) → Lp (q (H )⊗B( 2I )), x → x ⊗ eij and Qij : Lp (q (H )⊗B( 2I )) → Lp (q (H )), x =



xkl ⊗ ekl → xij .

k,l∈I

For any k, l ∈ I , we introduce the linear operator Lkl : Lp (q (H )⊗B( 2I )) → Lp (q (H )⊗B( 2I )), x⊗eij → δi=k,j =l sq (αi −αj )x⊗eij . For any i, j, j, l, k  , l  ∈ I and any x ∈ Lp (q (H )), we have

Lkl Lk  l  (x ⊗ eij ) = δi=k  ,j =l  Lkl sq (αi − αj )x ⊗ eij = δi=k  ,j =l  δi=k,j =l sq (αi − αj )2 x ⊗ eij

= δi=k,j =l Lk  l  sq (αi − αj )x ⊗ eij = Lk  l  Lkl (x ⊗ eij ).

262

5 Locally Compact Quantum Metric Spaces and Spectral Triples

So the operators Lkl commute. Moreover, we have Dα,q =



Lkl .

(5.100)

k,l∈I

For any k, l ∈ I , note that Lkl = Jkl Lsq (αk −αl ) Qkl .

(5.101)

Note that sq (αk − αl ) is selfadjoint. So by McIntosh and Monniaux [165, Remark 2.25] it is bisectorial of type 0 and admits a bounded bisectorial H∞ functional calculus for all θ ∈ (0, π2 ) (with Kθ = 1). By adapting [128, Proposition 8.4], the operator Lsq (αk −αl ) is bisectorial of type 0 on Lp (q (H )) and has bounded bisectorial H∞ functional calculus. By (5.101), the operator Lkl is also bisectorial of type 0 on Lp (q (H )⊗B( 2I )) and admits a bounded bisectorial  H∞ functional 21 calculus. It is easy to check that for any λ ∈ C\R, we have R(λ, k,l∈I Lkl ) =  Cauchy integral formula yields for any θ ∈ k,l∈I Jkl R(λ, Lsq (αk −αl ) )Qkl . Then the   ± (0, π2 ) and any f ∈ H∞ ( ) that f ( k,l∈I Lkl ) = k,l∈I Jkl f (Lsq (αk −αl ) )Qkl . θ 0 Thus by (5.100), we deduce that Dα,q is bisectorial and admits a bounded bisectorial H∞ functional calculus. The study of the operator (5.58) is similar and the verification is left to the reader. Here the assumption is that G is a finite group.

21 We

have



⎛ Jkl R(λ, Lsq (αk −αl ) )Qkl ⎝λ −

k,l



⎞ Jij Lsq (αi −αj ) Qij ⎠

i,j

=



Jkl λR(λ, Lsq (αk −αl ) )Qkl −

k,l

=

 

Jkl R(λ, Lsq (αk −αl ) )Qkl Jij Lsq (αi −αj ) Qij

k,l,i,j

Jkl λR(λ, Lsq (αk −αl ) )Qkl −

k,l

=

 

δk=i δl=j Jkl R(λ, Lsq (αk −αl ) )Lsq (αi −αj ) Qij

k,l,i,j

Jkl R(λ, Lsq (αk −αl ) )(λ − Lsq (αk −αl ) )Qkl

k,l

=



Jkl IdLp (q (H )) Qkl = IdLp (q (H )⊗B( 2 )) , I

k,l

and similarly the other way around.

Appendix A

Appendix: Lévy Measures and 1-Cohomology

Abstract In this short appendix, we describe a relation between 1-cohomology and Lévy-Khintchine decompositions in the case of an abelian group. This observation allows us to obtain with the Lévy measure an explicit description of the 1-cocycle associated to a markovian semigroup of Fourier multipliers.

The following observation describes the link between Lévy-Khintchine decompositions and 1-cocycles. Note that examples of explicit Lévy measures are given in [33, p. 184]. Let G be a locally compact abelian group. Recall that a 1-cocycle is defined by a strongly continuous unitary or orthogonal representation π : G → B(H ) on a complex or real Hilbert space H and a continuous map b : G → H such that b(s+t) = b(s)+πs (b(t)) for any s, t ∈ G. We also recall the following fundamental connection between continuous functions ψ : G → R which are conditionally of negative type in the sense of [34, 1.8 Definition p. 89], [33, 7.1 Definition] and quadratic forms q : G → R+ together with Lévy measures μ from [33, 18.20 Corollary p. 184]. If ψ is such a function satisfying ψ(0) = 0, there exist a unique continuous quadratic form q : G → R+ and a unique positive symmetric measure % − {0} such that μ on G  1 − Re s, χ dμ(χ), s ∈ G. (A.1) ψ(s) = q(s) + % G−{0}

Conversely, any continuous function of this type is conditionally of negative type and ψ(0) = 0. Note that a quadratic form is a map satisfying q(s + t) + q(s − t) = 2q(s) + 2q(t) for any s, t ∈ G [33, 7.18 Definition]. The quadratic form can be computed by q(s) = limn→∞ ψ(ns) , and μ is the Lévy measure associated with ψ. n2 In the following result, note that πs is the multiplication operator by the function

s, ·G,Gˆ .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Arhancet, C. Kriegler, Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers, Lecture Notes in Mathematics 2304, https://doi.org/10.1007/978-3-030-99011-4

263

264

A Appendix: Lévy Measures and 1-Cohomology

Proposition A.1 Let G be a locally compact abelian group. Let ψ : G → R be a continuous conditionally of negative type function such that ψ(0) = 0 and % − {0} then limn→∞ ψ(ns) = 0 for any s ∈ G. If μ is the Lévy measure of ψ on G n2 1 % − {0}, μ), b : G → H , s → (χ → 1 − s, χ) and π : G → B(H ), H = L 2 (G 2 s → (f → s, ·G,Gˆ f ) define a 1-cocycle on G such that ψ(s) = b(s)2H ,

s ∈ G.

Proof For any s ∈ G, note that |1 − s, χ|2 = (1 − s, χ)1 − s, χ = 1 − s, χ − s, χ + 1 = 2 − 2 Re s, χ. (A.2) For any s ∈ G, using (A.1) in the last equality, we deduce that b(s)2 2

1 ˆ L (G−{0}, 2 μ)

=

1 2

=

1 2

(A.2)

=

  

ˆ G−{0}

ˆ G−{0}

|(b(s))(χ)|2 dμ(χ) |1 − s, χ|2 dμ(χ)

ˆ G−{0}

1 − Re s, χ dμ(χ) = ψ(s).

% − {0}, we have Moreover, for any s, t ∈ G and any χ ∈ G (b(s + t))(χ) = 1 − s + t, χ = 1 − s, χ t, χ = 1 − s, χ + s, χ(1 − t, χ)

= b(s)(χ) + πs (b(t))(χ) = b(s) + πs (b(t)) (χ).

Hence, we conclude that b(s + t) = b(s) + πs (b(t)).

 

In particular, by restriction of scalars, we can consider the real Hilbert space % − {0}, 1 μ)|R , which comes with real scalar product f, gH|R = H|R = L2 (G 2

Re f, Re gH + Im f, Im gH . Then we take b : G → H|R , s → (χ → 1 − s, χ) = 0 for all and π : G → B(H|R ), s → (f → s, ·G,Gˆ f ). Thus if limn→∞ ψ(ns) n2 s ∈ G, then we obtain a 1-cocycle on G as we considered in Proposition 2.3 suitable for our markovian semigroups of Fourier multipliers. In the case of an arbitrary continuous function ψ : G → R+ is with ψ(0) = 0 of conditionally negative type, we can use [129, Lemma 3.1] and [131, Lemma B1] on the quadratic part q of ψ with a direct sum argument to obtain a fairly concrete Hilbert space and an associated 1-cocycle.

References

1. G. Alexopoulos, An application of homogenization theory to harmonic analysis: harnack inequalities and Riesz transforms on Lie groups of polynomial growth. Canad. J. Math. 44(4), 691–727 (1992) 2. E.M. Alfsen, Compact Convex Sets and Boundary Integrals. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57 (Springer, New York, 1971) 3. E.M. Alfsen, F.W. Shultz, Geometry of state spaces of operator algebras, in Mathematics: Theory & Applications (Birkhauser Boston, Inc., Boston, 2003) 4. A. Amenta, New Riemannian manifolds with Lp -unbounded Riesz transform for p > 2. Math. Z. 297(1–2), 99–112 (2021) 5. J.-P. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces. Duke Math. J. 65(2), 257–297 (1992) 6. N. Arcozzi, Riesz transforms on compact Lie groups, spheres and Gauss space. Ark. Mat. 36(2), 201–231 (1998) 7. N. Arcozzi, X. Li, Riesz transforms on spheres. Math. Res. Lett. 4, 401–412 (1997) 8. W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, 2nd edn. Monographs in Mathematics, vol. 96 (Birkhäuser/Springer Basel AG, Basel, 2011) 9. C. Arhancet, Noncommutative Figà-Talamanca-Herz algebras for Schur multipliers. Integr. Equ. Oper. Theory 70(4), 485–510 (2011) 10. C. Arhancet, On Matsaev’s conjecture for contractions on noncommutative Lp -spaces. J. Operator Theory 69(2), 387–421 (2013) 11. C. Arhancet, On a conjecture of Pisier on the analyticity of semigroups. Semigroup Forum 91(2), 450–462 (2015) 12. C. Arhancet, Dilations of semigroups on von Neumann algebras and noncommutative Lp spaces. J. Funct. Anal. 276(7), 2279–2314 (2019) 13. C. Arhancet, Dilations of markovian semigroups of Fourier multipliers on locally compact groups. Proc. Amer. Math. Soc. 148(6), 2551–2563 (2020) 14. C. Arhancet, Erratum to “Dilations of markovian semigroups of Fourier multipliers on locally compact groups”. To appear in Proc. Amer. Math. Soc. 15. C. Arhancet, Dilations of markovian semigroups of measurable Schur multipliers. Preprint online on https://arxiv.org/abs/1910.14434 16. C. Arhancet, Quantum information theory and Fourier multipliers on quantum groups. Preprint online on https://arxiv.org/abs/2008.12019

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Arhancet, C. Kriegler, Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers, Lecture Notes in Mathematics 2304, https://doi.org/10.1007/978-3-030-99011-4

265

266

References

17. C. Arhancet, C. Kriegler, Projections, multipliers and decomposable maps on noncommutative Lp -spaces. Submitted, Preprint online on https://arxiv.org/abs/1707.05591 18. N.H. Asmar, B.P. Kelly, S. Montgomery-Smith, A note on UMD spaces and transference in vector-valued function spaces. Proc. Edinburgh Math. Soc. (2) 39(3), 485–490 (1996) 19. J. Assaad, E.M. Ouhabaz, Riesz Transforms of Schrödinger Operators on Manifolds. J. Geom. Anal. 22(4), 1108–1136 (2012) 20. P. Auscher, On necessary and sufficient conditions for Lp -estimates of Riesz transforms associated to elliptic operators on Rn and related estimates. Mem. Amer. Math. Soc. 186(871), xviii+75 pp. (2007) 21. P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, P. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on Rn . Ann. Math. (2) 156(2), 633– 654 (2002) 22. P. Auscher, A. Axelsson, A. McIntosh, On a quadratic estimate related to the Kato conjecture and boundary value problems, in Harmonic Analysis and Partial Differential Equations. Contemporary Mathematics, vol. 505 (American Mathematical Society, Providence, 2010), pp. 105–129 23. A. Axelsson, S. Keith, A. McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators. Invent. Math. 163(3), 455–497 (2006) 24. N. Badr, E. Russ, Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs. Publ. Mat. 53(2), 273–328 (2009) 25. D. Bakry, Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée. (French) [A study of Riesz transforms in Riemannian manifolds with minorized Ricci curvature]. Séminaire de Probabilités, XXI. Lecture Notes in Mathematics, vol. 1247 (Springer, Berlin, 1987), pp. 137–172 26. D. Bakry, I. Gentil, M. Ledoux, Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften, vol. 348 (Springer, Berlin, 2014) 27. L. Bandara, Functional calculus and harmonic analysis in geometry. São Paulo J. Math. Sci. 15(1), 20–53 (2021) 28. R. Banuelos, The foundational inequalities of D. L. Burkholder and some of their ramifications. Illinois J. Math. 54(3), 789–868 (2010) 29. R. Banuelos, M. Kwasnicki, On the p -norm of the discrete Hilbert transform. Duke Math. J. 168(3), 471–504 (2019) 30. R. Banuelos, G. Wang, Sharp inequalities for martingales with applications to the BeurlingAhlfors and Riesz transforms. Duke Math. J. 80(3), 57–600 (1995) 31. B. Bekka, P. de la Harpe, Pierre, A. Valette, Kazhdan’s Property (T). New Mathematical Monographs, vol. 11 (Cambridge University Press, Cambridge, 2008) 32. J.V. Bellissard, M. Marcolli, K. Reihani, Dynamical Systems on Spectral Metric Spaces. Preprint online on https://arxiv.org/abs/1008.4617 33. C. Berg, G. Forst, Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 87 (Springer, New York, 1975) 34. C. Berg, J. Christensen, P. Ressel, Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions (Springer, New York, 1984) 35. J. Bergh, J. Löfström, Interpolation Spaces. An Introduction (Springer, Berlin, 1976) 36. E. Berkson, T.A. Gillespie, P.S. Muhly, Generalized analyticity in UMD spaces. Ark. Mat. 27(1), 1–14 (1989) 37. R. Bhatia, Positive Definite Matrices. Princeton Series in Applied Mathematics (Princeton University Press, Princeton, 2007) 38. D. Blecher, C. Le Merdy, Operator Algebras and Their Modules-An Operator Space Approach. London Mathematical Society Monographs. New Series, vol. 30 (Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford, 2004) 39. V.I. Bogachev, Measure Theory, vol. II (Springer, Berlin, 2007) 40. N. Bouleau, F. Hirsch, Dirichlet forms and analysis on Wiener space, in De Gruyter Studies in Mathematics, vol. 14 (Walter de Gruyter & Co., Berlin, 1991)

References

267

41. M. Bo˙zejko, Positive and Negative Definite Kernels on Discrete Groups. Lectures at Heidelberg University (1987) 42. M. Bo˙zejko, Bessis-Moussa-Villani conjecture and generalized Gaussian random variables. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11(3), 313–321 (2008) 43. M. Bo˙zejko, R. Speicher, Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces. Math. Ann. 300(1), 97–120 (1994) 44. M. Bo˙zejko, B. Kümmerer, R. Speicher, q-Gaussian processes: non-commutative and classical aspects. Comm. Math. Phys. 185(1), 129–154 (1997) 45. M. Bo˙zejko, S.R. Gal, W. Mlotkowski, Positive definite functions on Coxeter groups with applications to operator spaces and noncommutative probability. Comm. Math. Phys. 361(2), 583–604 (2018) 46. N.P. Brown, N. Ozawa, C ∗ -algebras and finite-dimensional approximations. Graduate Studies in Mathematics, vol. 88 (American Mathematical Society, Providence, 2008) 47. A.P. Calderon, A. Zygmund, On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952) 48. J. Cameron, R.R. Smith, Bimodules in crossed products of von Neumann algebras. Adv. Math. 274, 539–561 (2015) 49. A.L. Carey, V. Gayral, A. Rennie, F.A. Sukochev, Index theory for locally compact noncommutative geometries. Mem. Amer. Math. Soc. 231(1085) (2014) 50. G. Carron, T. Coulhon, A. Hassell, Riesz transform and Lp -cohomology for manifolds with Euclidean ends. Duke Math. J. 133(1), 59–93 (2006) 51. M. Caspers, Harmonic analysis and BMO-spaces of free Araki-Woods factors. Studia Math. 246(1), 71–107 (2019) 52. M. Caspers, Riesz transforms on compact quantum groups and strong solidity. Preprint online on https://arxiv.org/abs/2011.01609 53. A. Chamseddine, C. Consani, N. Higson, M. Khalkhali, H. Moscovici, G. Yu (eds.). Advances in Noncommutative Geometry. On the Occasion of Alain Connes’ 70th Birthday (Springer, Berlin, 2019) 54. I. Chatterji, Introduction to the rapid decay property, in Around Langlands Correspondences. Contemporary Mathematics, vol. 691 (American Mathematical Society, Providence, 2017), pp. 53–72 55. L. Chen, T. Coulhon, B. Hua, Riesz transforms for bounded Laplacians on graphs. Math. Z. 294, 397–417 (2020) 56. F. Cipriani, Noncommutative potential theory: a survey. J. Geom. Phys. 105, 25–59 (2016) 57. F. Cipriani, J.-L. Sauvageot, Derivations as square roots of Dirichlet forms. J. Funct. Anal. 201(1), 78–120 (2003) 58. F. Cipriani, D. Guido, T. Isola, J.-L. Sauvageot, Spectral triples for the Sierpinski gasket. J. Funct. Anal. 266(8), 4809–4869 (2014) 59. A. Connes, Compact metric spaces, Fredholm modules, and hyperfiniteness. Ergodic Theory Dynam. Systems 9(2), 207–220 (1989) 60. A. Connes, Noncommutative Geometry (Academic Press, Inc., San Diego, 1994) 61. A. Connes, A short survey of noncommutative geometry. J. Math. Phys. 41(6), 3832–3866 (2000) 62. A. Connes, M. Marcolli, A Walk in the Noncommutative Garden. An Invitation to Noncommutative Geometry (World Scientific Publishing, Hackensack, 2008), pp. 1–128 63. T. Coulhon, Heat Kernels on Non-compact Riemannian Manifolds: A Partial Survey. Séminaire de Théorie Spectrale et Géométrie, No. 15, Année 1996–1997, Sémin. Théor. Spectr. Géom., vol. 15 (University Grenoble I, Saint-Martin-d’Hères, 1997), pp. 167–187 64. T. Coulhon, Heat Kernel and Isoperimetry on Non-compact Riemannian Manifolds. Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002). Contemporary Mathematics, vol. 338 (American Mathematical Society, Providence, 2003), pp. 65–99 65. T. Coulhon, Heat kernel estimates, Sobolev-type inequalities and Riesz transform on noncompact Riemannian manifolds, in Analysis and Geometry of Metric Measure Spaces. CRM Proc. Lecture Notes, vol. 56 (American Mathematical Society, Providence, 2013), pp. 55–65

268

References

66. T. Coulhon, X.T. Duong, Riesz transforms for 1  p  2. Trans. Amer. Math. Soc. 351, 1151–1169 (1999) 67. T. Coulhon, X.T. Duong, Riesz transform and related inequalities on noncompact Riemannian manifolds. Comm. Pure Appl. Math. 56(12), 1728–1751 (2003) 68. M. Cowling, I. Doust, A. McIntosh, A. Yagi, Banach space operators with a bounded H ∞ functional calculus. J. Austral. Math. Soc. Ser. A 60(1), 51–89 (1996) 69. M. Cwikel, N. Kalton, Interpolation of compact operators by the methods of Calderon and Gustavsson-Peetre. Proc. Edinburgh Math. Soc. 38 (1995) 70. E.B. Davies, J.M. Lindsay, Noncommutative symmetric Markov semigroups. Math. Z. 210(3), 379–411 (1992) 71. J. De Cannière, U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107(2), 455–500 (1985) 72. A. Defant, K. Floret, Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176 (North-Holland Publishing Co., Amsterdam, 1993) 73. B. Devyver, Heat kernel and Riesz transform of Schrödinger operators. Ann. Inst. Fourier (Grenoble) 69(2), 457–513 (2019) 74. J. Dieudonné, Treatise on Analysis, vol. II. Enlarged and corrected printing. Translated by I. G. Macdonald. With a loose erratum. Pure and Applied Mathematics, 10-II (Academic Press [Harcourt Brace Jovanovich, Publishers], New York, 1976) 75. S. Dirksen, B. de Pagter, D. Potapov, F. Sukochev, Rosenthal inequalities in noncommutative symmetric spaces. J. Funct. Anal. 261(10), 2890–2925 (2011) 76. K. Domelevo, S. Petermichl, Sharp Lp estimates for discrete second order Riesz transforms. Adv. Math. 262, 932–952 (2014) 77. S.K. Donaldson, D.P. Sullivan, Quasiconformal 4-manifolds. Acta Math. 163(3–4), 181–252 (1989) 78. R.M. Dudley, Real Analysis and Probability (CRC Press, Boca Raton, 2018) 79. N. Dungey, Riesz transforms on a solvable Lie group of polynomial growth. Math. Z. 251(3), 649–671 (2005) 80. E.G. Effros, M. Popa, Feynman diagrams and Wick products associated with q-Fock space. Proc. Natl. Acad. Sci. USA 100(15), 8629–8633 (2003) 81. E. Effros, Z.-J. Ruan, Operator Spaces (Oxford University Press, Oxford, 2000) 82. L.B. Efraim, F. Lust-Piquard, Poincaré type inequalities on the discrete cube and in the CAR algebra. Probab. Theory Related Fields 141(3–4), 569–602 (2008) 83. M. Egert, On Kato’s conjecture and mixed boundary conditions. PhD (2015) 84. T. Eisner, B. Farkas, M. Haase, R. Nagel, Operator Theoretic Aspects of Ergodic Theory. Graduate Texts in Mathematics, vol. 272 (Springer, Cham, 2015) 85. K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194 (Springer, New York, 2000) 86. T. Ferguson, T. Mei, B. Simanek, H ∞ -calculus for semigroup generators on BMO. Adv. Math. 347, 408–441 (2019) 87. I. Forsyth, B. Mesland, A. Rennie, Dense domains, symmetric operators and spectral triples. New York J. Math. 20, 1001–1020 (2014) 88. V. Gayral, J.M. Gracia-Bondía, B. Iochum, T. Schücker, J.C. Varilly, Moyal planes are spectral triples. Comm. Math. Phys. 246(3), 569–623 (2004) 89. S. Ghorpade, B.V. Limaye, A course in multivariable calculus and analysis, in Undergraduate Texts in Mathematics (Springer, New York, 2010) 90. J.M. Gracia-Bondía, J.C. Varilly, H. Figueroa, Elements of noncommutative geometry, in Birkhäuser Advanced Texts: Basler Lehrbücher (Birkhäuser Boston, Inc., Boston, 2001) 91. L. Grafakos, Best bounds for the Hilbert transform on Lp (R1 ). Math. Res. Lett. 4(4), 469–471 (1997) 92. L. Grafakos, T. Savage, Best bounds for the Hilbert transform on Lp (R1 ): a corrigendum. Math. Res. Lett. 22(5), 1333–1335 (2015) 93. U. Haagerup, An Example of a nonnuclear C*-Algebra, which has the metric approximation property. Invent. Math. 50(3), 279–293 (1978/79)

References

269

94. U. Haagerup, Group C*-algebras without the completely bounded approximation property. J. Lie Theory 26(3), 861–887 (2016) 95. U. Haagerup, S. Knudby, A Lévy-Khinchin formula for free groups. Proc. Amer. Math. Soc. 143(4), 1477–1489 (2015) 96. U. Haagerup, J. Kraus, Approximation properties for group C ∗ -algebras and group von Neumann algebras. Trans. Amer. Math. Soc. 344(2), 667–699 (1994) 97. U. Haagerup, M. Junge, Q. Xu, A reduction method for noncommutative Lp -spaces and applications. Trans. Amer. Math. Soc. 362(4), 2125–2165 (2010) 98. M. Haase, The functional calculus for sectorial operators, in Operator Theory: Advances and Applications, vol. 169 (Birkhäuser Verlag, Basel, 2006) 99. M. Haase, Lectures on Functional Calculus. 21st International Internet Seminar (2018). https://www.math.uni-kiel.de/isem21/en/course/phase1 100. F. Hansen, G.K. Pedersen, Jensen’s inequality for operators and Löwner’s theorem. Math. Ann. 258(3), 229–241 (1981/82) 101. S. Haran, Riesz potentials and explicit sums in arithmetic. Invent. Math. 1013, 697–703 (1990) 102. S. Haran, Analytic potential theory over the p-adics. Ann. Inst. Fourier (Grenoble) 43(4), 905–944 (1993) 103. A. Hassell, P. Lin, The Riesz transform for homogeneous Schrödinger operators on metric cones. Rev. Mat. Iberoam. 30(2), 477–522 (2014) 104. M. Hellmich, Decoherence in Infinite Quantum Systems. PhD thesis, University of Bielefeld, Bielefeld (2009) 105. F. Hirsch, Opérateurs carré du champ (d’après J. P. Roth) (French). Séminaire Bourbaki, 29e année (1976/77), Exp. No. 501. Lecture Notes in Mathematics, vol. 677 (Springer, Berlin, 1978), pp. 167–182 106. S. Hofmann, M. Lacey, A. McIntosh, The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds. Ann. Math. (2) 156(2), 623–631 (2002) 107. T. Hytönen, A. McIntosh, P. Portal, Kato’s square root problem in Banach spaces. J. Funct. Anal. 254(3), 675–726 (2008) 108. T. Hytönen, A. McIntosh, P. Portal, Holomorphic functional calculus of Hodge-Dirac operators in Lp. J. Evol. Equ. 11(1), 71–105 (2011) 109. T. Hytönen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach Spaces, vol. I. Martingales and Littlewood-Paley Theory (Springer, Berlin, 2016) 110. T. Hytönen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach Spaces, vol. II. Probabilistic Methods and Operator Theory (Springer, Berlin, 2018) 111. T. Iwaniec, G. Martin, Quasiregular mappings in even dimensions. Acta Math. 170(1), 29–81 (1993) 112. T. Iwaniec, G. Martin, Riesz transforms and related singular integrals. J. Reine Angew. Math. 473, 25–57 (1996) 113. S. Janson, Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics, vol. 129 (Cambridge University Press, Cambridge, 1997) 114. B. Jaye, F. Nazarov, M.C. Reguera, X. Tolsa, The Riesz transform of codimension smaller than one and the Wolff energy. Mem. Amer. Math. Soc. 266(1293), v+97 pp. (2020) 115. R. Jiang, Riesz transform via heat kernel and harmonic functions on non-compact manifolds. Adv. Math. 377, 50 pp. (2021). Paper No. 107464 116. M. Junge, Doob’s inequality for non-commutative martingales. J. Reine Angew. Math. 549, 149–190 (2002) 117. M. Junge, Fubini’s theorem for ultraproducts of noncommmutative Lp -spaces. Canad. J. Math. 56(5), 983–1021 (2004) 118. M. Junge, H.H. Lee, q-Chaos. Trans. Amer. Math. Soc. 363, 5223–5249 (2011) 119. M. Junge, T. Mei, Noncommutative Riesz transforms–a probabilistic approach. Amer. J. Math. 132(3), 611–680 (2010) 120. M. Junge, M. Perrin, Theory of Hp -spaces for continuous filtrations in von Neumann algebras. Astérisque No. 362 (2014)

270

References

121. M. Junge, Z.-J. Ruan, Approximation properties for noncommutative Lp -spaces associated with discrete groups. Duke Math. J. 117(2), 313–341 (2003) 122. M. Junge, Z.-J. Ruan, Decomposable maps on non-commutative Lp -spaces, in Operator Algebras, Quantization, and Noncommutative Geometry. Contemporary Mathematics, vol. 365 (American Mathematical Society, Providence, 2004), pp. 355–381 123. M. Junge, D. Sherman, Noncommutative Lp modules. J. Operator Theory 53(1), 3–34 (2005) 124. M. Junge, Q. Xu, Noncommutative Burkholder/Rosenthal inequalities. II. Applications. Israel J. Math. 167, 227–282 (2008) 125. M. Junge, Q. Zeng, Noncommutative Bennett and Rosenthal inequalities. Ann. Probab. 41(6), 4287–4316 (2013) 126. M. Junge, Q. Zeng, Noncommutative martingale deviation and Poincaré type inequalities with applications. Probab. Theory Related Fields 161(3–4), 449–507 (2015) 127. M. Junge, Q. Zeng, Subgaussian 1-cocycles on discrete groups. J. Lond. Math. Soc. (2) 92(2), 242–264 (2015) 128. M. Junge, C. Le Merdy, Q. Xu, H ∞ functional calculus and square functions on noncommutative Lp -spaces. Astérisque No. 305 (2006) 129. M. Junge, T. Mei, J. Parcet, Smooth Fourier multipliers on group von Neumann algebras. Geom. Funct. Anal. 24(6), 1913–1980 (2014) 130. M. Junge, C. Palazuelos, J. Parcet, M. Perrin, Hypercontractivity in group von Neumann algebras. Mem. Amer. Math. Soc. 249(1183) (2017) 131. M. Junge, T. Mei, J. Parcet, Noncommutative Riesz transforms–dimension free bounds and Fourier multipliers. J. Eur. Math. Soc. (JEMS) 20(3), 529–595 (2018) 132. M. Junge, S. Rezvani, Q. Zeng, Harmonic analysis approach to Gromov-Hausdorff convergence for noncommutative tori. Comm. Math. Phys. 358(3), 919–994 (2018) 133. R.V. Kadison, A representation theory for commutative topological algebra. Mem. Amer. Math. Soc. (7), 39 (1951) 134. R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. I. Elementary theory. Reprint of the 1983 original. Graduate Studies in Mathematics, 15 (American Mathematical Society, Providence, 1997) 135. N.J. Kalton, S. Montgomery-Smith, Interpolation of Banach spaces, in Handbook of the Geometry of Banach Spaces, vol. 2 (North-Holland, Amsterdam, 2003), pp. 1131–1175 136. T. Kato, Perturbation Theory for Linear Operators, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Band 132 (Springer, Berlin, 1976) 137. F. King, Hilbert Transforms, vol. 1. Encyclopedia of Mathematics and Its Applications, 124 (Cambridge University Press, Cambridge, 2009) 138. F. King, Hilbert Transforms, vol. 2. Encyclopedia of Mathematics and its Applications, 125 (Cambridge University Press, Cambridge, 2009) 139. R.P. Kostecki, W∗ -algebras and noncommutative integration. Preprint online on https://arxiv. org/abs/1307.4818 140. P.C. Kunstmann, L. Weis, Maximal Lp -regularity for parabolic equations, Fourier multiplier theorems and H ∞ -functional calculus, in Functional Analytic Methods for Evolution Equations. Lecture Notes in Mathematics, vol. 1855 (Springer, Berlin, 2004), pp. 65–311 141. K. Landsman, Notes on Noncommutative Geometry. https://www.math.ru.nl/~landsman/ notes.html 142. L. Larsson-Cohn, On the constants in the Meyer inequality. Monatsh. Math. 137(1), 51–56 (2002) 143. F. Latrémolière, Quantum locally compact metric spaces. J. Funct. Anal. 264(1), 362–402 (2013) 144. F. Latrémolière, Convergence of fuzzy tori and quantum tori for the quantum GromovHausdorff propinquity: an explicit approach. Münster J. Math. 8(1), 57–98 (2015) 145. F. Latrémolière, The dual Gromov-Hausdorff propinquity. J. Math. Pures Appl. (9) 103(2), 303–351 (2015) 146. F. Latrémolière, Equivalence of quantum metrics with a common domain. J. Math. Anal. Appl. 443(2), 1179–1195 (2016)

References

271

147. F. Latrémolière, Quantum metric spaces and the Gromov-Hausdorff propinquity, in Noncommutative Geometry and Optimal Transport. Contemporary Mathematics, vol. 676 (American Mathematical Society, Providence, 2016), pp. 47–133 148. F. Latrémolière, The quantum Gromov-Hausdorff propinquity. Trans. Amer. Math. Soc. 368(1), 365–411 (2016) 149. F. Latrémolière, Heisenberg Modules over Quantum 2-tori are metrized quantum vector bundles. Preprint online on https://arxiv.org/abs/1703.07073 150. F. Latrémolière, The Gromov-Hausdorff propinquity for metric Spectral Triples. Preprint online on https://arxiv.org/abs/1811.10843 151. C. Le Merdy, H ∞ -Functional Calculus and Applications to Maximal Regularity. Semigroupes d’opérateurs et calcul fonctionnel (Besançon, 1998). Publ. Math. UFR Sci. Tech. Besançon, vol. 16 (Univ. Franche-Comté, Besançon, 1999), pp. 41–77 152. N. Lohoué, S. Mustapha, Sur les transformées de Riesz sur les espaces homogènes des groupes de Lie semi-simples. Bull. Soc. Math. de France 128(4), 485–495 (2000) 153. F. Lust-Piquard, Inégalités de Khintchine dans Cp (1 < p < ∞). (French) [Khinchin inequalities in Cp (1 < p < ∞)]. C. R. Acad. Sci. Paris Sér. I Math. 303(7), 289–292 (1986) 154. F. Lust-Piquard, Riesz transforms associated with the number operator on the Walsh system and the fermions. J. Funct. Anal. 155(1), 263–285 (1998) 155. F. Lust-Piquard, Riesz transforms on deformed Fock spaces. Comm. Math. Phys. 205(3), 519–549 (1999) 156. F. Lust-Piquard, Dimension free estimates for discrete Riesz transforms on products of abelian groups. Adv. Math. 185(2), 289–327 (2004) 157. F. Lust-Piquard, Riesz transforms on generalized Heisenberg groups and Riesz transforms associated to the CCR heat flow. Publ. Mat. 48(2), 309–333 (2004) 158. F. Lust-Piquard, G. Pisier, Noncommutative Khintchine and Paley inequalities. Ark. Mat. 29(2), 241–260 (1991) 159. Z.M. Ma, M. Röckner, Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext (Springer, Berlin, 1992) 160. J. Maas, J. van Neerven, Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces. J. Funct. Anal. 257(8), 2410–2475 (2009) 161. P. Malliavin, Stochastic Analysis (Springer, Berlin, 1997) 162. C. Martinez Carracedo, M. Sanz Alix, The theory of fractional powers of operators, in NorthHolland Mathematics Studies, vol. 187 (North-Holland Publishing Co., Amsterdam, 2001) 163. A. Mas, X. Tolsa, Variation for the Riesz transform and uniform rectifiability. J. Eur. Math. Soc. (JEMS) 16(11), 2267–2321 (2014) 164. J. Mateu, X. Tolsa, Riesz transforms and harmonic Lip1 -capacity in Cantor sets. Proc. London Math. Soc. (3) 89(3), 676–696 (2004) 165. A. McIntosh, S. Monniaux, Hodge-Dirac, Hodge-Laplacian and Hodge-Stokes operators in Lp spaces on Lipschitz domains. Rev. Mat. Iberoam. 34(4), 1711–1753 (2018) 166. H.P. McKean, Geometry of differential space. Ann. Probab. 1, 197–206 (1973) 167. R.E. Megginson, An introduction to Banach space theory, in Graduate Texts in Mathematics, vol. 183 (Springer, New York, 1998) 168. F.G. Mehler, Über die Entwicklung einer Function von beliebig vielen Variablen nach Laplaschen Functionen höherer Ordnung. Crelles J. 66, 161–176 (1866) 169. T. Mei, É. Ricard, Free Hilbert transforms. Duke Math. J. 166(11), 2153–2182 (2017) 170. P.A. Meyer, Démonstration probabiliste de certaines inégalités de Littlewood-Paley. II. L’opérateur carré du champ (French). Séminaire de Probabilités, X (Première partie, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975). Lecture Notes in Mathematics, vol. 511 (Springer, Berlin, 1976), pp. 142–161 171. P.A. Meyer, Transformations de Riesz pour les lois gaussiennes (French) [Riesz transforms for Gaussian laws]. Seminar on Probability, XVIII. Lecture Notes in Mathematics, vol. 1059, (Springer, Berlin, 1984), pp. 179–193

272

References

172. A. Naor, Discrete Riesz transforms and sharp metric Xp inequalities. Ann. Math. (2) 184(3), 991–1016 (2016) 173. F. Nazarov, X. Tolsa, A. Volberg, On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1. Acta Math. 213(2), 237–321 (2014) 174. J.v. Neerven, The Adjoint of a Semigroup of Linear Operators. Lecture Notes in Mathematics, vol. 1529 (Springer, Berlin, 1992), x+195 pp. 175. J.v. Neerven, Stochastic Evolution Equations. ISEM Lecture Notes (2007/08) 176. J.v. Neerven, R. Versendaal, Lp -analysis of the Hodge-Dirac operator associated with Witten Laplacians on complete riemannian manifolds. J Geom. Anal., 1–30 (2017) 177. A. Nou, Non injectivity of the q-deformed von Neumann algebra. Math. Ann. 330(1), 17–38 (2004) 178. D. Nualart, The Malliavin Calculus and Related Topics, 2nd edn. (Springer, Berlin, 2006) 179. A. Ose¸kowski, I. Yaroslavtsev, The Hilbert transform and orthogonal martingales in Banach spaces. Preprint online on https://arxiv.org/abs/1805.03948 180. E.M. Ouhabaz, Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series, vol. 31 (Princeton University Press, Princeton, 2005) 181. N. Ozawa, About the QWEP conjecture. Internat. J. Math. 15(5), 501–530 (2004) 182. N. Ozawa, M. Rieffel, Hyperbolic group C ∗ -algebras and free-product C ∗ -algebras as compact quantum metric spaces. Canad. J. Math. 57(5), 1056–1079 (2005) 183. T.W. Palmer, Banach Algebras and the General Theory of *-Algebras, vol. 2. Encyclopedia of Mathematics and its Applications 79 (Cambridge University Press, Cambridge, 2001) 184. J. Parcet, K.M. Rogers, Twisted Hilbert transforms vs Kakeya sets of directions. J. Reine Angew. Math. 710, 137–172 (2016) 185. V. Paulsen, Completely Bounded Maps and Operator Algebras (Cambridge University Press, Cambridge, 2002) 186. S.K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov. Studia Math. 44, 165–179 (1972) 187. G. Pisier, Riesz Transforms: A Simpler Analytic Proof of P.-A. Meyer’s Inequality. Séminaire de Probabilités, XXII. Lecture Notes in Mathematics, vol. 1321 (Springer, Berlin, 1988), pp. 485–501 188. G. Pisier, Regular operators between non-commutative Lp -spaces. Bull. Sci. Math. 119(2), 95–118 (1995) 189. G. Pisier, Introduction to Operator Space Theory (Cambridge University Press, Cambridge, 2003) 190. G. Pisier, Grothendieck’s theorem, past and present. Bull. Amer. Math. Soc. (N.S.) 49(2), 237–323 (2012) 191. G. Pisier, Q. Xu, Non-commutative Lp -spaces, in Handbook of the Geometry of Banach Spaces, vol. II, ed. by W.B. Johnson, J. Lindenstrauss (Elsevier, Amsterdam, 2003), pp. 1459– 1517 192. N. Randrianantoanina, Hilbert transform associated with finite maximal subdiagonal algebras. J. Austral. Math. Soc. Ser. A 65(3), 388–404 (1998) 193. N. Randrianantoanina, Non-commutative martingale transforms. J. Funct. Anal. 194(1), 181– 212 (2002) 194. J.G. Ratcliffe, Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149 (Springer, New York, 1994) 195. M. Reed, B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edn. (Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980) 196. M.A. Rieffel, Metrics on state spaces. Doc. Math. 4, 559–600 (1999) 197. M.A. Rieffel, Compact quantum metric spaces, in Operator Algebras, Quantization, and Noncommutative Geometry. Contemporary Mathematics, vol. 365 (American Mathematical Society, Providence, 2004), pp. 315–330

References

273

198. M.A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces. Appendix 1 by Hanfeng Li. Gromov-Hausdorff distance for quantum metric spaces. Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance. Mem. Amer. Math. Soc. 168(796), 1–65 (2004) 199. M. Riesz, Sur les fonctions conjuguées. (French). Math. Z. 27(1), 218–244 (1928) 200. L. Roncal, P.R. Stinga, Transference of fractional Laplacian regularity, in Special Functions, Partial Differential Equations, and Harmonic Analysis. Springer Proceedings in Mathematics & Statistics, vol. 108 (Springer, Cham, 2014), pp. 203–212 201. L. Roncal, P.R. Stinga, Fractional Laplacian on the torus. Commun. Contemp. Math. 18(3), 1550033 (2016) 202. J.-P. Roth, Opérateurs dissipatifs et semi-groupes dans les espaces de fonctions continues. (French. English summary). Ann. Inst. Fourier (Grenoble) 26(4), ix, 1–97 (1976) 203. E. Russ, Riesz transforms on graphs for 1  p  2. Math. Scand. 87(1), 133–160 (2000) 204. E. Russ, Racines carrées d’opérateurs elliptiques et espaces de Hardy. (French) [Square roots of elliptic operators and Hardy spaces]. Confluentes Math. 3(1), 1–119 (2011) 205. J.-L. Sauvageot, Tangent bimodule and locality for dissipative operators on C*-algebras, in Quantum Probability and Applications IV (Rome, 1987). Lecture Notes in Mathematics, vol. 1396 (Springer, Berlin, 1989), pp. 322–338 206. J.-L. Sauvageot, Quantum Dirichlet forms, differential calculus and semigroups, in Quantum Probability and Applications, V (Heidelberg, 1988). Lecture Notes in Mathematics, 1442 (Springer, Berlin, 1990), pp. 334–346 207. K. Schmüdgen, Unbounded self-adjoint operators on Hilbert space, in Graduate Texts in Mathematics, vol. 265 (Springer, Dordrecht, 2012) 208. A. Sitarz, A friendly overview of noncommutative geometry. Acta Phys. Polon. B 44(12), 2643–2667 (2013) 209. P. Sjögren, An estimate for a first-order Riesz operator on the affine group. Trans. Amer. Math. Soc. 351(8), 3301–3314 (1999) 210. E.M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory, in Annals of Mathematics Studies, vol. 63 (Princeton University Press, Princeton; University of Tokyo Press, Tokyo, 1970) 211. E.M. Stein, Some results in harmonic analysis in Rn , for n → ∞. Bull. Amer. Math. Soc. (N.S.) 9(1), 71–73 (1983) 212. S. Stratila, Modular Theory in Operator Algebras. Translated from the Romanian by the author. Editura Academiei Republicii Socialiste România, Bucharest (Abacus Press, Tunbridge Wells, 1981) 213. R.S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52(1), 48–79 (1983) 214. F. Sukochev, D. Zanin, The Connes character formula for locally compact spectral triples. Preprint online on https://arxiv.org/abs/1803.01551 215. V. Sunder, An Invitation to von Neumann Algebras. Universitext (Springer, New York, 1987) 216. M. Takesaki, Theory of operator algebras. II, in Encyclopaedia of Mathematical Sciences, vol. 125. Operator Algebras and Non-commutative Geometry, 6 (Springer, Berlin, 2003) 217. P. Tchamitchian, The Solution of Kato’s Conjecture (after Auscher, Hofmann, Lacey, McIntosh and Tchamitchian). Journées “Équations aux Dérivées Partielles” (Plestin-les-Grèves, 2001), Exp. No. XIV (Univ. Nantes, Nantes, 2001), 14 pp. 218. A.F.M. ter Elst, D.W. Robinson, A. Sikora, Heat kernels and Riesz transforms on nilpotent Lie groups. Colloq. Math. 74(2), 191–218 (1997) 219. B. Thaller, The Dirac equation, in Texts and Monographs in Physics (Springer, Berlin, 1992) 220. X. Tolsa, Principal values for Riesz transforms and rectifiability. J. Funct. Anal. 254(7), 1811– 1863 (2008) 221. A. van Daele, Continuous Crossed Products and Type III von Neumann Algebras. London Mathematical Society Lecture Note Series, vol. 31 (Cambridge University Press, Cambridge, 1978)

274

References

222. J.C. Varilly, Dirac operators and spectral geometry. Lecture notes on noncommutative geometry and quantum groups edited by P. M. Hajac. https://www.mimuw.edu.pl/~pwit/ toknotes/ 223. C. Villani, Optimal transport. Old and new, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338 (Springer, Berlin, 2009) 224. D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms. Invent. Math. 132(1), 189–227 (1998) 225. N. Weaver, Lipschitz algebras and derivations of von Neumann algebras. J. Funct. Anal. 139(2), 261–300 (1996) 226. N. Weaver, Lipschitz Algebras, 2nd edn. (World Scientific Publishing Co. Pte. Ltd., Hackensack, 2018) 227. Q. Xu, Non-commutative Lp -spaces. Script not available

Index

1-cocycle law, 33, 263 Bp = (IdLp (q (H )) ⊗ Ap )Ran ∂α,q,p , 205 ∗ BRp = (∂α,q,p ∂α,q,p ∗ )Ran ∂α,q,p (Rp ), 246 R-bisectoriality of Hodge-Dirac operator Dα,q,p , 169 of Hodge-Dirac operator Dψ,q,p , 146 R-boundedness, 26   of t∂α,q,p (Id + t 2 Ap )−1 : t > 0 , 167   of t∂ψ,q,p (Id + t 2 Ap )−1 : t > 0 , 138 aD Lipschitz norm, 219 lower semicontinuous, 220 seminorm, 219 x,α,p Schur case, 202 x,p Fourier case, 186 Leibniz property, Fourier case, 188 lower semicontinuous, Fourier case, 190 lower semicontinuous, Schur case, 200 Schur case, 199 seminorm, Fourier case, 187 seminorm, Schur case, 200 P,G , 42 Pp,,G , 42 CBAP, 45 CCAP, 45, 89, 190, 195, 234, 258 C∗ -algebra reduced group, 32 H∞ functional calculus, 26, 27, 145 of Dψ,q,p , independence of group and cocycle, 156 of Dα,1,p , Schur II case, 261

of Dα,q , Schur II case, 262 of Dψ,1,p , Fourier II case, 259 of Dα,q,p , independence of markovian semigroup, 178, 179 of full Hodge-Dirac operator Dα,1,p , 16 of full Hodge-Dirac operator Dα,q,p , 172 of full Hodge-Dirac operator Dψ,q,p , 155 of Hodge-Dirac operator Dα,q,p , 170 of Hodge-Dirac operator Dψ,1,p , 12 of Hodge-Dirac operator Dψ,q,p , 148 of markovian semigroups of crossed product Fourier multipliers, 145 of markovian semigroups of Fourier multipliers, 145 of markovian semigroups of Schur multipliers, 169 p L0 (M), 32 MI,fin , 16 q-Fock space, 34 Amenable, 226 AP, 44, 86, 88, 226, 234 Beurling-Ahlfors transform, 3 Bilinear form Cauchy-Schwarz inequality, 38 closable, 40 closed, 38 closure, 41 Carré du champ extension, Fourier case, 91

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Arhancet, C. Kriegler, Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers, Lecture Notes in Mathematics 2304, https://doi.org/10.1007/978-3-030-99011-4

275

276 Fourier case, 56 link with gradient, Fourier case, 60, 91 link with gradient, Schur case, 62 Meyer’s problem, classical case, 9 Meyer’s problem, Fourier case, 60, 92 Meyer’s problem, Schur case, 17, 124, 126 Meyer’s problem, Schur II case, 256 Schur case, 61 Complex interpolation, 28 Coproduct crossed, 43 Core, 24, 27, 222 of gradient, Fourier case, 141 of gradient, Schur case, 168 weak*, 25

FH property, 215 Fourier multiplier crossed product, 43, 141, 259 Fractional laplacian, 13 Fractional powers, 28

Gap Gapα , Schur case, 195, 211, 213, 214, 216 Gapψ , Fourier case, 19, 189, 213–216 Gauss projection Fourier case, 67, 77, 152 over Lp (, Lp (X, S p )), 103 Schur case, 109, 116, 172 Gauss space Gaussq,p, (Lp (VN(G))), 65 p Gaussq,p (SI ), 108 ψ,q,p , 160 Gradient (∂α,q,p∗ )∗ Ran ∂α,q,p , 167 (∂ψ,q,p∗ )∗ Ran ∂ψ,q,p , 139 (∂ψ,q,p∗ )∗ over Lp (q (H ) α G), 135 Ap = (∂ψ,q,p∗ )∗ ∂ψ,q,p , 136 Ap = (∂α,q,p∗ )∗ ∂α,q,p , 166 Bp = (IdLp (q (H )) ⊗ Ap )Ran ∂α,q,p , 167 Bp = ∂ψ,q,p (∂ψ,q,p∗ )∗ Ran ∂ψ,q,p , 144 p ∂α,1,p over SI , 16 ∂α,q,∞ over B( 2I ), 122, 241, 251 p ∂α,q,p over SI , 17, 122 ∗ over Lp ( (H )⊗B( 2 )), 121 ∂α,q q I p ∂α.q over SI , 62 ˆ 10 ∂ψ,1,p over Lp (G), ∂ψ,q,∞ over VN(G), 86, 226, 234 ∂ψ,q,p over Lp (VN(G)), 17, 84 ∂ψ,q over Lp (VN(G)), 58

Index ∗ over Lp ( (H )  G), 81 ∂ψ,q q α trace formula, Fourier case, 87 trace formula, Schur case, 100, 123 Group of automorphisms, 94, 258, 260 Coxeter, 14 cyclic, 15 discrete Heisenberg, 15 free, 15, 217 Lie, 6

Hilbert transform, 1 discrete, 4 p on Lp (R, Lp (, Lp (VN(Hdisc ), SI ))), 96 Hodge decomposition for Dα,q,p , Schur case, 172, 173 for Dψ,q,p , Fourier case, 155, 156 for Dα,q,p , Schur case, 170 for Dψ,q,p , Fourier case, 150 Hodge-Dirac operator Dα,q,p , Schur case, 169 Dψ,1,p , Fourier case, 11 Dψ,q,p , Fourier case, 146 Dα,1,∞ , Schur II case, 260 Dα,1,p , Schur II case, 257, 260 Dα,q,p , Schur II case, 250, 251 Dψ,1,∞ , Fourier II case, 258 Dψ,1,p , Fourier II case, 257, 258 Dψ,q,p , Fourier II case, 232, 234 full, Dα,q,p , Schur case, 171, 240 full, Dψ,q,p , Fourier case, 150, 224 on Lp (VN(G)) ⊕ ψ,q,p , Fourier case, 163 Homomorphism π : A → B(X), 219, 241 Hypercube, 7

Independent von Neumann algebras, 37 Inequality noncommutative Hölder, 29 Intertwining formula, 99 Isonormal process, 10, 36

Jordan-Lie algebra, 184

Khintchine inequality Fourier case, 68, 78 Schur case, 110, 117

Left multiplication operator La , 224, 234, 241, 251

Index Leibniz property, 184 Leibniz rule of gradient, Fourier case, 58 of gradient, Schur case, 62 Length of a group element block length, 15 colour length, 15 word length, 14 Lévy-Khintchine decomposition, 263 Lip-norm, 183 Lipschitz pair, 185 unital, 182, 184 Lipschitz triple, 185 Local state, 185 Matrices truncated, 33, 127, 200, 209 Noncommutative Lp -space, 28 Hilbertian valued, 46 Operator bisectorial, 26, 27 compact, 28 number, 7 R-sectorial, 26 sectorial, 25 selfadjoint, 31 Operator concave function, 116 Order unit space, 182, 203 Ornstein-Uhlenbeck semigroup, 4 Pisier formula for Gaussian projection, 97 Projection onto Ker A, 26 Quadratic form, 263 Quantum compact metric space, 19, 183 Fourier case, 190 Leibniz, Fourier case, 190 Leibniz, Schur case, 206 Schur case, 206 Quantum locally compact metric space, 19, 186 boundedness condition, 186 Leibniz, Schur case, 206 Schur case, 206

Rapid decay of a group, 194

277 Representation G → Aut (M), 10, 41 second quantization, 58 Riesz transform classical, dimension free, 2 classical on Rn , 2 directional, Schur case, 129, 132 Fourier case, −1  q  1, 81 Fourier case, q = 1, 81 Fourier case, constants depending on p, 81 1

Fourier case, full space dom Ap2 , 84 hypercube case, 7 manifold case, 5 Meyer’s problem, Fourier case, 60, 92 Meyer’s problem, Schur case, 124, 126 Ornstein-Uhlenbeck case, 4 Schur case, 107 Schur case, −1  q  1, 118 Schur case, q = 1, 16, 101, 103 Schur case, constants depending on p, 119 1

Schur case, full space dom Ap2 , 121 sphere case, 5 Rosenthal inequality, 36, 113

Schoenberg’s Theorem Fourier case, 32 Schur case, 34 Schur product, 33 Semigroup on abelian groups, 13 on discrete Heisenberg group, 15 on finitely generated groups, 14 heat Schur, 210 heat, on Tn , 215 markovian of Fourier multipliers, 11, 31 markovian of Fourier multipliers, crossed product extension, 137, 139 markovian of Herz-Schur multipliers, 212 markovian of multipliers of donut type, 215 markovian of Schur multipliers, 16, 33 Poisson Schur, 210 Spectral triple, 17 compact (Banach), 219 even, 224 even, Fourier I case, 230 even, Schur I case, 249 for Hodge-Dirac operator Dψ,q,p , Fourier I case, 19, 226 locally compact (Banach), 223 locally compact, for Hodge-Dirac operator Dα,q,p , Schur I case, 247

278 T Kazhdan’s property, 215 Trigonometric polynomials span {λs : s ∈ G}, 32

Unitary fundamental, 42

Index Von Neumann algebra VN(G) crossed product, 41 group, 11, 32 of q-Gaussians q (H ), 17, 35 Walsh function, 7 Weakly amenable, 11, 44, 136, 190, 195, 234

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