Finite approximation and continuous change in spectral geometry

Proefschrift

ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de rector magnificus prof. dr. J.H.J.M. van Krieken, volgens besluit van het college van decanen in het openbaar te verdedigen op dinsdag 30 maart 2021 om 12.30 uur precies

door

Abel Boudewijn Stern geboren op 1 november 1989 te Amsterdam Promotor: dr. W.D. van Suijlekom

Manuscriptcommissie: prof. dr. N.P. Landsman dr. F. Arici Universiteit Leiden prof. dr. J.W. Barrett University of Nottingham, Verenigd Koninkrijk dr. H.B. Posthuma Universiteit van Amsterdam prof. dr. R. Wulkenhaar WWU Münster, Duitsland

Gedrukt door: de Toekomst, Hilversum – detoekomst.nl voor Tera Fopma

Contents

Summary v

I Finite spectral geometry 1

1 Introduction 3 1.1 Mathematical preliminaries ...... 6

2 Finite-rank approximation of spectral zeta residues 25 2.1 Introduction ...... 25 2.2 Zeta residues as normal functionals ...... 28 2.3 Localization ...... 35 2.4 Final remarks and suggestions ...... 37

3 Reconstructing manifolds from truncated spectral triples 39 3.1 Introduction ...... 39 3.2 Truncated spectral geometries and point reconstruction ...... 41 3.3 The metric space of localized states ...... 44 3.4 The PointForge algorithm: associating a finite metric space . . . 55 3.5 Example: 푆2 ...... 58 3.6 Embedding a distance graph in ℝ푛 ...... 64 3.7 Final remarks ...... 68

4 Computer simulations of truncated noncommutative geometries 69 4.1 Introduction ...... 69 4.2 The Heisenberg relation in simulations ...... 74 4.3 An alternative analytic solution to the Heisenberg relation . . . . . 83

iii Contents

4.4 Conclusions ...... 87 4.A Representation of 푌,훾,퐷푆2 ...... 89

II Schatten classes for commutative Hilbert modules 91

5 Introduction 93 5.1 Preliminaries ...... 96

6 Schatten classes for Hilbert 퐶0(푋)-modules 105 6.1 Introduction and definition ...... 105 6.2 The Schatten class on the standard module ...... 108 6.3 Properties of the Schatten classes on Hilbert modules ...... 114 6.4 The Hilbert module of Hilbert–Schmidt operators ...... 116 6.5 The trace class and the trace ...... 118

7 Applications of Schatten classes 121 7.1 The Fredholm determinant ...... 121 7.2 The zeta function ...... 125 7.3 Summability of unbounded Kasparov cycles ...... 129

8 Outlook: the noncommutative case 143 8.1 Ideals inside a smaller endomorphism algebra ...... 144 8.2 The bivariant Chern character ...... 146

Bibliography 151

Samenvatting 163

Acknowledgements 167

iv Summary

This dissertation explores two aspects of spectral geometry: that of finite approximation and that of continuous change.

Finite approximation The main theme of Part I is that the spectral description of geometry, like the point-set one, should include a natural way to represent partial knowledge. This would both be very desirable from the side of applications (such as to quantum field theory and computer science) and be worth seeking as a further enrichment of the dictionary between noncommutative and differential geometry. The proposed candidate is thatof truncations of Dirac-type spectral triples, selected both for its physical intuitiveness and for its mathematical simplicity. The body of this part comprises three chapters, each a freestanding scientific article. The first, Chapter 2, addresses the relation between purely spectral asymptotic invariants (that is, geometric invariants determined uniquely by the asymptotics of the spectrum of any Laplace-type operator) and the asymptotics of functionals of the truncated spectrum. The requisite technical innovation is a careful balance between the asymp- totic short-time behaviour of the heat trace and its exponential decay in the operator spectrum. Chapter 3 addresses the issue of reconstructing a Riemannian manifold from succes- sively larger truncations of any associated Dirac-type spectral triple. A new notion of quantitatitve dispersion (roughly, under the Riesz representation theorem, correspond- ing to a statistical variance) is introduced to present a further balancing consideration, that of localization of states. This is then combined with time-spectrum balance of the previous chapter to show that from these truncations one can reconstruct a metric space that increasingly approximates the manifold itself. Computability being a guiding

v Summary

principle, a conjectural assumption about the equivalence of certain metrics on the localized states is applied to propose a computer algorithm to isometrically visualise the approximating metric spaces. Finally, given the vast generality of the notion of spectral triples and the difficulty in recognizing them from any given truncation, Chapter 4 asks whether we can recognize whether a given operator system spectral triple is the truncation of a Dirac-type spectral triple, as a truncated variant of Connes’ reconstruction theorem. The higher Heisen- berg relation in noncommutative geometry is used to present an asymptotic algebraic constraint on the truncated operator. A numerical investigation of this constraint in the case of the sphere then results in the analytic discovery of a pseudodifferential operator on the sphere that, surprisingly, better exhibits the properties of a commutative spectral triple in its truncations. The results of the previous articles are then applied to compare these, no longer Dirac-type but still approximately commutative, truncations of spectral triples.

Continuous change Part II extends the concept of Schatten-class operators to (countably generated) Hilbert modules over commutative C∗-algebras. Two main ingredients guide the construction. First, we realize that there are two extremal generalizations of the Schatten-class criterion to operators on the standard module, namely, either that a family be continuous (and vanishes at infinity) in Schatten norm or that its pointwise Schatten norm be continuous (and vanish at infinity). Each has desirable properties, but a priori neither is obviously the correct choice. A refined understanding of the Schatten-norm convergence of finite matrix truncations of Schatten-class operators on the standard Hilbert space is then developed to show that our considered generalizations do in fact coincide. As a corollary the main, and very desirable, properties of the Schatten classes on Hilbert spaces are extended to those on the standard Hilbert module over a commutative C∗-algebra. Second, Kasparov’s stabilization theorem allows us to view all countably generated Hilbert modules over a C∗-algebra as complementable submodules of its standard module. The pullbacks of the Schatten classes on the standard module under this identification form two-sided ideals and are therefore invariant under the choice of isomorphic submodules. Moreover, because the Schatten norms are unitarily invariant, the two-sided ideal thus obtained canonically inherits a Banach space structure and is identified with the set of adjointable operators whose characterwise Schatten normslie in the base algebra.

vi Summary

As direct applications, the Fredholm determinant and the operator zeta function ob- tain their generalization to the setting of countably generated Hilbert modules over commutative C∗-algebras, while retaining essential properties such as multiplicativity (of the former) and holomorphicity. The relation between traces and geometric invariants, already central to Chapter 2 of the thesis, should generalize to geometric correspondences, that is, Hilbert bimodules over commutative C∗-algebras equipped with a suitable selfadjoint regular operator. An important step in that direction is a definition of the relevant class of finitely summable cycles, so as to give rise to at least a spectral notion of (fiber) dimension and allow application of the preliminary theory of operator zeta functions on Hilbert modules developed earlier in the thesis. This leads to the conjecture that the new notion of summability be additive under the unbounded Kasparov product, followed by the presentation of some important supporting examples. Operators on Hilbert modules over noncommutative C∗-algebras exhibit deeply con- trasting properties with respect to the trace. Chapter 8 presents the difficulty in the form of a highly problematic and basic example. As a possible approach to a solution, it goes on to discuss the idea that one should not work in the space of all adjointable operators, but rather with a subalgebra. The natural setting would then become that of inner-product bimodules.

vii

Part I

Finite spectral geometry

Chapter 1

Introduction

Now we see as through a glass, darkly … Now what I know is incomplete.

1 Corinthians 13:12

The spectral picture of geometry begets its naturality from deep ties to physics, both classical – through the language of vibrations and heat flow – and quantum – through its natural accomodation of noncommutativity. Both aspects call for a coherent approach to approximate geometry, to accomodate the limited physical information accessible by ourselves as well as the viewpoints of renormalization and effective field theory. The present work engages the issue through a study of truncations of Dirac-type spectral triples.

Sketches and building blocks

Both construction and observation of shapes in the physical world proceed successively, in stages of increasing refinement. The knowledge of the final shape that each intermediate stage represents is finite and incomplete. It is, however, nontrivial. The theory of spectral geometry, in contrast to that of metric spaces, reflects this notion only rather opaquely. It is this opacity that we aim to alleviate by introducing the spectral analogon of partial geometric knowledge.

3 1. Introduction

Spectral truncations

The basic objects of noncommutative geometry are the spectral triples (퐴,퐻,퐷), which represent a Riemannian manifold whenever the triple is of Dirac type, that is, when 퐴 is the algebra of its smooth functions and 퐷 is a Dirac-type operator on a Hermitian bundle, both represented on the Hilbert space 퐻 of its sections. Spectral triples come with a natural momentum cutoff, the projection 푃훬 onto the [−훬,훬]-eigenspaces of 퐷 that yields the truncation (푃훬퐴푃훬,푃훬퐻,푃훬퐷) [AB84; DLM14b; CS20]. Because 푃훬 commutes with 퐷, this truncation preserves all symmetries – including the isometries of an underlying Riemannian manifold.

Chapters 2 and 3 investigate a natural framework for incremental observation in spectral geometry, by connecting metric geometry and its invariants to truncations of Dirac-type spectra. Chapter 4 explores the issue of recognizing those objects that are truncations of Dirac-type spectral triples and thereby takes a first step towards a spectral account of cumulative construction.

Nonperturbative regularization in quantum gravity

A common starting point for several candidate theories of quantum gravity is the path integral over the space of all Lorentzian metrics (on a given smooth background manifold, say), with respect to some gravitational action. To obtain effective field theories from this paradigm, one introduces some type of regularization, restricting the integral to metrics constrained by some cutoff parameter 훬 – ideally one that preserves the appropriate symmetries. These remaining metrics are then the building blocks that constitute the geometries considered at given scale. One salient example is that of Causal Dynamical Triangulations, where the Regge calculus is used to regularize the space of Lorentzian metrics in terms of flat simplicial manifolds.

In noncommutative geometry, the spectral action principle guides an approach to Eu- clidean quantum gravity that is naturally regularized by spectral truncation [CL90; Con96; Bar07; CCM07]. The construction of observables for such a theory then re- quires a metric understanding of truncated spectral triples (again, see Chapters 2, 3), and the parametrization of a suitable domain for the path integral requires a demarcation of the truncations of Dirac-type (that is, commutative) spectral triples (see Chapter 4) within the broader set of operator system spectral triples.

4 1. Introduction

Digital representation of geometric objects In computer science, the discrete and finite nature of our machines necessitate ap- proximation and precision management when dealing with geometric objects. This often leads to discrete representations thereof through e.g. polygonal meshes, which exemplifies the metric-spaces approach to partial geometric knowledge. In contrast, the philosophy of spectral geometry in its most basic form, the Fourier transform, under- pins many important applications such as the lossy compression methods of the JPEG and MP3 standards. Laplacian-based approaches to shape recognition and manipulation have been the subject of persistent interest in the fields of computer graphics (following e.g. [Lev06]) and manifold learning (such as [BN02]). A true understanding of truncated spectral geometry would provide a consistent framework for methods in this direction. As a first step, the method of [BN07] to recapture part of the Laplacian spectrumand eigenfunctions from a sufficiently dense finite metric subspace finds its dual counterpart in the algorithm of Chapter 3 that reconstructs such metric subspaces from finite spectral data.

Spectral timescale and localization An important ingredient in the relation between truncated Dirac-type spectral triples (and, in particular, the truncated spectrum of generalized Laplacians 훥) and Rieman- nian geometry is the short-time asymptotic behaviour of the heat kernel, see Section 1.1. It is well-known that the heat equation 휕푡푢(푥,푡) = −훥푢 has a fundamental solution 퐾(푡,푥,푦) ∫ 퐾(푡,⋅,푦)푢(푦)푑푦 = 푒−푡훥푢 퐾(푡,푥,푦) which satisfies 푀 , and the behaviour of as 푡 → 0 captures much (indeed, all) of the metric geometry of a Riemannian manifold 푀. In order to connect to partial spectra, we need to balance the exponential decay, in the spectrum of 훥, of 퐾 with its short-time asymptotics. The timescale at which this −1 balance is preserved, as first seen in Chapter 2, is that of 푡훬 = 푂(훬 log훬) – where 훬 is the relevant cut-off of the spectrum of 훥. Interestingly, the principle of heat flow on an appropriate timescale underlies not only our Chapters 2, 3 – as a bridge from spectral to metric data – but also – in the other direction – many of the applications of the Laplacian in computer science discussed above.

At the spectral timescale 푡훬, the heat kernel is not only well approximated by that of the truncated Laplace-type operator 푃훬훥, but is also optimally localized given that condition. In Section 3.3 we show that the probability measure on 푀 associated to the fiberwise square norm of the vector 푃훬퐾(푡훬,푥,⋅) has second Wasserstein distance

5 1.1. Mathematical preliminaries

푂(훬̃ −2) to 푥 ∈ 푀, providing an asymptotic counterpart to the points of 푀 in terms of truncations of its Dirac-type spectral triple.

The continuum nature of Connes’ reconstruction theorem Somewhat orthogonal to the issue of connecting metric and spectral geometry is that of demarcating spectral geometry inside the larger realm of noncommutative geometry. Connes’ reconstruction theorem [Con13] (cf. pp. 20 below) provides algebraic criteria to determine when a spectral triple is commutative, that is, of Dirac type. These criteria, however, do not coexist well with truncation. It is very much an open problem to find conditions that, if only asymptotically, ensure that a given operator system spectral triple corresponds to the truncations of a Dirac-type spectral triple. Chapter 4 investigates a first approach, employing the higher Heisenberg relation of [CCM14] as a version of the crucial orientability axiom. Both the higher Heisenberg relation - as an index constraint - and the results of Chapter 2 relate truncations of spectral triples to K-theory invariants. The fascinating result of [LS17] shines a different light on that relation, and combining the two presents an interesting avenue for further work.

1.1 Mathematical preliminaries

The following chapters are concerned with the geometric properties of truncated, Dirac- type spectral triples. The basic philosophy is that the geometric information about 푀 that is present in such a spectral triple should be recoverable, asymptotically, from its truncations. The most important technique that connects a Dirac-type spectral triple to its truncations is the asymptotic expansion of the heat kernel of generalized Laplacians. The interplay between that expansion and the (metric and topological parts of) Connes’ reconstruction theorem constitutes the central theme of Chapters 2 and 3.

Dirac-type spectral triples This section introduces the topic of Dirac-type operators on Hermitian bundles over smooth Riemannian manifolds. For further reading and context, see [BGV04; GVF01].

Definition 1.1.1. A second-order differential operator 훥 on a vector bundle 퐸 over a smooth Riemannian manifold 푀 is a generalized Laplacian if its principal symbol ∞ equals the metric. That is, for all 푓 ∈ 퐶 (푀), the section 휎2(훥,푑푓) of End퐸 is given

6 1.1. Mathematical preliminaries

by 1 휎 (훥,푑푓) def= − [[훥,푓∗],푓] = (푑푓,푑푓), 2 2 ∞ (푑푓,푑푓) 퐶 (푀) 푥 ↦ (푑푓 ,푑푓 ) ∗ where is the element of given by 푥 푥 푇푥푀.

The definition sets up a strong expectation that the Riemannian structure of 푀 is fully captured by such an (elliptic) operator 훥 and the action of 퐶∞(푀) on 훤∞(퐸). This (and more) is indeed the case. The precise statement will be part of Connes’ reconstruction theorem, below.

Definition 1.1.2. A Dirac-type operator 퐷 on a vector bundle 퐸 over a smooth manifold 푀 is a first-order differential operator on 퐸 such that 퐷2 is a generalized Laplacian.

Proposition 1.1.3. The principal symbol of a Dirac-type operator induces a Clifford module structure on 퐸. That is to say, for 푓 ∈ 퐶∞(푀), the section

푐(푑푓) def= 푖[퐷,푓] of End퐸 satisfies 푐(푑푓)푐(푑푔) + 푐(푑푔)푐(푑푓) = 2(푑푓,푑푔).

Proof. Because 퐷 is of order 1, the commutator [퐷,푓] is an endomorphism of 퐸 and so commutes with 퐶∞(푀). Therefore, we have 퐷푓[퐷,푔] + [퐷,푔]퐷푓 = 퐷[퐷,푔]푓 + [퐷,푔]퐷푓 = [퐷2,푔]푓 and similarly 푓퐷[퐷,푔] + [퐷,푔]푓퐷 = 푓퐷[퐷,푔] + 푓[퐷,푔]퐷 = 푓[퐷2,푔] so that [퐷,푓][퐷,푔] + [퐷,푔][퐷,푓] = [[퐷2,푔],푓], which equals −2(푑푓,푑푔) by polar decomposition of 휎2(훥,⋅).

∞ Corollary 1.1.4. For 푓 ∈ 퐶 (푀), one has sup ∥푑푓푥∥ ∗ = ∥[퐷,푓]∥ which 푥∈푀 푇푥푀 훤(End퐸) is in turn, by smooth extension of extremizing elements of 퐸, equal to ∥[퐷,푓]∥ . 퐵(퐿2(퐸))

The principal symbol of 퐷 is therefore nothing more or less than a Clifford module structure. In fact, there exists a canonical connection ∇ on 퐸 associated to 퐷2 such that 퐷 = −푖푐 ∘ ∇ up to a bundle endomorphism, as follows.

Proposition 1.1.5. Let 훥 be a generalized Laplacian and 훥0 be the Laplacian on func- 1 tions. Then, the map ∇∶ 훤(퐸) × 훤(푇푀) → 훤(퐸) defined by ∇(푑푓)# 푠 = 2 (훥0(푓) − [훥,푓])푠 is an affine connection on 퐸. Moreover, up to a bundle endomorphism of 퐸, the associated connection Laplacian on 퐸 is equal to 훥.

7 1.1. Mathematical preliminaries

Corollary 1.1.6. The operator 퐷′ = −푖푐 ∘ ∇ is a first-order Dirac-type differential oper- ator on 퐸. In particular, 퐷 − 퐷′ is a bundle endomorphism of 퐸.

Remark 1.1.7. Any compact, orientable Riemannian manifold carries a Dirac-type operator on some Hermitian vector bundle. For spinc manifolds the most natural choice of Dirac-type operator is given by the Dirac operator 퐷 = −푖(1 ⊗ 푐) ∘ ∇, where ∇ is a Clifford connection on the chosen spinor bundle – if the manifold is in factspin, then the connection can be determined uniquely if one demands that it commute with the charge conjugation operator. On spinc manifolds equipped with a spinor bundle 푆, each bundle 퐸 carrying a Dirac- type operator 퐷퐸 is of the form 푆 ⊗ 퐹, where 퐹 is a Hermitian vector bundle carrying a 퐹 퐹 Hermitian connection ∇ , and 퐷퐸 is of the form 퐷푆 ⊗ 1 − 푖푐 ⊗ ∇ , plus possibly an endomorphism of 퐸. That is to say, the rank of 푆 is minimal among those carrying Dirac-type operators, and each Dirac-type operator is just a twist of 퐷푆 - see [Ply86].

The interplay between the algebra 퐶∞(푀) and a Dirac-type operator 퐷 is realized on the Hilbert space 퐿2(퐸;푀) given by the completion of 훤(퐸) with respect to the inner product induced by the Hermitian structure on 퐸 and the volume form on 푀. This triple is in fact sufficient to recover the geometric data used to define it, such asthe smooth, Riemannian manifold 푀, the bundle 퐸 → 푀 with its Hermitian structure, and of course the differential operator 퐷.

Definition 1.1.8 (Dirac-type spectral triples). A Dirac-type spectral triple (associated to the compact, Riemannian manifold 푀) consists of the algebra 퐶∞(푀) and the Dirac-type operator 퐷 on a Hermitian vector bundle 퐸 over 푀, both viewed in their representation on the Hilbert space 퐿2(퐸;푀).

By Proposition 1.1.3, Corollary 1.1.14 (below) and the discussion preceding Propo- sition 1.1.13 (below), a Dirac-type spectral triple is a special case of the following definition.

Definition 1.1.9 (Spectral triples). Let 퐴 be a unital C∗-algebra represented on a Hilbert space 퐻, and let 퐷 be a selfadjoint, possibly unbounded, operator on 퐻. Let 풜 ⊂ 퐴 be a dense ∗-subalgebra. Then, (풜,퐻,퐷) is said to be a spectral triple whenever

• The resolvent (퐷 + 푖)−1 is compact.

• [퐷,푎] is bounded for all 푎 ∈ 풜.

8 1.1. Mathematical preliminaries

Connes’ reconstruction theorem (Theorem 1.1.39, below) shows that each commutative spectral triple is (unitarily equivalent to) a Dirac-type spectral triple. That is to say, given a spectral triple (풜,퐻,퐷), there is a known list of algebraic conditions on 풜,퐷 that allow one to determine whether it is unitarily equivalent to a Dirac-type spectral triple. As such, the Dirac-type spectral triples form a reasonably general and well-understood spectral picture of Riemannian geometry.

Truncated spectral triples In quantum field theory, the scaling behaviour of classical field theories and thepos- tulated infinite granularity of space(time) often conspire to introduce (ultraviolet) di- vergences. This necessitates a careful balancing act, regularizing the scaling behaviour, that is then combined with experimental evidence to produce a renormalized theory with physical predictions. A common way to regularize the theory is to introduce a momentum cutoff, restricting the considered interactions to those whose momentum is bounded by a cutoff parameter 훬, and then investigating the scaling with 훬. In the fermionic sector1 of gauge theories, this was expressed in [AB84] as a trunca- tion 퐷 ↦ 푃훬퐷푃훬 of the relevant Dirac operator, where 푃훬 = 1[−훬,훬](퐷) is the spectral projection onto the eigenspaces of eigenvalue |휆| ≤ 훬 of 퐷. Such a truncation is very natural: one does not need to make additional choices (unlike when discretizing spacetime2, for instance), the high-momentum interactions that are neglected are naturally difficult to observe, and all symmetries of the original Dirac operator (including, notably, those induced by isometries) are preserved because they commute with 푃훬. Thirty years later, after the birth of noncommutative geometry, [DLM14b] investigated what the geometric consequences of such a truncation would be, when interpreted in the context of that discipline. Viewed from the perspective of comparison to the non-truncated geometry, the relevant objects are of the following form.

Definition 1.1.10. Let (풜,퐻,퐷) be a spectral triple, let 훬 ∈ ℝ+ and let 푃훬 = 1[−훬,훬](퐷) ∈ 퐵(퐻). Then, (푃훬풜푃훬,푃훬퐻,푃훬퐷푃훬) is said to be a truncated spectral triple and a spectral truncation of (풜,퐻,퐷).

1And in far greater generality: see Section 1.1. 2In the sense that the only parameter here is 훬, whereas discretizations require a choice of triangulation or other decomposition into building blocks. It should be noted, however, that such geometric choices may exhibit a degree of universality, making them less relevant from a physical point of view (cf. e.g. [JL13; CS12]).

9 1.1. Mathematical preliminaries

It is important to realize that 푃훬풜푃훬 is no longer naturally an algebra, because 푃훬 is usually far from commuting with 풜 (compare Proposition 1.1.3). This puts us firmly outside the realm of spectral triples per se, and into the realm of operator system spectral triples as introduced in [CS20]. The general theory developed there is very much in line with the physics angle discussed here, and aims in particular to bring its increased3 flexibility to bear on quantum gravity. The central objects of study of the following chapters are the spectral truncations of Dirac-type spectral triples. Chapters 2 and 3 explore how the geometry of 푀 relates to truncations of Dirac-type spectral triples (퐶∞(푀),퐿2(퐸;푀),퐷). Chapter 4, in contrast, probes whether one can recognize spectral truncations of Dirac-type spectral triples algebraically.

The spectral function of a generalized Laplacian The interplay between the geometric properties encoded by the principal symbol of a generalized Laplacian and the spectral properties of its closure 훥 in 퐿2(퐸;푀) is largely mediated through the heat kernel and its asymptotic expansion. At the heart of Chap- ter 2 lies a balance between the exponential decay of the (trace of) the associated heat operator in the spectrum of 훥 and its well-known short-time divergence. Additional control over the exponential off-diagonal decay of the heat kernel, balanced with both, then allows for technical control over the tension between the local geometry of 푀 and the global invariants provided by the spectrum of 훥, as figures prominently in Chapter 3. Definition 1.1.11 (Heat kernel). Let 푀 be a compact, orientable Riemannian mani- fold and let 훥 be a generalized Laplacian acting on a smooth bundle 퐸 → 푀.A heat ∗ kernel of 훥 is a smooth section 푝푡(푥,푦) of the bundle 퐸 ⊠ 퐸 over 푀 × 푀 × ℝ>0, such that

• 푝푡(푥,푦) satisfies the heat equation

(휕푡 + 훥푥)푝푡(푥,푦) = 0.

• For all smooth sections 푠 of 퐸, one has

lim∫ 푝푡(푥,푦)푠(푦)푑vol(푦) = 푠(푥), 푡→0 푀 3As compared to Riemannian geometry.

10 1.1. Mathematical preliminaries

uniformly in 푥 ∈ 푀.

The second condition guarantees uniqueness, if a heat kernel exists. −푡훥 Note that 푝푡(푥,푦) is the integral kernel of the operator 푒 . Let 휌 be the injectivity radius of 푀 and let 휒 be a smooth, monotone decreasing 2 2 function on ℝ+ such that 휒(푠 ) = 1 for 0 ≤ 푠 < 휌/2 and 휒(푠 ) = 0 for 푠 > 휌. Theorem 1.1.12 (Asymptotics of the heat kernel). Let 푀 be a compact, orientable Riemannian manifold and let 훥 be a generalized Laplacian acting on a smooth bundle 퐸 → 푀. Then, there exists a heat kernel 푝푡(푥,푦) of 훥. ∞ ∗ Moreover, there exist unique smooth sections 훷푘 ∈ 훤 (푀 × 푀,퐸 ⊠ 퐸 ) such that asymptotically ∞ −푚/2 −푑(푥,푦)2/4푡 2 푘 푝푡(푥,푦) ∼ (4휋푡) 푒 휒(푑(푥,푦) )∑푡 훷푘(푥,푦), 푘=0 in the sense that 푁 푙 −푚/2 −푑(푥,푦)2/4푡 2 푘 푁−푚/2−푙 sup ∥휕푡 (푝푡(푥,푦) − (4휋푡) 푒 휒(푑(푥,푦)) ∑푡 훷푘(푥,푦))∥ = 푂(푡 ), 푥,푦∈ℐ 푘=0 for all 푁 > 푚/2 and all 푙 ≥ 0. Here, 훷0 is the parallel transport map associated to the connection determined by 훥.

Proof sketch, as in [BGV04, Theorem 2.30]. Construct a formal solution to the heat −푚/2 −푑(푥,푦)2/4푡 2 ∞ 푘 equation by termwise solving (휕푡 +훥푥)(4휋푡) 푒 휒(푑(푥,푦)) ∑푘=0 푡 훷푘(푥,푦) = 0 in order to obtain the 훷푘. Then show that the partial sum

푁 푁 def −푚/2 −푑(푥,푦)2/4푡 2 푘 푘푡 (푥,푦) = (4휋푡) 푒 휒(푑(푥,푦)) ∑푡 훷푘(푥,푦) 푘=0

푁 def 푁 constitutes a parametrix of the heat equation, meaning that 푟푡 (푥,푦) = (휕푡 +훥푥)푘푡 (푥,푦) 푙 ∗ 푁−푚/2−푙/2 푙 lies in 훤 (ℝ≥0 × 푀 × 푀,퐸 ⊠ 퐸 ) and is 푂(푡 ) in 퐶 norm, for 푁 > 푙 + 푚/2, and that 푁 lim∫ 푘푡 (푥,푦)푠(푦)푑vol(푦) = 푠(푥) 푡→0 푀 for sections 푠 of 퐸. Fix 푁 > 푚/2 + 1 and define the iterated convolution products 푙 def 푁 푁 ∗푙−1 ∞ 푙 푙 푞푡 = 푘푡 ∗ (푟푡 ) in the variable 푡. Then, the series ∑푙=0(−1) 푞푡(푥,푦) will converge

11 1.1. Mathematical preliminaries

2 푝 in 퐶 norm, its limit 푝푡(푥,푦) will be a heat kernel, and the remainder 휕푡 (푝푡(푥,푦) − 푁 푝 ∞ 푙 푙 푁−푚/2−푝−푞/2+1 푞 푘푡 (푥,푦)) = 휕푡 ∑푙=1(−1) 푞푡 will be 푂(푡 ) in 퐶 norm.

Let 푃푡 be the operator defined by the integral kernel 푝푡(푥,푦). One can use the facts that, due to the heat equation, 푃푡 commutes with 훥 and that lim푡→0 푃푡푠 → 푠 to show that 훥 is essentially selfadjoint whenever it is symmetric, with respect to a Hermitian structure on 퐸 that defines the Hilbert space 퐿2(퐸;푀). For brevity, we will denote its closure by 훥 as well.

Proposition 1.1.13. Let 훥 be the selfadjoint closure of a symmetric generalized Lapla- 2 −푡훥 cian on 퐿 (퐸;푀) with heat kernel 푝푡(푥,푦). Then, we have 푃푡 = 푒 . In particular, if −푡휆 퐸휆 is the projection on the 휆-eigenspace of 훥, we have 푃푡 = ∑휆 푒 퐸휆 in norm.

Proof. Note that the operators 푃푡 form a semigroup by uniqueness of the heat kernel. Thus, there is an increasing sequence of real numbers 휆푖 such that the eigenvalues of 2 −휆푖푡 the positive operators 푃푡 = 푃푡/2 are of the form 푒 . Now, for eigenspinors 휓 of 푃푡 −휆푡 −푡훥 with eigenvalue 푒 , we have −휆휓 = 휕푡푃푡휓|푡=0 = −훥휓, so that in fact 푃푡 = 푒 .

This exponentially decaying series expression provides an essential link between the spectral truncations of 훥 and the geometric properties of 푀 encoded in its symbol.

Corollary 1.1.14 (Asymptotic trace of the heat kernel). Let 퐸 be equipped with a Hermitian structure such that the generalized Laplacian 훥 is symmetric. Then, the −푡훥 operator 푃푡 = 푒 is of trace class for all 푡 > 0, and its trace,

푃 = ∫ 푝 (푥,푥)푑푥, tr 푡 tr퐸푥 푡 푀 satisfies ∞ 푃 ∼ (4휋푡)−푚/2 ∑푡푘 ∫ 훷 (푥,푥)푑 (푥). tr 푡 tr퐸푥 푘 vol 푘=0 푀

Proof sketch. That 푃푡 is of trace class follows from the fact that the kernel of its square root, 푃푡/2, is square-integrable. The asymptotic expansion of the trace follows from the restriction of the expansion of 푝푡(푥,푦) to the diagonal.

Corollary 1.1.15. The operator 훥 has compact resolvent.

12 1.1. Mathematical preliminaries

Proof. Because 푒−푡훥 is in the trace class, the spectrum of 훥 must be discrete, with finite multiplicity, and accumulate only at ∞.

Corollary 1.1.16 (The zeta function). Let 퐸휆 be the projections onto the 휆 eigenspaces −푠 −푠 def −푠 of 훥. Then, define the one-parameter semigroups 훥 by 훥 = ⨁휆>0 휆 퐸휆, which are clearly bounded (and compact) for ℜ푠 > 0. Then, for ℜ푠 > 푚/2, the operator 훥−푠 is of trace class. Moreover, the zeta function, defined for ℜ푠 > 푚/2 as 휁(훥,푠) def= tr훥−푠 extends meromorphically to ℂ, with simple poles at {푚/2,푚/2 − 1,…,1} whenever 푚 is even, and at {푚/2,푚/2 − 1,…} whenever 푚 is odd, and with residues

∫ 훷푘(푥,푥)푑vol(푥) res 휁(훥,푠) = 푀 푠=푚/2−푘 (4휋)푚/2훤(푚/2 − 푘) whenever 푚/2 − 푘 is indeed a pole.

−푠 Proof. By the Jensen-Cahen theorem [HR64, Theorem 2], the series ∑휆>0 rk퐸훬휆 is uniformly convergent on angular regions in the half-plane that forms its domain, so that it is holomorphic there.

def −푡휆 def Let 푔(푡) = ∑휆>0 푒 , so that 푔(푡) = tr푃푡 −rk퐸0. Consider the Mellin transform 푓(푠) = ∫∞ 푡푠−1푔(푡) ℜ푠 > 푚/2 0 , which is well-defined for due to the small-time asymptotics of 푔(푡), its positivity and its monotone decay. Then, as in [FGD95, Theorem 3], 푓(푠) 푐 has the singular expansion 푓(푠) = ∑ 푘 , where 푠 = 푚/2 − 푘 and 푐 is the coefficient 푘 푠−푠푘 푘 푘 of 푡푘−푚/2 in the asymptotic expansion of the heat trace. Moreover, by the Fubini- 푓(푠) = ∑ 퐸 ∫∞ 푒−휆푡푡푠−1푑푡 = 훤(푠)∑ 휆−푠 퐸 = Tonelli theorem we may write 휆>0 rk 휆 0 휆>0 rk 휆 훤(푠)tr훥−푠. We must therefore conclude that 휁(훥,푠) = 푓(푠)/훤(푠) continues analytically to a meromorphic function on ℂ. Moreover, by the knowledge of the poles of 푓(푠) and 훤(푠), the poles of 휁(훥,푠) must all be simple and must be located at 푠푘 = 푚/2 − 푘, for 푘 ∈ ℤ≥0, unless 푠푘 is a nonpositive integer.

Proposition 1.1.17 (Identification of the first asymptotic heat coefficients). We have

훷 (푥,푥) = 퐸, 훷 (푥,푥) = 퐸 ⋅ 푅(푥)/6, tr퐸푥 0 rk tr퐸푥 1 rk where 푅 is the scalar curvature of the Riemannian metric 휎(훥) on 푀.

13 1.1. Mathematical preliminaries

훷 (푥,푥) = Proof sketch, cf. [Ros97]. Clearly 0 id퐸푥 . If we locally trivialize the bundle 퐸 in Riemannian normal coordinates around 푥, then we have 훷0(exp(0),exp(푦)) = 푚 √det푔exp푦 for sufficiently small 푦 ∈ ℝ , so that 훷0 is expressible as a Taylor series in universal polynomials of the components of the Riemann tensor and its covariant derivatives by a theorem due to Cartan, cf. [Ber03, Proposition 67]. By construction, then, all 훷푘 are expressible in that fashion, so that tr훷1(푥,푥) is expressible as a universal 훼2 polynomial in the Riemann tensor. Now, note that the endomorphisms 훷푘 associated 2 −2푘 to the generalized Laplacian 훼 훥 (with respect to the rescaled metric) must equal 훼 훷푘, whereas the 푘th iterated covariant derivatives of the Riemann tensor scale as 훼−2−푘. This allows us to conclude that tr훷1 cannot contain constant or derivative terms, so that it is a universal linear polynomial in the Riemann tensor. It must, therefore, be proportional to the scalar curvature. Explicit calculation on e.g. 푛-spheres completes the proof.

Chapter 2 discusses how to approximate these coefficients of the heat trace asymptotics using only a finite part of the spectrum of 훥.

def Corollary 1.1.18 (Weyl’s law). The counting function 푁(훬) = #{∑휎(훥)∋휆<훬 tr퐸휆} satisfies vol(푀)rk(퐸) 푁(훬) ∼ 훬푚/2 (4휋)푚/2훤(푚/2 + 1) as 훬 → ∞.

Proof. The Wiener-Ikehara theorem [Wie88, Theorem 19.16] shows that, whenever the function 푔 is positive and the integral 푓(푠) = ∫ 푥−푠푑푔(푥) converges for ℜ푠 > 1, ℝ+ 푚/2 we have 푔(푥) ∼ 푥 ⋅ lim푠→1(푠 − 1)푓(푠). Now write 푔(푥) = 푁(푥 ), so that res푠=1 푓(푠) = 푚푠 res푠=1 휁(훥, 2 ).

States and Gelfand duality

Let TopHCpt be the category of compact Hausdorff spaces and continuous maps, and ∗ ∗ let 퐶 AlgCom be the category of unital, abelian C -algebras and ∗-homomorphisms. Abstractly speaking, Gelfand duality is the statement that a specific pair 퐶,훤 of functors implements an equivalence between these categories. For further details on the topic of this section, see [GVF01, Chapter 1] and the references therein.

14 1.1. Mathematical preliminaries

∗ Definition 1.1.19. The (contravariant) functor 퐶∶ TopHCpt → 퐶 AlgCom sends a compact topological space 푋 to the unital, abelian C∗-algebra 퐶(푋) of ℂ-valued con- tinuous functions. On morphisms, 퐶 sends a continuous map 휙∶ 푋 → 푌 to the ∗-homomorphism 휙∗ from 퐶(푌) to 퐶(푋) given by 푓 ↦ 푓 ∘ 휙.

Now, points 푥 ∈ 푋 correspond to elements of Hom(∗,푋), and so under the functor 퐶 correspond to evaluation morphisms ev푥 ∶ 퐶(푋) → 퐶(∗) = ℂ. The construction of the functor 훤 proceeds from the realization that such ℂ-valued homomorphisms (characters), endowed with the weak ∗-topology, correctly capture the notion of point ∗ in the setting of 퐶 AlgCom. For the purposes of Chapter 3, we will first work with the larger subspace of the topo- logical dual 퐴∗ consisting of states (instead of just the characters), so as to gain a more ∗ flexible picture of the dictionary between TopHCpt and 퐶 AlgCom. Definition 1.1.20. Let 퐴 be a C∗-algebra. A bounded linear map 휙∶ 퐴 → ℂ is said to be a state of 퐴 if 휙(1) = 1 and 휙(푎∗푎) ≥ 0 for all 푎 ∈ 퐴. The convex set of all such states is denoted by 푆(퐴).

Indeed, all states of 퐶(푋) are convex combinations of evaluation maps, in the following sense. Theorem 1.1.21 (Riesz representation theorem). Let 휙 be a state of 퐶(푋), for 푋 a compact Hausdorff space. Then there exists a regular Borel probability measure 휇휙 on 푋 such that

휙(푓) = ∫ 푓푑휇휙 푋

Proof. See [Con90, Appendix C]. Corollary 1.1.22. The extreme points of the convex set 푆(퐶(푋)) are evaluation func- tionals {ev푥 ∣ 푥 ∈ 푋}.

Proof. Let 휇 be the measure corresponding to an extreme point of the convex set of states. def First, assume 퐾 ↪ 푋 has measure 0 < 휇(퐾) < 1. Then, let 휇0(푌) = 휇(푌 ∩ 푋)/휇(퐾) def and 휇1(푌) = 휇(푌⧵퐾)/휇(푋⧵퐾) and observe that 휇 = 휇(퐾)⋅휇0 +휇(푋⧵퐾)⋅휇1. Thus, 휇 takes values in {0,1} whenever it is an extreme point.

15 1.1. Mathematical preliminaries

Let now 푥,푦 be both in the support of 휇 and let 퐾푥,퐾푦 be disjoint open sets containing 푥,푦 respectively. Now, by additivity and positivity of 휇, either 휇(퐾푥) = 0 or 휇(퐾푦) = 0, contradicting the assumption. We conclude that supp(휇) is a single point 푥 ∈ 푋, so ∫ 푓푑휇 = 푓(푥) = (푓) that 푋 ev푥 .

That is to say, we can recognize the points of 푋 purely in terms of the C∗-algebra 퐶(푋). What about its topology?

Definition 1.1.23. Let 퐴 be a unital, abelian C∗-algebra. Its Gelfand spectrum 퐴̂is the set of extreme points of 푆(퐴), equipped with the weak ∗-topology.

Theorem 1.1.24. Let 퐴 be a unital, abelian C∗-algebra. Then 퐴̂is compact and Haus- dorff.

Proof. First, note that the weak ∗-topology on 퐴∗ is completely Hausdorff: after all, the evaluation functionals {ev푣 ∣ 푣 ∈ 풳} are continuous and separate points. Moreover, by Banach-Alaoglu (see [Con90, p. V.3.1]) its unit ball is compact.

Now let us show that 퐴̂is a closed subspace (which is contained in that unit ball by Cauchy-Schwartz). To that end, we will show that 퐴̂consists precisely of the multi- plicative elements of 퐴∗. ̂ Assume 휙 ∈ 퐴 is multiplicative and that there exist 휙1,휙2,휆1,휆2 with 휙 = 휆1휙1 + 휆2휙2. ∗ In particular, 휆1휙1 ≤ 휙. By Cauchy-Schwartz for the functional (푎,푏) → 휙1(푎 푏), then, we see that ker휙 ⊂ ker휆1휙1, so that we must have 휙1 = 휙. That is to say, 휙 must be an extreme point. ∗ Conversely, suppose 휙 ∈ 퐴 is not multiplicative. Then the (GNS) representation 휋휙 of 퐴 by multiplication on 퐴/ker휙, with the inner product induced by 휙, cannot be one- ∗ dimensional: there exist 푎,푏 in 퐴 such that ⟨휋휙(푎)(1퐴),휙휙(푏)(1퐴)⟩ = 휙(푎 푏), which ∗ by assumption on 푎,푏 does not equal 휙(푎 )휙(푏) = ⟨휙휙(푎)(1퐴),1퐴⟩⟨1퐴,휋휙(푏)(1퐴)⟩, so that 1퐴 cannot be an orthonormal basis even though it is normalized. Thus, by Schur’s lemma there exist a projection 푃 commuting with 휋휙. Then, the linear functionals

푎 ↦ ⟨1퐴,푃휙휙(푎)1퐴⟩ and 푎 ↦ ⟨1퐴,(1 − 푃)휋휙(푎)1퐴⟩ are positive and sum to 휙, so that 휙 cannot be an extreme point of the state space. Now, the weak ∗-limit of multiplicative states is multiplicative, so that the set of extremal states is a closed subset of a compact Hausdorff space.

16 1.1. Mathematical preliminaries

Corollary 1.1.25. Let 퐴 be a unital, abelian C∗-algebra and let 퐴̂be its space of extremal states with the weak ∗-topology. Then the map ev∶ 퐴 → 퐶(퐴)̂ is an isomorphism of C∗-algebras.

Proof. As 퐴̂consists of multiplicative ∗-functionals, the map ev is itself a ∗-homo- morphism. It is injective because the states are the convex hull of the extremal states by Krein-Milman and separate 퐴. Thus, ev(퐴) is a closed subalgebra of 퐶(퐴)̂ that separates points. By Stone-Weierstrass (the proof of which, as in [Con90, p. V.8.1], is similar to the arguments above: roughly, the unit ball of the set of functionals vanishing on such a closed subalgebra must be the convex hull of its extreme elements, which must then correspond to points of 퐴̂) we conclude that ev is surjective.

Corollary 1.1.26. Let 푋 be a compact Hausdorff space. Then the map ev∶ 푋 → 퐶(푋)̂ is a homeomorphism.

Proof. That ev is surjective was already shown, and it is injective by Urysohn’s lemma. If 푥 → 푥 → ∗ 푛 , then ev푥푛 ev푥 by definition of the weak -topology. Thus, ev is a continuous bijection of compact Hausdorff spaces and is therefore a homeomorphism.

∗ Definition 1.1.27. The Gelfand spectrum is the covariant functor 훤∶ 퐶 AlgCom → ∗ ̂ TopHCpt that maps objects 퐴 ∈ 퐶 AlgCom to their extremal state space 퐴 and morphisms 휓∶ 퐴 → 퐵 to the pullback 휓∗ ∶ 퐵̂ → 퐴,휒̂ ↦ 휒 ∘ 휓.

Gelfand duality now follows from Corollary 1.1.25 and 1.1.26: Theorem 1.1.28 (Gelfand duality). The functors 퐶,훤 form an equivalence of cate- gories.

This is, roughly, the topological statement at the heart of spectral geometry. For the metric part, we need to introduce the Connes metric on 푆(퐶(푋)).

The Connes metric

Definition 1.1.29 (Connes metric). Let (퐶∞(푀),퐿2(퐸;푀),퐷) be a Dirac-type spec- tral triple. Then, the Connes metric on the space 푆(퐶(푀)) of states of 퐶(푀) is given by 푑(휙,휓) def= sup{∣휙(푓) − 휓(푓)∣∣푓 ∈ 퐶∞(푀),∥[퐷,푓]∥ ≤ 1}.

17 1.1. Mathematical preliminaries

Definition 1.1.30 (Wasserstein4 distance). Let 푀 be a compact Riemannian manifold and let 휇,휈 be probability measures. Then the 푝-th Wasserstein distance between 휇 and 휈 is 1/푝 def 푝 푊푝(휇,휈) = ( inf ∫ 푑(푥,푦) 푑휌(푥,푦)) , 휌∈훱(휇,휈) 푀×푀 where 훱(휇,휈) is the set of probability measures 휌 on 푀×푀 such that 휌[퐴×푀] = 휇[퐴], 휌[푀 × 퐴] = 휈[퐴], for all measurable 퐴 ⊂ 푀.

As, for atomic measures 훿푥, 훿푦 supported at 푥,푦 in 푀, we have 훱(훿푥,훿푦) = {훿(푥,푦)}, the following proposition follows immediately from the definition.

Proposition 1.1.31. With respect to the metric 푊1, the embedding 푥 → 훿푥 of 푀 into its space of probability measures is an isometry.

The relation between the Connes metric and the first Wasserstein distance is cemented by the following important theorem. Theorem 1.1.32 (Kantorovich-Rubinstein duality). Let 휇,휈 be probability measures on 푀. Then,

푊1(휇,휈) = sup{∫ 푓푑(휇 − 휈)∣푓 ∈ 퐶(푀),Lip푓 ≤ 1}, 푀

0 ≤ 푓 ≤ ∞ |푓(푥)−푓(푦)| where Lip denotes the Lipschitz constant sup푥,푦∈푀 푑(푥,푦) .

Proof. See [Vil09, Theorem 5.10].

In particular, by density of the inclusion 퐶∞(푀) ⊂ 퐶(푀), we now see that Dirac-type operators carry not just the Riemannian metric in their symbol but, rather elegantly, the geodesic distance as well: Corollary 1.1.33 (Connes metric, commutative case). For states 휙,휓 of the algebra 퐶(푀) of continuous functions on a compact Riemannian manifold 푀 with associated probability measures 휇휙,휇휓, the Connes metric is equal to the first Wasserstein distance:

푑(휙,휓) = 푊1(휇휙,휈휓).

4This metric is known under various other names, especially for the case 푝 = 1, including ‘1-Wasserstein’, ‘Monge-Kantorovich’ and ‘Earth Mover’s’. See e.g.[Vil09, Chapter 6], where it is argued that the ‘Wasserstein’ nomenclature is unfortunate; it is, however, the most common.

18 1.1. Mathematical preliminaries

Corollary 1.1.34. Let (퐶∞(푀),퐿2(퐸;푀),퐷) be a Dirac-type spectral triple. Then the metric space 푀 is isometric to the Gelfand dual 퐶(푀)̂ when the latter is equipped with the Connes metric.

∗ The Riemannian metric on 푀 is given by the map (푓,푔) ↦ (푑푓,푑푔)푇∗푀 = [퐷,푓] [퐷,푔]. We now see that it is in fact already fully determined by the seminorm 푓 ↦ ∥[퐷,푓]∥ on 퐶∞(푀).

Corollary 1.1.35. Let 퐷1,퐷2 be Dirac-type operators on Hermitian bundles 퐸1,퐸2 over Riemannian manifolds 푀1,푀2. Then 푀1,푀2 are isometric if and only if there exists an isomorphism 휙∶ 퐶(푀1) → 퐶(푀2) such that ∥[퐷1,푓]∥ = ∥[퐷2,휙(푓)]∥ for all 푓 ∈ 퐶(푀1).

Proof. The ‘if’ part is an immediate corollary, and the ‘only if’ part follows from the uniqueness of the Levi-Civita connection and the fact that ‖[퐷,푓]‖2 = ‖∇푓‖2.

Chapter 3 is occupied with approximating points (as elements of 퐶(푀)̂ ⊂ 푆(퐶(푀))) and the Connes distance between them in terms of truncations of Dirac-type spectral triples, so as to recover the metric space 푀.

The reconstruction theorem Dirac-type spectral triples are the prime example of commutative spectral triples. Indeed, Connes’ reconstruction theorem (Theorem 1.1.39 below) shows that they are the only examples thereof and are therefore uniquely characterized by the following properties.

Definition 1.1.36. A spectral triple (풜,퐻,퐷) is said to be commutative whenever the ∗ -algebra 풜 is a commutative and there exists some integer 푝 such that

1. Dimension: The 푛-th singular value, with multiplicity, of (퐷 + 푖)−1 is 푂(푛−1/푝).

2. First-order: For all 푓,푔 ∈ 풜, [[퐷,푓],푔] = 0.

3. Regularity: For 푎 ∈ 풜, all iterated commutators of 푎 and [퐷,푎] with |퐷| are bounded.

19 1.1. Mathematical preliminaries

0 푝 4. Orientability: There exists a Hochschild cycle ∑푖 푐푖푎푖 ⊗ ⋯ ⊗ 푎푖 in 푍푝(풜,풜) such that 0 1 푝 ∑푐푖푎푖 [퐷,푎푖 ]⋯[퐷,푎푖 ] = 훾, 푖 where 훾 = 1 if 푝 is odd and 훾 is a grading on 퐻 that anticommutes with 퐷 otherwise.

푚 5. Finiteness and absolute continuity: The 풜-module 퐸 = ⋂푚 dom퐷 is finite and projective, and there exists an 풜-linear, 풜-valued inner product on 퐸 such that

⟨휉,푎휂⟩ = res tr푎⟨휉,휂⟩ |퐷|−푝−푠, ℂ 푠=0 풜 for all 푎 ∈ 풜.

In Dirac-type spectral triples, the first axiom corresponds to Weyl’s asymptotics (Corol- lary 1.1.18), the second to the fact that 퐷 is a first-order differential operator, the third to the fact that 푎 and [퐷,푎] are smooth sections of the endomorphism bundle for all 푎 ∈ 풜, the fourth to the existence of a Riemannian volume form, and the fifth to the fact that 퐻 is the completion of the smooth sections 훤∞(퐸;푀) with respect to a Hermitian structure on 퐸.

Definition 1.1.37. Two spectral triples (풜0,퐻0,퐷0), (풜1,퐻1,퐷1) are said to be ∗ unitarily isomorphic if there exists a unitary 푈∶ 퐻0 → 퐻1 such that 푈풜0푈 = 풜1, ∗ 푈dom퐷0 = dom퐷1 and 푈퐷0푈 = 퐷1. ∞ ∞ Remark 1.1.38. When 퐶 (푀) ≃ 풜0 ≃ 풜1 ≃ 퐶 (푁), the isomorphism with 풜1 given by 푎 ↦ 푢푎푢∗ is necessarily given by the pullback through a diffeomorphism 휙∶ 푁 → 푀, as in [Mrč05]. If (풜0,퐻0,퐷0) and (풜1,퐻1,퐷1) are Dirac-type spectral triples, the oper- 2 2 ∗ 2 2 ∗ ator |푑푎| = [[퐷0,푎],푎] lies in 풜0 for all 푎 ∈ 풜0, so that 휙 (|푑푎|푀) = 푢[[퐷0,푎],푎])푢 = 2 ∗ ∗ ∗ 2 [[퐷1,휙 (푎)],휙 (푎)] = |푑휙 (푎)|푁 so that 휙 must be a Riemannian isometry. Moreover, using Gelfand duality and the Serre-Swan theorem, we can prove that 푢 must in fact be the pullback by a bundle isomorphism that covers 휙 and intertwines the Dirac-type operators.

Theorem 1.1.39 (Connes’ reconstruction theorem, [Con13]5). If a spectral triple (풜,퐻,퐷) is commutative (and only then), there exist a compact, oriented, smooth

5 For reference, [Con13] contains the proof, initiated by Rennie and Varilly in [RV06], that 풜 is of the form 퐶∞(푀); for proof that under these assumptions 퐷 is then a Dirac-type operator, cf. [GVF01, Chapter 11].

20 1.1. Mathematical preliminaries

∞ 2 manifold 푀 and a Dirac-type spectral triple (퐶 (푀),퐿 (퐸;푀),퐷푀) that is unitarily isomorphic to (풜,퐻,퐷).

Note that the criteria of Definition 1.1.36 do not translate easily tothe truncations of a commutative spectral triple. This leads to the question whether there is a version of these axioms that does, so as to detect whether a given operator system spectral triple is the truncation of a Dirac-type spectral triple. In Chapter 4 we investigate the cycle for the orientability axiom that is provided by the higher Heisenberg equation [CCM14] in a first step towards such detection.

The orientability axiom and the higher Heisenberg equation The orientability axiom for commutative spectral triples, as in Definition 1.1.36 is essential to the proof of the reconstruction theorem, where it is used to construct a smooth atlas of 푀. It is so called because it expresses the grading (or the identity, in the odd case) as the Clifford action of a differential form of top degree. In [CCM14], Chamseddine, Connes and Mukhanov prove that, in certain circum- stances, there exist a very special representative of the orientability axiom. This relates to a specific choice of generators of the coordinate algebra 퐶∞(푀), which satisfy a version of the orientability axiom that is analogous to the Heisenberg commutation relation [푥,푝] = 푖ℏ, as follows.

Theorem 1.1.40 (One-sided higher Heisenberg equation). If (풜,퐻,퐷) is a Dirac-type + spectral triple of dimension 푚 ∈ 2ℕ, then there exists an element 푌 ∈ 풜 ⊗ Cl푚 with 푌2 = 1, 푌∗ = 푌 such that

푚 tr 푌[퐷 ⊗ 1 + ,푌] = 푚!훾, 퐵(퐻) Cl푚 if and only if 푀 is diffeomorphic to a disjoint sum of spheres, each of which hasunit volume with respect to the Riemannian metric on 푀. + Here Cl푚 is the real (Clifford) algebra generated by 1,푒1,…,푒푚, with the relation 푒푖푒푗 + 푚 푒푗푒푖 = 2훿푖푗 (which is, as a vector space, canonically isomorphic to ⋀ℝ ), and the + 퐵(퐻)-valued trace tr ∶ 퐵(퐻) ⊗ Cl → 퐵(퐻) gives 푚 times the coefficient of 1 + . 퐵(퐻) 푚 Cl푚

The construction of a smooth atlas using the orientability axiom for commutative spectal triples is here illustrated by the fact that, by its algebraic properties, 푌 must be a covering of the sphere: its components, viewed as maps into ℝ푚+1, therefore

21 1.1. Mathematical preliminaries

locally form a coordinate system. The sphere being simply connected, this is also what determines the topology of 푀 as a disjoint sum of spheres. The reason that each disjoint summand must be of unit volume is that the Riemannian volume form on 푀 is now equal to the pullback, under 푌, of the volume form on 푆푚, so that the integral evaluates to deg푌 times the volume of the standard 푚-sphere. More −푚−푠 abstractly, the top zeta residue, res푠=0 tr퐷 , which is proportional to the volume 푚 by Proposition 1.1.17, now equals res푠=0 tr훾tr풜 푌[퐷 ⊗ 1,푌] /푚!, and the latter is proportional to the pairing between the Chern character of 퐷 and the Chern character of the idempotent 푒 = (푌+1)/2, so that it is proportional to the integer Index푒(퐷⊗1)푒. The authors of [CCM14] go on to show that, for spectral triples associated to a spin Dirac operator, the statement can be slightly modified so as to hold in much greater generality, including for all 4-dimensional compact, oriented, connected spin manifolds.

The spectral action The hypothetical (Wick rotated) path integral in pure Euclidean quantum gravity takes the form 풵 = ∫풟푔exp(−푆[푔]), where the Euclidean Einstein-Hilbert action 푆[푔] equals the total scalar curvature asso- ciated to 푔 and the integral is over the space of four-dimensional, compact Riemannian manifolds without boundary – or perhaps of a fixed background manifold 푀. In noncommutative geometry, the natural analogon of a compact Riemannian manifold (albeit with extra structure) is a Dirac-type spectral triple. The heat trace asymptotics of Corollary 1.1.14 allow one to express the Euclidean Einstein-Hilbert action in terms of the spectrum of the Dirac-type operator 퐷. The spectral action principle of [CC97] now conjectures that the physical action (of any6 reasonable field theory, including gravity) should likewise depend only on the spectrum of 퐷 and, in particular, that the natural (bosonic) physical action functional 푆[퐷] should be of the form

푆[퐷] = tr(푓(퐷/훬)), for some positive, even function 푓, whose moments determine the bare couplings. For an in-depth overview of this program, see [CM08, Chapters 11-19].

6Including the Standard Model. The surprisingly natural treatment of gauge theories in noncommuta- tive geometry, through the inner automorphisms of almost-commutative finite spectral triples, is beyond the scope of this thesis: see [CM08, Chapter 11] and [Sui15].

22 1.1. Mathematical preliminaries

The suppression of higher eigenvalues through the function 푓, parametrized by 훬, is a smoothed version of the finite-mode regularization discussed in Section 1.1 –this smoothed version is however a priori still strongly dependent on the higher eigenvalues of 퐷. Computer simulations of the regularized path integral (see e.g. [BG16]) rather imply working with the finite-dimensional truncations 푃훬퐷. To bridge the gap, the results of Chapter 2 can be interpreted as the construction of admissible7 functions 푓 with compact support: see Theorem 2.2.3. A further important question is that of observables of such a theory. To that end, the results of Chapter 2 can be applied immediately to transfer the spectral observables −푠 res푠=푚/2−푘 tr푎훥 , for 푎 ∈ 퐶(푀), to the truncated realm. Moreover, the algorithm of Chapter 3 allow one to interpret the integrands of the truncated path integral as metric spaces in their own right, so that we can translate observables from the purely metric realm (including e.g. the Gromow-Hausdorff distance to a fixed comparison space) to the setting of the spectral action. When constructing the (truncated) spectral path integral we are faced with the choice of a domain of integration. For concreteness, let us say that a background manifold 푀 and a energy scale 훬 were fixed, so that we want to integrate exp(−푆[퐷훬]) over the space of ‘truncated Dirac operators’ {퐷훬}. Which finite-rank operators 퐷훬 should then be considered? The original integral over Riemannian metrics suggests we should restrict to truncations 퐷훬 = 푃훬퐷 of Dirac-type operators 퐷. The question is then how to implement this restriction in the path integral, keeping in mind the difficulty of the corresponding recognition problem as discussed in Section 1.1. The approach laid out in [CM08, Chapter 18.4] is to implement the higher Heisenberg equation of [CCM14] by adding the corresponding term to the action as a dynamical constraint. To that end, Chapter 4 investigates the behaviour of such a dynamical constraint when 푀 = 푆1 or 푀 = 푆2.

7In the sense of producing the desired asymptotics of tr(푓(퐷/훬)) as 훬 → ∞.

23

Chapter 2

Finite-rank approximation of spectral zeta residues1

We employ the asymptotic expansion of the heat trace to express all residues of spectral zeta functions as regularized sums over the spectrum. The method extends to those spectral zeta functions that are localized by a bounded operator.

2.1 Introduction

The spectral theory of elliptic operators presents a major connection between func- tional analysis and differential geometry. It provides a number of interesting relations between the spectrum with multiplicities (which is the complete unitary invariant of self-adjoint operators) and the symbol (which is directly tied to the local expression of pseudodifferential operators). Thereby, it shows how the local structure ofsuchan operator influences its global properties, and vice versa. Of particular interest isthe relation between the symbol of an elliptic operator and its spectral asymptotics that is conveyed by the spectral zeta residues. This chapter is concerned with expressing the spectral zeta residues as a limit of partial sums of particular functions over the spectrum. Togetherwith the well-known relations between the zeta residues, the symbol, and the heat expansion, this bridges a gap between the continuum setup of spectral geometry and the finite objects in combinatorial geometry and computer science. For instance, this method allows one to approximate

1This work was published as [Ste19b].

25 2.1. Introduction

the scalar curvature on a compact Riemannian manifold using only a finite part of its Laplacian spectrum. We will first provide some background material in Section 2.1 and then introduce the main topic and results of this chapter in Section 2.1.

Background: spectral zeta residues and Weyl’s law In 1949, Minakshisundaram and Pleijel showed [MP49] that the zeta function

휁(훥,푠) def= tr훥−푠 of the Laplace operator on a compact Riemannian 푑-dimensional manifold can be meromorphically extended to the complex plane, with simple poles occurring in the points 푑/2 − ℤ≥0 ⊂ ℝ. The residues at these poles (which are proportional to the so- called heat kernel coefficients) relate the spectrum of the Laplace operator, itself an isometry invariant, to other known isometry invariants of the manifold, such as its volume and scalar curvature. More generally, an elliptic pseudodifferential operator defines a spectral zeta function, by the work of Seeley [See67], and its residues relate geometric information to the operator spectrum in the following way. If 훥 is a positive elliptic classical pseudodifferential operator of order 푚 ∈ ℝ≥0 and 푘 is any nonnegative integer, the residue at 푠 = (푑−푘)/푚 of 휁(훥,푠) equals the Wodzicki residue

1 (푘−푑)/푚 푛−1 ∫ tr푎−푛 (푥,휉)푑 휉푑푥, 푚 푆∗푀 where 푎−푛(푥,휉) is the homogenous term of order −푛 in the decomposition of the symbol of 훥 and 푆∗푀 ⊂ 푇∗푀 is the cosphere bundle, cf. [Wod87; Gil95; Con94]. In the inversely Mellin transformed picture, the residues correspond to integrals of terms in the asymptotic expansion of the integral kernel of 푒−푡훥 along the diagonal, as 푡 → 0+. One wonders which conclusions about the operator spectrum can be drawn from the zeta residues, apart from the rather opaque one provided by their definition. The most well-known result in this direction is provided by the Wiener-Ikehara theorem, which relates the first residue of the zeta function to the asymptotics of the number 푁(훬) of eigenvalues smaller than 훬. If 훼 is a monotone increasing function and the zeta function ∞ 휁(푧) = ∫ 휆−푧푑훼(휆) 1

26 2.1. Introduction

converges for ℜ푧 > 1 and can be meromorphically extended to ℜ푧 ≥ 1 with a simple pole at 푧 = 1, then the Wiener-Ikehara theorem states that 훼(훬) ∼ 훬res푧=1 휁(푧) as 훬 → ∞. By Seeley’s results on complex powers of elliptic operators, we may apply this theory to the spectral zeta function of a degree 푚 elliptic pseudodifferential operator on a 푑-dimensional manifold. As in [MP49], this yields Weyl’s famous law

푑/푚 푑/푚 푁(훬) = res푠=푑/푚 휁(훥,푠)훬 + 표(훬 ).

Note that this restriction only uses the first residue of the zeta function: the zeta series tr훥−푠 is absolutely convergent for ℜ푠 > 푑/푚.

The problem of improving on the accuracy in Weyl’s law has attracted much attention over the last century. That is, one wonders whether we can obtain an asymptotic 푑/푚 expansion of 푁(훬) − res푠=푑/푚 휁(훥,푠)훬 for specific classes of operators. However, sharp bounds that depend only on the spectral zeta function of 훥 have not yet been produced, even in the well-studied case of the Laplacian on flat tori. It would seem natural that the Wiener-Ikehara result could be extended to relate the asymptotic expansion of 푁(훬) to the location of the poles of the zeta function. However, this approach is limited by the difficulties of inverse Mellin and Laplace transforms, see e.g. [Ara96], [Ber03, p. 9.7.2]. The lower poles of the zeta function can therefore not yet be related to the asymptotics of 푁(훬). However, we will explain in the present chapter how to relate their residues to the asymptotics of other functionals of the operator spectrum.

Zeta residues as a resummation of the spectrum

In the converse direction to Weyl’s law, we ask which conclusions one can draw about the residues, given access to increasingly large finite subsets of the operator spectrum. 2 For the first residue, Weyl’s law gives rise to the Dixmier trace formula: if 휆0 ≤ 휆1 ≤ ⋯ are the eigenvalues of 훥, then

∑푁 휆−푑/푚 ∑ 휆−푑/푚 res 휁(훥,푠) = lim 푖=1 푖 = lim 휆<훬 . 푠=푑/푚 푁→∞ log푁 훬→∞ 푑 푚 log훬

2As restricted to the ideal on which this formula converges. For a treatment of the Dixmier trace on the larger ideal 퐿(1,∞), see [Car+07].

27 2.2. Zeta residues as normal functionals

The Dixmier trace is singular: it vanishes if tr훥 < ∞. However, if 훥푗 is a sequence of def finite rank operators of operator norm ‖훥푗‖ = 훬푗 → ∞ such that

−푠0 −푠0 tr훥푗 − ∑ 휆 = 표(log훬푗), 0<휆<훬푗

−푠0 −푠0 the Dixmier trace of 훥 clearly equals lim푗→∞ tr훥푗 /log훬푗. It would seem plausible that this result can be extended to further poles, i.e. that there are also functionals 푐푘 on the finite rank operators such that

lim 푐 (훥 ) = res 휁(훥,푠), 푗→∞ 푘 푗 푠=푠푘 if 휁(훥,푠) has poles at 푠0 > ⋯ > 푠푘 and the finite-rank 훥푗 converge to 훥 in some appropriate sense. The main question, then, is whether we can write these functionals down explicitly. The existence of such asymptotic residue functionals is shown by our Theorem 2.2.3, and explicit expressions follow from the conditions of Propositions 2.2.7 and 2.2.8. Corollary 2.2.9 uses the theorem to improve on the convergence of the Dixmier trace formula, and Corollary 2.2.11 exhibits the resulting expression for the second pole. Finally, Theorem 2.3.1 is a simple extension of our treatment to the localized residue ℎ훥−푠 ℎ traces res푠=푠푘 tr , for any bounded operator .

2.2 Zeta residues as normal functionals

We can ask how spectral zeta residues relate to finite subsets of the operator spectrum without referring to the original setting of differential geometry. Wewill henceforth con- sider the question in such generality, but we will need to restrict ourselves to operators whose zeta functions share an essential property with the Minakshisundaram-Pleijel zeta function.

Definition 2.2.1. A positive, invertible3, self-adjoint, unbounded operator 훥 with compact resolvent is said to be spectrally elliptic if the ‘heat trace’ tr푒−푡훥 admits an ∞ −푠푖 + asymptotic expansion ∑푖=0 푐푖푡 as 푡 → 0 , where the 푠푖 are decreasingly ordered reals. We say that 훱 = {푠푖} is the heat spectrum of 훥.

3If not, just restrict to the complement of the kernel.

28 2.2. Zeta residues as normal functionals

−푠 The definition implies that there is some 푠0 ∈ ℝ such that 훤(푠)tr훥 converges for ℜ푠 > 푠0 and can be analytically continued to a meromorphic function 훤(푠)휁(훥,푠) whose poles are all simple and located in 훱 ⊂ (−∞,푠0]. This particular terminology was chosen because the motivating example of such opera- tors are the positive elliptic differential4 operators, cf. [See67; DG75]. Indeed, for a positive elliptic differential operator of order 푚 on a 푛-dimensional manifold, the heat 푛 푛−1 1 spectrum is contained in 푚 , 푚 ,…, 푚 . Definition 2.2.1 suggests that we will in fact be technically concerned with theasymp- totic heat trace coefficients 푐푖, and the reader who is more familiar with that terminology may rest assured that they are indeed what we are talking about. However, because zeta residues are mathematically the more general notion, the text will mainly refer to the coefficients by that name. Our main theorem shows how the zeta residues of a spectrally elliptic operator are related to restrictions of its spectrum. We will provide the proof in Section 2.2, below. 휁(훥,푠) 푠 ∈ 훱 The precise asymptotic formula for the residue res푠=푠푘 at some pole 푘 de- pends only on the location of the ‘previous’ poles, that is, on the set 훱 ∩ [푠푘,∞). The Dixmier trace, for instance, requires knowledge of 푠0 and of the fact that no poles 푠−1 > 푠0 exist. For a given finite set {푠푖} of such poles, all operators with zeta residues contained in this set can be considered simultaneously. Hence the following definition.

푘 푘 Definition 2.2.2. For any finite set {푠푖}푖=0 of decreasingly ordered reals, let 풟({푠푖}푖=0) be the set of all spectrally elliptic operators 훥 whose heat spectrum 훱 satisfies 훱 ∩ 푘 −푠푖 −푠푘+1 [푠푘,∞) ⊂ {푠푖}, i.e., whose heat trace admits an asymptotic expansion ∑푖=0 푐푖푡 +푂(푡 ) + as 푡 → 0 , for some 푠푘+1 < 푠푘.

If 휎(훥) is the (necessarily point) spectrum of some spectrally elliptic operator 훥 and 퐹 is any Borel measurable function, we will denote the summation of 퐹 over the strictly positive eigenvalues smaller than 훬 by

def tr훬 퐹(훥) = ∑ 퐹(휆). 훬>휆∈휎(훥)

4The classical elliptic pseudodifferential operators may have logarithmic terms in the heat expansion, leading to double poles of 훤(푧)휁(푧) (e.g. at negative integers), and are thus excluded in general. However, see remark 2.2.6.

29 2.2. Zeta residues as normal functionals

푘 Theorem 2.2.3. For any finite set {푠푖}푖=0 of decreasingly ordered reals there exists a 푘 function 퐹 such that, for all 훥 ∈ 풟({푠푖}푖=0), res 휁(훥,푠)훤(푠) = lim 휖(훬)푠푘 tr 퐹(훥휖(훬)), (2.1) 푠=푠푘 훬→∞ 훬

def where 휖(훬) = 푚log훬/훬 for any 푚 > 푠0 − 푠푘.

The problem posed in the introduction is then trivially resolved by the following corol- lary, together with an explicit choice of 퐹.

Corollary 2.2.4. Let {훥푗} be any sequence of finite rank positive operators of norm def ‖훥푗‖ = 훬푗 such that, as 푗 → ∞, 훬푗 → ∞ and

퐹(훥휖(훬 )) − 퐹(훥 휖(훬 )) = 표((휖(훬 )−푠푘 ). tr훬푗 푗 tr 푗 푗 푗 Then, res 휁(훥,푠)훤(푠) = lim 휖(훬 )푠푘 tr퐹(훥 휖(훬 )). 푠=푠푘 푗→∞ 푗 푗 푗

푘 Remark 2.2.5. A uniform bound over 풟({푠푖}푖=0) on the rate of convergence is too much to ask: it depends on the complete spectrum of 훥. However, the convergence rate −1 will be shown to always be 푂(훬 ) if the next pole is at 푠푘 −1. For examples of arbitrary error for given 훬, see for instance [Col87; Loh96]. For comparison, see remark 2.4.1 on page 37. Remark 2.2.6. The exclusion of logarithmic terms in the heat expansion from Defini- tion 2.2.1 corresponds to an exclusion of higher order poles of 훤(푠)휁(훥,푠). However, Theorem 2.2.3 is unchanged if higher order poles are allowed, as long as they lie below 푠푘. Moreover, the approach can probably be modified to accomodate such logarithmic terms, at the expense of brevity and asymptotic rate of convergence.

Asymptotic series for zeta function residues For any spectrally elliptic operator 훥, the function 휁(훥,푠)훤(푠) is the Mellin transform of 푒−푡훥. Therefore, the asymptotic expansion of 푒−푡훥

푘 −푡훥 −푠푖 −푠푘 tr푒 ∼ ∑푐푖푡 + 표(푡 ) 푖=0 has coefficients 푐 = 휁(훥,푠)훤(푠). 푖 res푠=푠푖

30 2.2. Zeta residues as normal functionals

The proof of Theorem 2.2.3 relies on the asymptotic expansion of a Mellin convolution integral with tr푒−푡훥. This strategy was inspired by the spectral action principle of [CM08, Thm 1.145] and the approach of [Ara96]. However, here we have specifically constructed the function we convolve with such that its first moments vanish, which allows us to recover the zeta residue in the leading term.

Proof of Theorem 2.2.3. The proof proceeds in two elementary steps. First, we show (Proposition 2.2.7, below) that for suitable functions 푓 one has

∞ −휖푡훥 −푠푘 −푠푘+1 ∫ tr푒 푓(푡) = 푐푘휖 + 푂(휖 ) 0 as 휖 → 0. This part requires the asymptotic expansion of tr푒−푡훥, implying in particular that the poles of 훤(푠)휁(훥,푠) must be simple and discrete (i.e. 훥 has simple dimension spectrum disjoint with the negative integers) and lie on the real line. Then, we use the Laplace transform 퐹 to rewrite the integral as a trace

∞ ∫ tr푒−휖푡훥푓(푡)푑푡 = tr퐹(휖훥), 0 and we prove in Proposition 2.2.8 that there is a choice 휖(훬) = 훬−1 log훬푚 such that 퐹 decays as −푠푘+1 (tr−tr훬)퐹(훥휖(훬)) = 푂(휖 ), so that we obtain the useful asymptotic behaviour

−푠푘 −푠푘+1 tr훬 퐹(훥휖(훬)) = 푐푘휖(훬) + 푂(휖 ).

This second part relies on the finite summability of 훥, that is, on the fact that tr푒−푡훥 = 푂(푡−푠0 ) as 푡 → 0+.

The following asymptotic estimate is completely straightforward.

Proposition 2.2.7. Let 푓 be a piecewise continuous function supported in (1,∞) that −푚 푘 is 푂(푡 ) for all 푚 ∈ ℝ towards ∞. Let 훥 ∈ 풟({푠푖}푖=0). Then, the Mellin convolution integral of 푓 with tr푒−푡훥 has an asymptotic expansion

∞ 푘 ∞ −휖푡훥 −푠푖 −푠푖 −푠푘 ∫ tr푒 푓(푡)푑푡 = ∑푐푖휖 ∫ 푡 푓(푡)푑푡 + 표(푡 ). 0 푖=0 0

31 2.2. Zeta residues as normal functionals

∞ 푓 ∫ 푡−푠푖 푓(푡)푑푡 푖 ≠ 푘 Thus, if we simply choose so that its moments 0 vanish for and ∞ ∫ 푡−푠푘 푓(푡)푑푡 = 1 푐 normalize it so that 0 , we gain access to the coefficient 푘. By asymptotic decay of such a function 푓 towards ∞, it has absolutely convergent Laplace transform 퐹. As ∣푓(푡)tr푒−휖푡훥∣ is absolutely integrable as well, we can apply the Fubini-Tonelli theorem to obtain ∞ ∞ ∫ tr푒−휖푡훥푓(푡)푑푡 = ∑ ∫ 푒−휆휖푡푓(푡)푑푡 0 휆∈휎(훥) 0 = tr퐹(훥휖), where 휎(훥) is the spectrum (with multiplicities) of 훥. Therefore, Proposition 2.2.7 leads us to the conclusion that

−푠푘 −푠푘+1 tr퐹(휖훥) = 푐푘휖 + 푂(휖 ). (2.2)

The following proposition shows that we retain the same asymptotics if we replace the trace of 퐹(휖훥) by a sum over a finite part of the spectrum, provided we scale 휖 accordingly. Proposition 2.2.8. If 푓 and 훥 are as in Proposition 2.2.7 and additionally ∞ ∫ 푡−푠푖 푓(푡)푑푡 = 0 (for all 푖 < 푘), 0 then the Laplace transform 퐹 satisfies

∑ 퐹(휆log훬푚/훬) = 표((log훬푚/훬)−푠푘 ) 훬<휆∈휎(훥) for all 푚 > 푠0 − 푠푘

Proof. By the fact that supp푓 ⊂ [1,∞)], its Laplace transform decays as 퐹(푠) = 푂(푠−1푒−푠) −휖(휆−훬)푡 towards ∞. As ∑휆>훬 푒 is monotone decreasing in 푡, we see that

∑퐹(휖휆) = 푂(∑(휖훬)−1푒−휖휆) = 푂((휖훬)−1푒−휖훬 ∑푒−휖(휆−훬)푡) 휆>훬 휆>훬 휆>훬 for any 푡 ≤ 1, as 훬휖 → ∞. This, in turn, is 푂((휖훬)−1푒−휖훬(1−푡)휖−푠0 ). With 휖(훬) = −1 −푠푘 푚훬 log훬 for any 푚 > (푠0 − 푠푘), the remainder ∑휆>훬 퐹(휖휆) will be 표(휖 ).

This completes the proof of Theorem 2.2.3 that started on page 31.

32 2.2. Zeta residues as normal functionals

Explicit formulas for the first two poles One reason to look for series that converge to zeta residues is to obtain geometric information from finite-dimensional approximations of the Laplacian on a Riemannian manifold. For instance, if 훥 is the Laplacian, the first two residues are proportional to the volume and the total scalar curvature, respectively. The first pole is a classical object of study. Its residue is expressed by Dixmier’s singular trace, and is used, for instance, for the zeta regularization of divergent series. It is connected to counting asymptotics by the Wiener-Ikehara theorem, and for a Laplace operator on a compact Riemannian manifold it is proportional to the volume. Our Theorem 2.2.3 provides the following formula for the first residue.

Corollary 2.2.9. If 훥 is a spectrally elliptic operator with heat spectrum bounded from above by 푠0 ∈ ℝ, then if 휖(훬) = log훬/훬,

휖(훬)푠0 푒−훥휖(훬) 푠0 훤(푠0)res푠=1 휁(훥 ,푠) = lim tr훬 . 훬→∞ 푒훤(1 − 푠0,1) 1 + 훥휖(훬)

Proof. Use 푓 = [푡 ≥ 1]푒−푡, with Laplace transform 퐹(푠) = 푒−1−푠/(1 + 푠), in Proposition ∞ ∫ 푡−푠0 푓(푡)푑푡 2.2.7. Then, divide by 0 to satisfy the conditions of Proposition 2.2.8.

Remark 2.2.10. The present series converges faster in general than the logarithmic trace suggested by Weyl’s asymptotic formula; the remainder is 푂(휖푠0−푠1 ) = 푂((훬/log훬)푠1−푠0 ), whereas for e.g. the Laplacian on the circle the logarithmic trace of 훥−1/2 convergences −1 푁 1 훾+푂(푁 ) only as ∑푛=1 푛log푁 − 1 = log푁 .

Now, the second pole is of particular interest because it is the first pole for which no asymptotic residue formula was previously known and because it provides a way to calculate the total scalar curvature of a Riemannian manifold from partial spectra of the Laplacian.

Corollary 2.2.11. If 훥 is a spectrally elliptic operator with heat spectrum {푠0,푠0 −1,…}, then if 휖(훬) = 2log훬/훬,

휖(훬)푠0−1 2푠0 푒−2훥휖(훬) 푒−훥휖(훬) 휁(훥,푠)훤(푠) = ( − ). res푠=푠1 lim tr훬 훬→∞ 푒훤(2 − 푠0,1) 1 + 2훥휖(훬) 1 + 훥휖(훬)

33 2.2. Zeta residues as normal functionals

Proof. Use 푓 = [푡 ≥ 1]푒−푡 − 2푠0−1[푡 ≥ 2]푒−푡/2, which clearly satisfies the conditions of ∞ ∫ 푡−푠0 푓(푡)푑푡 Proposition 2.2.7 and has vanishing moment 0 . Therefore, if we divide ∞ ∫ 푡1−푠0 푓(푡)푑푡 by the factor 0 , the result will be as in Proposition 2.2.8. The Laplace transform of 푓 is 퐹(푠) = 푒−1−푠/(1 + 푠) − 2푠0 푒−1−2푠/(1 + 2푠).

Example 2.2.12 (The total scalar curvature of the sphere 푆2). Let 퐹 be the residue functional given by exp(−푠) 2exp(−2푠) 퐹(푠) ∶= ( − ), 1 + 푠 1 + 2푠 let 푞푛 = 푛(푛+1) be the 푛th eigenvalue of the Laplacian on the sphere and let 푝푛 = 2푛+1 −1 be its multiplicity. As before, write 휖푚 = 2푞푚 log푞푚. We will use the Euler-Maclaurin formula to estimate the series from Corollary 2.2.11,

푚 푆(푚) ∶= ∑푝푛퐹(휖푚푞푛). 푛=0 Sufficient accuracy will turn out to require only the first nontrapezoidal correction term. The Euler-Maclaurin estimate reads 푚 푚 1 퐵2 휕 푆(푚) = ∫ 푝푛퐹(휖푚푞푛)푑푛 + ( + )푝푛퐹(휖푚푞푛)∣ + 푅3(푚), 0 2 2 휕푛 0

1 where 퐵2 = 6 , and we will employ the usual rough bound

3 휁(3) 휕 (푝푛퐹(휖푚푞푛)) |푅3(푚)| ≤ 3 푚 sup ∣ 3 ∣. (2휋) 0≤푛≤푚 휕푛

The integral in the first term can be taken analytically,

푚 푞 푚 −1 ∫ 푝푛퐹(휖푚푞푛)푑푛 = ∫ 퐹(푞,휖푚)푑푞 = 푒휖푚 (퐸1(1 + 2휖푚푞푚) − 퐸1(1 + 휖푚푞푚)), 0 0

−2 and is 푂(푞푚 log(1 + 푞푚/(2log푞푚)). −1 −푠 1 1 Because 퐹(푠) = 푂(푠 푒 ) as 푠 → ∞, the order zero boundary term 2 + 2 푝푀(퐹(휖푚푞푚)) 1 −2 is 2 + 푂(푝푚푞푚 /log푞푚). The first correction term to the trapezoidal approximation equals 1 1 ( − 2 + 2퐹(휖 푞 ) + 휖 푝2 퐹(1)(휖 푞 )) = − + 푂(푞−2 log푞 ). 12 푚 푚 푚 푚 푚 푚 6 푚 푚

34 2.3. Localization

Finally, the third derivative of 푝푛퐹(휖푚푞푛) with respect to 푛, whose supremum appears in the error estimate, is

(1) 2 2 (2) 3 4 (3) 12휖푚퐹 (휖푚푞푛) + 12휖푚푝푛퐹 (휖푚푞푛) + 휖푚푝푛퐹 (휖푚푞푛).

−1 All derivatives of 퐹 are bounded on [0,∞), so these terms are all 푂(푞푚 log푞푚) uniformly −1 in 푛. Thus, 푅3(푚) = 푂(푚 log푞푚). Combining these facts with Corollary 2.2.11, we see that the nonleading residue for the zeta function 휁푆2 is

−푠 1 1 −1 1 2 ⋅ 4휋 res 휁 2 = lim ( − + 푂(푚 log푞 )) = = . 푠=0 푆 푚→∞ 2 6 푚 3 6 ⋅ (4휋)푑/2

Of course, one could apply Euler-Maclaurin directly to the sum that defines the heat trace on 푆2; since the corresponding limits 푚 → ∞ can termwise be identified as power series in 푡, one can reproduce the asymptotics as 푡 → 0 afterwards. In the present formulation, however, we need only take a single limit in 푚 and have expressed the residue as a single universal series in the eigenvalues, with an asymptotic bound on the error over all of 풟(푠0,푠0 − 1).

2.3 Localization

If 훥 is spectrally elliptic, Theorem 2.2.3 allows us to calculate its zeta residues as a series in its eigenvalues. However, the classical theory of elliptic pseudodifferential operators assigns to them not just the total zeta residues, but rather a set of zeta densities. To be precise, the somewhat simpler situation for elliptic differential operators is as follows. Let 푀 be a Riemannian manifold of dimension 푛 equipped with a smooth Hermitian vector bundle 퐸, and let 훥 be a positive, self-adjoint, elliptic differential operator of order 푚 > 0, acting inside the Hilbert space of square-integrable sections of 퐸. Then, the following localized asymptotics are available.

−푡훥 + 1. The operator 푒 is given by a smooth kernel 푘(푥,푦,푡) ∶ 퐸푦 → 퐸푥, and as 푡 → 0 its restriction to the diagonal admits an asymptotic expansion in smooth sections 푘푗 of End퐸, ∞ (푗−푛)/푚 푘(푥,푥,푡) ∼ ∑푘푗(푥)푡 . 푗=0

35 2.3. Localization

2. For any continuous section ℎ of End퐸 and any 푗 ≥ 0, the following residues exist and satisfy

−푠 res푠=(푗−푛)/푚 훤(푠)trℎ훥 = ∫ tr(ℎ(푥)푘푗(푥))푑vol푀(푥). 푀

We say that ℎ localizes the residue trace, and we are interested in expressing the localized residues in a fashion similar to Theorem 2.2.3. We will solve this localization problem in a slightly more general setting, along the lines of the previous treatment of the zeta function residues. Let 훥 be a spectrally elliptic operator on a Hilbert space ℋ and let ℎ ∈ ℬ(ℋ) be + bounded. Then, there exist 푠0 > ⋯ > 푠푘+1 such that, as 푡 → 0 , 푘 −푡훥 −푠푘 −푠푘+1 trℎ푒 − ∑푡 푐푖(ℎ) = 푂(푡 ). 푖=0

푘 As before, let {푠푖}푖=1 be a set of decreasingly ordered reals and let 퐹 be the Laplace transform of a piecewise continuous function 푓 supported in [1,∞), which is 푂(푡−푚) ∞ 푚 ∈ ℝ ∞ ∫ 푡−푠푖 푓(푡)푑푡 = 0 푖 < 푘 for all towards , satisfies 0 for all , and is normalized to ∞ ∫ 푡−푠푘 푓(푡)푑푡 = 1 satisfy 0 . 푘 Theorem 2.3.1. Let 훥 ∈ 풟({푠푖}푖=1) be a spectrally elliptic operator on a Hilbert space ℋ and let ℎ ∈ ℬ(ℋ) be bounded. Then, for any orthonormal basis {휙휆}휆∈휎(훥) of ℋ diagonalizing 훥, we have

res 훤(푠)trℎ훥−푠 = lim 휖(훬)푠푘 ∑ ⟨ℎ휙 ,휙 ⟩퐹(휆휖(훬)), 푠=푠푘 훬→∞ 휆 휆 훬>휆∈휎(훥)

def −1 푚 where 휖(훬) = 훬 log훬 for any 푚 > 푠푘 − 푠0.

Proof. As before, Proposition 2.2.7 holds for any such function 푓. That is, ∞ −휖푡훥 −푠푘 −푠푘+1 ∫ trℎ푒 푓(푡)푑푡 = 푐푘(ℎ)휖 + ‖ℎ‖op푂(휖 ). 0 −푡휆 −푡훥 1 Moreover, ∑휆∈휎(훥) |푓(푡)푒 ⟨ℎ휙휆,휙휆⟩| ≤ |푓(푡)|‖ℎ‖op tr푒 is in 퐿 (0,∞). Therefore, by dominated convergence, ∞ −휖푡훥 ∫ trℎ푒 푓(푡)푑푡 = ∑ ⟨ℎ휙휆,휙휆⟩퐹(휖휆) 0 휆∈휎(훥)

36 2.4. Final remarks and suggestions

and by Proposition 2.2.8, the rest term decays like

−푠푘+1 ∑ 퐹(휆휖(훬))⟨ℎ휙휆,휙휆⟩ = ‖ℎ‖op푂((휖(훬)) ). 훬<휆∈휎(훥)

훤(푠) ℎ훥−푠 = 푐 (ℎ) We have res푠=푠푘 tr 푘 and the conclusion follows.

2.4 Final remarks and suggestions

The original motivation behind this chapter was to confirm the point of view that finite- rank cutoffs of spectral triples can carry geometric information in noncommutative geometry. The following remarks all proceed in that direction.

Remark 2.4.1. Our Theorem 2.2.3 provides a partial counterweight to a classical result by Colin de Verdière [Col87]. On the one hand, he showed that a finite set of Laplace eigenvalues carries no information on the metric if no pointwise bounds on the sectional curvature are imposed. On the other hand, we now see that for each metric and each desired accuracy, there is a bound 훬 such that the set of eigenvalues smaller than 훬, with multiplicity, allow computation of all zeta residues (and hence, of the associated global invariants of the metric) up to that accuracy. The local version, using Theorem 2.3.1, of this statement is that for each metric and each 휖 there is a finite-rank projection 푃훬 = [0,훬](훥푔) such that the cutoff matrix 푃훬훥푔푃훬 together with the cutoff 푃훬퐶(푀)푃훬 of the function algebra yield the residues of tr푎훥−푠 up to an error of ‖푎‖휖.

Remark 2.4.2. The method of Theorem 2.3.1 may be applied to obtain a reasonable definition of scalar curvature in finite-dimensional noncommutative geometry. Let (퐴,퐻,퐷) be a spectral triple such that 퐷2 is spectrally elliptic with heat spectrum {(푗−푛)/2} 푗∈ℤ>0 . In [CM08] the scalar curvature functional of such a spectral triple was defined to be the map

def −푠 푎 ↦ 푅(푎) = res푠=푛−2 훤(푠)tr푎퐷 .

If 퐻 is finite-dimensional but a dimension spectrum of 퐷 is specified in advance (e.g. by modelling considerations), this residue always vanishes and Theorem 2.3.1 suggests it should perhaps be replaced by

퐹(휆log훬푚/훬) 푎 ↦ 푅 (푎) def= ∑ ⟨푎휙 ,휙 ⟩ , 훬 휆 휆 푚 푛/2−1 휆∈휎(퐷2) (log훬 /훬)

37 2.4. Final remarks and suggestions

def 2 휌 2 where 훬 = ‖퐷 ‖ for any 0 < 휌 < 1, {휙휆 ∣ 퐷 휙휆 = 휆휙휆} is an orthonormal basis of 퐻 and 퐹 is as in Corollary 2.2.11. Mutatis mutandis, the same applies to the volume and other spectral invariants.

Remark 2.4.3. The calculation of the residues of localized zeta functions, Theorem 2.3.1, can be combined with the local index formula of Connes and Moscovici [CM95] for the Chern character of the Fredholm module associated to a spectral triple (퐴,퐻,퐷), in order to estimate some KK-theoretic index pairings numerically. If the square of the operator 퐷 is spectrally elliptic, the index pairings can be expressed as a series in the 2 (푘1) (푘푛) spectrum of 퐷 and the coefficients tr휋휆푎0[퐷,푎1] ⋯[퐷,푎푛] 휋휆, where 휋휆 projects onto the eigenspace associated with the eigenvalue 휆, the commutators are defined (푘+1) def 2 (푘) recursively as [퐷,푎] = [퐷 ,[퐷,푎] ] and 푎0,…,푎푛 ∈ 퐴. A similar statement holds in the presence of a grading.

38 Chapter 3

Reconstructing manifolds from truncated spectral triples1

We explore the geometric implications of introducing a spectral cut-off on compact Rie- mannian manifolds. This is naturally phrased in the framework of non-commutative geometry, where we work with spectral triples that are truncated by spectral projec- tions of Dirac-type operators. We associate a metric space of ‘localized’ states to each truncation. The Gromov-Hausdorff limit of these spaces is then shown to equal theun- derlying manifold one started with. This leads us to propose a computational algorithm that allows us to approximate these metric spaces from the finite-dimensional truncated spectral data. We subsequently develop a technique for embedding the resulting metric graphs in Euclidean space to asymptotically recover an isometric embedding of the limit. We test these algorithms on the truncated sphere and a recently investigated perturbation thereof.

3.1 Introduction

A natural notion of scale is a major asset to any geometric theory with ties to the physical world. After all, our geometric knowledge of objects appearing around us is finite, being limited by our observational power which fails us at high energy scales. Additionally, the appearance of divergences in e.g. quantum field theory, especially when combined with gravity, is closely tied to bridging the gap between finite and infinite, or discrete and

1This work was written jointly with Lisa Glaser and published as [GS21]

39 3.1. Introduction

continuum, models. Moreover, computational representation and analysis of geometric models strongly relies on our ability to extract from our model what is relevant and computationally feasible. The field of noncommutative geometry has had close ties to physics ever sinceits inception, and yet lacks a consistent treatment of scale in the sense imagined. The aim of this chapter is to ameliorate this situation, by constructing a natural metric counterpart to the finite objects that are here referred toas truncations of (commutative) spectral triples, and aiming to show that these do indeed carry enough information to describe their continuum limit in arbitrary detail. For a wide-ranging and systematic approach to such truncations, see [CS20]. Admittedly, finite-dimensional objects in noncommutative geometry have enjoyed enduring attention. General finite spectral triples have been classified [GP95; Ćać11; CC08] and parametrized [Bar15], and the Connes metric on these spaces has been studied in depth, cf. [IKM01]. However, this framework seems to lack simultaneous presence of 1) a natural link to the continuum in terms of metric spaces and 2) a natural link to the continuum in terms of spectral triples. When representation-theoretic knowledge of the continuum is available, the framework of fuzzy spaces such as those of [GP95; Bal+02; DO03] seems to provide at least a natural link to the continuum in terms of spectral triples, and even some metric knowledge is available there [SS16]. However, it might be said that the construction of a ‘fuzzy’ version of a manifold is somewhat ad hoc from a Riemannian viewpoint, and at least the framework has not yet seen successful extension to a reasonably large class of manifolds. Truncations of spectral triples (see Section 3.2 and beyond) provide the advantage of a natural scale parameter and a natural symmetry-preserving correspondence between different scales. The natural framework in which to study these truncations themselves as metric spaces is that of state spaces, as in [CS20], which can then be equipped either with the Connes metric associated to the truncation of a spectral triple itself or with the pullback of the Connes metric on the full spectral triple. An early and interesting study of the topological and metric properties of such spaces can be found in [DLM14b]. More recently, [Sui20] investigates the question of Gromov- Hausdorff convergence of such truncated spaces under the truncated metric; see [Ber19] for the example of the torus. By contrast, here we are interested in localized states, in order to recover e.g. a manifold 푀, rather than its metric space of probability measures. Chapter 4 investigates the relevance of the higher Heisenberg equation of [CCM15] in the framework of truncated spectral triples (see Section 3.6).

40 3.2. Truncated spectral geometries and point reconstruction

Arguably the main mathematical result of this chapter is Corollary 3.3.10, which shows that the ‘localized’ states of Section 3.3, when equipped with the pullback metric, recover the compact Riemannian manifold one started with in the Gromov-Hausdorff limit. In order to make the result more concrete and link back to the ‘computational repre- sentation’ alluded to above, Section 3.4 is devoted to the description of an algorithm to approximate (finite subsets of) these metric spaces from the raw datum of thetrunca- tion of a spectral triple. This allows us to test some of the results of Section 3.3 on the example of the sphere in Section 3.5. Finite non-Euclidean metric spaces, as obtained by our algorithm, do not necessar- ily lend themselves to easy visualization or comparison by standard computational techniques. In order to gain traction in this direction, we propose to look for new (asymptotically, locally isometric) embedding techniques and present a candidate ap- proach in Section 3.6. This allows us to visualize the metric results of Section 3.5 and – as originally inspired the technique – compare the truncation of the spectral triple for 푆2 and its perturbation as in [GS20].

3.2 Truncated spectral geometries and point reconstruction

In noncommutative geometry one describes a compact Riemannian manifold 푀 in ∞ 2 terms of an associated spectral triple (퐶 (푀),퐿 (푀,풮),퐷풮), where 풮 is a Hermitian Clifford module bundle over 푀 and 퐷풮 is a Dirac-type operator on 풮. Connes’ reconstruction theorem [Con13] shows that this association is a bijection: one can fully reconstruct the underlying manifold 푀 and the chosen Hermitian Clifford module bundle 풮 from a triple (퐴,퐻,퐷) that satisfies the axioms ofa commutative spectral triple. Of particular interest for the present chapter is the way one can recover the Kantorovich- Rubinstein2 distance between probability measures on 푀 from the interplay of the ∞ 2 algebra 퐶 (푀) and the Dirac operator 퐷 = 퐷풮, acting on the Hilbert space 퐿 (푀,풮). Probability measures correspond to states on the C∗-algebra 퐶(푀) ⊃ 퐶∞(푀), which carries the topological, as opposed to differentiable, information about 푀. Moreover, a function 푓 ∈ 퐶∞(푀) has Lipschitz constant 푘 if and only if ∥[퐷,푓]∥ ≤ 푘 (as operators

2This metric is known under various names including ‘1-Wasserstein’, ‘Monge-Kantorovich’ and ‘Earth Mover’s’. See e.g.[Vil09, Chapter 6].

41 3.2. Truncated spectral geometries and point reconstruction

2 on 퐻 = 퐿 (푀,풮푀)). Thus, Kantorovich-Rubinstein duality allows us to write

푑(휔1,휔2) = sup {|휔1(푎) − 휔2(푎)| ∣ ‖[퐷,푎]‖ ≤ 1} . (3.1) 푎∈퐶∞(푀)

When the states 휔 of 퐶(푀) are pure, they correspond to the atomic measures on single points, and we can thus recover 푀 with its metric. This chapter answers the question as to how we can understand this recovery of 푀 from the perspective of finite- dimensional parts of the representation of 풜 and 퐷 on 퐻. That is, we will construct natural counterparts to the ingredients above in the setting of truncations of spectral triples, and show that the metric space 푀 can be recovered as an asymptotic limit thereof.

Truncated spectral geometries From a mathematical perspective, it is desirable to be able to describe the infinite- dimensional datum of a spectral triple as a limit of finite-dimensional data of increasing precision, just like one can describe a compact Riemannian manifold as a Gromov- Hausdorff limit of finite metric spaces (by, for instance, equipping suitably densefinite subsets with the induced metric). From a physical perspective, the same desire results from the view that one should be able to gain at least some information about the geometry by probing it at finite energies. A natural way to introduce such a ‘finite-energy cutoff’, that is, truncation, ofthe geometric data (퐴,퐻,퐷) is to pick a scale 훬, and define the corresponding spectral projection of 퐷, def 푃훬 = 휒[−훬,훬](퐷) to generate the finite-dimensional data

def (퐴훬,퐻훬,퐷훬) = (푃훬퐴푃훬,푃훬퐻,푃훬퐷) . (3.2)

By compactness of the resolvent of 퐷, one might as well write 푃푁 for the projection onto the first 푁 eigenspaces ordered in absolute value and work with 푃푁 instead. However, this would require keeping track of the relation between 푁 and 훬 in the analysis to follow and thus complicate notation. A further advantage of using 푃훬 is it is physically more intuitive to work with the energy scale rather than with the number of modes considered.

42 3.2. Truncated spectral geometries and point reconstruction

The study of truncations of spectral triples and their convergence was initiated in [DLM14b], while the structure of the truncations as operator systems was explored in depth by Connes and van Suijlekom in [CS20] and [Sui20]. Optimal transport in the framework of noncommutative geometry has seen widespread inquiry. See e.g. [BLS94; DM99] for the case of lattices, [IKM01] for finite spectral triples, [Mar06] for the context of gauge theory, [Wal12; ŻS01; DAn16; DLM14a] for several quantum mechanically inspired examples and deformations, [Sui20; DLM14b] for truncations of the circle, and notably [Rie99; Rie04] for generalization of the inducing Lipschitz seminorm. This is the setting in which we wonder what counterpart to the duality (퐴,퐻,퐷) ↔ 푀, provided by the reconstruction theorem, can be found at the level of (퐴훬,퐻훬,퐷훬).

Point reconstruction We aim to refine the reconstruction of 푀 from its spectral triple (퐴,퐻,퐷) through an understanding of the metric information contained in the truncations (퐴훬,퐻훬,퐷훬). The full manifold should then emerge asymptotically, such as through a Gromov- Hausdorff limit of the objects corresponding to the truncations.

The vector states of 퐶(푀) that are induced by elements of 퐻훬 appear naturally as vector states of 퐴훬 as well, so that we have access to probability measures on 푀 directly in the truncated setting. This, together with the distance formula (3.1), is the main ingredient of our approach: we will identify states that correspond to points of 푀 in a suitable (asymptotic) sense, such that (3.1) asymptotically recovers the corresponding geodesic distance. Our ‘proxy’ approach to state localization, as discussed in section 3.3, was inspired by the notion of quasi-coherent states on fuzzy spaces defined in [SS16]. However, ∞ as we aim just for the induced metric geometry on 푀 and view (퐶 (푀)훬,퐻훬,퐷훬) rather as a finite observation of a spectral triple than as a quantization thereof, we will not construct coherent states in any quantum-mechanical sense but rather aim for localization only. Moreover, as discussed below, this ‘proxy’ approach is merely introduced to gain computational feasibility, at the expense of requiring identification of an embedding 휙 ∶ 푀 → ℝ푛. See also the end of Section 3.3. After we define localized states, we prove existence of the desired objects: a sequence of metric spaces associated to the truncations (퐴훬,퐻훬,퐷훬) that do indeed converge to 푀 in the Gromov-Hausdorff sense. Then, Section 3.4 proposes an algorithm to construct

43 3.3. The metric space of localized states

these metric spaces computationally, in order to make actual examples amenable to computer simulation. In Section 3.6 we propose a simple algorithm to obtain approximately locally isometric embedding of the resulting finite metric spaces into Euclidean space. These should asymptotically converge to an isometric embedding of 푀 itself, and allow us to view the resulting finite metric spaces and investigate them more easily.

3.3 The metric space of localized states

∞ Given a truncation of a commutative spectral triple (퐶 (푀)훬,퐻훬,퐷훬), we aim to construct a finite metric space that describes 푀 to the level of accuracy that the truncated spectrum will allow. ∞ We will identify a subset of the (vector) states of 퐶 (푀)훬, consisting of those states that are localized in a suitable sense and, therefore, correspond approximately to points of 푀. The Connes metric on these vector states will then turn this subset into a metric space. The guiding demand for this construction will be that the resulting metric spaces should asymptotically (as 훬 → ∞) converge, in the Gromov-Hausdorff sense, to the metric space 푀. Now, the pure states of 퐶(푀) – that is, actual points in the metric space 푀 we are approximating – do not necessarily extend to 퐶(푀)훬. We do, however, have access to vector states induced by 푣 ∈ 퐻훬, which can be applied to either because both 퐶(푀) and 퐶(푀)훬 are subsets of 퐵(퐻).

Definition 3.3.1. ℙ(퐻훬) is the projective space over 퐻훬.

Each element v of ℙ(퐻훬) corresponds to a positive linear functional of norm 1 on 퐶(푀) given by 푎 ↦ ⟨푣,푎푣⟩/⟨푣,푣⟩ (for any representative 푣 of v). By the Riesz Representation theorem, such functionals correspond uniquely to probability measures on 푀.

Definition 3.3.2. For v ∈ ℙ(퐻훬), 휇푣 is the unique probability measure such that ⟨푣,푎푣⟩ = ∫ 푎(푥)푑휇 (푥) 푎 ∈ 퐶(푀) 푣 푀 푣 for all and any unit norm representative of v.

By this identification, the Kantorovich-Rubinstein metric on the probability measures of 푀 induces a metric 푑훬 on ℙ(퐻훬). It is an open conjecture that this metric can be

44 3.3. The metric space of localized states

∞ (asymptotically) computed using the data (퐶 (푀)훬,퐻훬,퐷훬): see Section 3.3 for a discussion.

We will say that v is localized when 휇푣 is sufficiently concentrated near a single point in 푀. In order to quantify this notion, we will introduce the dispersion functional 휂 on ℙ(퐻훬), below.

Let 푑훬 and 푑푀 denote the metrics on ℙ(퐻훬) and 푀, respectively, so that we may regard both as subsets of the space of probability measures on 푀 equipped with the Kantorovich-Rubinstein metric. We now wish to construct a subspace of (ℙ(퐻훬),푑훬) that is (Gromov-)Hausdoff close to (푀,푑푀).

Proposition 3.3.6 will show that there is a map 푏∶ ℙ(퐻훬) → 푀 such that |푑훬(v1,v2) − 푑푀(푏(v1),푏(v2))| = 푂(√휂(v1)+√휂(v2)); that is, our localized states can be identified al- most isometrically with points of 푀. Proposition 3.3.8 will then show that there is a cor- responding asymptotically inverse map 훷훬 ∶ 푀 → ℙ(퐻훬) such that 푑푀(푥,푏(훷훬(푥))) = ̃ −1 ̃ −2 푂(훬 ) and 휂(훷훬(푥)) = 푂(훬 ) uniformly in 푥. This leads to Corollary 3.3.10, which shows that there exist a subspace ℙ(퐻훬)휖2 of ℙ(퐻훬) that is 휖-close to 푀 in Gromov- Hausdorff distance, where 휖 = 푂(훬̃ −1). Finally, Section 3.3 discusses how these notions connect to the setting of truncated ∞ spectral triples (퐶 (푀)훬,퐻훬,퐷훬).

Localization: 휙 and the dispersion functional

Since elements of ℙ(퐻훬) correspond uniquely to probability measures on 푀, a natural way to measure the localization of such an element would be to take e.g. the variance of (isometrically embedded) position under this measure; that is, one would naturally ∈ ℙ(퐻 ) 퐸 [푑(푥,⋅)2] 퐸 define the dispersion of v 훬 to be inf푥∈푀 휇푣 , where 휇푣 denotes expec- tation values under 휇푣. In terms of the algebraic data, this quantity can be estimated ∗ 2 as sup ∞ {⟨푣,푎 푎푣⟩ − |⟨푣,푎푣⟩| ∣ ‖[퐷,푎]‖ ≤ 1}. However, the relevant non-convex 푎∈퐶훬(푀) double optimization problem – to find minima 푣 of this dispersion in high-dimensional 퐻 – is computationally extremely challenging except in the simplest cases. Therefore, we will require a proxy, 휙∶ 푀 → ℝ푛, for the extremizing element 푎 above, in the sense that the Euclidean distance 푑ℝ푛 (휙(푥),휙(푦)) on 푀 is bi-Lipschitz equivalent 푛 to the distance 푑푀(푥,푦) appearing in the variance. Thus, let 휙 ∶ 푀 → ℝ be a (not necessarily Riemannian) embedding. In other words, we take 푀 to be a compact embedded submanifold of ℝ푛, with a Riemannian metric not necessarily the one induced by the embedding. However, we

45 3.3. The metric space of localized states

will keep the notation 휙 to emphasize the arbitrary nature of the embedding, especially in connection with Section 3.4.

Definition 3.3.3. Let 휇 be a probability measure on 푀. Then, the dispersion 휂(휇) equals def 2 휂(휇) = ∫ 푑ℝ푛 (휙(푥),퐸휇 [휙]) 푑휇(푥) 푀

In probabilistic terms, 휂(휇) is just the trace of the covariance matrix of the vector-valued random variable 휙, under the probability measure 휇.

The 휙-barycenter of a localized state

An element v of ℙ(퐻훬) that is considered to be localized should be localized somewhere, that is, around some ‘barycenter’ 푥푣 ∈ 푀. In order to control the localization of 휇푣 around the point 푥푣 by the dispersion 휂(휇푣), it is important that 휙(푥푣) be close to 퐸 [휙] ℝ푛 휇푣 in . Hence, Definition 3.3.4. Let 휇 be a probability measure on 푀. Then a 휙-barycenter of 휇 is any point 푥 ∈ 푀 that minimizes 푑ℝ푛 (휙(푥),퐸휇 [휙]).

By compactness of 푀 and continuity of 푑ℝ푛 (휙(⋅),퐸휇 [휙]), there always exists a 휙- barycenter. Localized states are indeed concentrated near their 휙-barycenters, as the following lemma shows. That is, the dispersion 휂(휇) is a good proxy for the squared second Wasserstein 3 2 distance 푊2(휇,훿푥) between the measure 휇 and any given 휙-barycenter 푥 thereof.

Lemma 3.3.5. Any 휙-barycenter 푥 of a probability measure 휇 satisfies

2 def 2 푊2(휇,훿푥) = ∫ 푑푀(푧,푥) 푑휇(푧) = 푂(휂(휇)), 푀

4 uniformly in 휇, where 훿푥 denotes the Dirac measure centered on 푥. Moreover, any two 휙-barycenters of 휇 are within distance 푂(√휂(휇)), uniformly in 휇, of each other.

3See e.g. [Vil03] for an introduction to the measure-theoretic notions that are applied (without any hint of sophistication) in this section. 4That is, the relevant constant depends only on 휙 and 푀, not on 휇.

46 3.3. The metric space of localized states

Proof. By Chebyshev’s inequality, 휇({푥 ∈ 푀 ∣ ∥휙(푥) − 퐸휇 [휙]∥ ≥ 푡}) must be bounded −2 2 −2 2 by 푡 퐸휇 [∥휙 − 퐸휇 [휙]∥ ] = 푡 휂(휇). Therefore, if 푑ℝ푛 (휙(⋅),퐸휇 [휙]) ≥ 푡 on the support −2 of 휇, we see that 1 ≤ 푡 휂(휇). Therefore, we conclude that inf푥∈supp휇 푑ℝ푛 (휙(푥),퐸휇 [휙])

≤ √휂(휇). Any 휙-barycenter of 휇 must therefore, as a minimizer of 푑ℝ푛 (휙(⋅),퐸휇 [휙]) in 푀 ⊃ supp휇, also satify this inequality. Now, as a smooth embedding, 휙 is automatically bi-lipschitz. In particular, there exists 훽 such that 푑푀(푥,푦) ≤ 훽푑ℝ푛 (휙(푥),휙(푦)) uniformly in 푥,푦. We see, therefore, that any two 휙-barycenters of 휇 are at a distance at most 2훽√휂(휇).

2 2 2 ∫ 푑 (푧,푥) 푑휇(푧) ≤ 훽 ∫ 푑 푛 (휙(푧),휙(푥)) 푑휇(푧) Moreover, we conclude that 푀 푀 푀 ℝ for 푥 ∈ 푀 |∫ 푓2 − 푔2푑휇| ≤ ∫ (2|푔| + |푓 − 푔|)|푓 − 푔|푑휇 휙 푥 all . As 푀 푀 and all -barycenters of 휇 satisfy ∣푑ℝ푛 (휙(푧),휙(푥)) − 푑ℝ푛 (휙(푧),퐸휇 [휙])∣ ≤ √휂(휇), we can estimate the multi- 2 variate variance ∫푑ℝ푛 (휙(푧),휙(푥)) 푑휇(푧) by 휂(휇), up to an error that is bounded by

∫ (2푑 푛 (휙(푧),퐸 [휙]) + √휂(휇))√휂(휇)푑휇(푧) 푀 ℝ 휇 which is, by the classical Jensen inequal- ity and the definition of 휂(휇), bounded by 3휂(휇). The Lemma follows.

We now consider the implications of the above for the barycenters of probability mea- sures 휇푣, for v ∈ ℙ(퐻훬).

Proposition 3.3.6. There exists a map 푏∶ ℙ(퐻훬) → 푀 such that

|푑훬(v,w) − 푑푀(푏(v),푏(w))| = 푂(√휂(휇푣) + √휂(휇푤)) as 휂(휇푣),휂(휇푤) → 0, uniformly in v, w.

Proof. Let 푏 assign a choice of 휙-barycenter to each 휇푣, v ∈ ℙ(퐻훬).

Now let 훿푣,훿푤 be the Dirac measures centered on 푏(v), 푏(w). Recall that the distance 푑훬(v,w) is the Kantorovich-Rubinstein distance 푊1(휇푣,휇푤) between 휇푣 and 휇푤, and similarly 푑푀(푏(v),푏(w)) = 푊1(훿푣,훿푤).

Then, by the triangle inequality for the metric 푊1, we have ∣푊1(휇푣,휇푤) − 푊1(훿푣,훿푤)∣ ≤ 푊1(휇푣,훿푣) + 푊1(휇푤,훿푤).

47 3.3. The metric space of localized states

∫ |푓|푑휇 ≤ √∫ |푓|2푑휇 푊(⋅,⋅) ≤ Now, by the classical Jensen inequality 푀 푀 we have 1 푊2(⋅,⋅) and we conclude that

|푑훬(v,w) − 푑푀(푏(v),푏(w))| = ∣푊1(휇푣,휇푤) − 푊1(훿푣,훿푤)∣

≤ 푊1(휇푣,훿푣) + 푊1(휇푤,훿푤)

≤ 푊2(휇푣,훿푣) + 푊2(휇푤,훿푤)

= 푂(√휂(휇푣) + √휂(휇푤)), where the last line is Lemma 3.3.5.

Existence of localized states near any point Proposition 3.3.6 tells us that probability measures 휇 on 푀 of sufficiently small disper- sion correspond well to their 휙-barycenters. This holds in particular for the probability measures 휇푣 associated to v ∈ ℙ(퐻훬). We would now like to estimate the converse, i.e. to show that each point 푥 corresponds to an element v ∈ ℙ(퐻훬) whose probability measure is of small dispersion and such that 푏(v) is close to 푥. When we thus construct an asymptotically isometric embedding 푀 ↪ ℙ(퐻훬) that is asymptotically inverted by 푏, we would rightly be able to say there is a picture of 푀 inside ℙ(퐻훬). To simplify the asymptotic estimates we will introduce the notation 푂()̃ common in computer science:

Definition 3.3.7. Let 푋 be a set and consider functions 푓∶ 푋 × ℝ+ → ℂ, 푔∶ 푋 × ℝ+ → ℝ+. We say that 푓 = 푂(푔) uniformly when there exist finite 퐶,푟0 > 0 such that ̃ |푓(푥,푟)| ≤ 퐶푔(푥,푟) for all 푟 > 푟0 and all 푥 ∈ 푋. We say that 푓 = 푂(푔) uniformly when 푓 = 푂(푔|log푔|푠) uniformly for some 푠 ≥ 0. Proposition 3.3.8. Let 푀 be equipped with a Dirac-type operator 퐷 on a Hermitian vector bundle 휋∶ 풮 → 푀, and let ̃휋∶ ℙ(풮) → 푀 be its projectivized bundle. Then, there exists a family {훷훬}훬 of maps 훷훬 ∶ ℙ(풮) → ℙ(퐻훬) such that for all 휖 > 0, ̃ −1 • 푑훬(훷훬(푣),훷훬(푤)) = 푑푀(̃휋(푣),̃휋(푤))+ 푂(훬 ) uniformly. ̃ −2 • The dispersion 휂(휇) of the measure 휇 associated to 훷훬(푣) is 푂(훬 ) uniformly.

• The maps 훷훬 asymptotically invert 푏, in the sense that uniformly ̃ −1 ̃ −2 푑푀(̃휋(푣),푏(훷훬(푣))) = 푂(훬 ), 푑훬(훷훬(푣)),v) = 푂(√휂(휇푣) + 훬 ) whenever 푏(v) = ̃휋(푣).

48 3.3. The metric space of localized states

The proof depends on a balanced rescaling of the truncated heat flow and will be presented at the end of this section. To discuss the geometric consequences of Proposition 3.3.8, we will introduce notation for the small-dispersion subset into which 훷훬 maps.

Definition 3.3.9. Let 휖 > 0. Then, ℙ(퐻훬)휖 ⊂ ℙ(퐻훬) consists of those v ∈ ℙ(퐻훬) for which 휂(휇푣) < 휖.

As ℙ(퐻훬) is the projectivization of a finite-dimensional complex vector space – in par- ticular a compact, smooth (real) manifold – and 휂 is a smooth real function, presenting as a quartic polynomial on the standard real charts, each sublevel set ℙ(퐻훬)휖 is itself the interior of a smooth compact manifold with boundary. Corollary 3.3.10. As 훬 → ∞, there exists 휖 = 푂(훬̃ −1) such that the Gromov-Hausdorff distance between 푀 and the space ℙ(퐻훬)휖2 , equipped with the metric 푑훬, is 푂(휖).

Proof. Let 휖2 = sup 휂(휇 ). By the second part of Proposition 3.3.8, 휖2 = 푂(훬̃ −2). 푥∈푀 휙훬(푥)

Now, the map v → 휇푣 sends ℙ(퐻훬)휖2 isometrically into the space of probability measures on 푀 with the Kantorovich-Rubinstein metric 푊1, and the map 푥 ↦ 훿푥 sends 푀 isometrically into the same space.

For v ∈ ℙ(퐻훬)휖2 , there is a point 푥 = 푏(v) in 푀 such that 푊1(휇푣,훿푥) = 푂(휖) by Proposi- tion 3.3.6. ̃ −1 For 푥 ∈ 푀, there is an element v = 훷훬(푥) of ℙ(퐻훬)휖2 that satisfies 푑푀(푥,푏(v)) = 푂(훬 ) = 푂(휖) by the third part of Proposition 3.3.8. Let 훿푣 be the Dirac measure centered at 푏(v), so that 푊1(훿푣,훿푥) = 푑푀(푥,푏(v)). Now, 푊1(휇푣,훿푥) ≤ 푊1(휇푣,훿푣) + 푊1(훿푣,훿푥), and moreover 푊1(휇푣,훿푣) = 푂(√휂(휇푣)) = 푂(휖) by Proposition 3.3.6, so that indeed 푊1(휇푣,훿푥) = 푂(휖). We conclude that the Hausdorff distance between 푀 and ℙ(퐻훬)휖2 , as subsets of the space of probability measures on 푀, is 푂(휖).

Proof of Proposition 3.3.8 Recall the spectral triple (퐶∞(푀),퐿2(푀,풮),퐷) associated to 푀, where 풮 is a Hermi- tian Clifford bundle over 푀 and 퐷 is a Dirac-type operator on 풮.

We will define a localization map 퐹훬 ∶ 풮 → 푃훬퐻 such that

⟨퐹 (푣 ),푎퐹 (푤 )⟩ = (푣 ,푎 푤 ) + ‖푣 ‖‖푤 ‖푂(‖푎‖훬−2 + (푘)(푎)훬−푘) 훬 푥 훬 푥 푥 푥 푥 풮 푥 푥 Lip푥

49 3.3. The metric space of localized states

for all 휖 > 0, whenever 푣푥,푤푥 ∈ 풮푥 and 푎 ∈ 훤(End풮). If 훹푥푦 is the parallel transport map 풮 풮 (푘)(푎) def= ∥푎 − 훹∗ 푎 ∥/푑(푥,푦)푘 from 푦 to 푥, define the constant Lip푥 sup푦∶ 푑(푥,푦)≤휌 푥 푥푦 푦 .

To do so, we will take the element 푣푥 ∈ 풮푥 and use the short-time heat flow associated to 2 the Laplace-type operator 퐷 to obtain a smooth section 푦 ↦ 푝푡(푥,푦)(푣푥) of 풮, which then corresponds to an element of 퐻. The known estimates on heat asymptotics will allow us to bound the dispersion of 푝푡(푥,푦)(푣푥) for small 푡. Then, the fact that 푝푡 is 2 the heat kernel associated to 퐷 whereas 푃훬 is an associated projection, will allow us to control the behaviour of (1 − 푃훬)푝푡(푥,푦)(푣푥).

Definition 3.3.11. Let 푣푥 ∈ 풮푥, for 푥 ∈ 푀. Then 푝푡(푣푥) is the section 푦 ↦ 푝푡(푥,푦)(푣푥) −푡퐷2 of 풮, where 푝푡 is the integral kernel associated to the operator 푒 .

The following Lemma allows us to control the leading term in the short-time be- haviour of the heat flow 푝푡(푣푥). To that end, let ℎ푡(푥,푦) equal the scalar coefficient 2 푒−푑푀(푥,푦) /4푡(4휋푡)−푚/2 of the leading term in the asymptotics of the heat kernel. For 푥 ∈ 푀 and 푠 ∈ ℝ, let 퐵푠(푥) ⊂ 푀 be the metric ball of radius 푠 around 푥. Lemma 3.3.12. Let 푎 ∈ 훤(End풮). Then, we have for all 푠 smaller than the injectivity radius of 푀, and all 푣,푤 ∈ 훤(풮),

∗ ∫ ℎ푡(푥,푦)(푣푥,훹푥푦푎푦푤푥) 푑푦 =(푣푥,푎푥푤푥) ∫ ℎ푡(푥,푦)푑푦+ 풮 풮 퐵푠(푥) 퐵푠(푥) + ‖푣‖‖푤‖ (푘)(푎)푂(푡푘/2 + 푠−2푡(푘+2)/2), Lip푥 −4 2 ∫ ℎ푡(푥,푦)푑푦 =1 + 푂(푡 + 푠 푡 ), 퐵푠(푥) uniformly in 푣푥,푎,푥 ∈ 푀.

def 푘 ′ def Proof. For 푘 ≥ 0, consider the integral 푚푡,푠,푘(푥) = ∫ ℎ푡(푥,푦)푑 (푥,푦)푑푦. Let 푚푡,푠,푘 = 퐵푠(푥) 2 푘 (4휋푡)−푚/2 ∫ 푒−∥푦∥ /4푡 ∥푦∥ 푑푦 퐶 ∥푦∥≤푠 . There exists a global constant such that the pull- 2 back of the volume form on 푀 is bounded by 퐶∥푦∥ times the Euclidean volume form. Thus, pulling back our integral through the exponential map at 푥, we have ′ ′ |푚푡,푠,푘(푥) − 푚푡,푠,푘| ≤ 퐶푚푡,푠,푘+2. By Chebyshev’s inequality, we have

2 2 ′ −푚/2 −∥푦∥ /4푡 2푘 −푚/2 −4 −∥푦∥ /4푡 2푘+4 푚푡,푠,2푘 = (4휋푡) ∫푒 ∥푦∥ 푑푦 + 푂(푡 푠 ∫ 푒 ∥푦∥ 푑푦), ∁퐵푠(0)

50 3.3. The metric space of localized states

where ∁ denotes the complement. With Isserlis’ theorem to calculate the full Gaus- ′ 푘 −4 푘+2 푘 sian integrals, we see that 푚푡,푠,2푘 = 푐푘푡 + 푂(푠 푡 ) for all 푘. Thus,푚푡,푠,2푘(푥) = 푐푘푡 + 푂(푡푘+1) + 푂(푠−4푡푘+2). ∗ (푘) 푘 Now, estimate ∣(푣푥,훹푥푦푎푦푤푥) − (푣푥,푎푥푣푥) ∣ ≤ Lip (푎)푑(푥,푦) ‖푣푥‖‖푤푥‖. 풮 풮 푥 2 As 푚푡,푠,푘 ≤ 푚푡,푠,0푚푡,푠,2푘 by the classical Jensen’s inequality, we can use the simple estimate 푘/2 −2 푘/2 √푚푡,푠,0(푥)푚푡,푠,푘(푥) = 푂(푡 + 푠 푡 ) to conclude that the remaining error is itself 푂(‖푣 ‖‖푤 ‖ (푘)(푎)(푡푘/2 + 푠−2푡(푘+2)/2)) 푥 푥 Lip푥 uniformly.

푚/4 We are now in a position to show that the rescaled heat flow (2휋푡) 푝푡 ∶ 풮푥 → 퐻 is asymptotically isometric, in the following sense: Lemma 3.3.13. For 푎 ∈ 훤(End풮) and 푣,푤 ∈ 훤(푠), we have uniformly

−푚/2 ⟨푝푡(푣푥),푎푝푡(푤푥)⟩ =(2휋푡) (푣푥,푎푥푤푥)풮 + + ‖푣 ‖‖푤 ‖푂( (푘)(푎)푡(푘−푚)/2 + ‖푎‖푡(2−푚)/2) 푥 푥 Lip푥

Proof. It is well-known, see e.g. [BGV04, Theorem 2.30], that there exist a nonzero radius 푠 around 푥 such that for 푑푀(푥,푦) < 푠, one has 푝푡(푥,푦)(푣푥) = ℎ푡(푥,푦)(훹푥푦(푣푥) + 푂(푡)) as 푡 → 0, where 훹 is the parallel transport along the Clifford connection. There- fore,

2 ∗ (푝푡(푥,푦)(푣푥),푎푦푝푡(푥,푦)(푤푥)) = (ℎ푡(푥,푦)) (푣푥,훹푥푦푎푦푤푥) 풮 풮 2 + 푂(푡(ℎ푡(푥,푦)) ‖푎‖‖푣푥‖‖푤푥‖), uniformly in 푥. Moreover, for 푑푀(푥,푦) > 푠, one has 2 (푝푡(푥,푦)(푣푥),푎푦푝푡(푥,푦)(푤푥)) = 푂((ℎ푡(푥,푦)) ‖푎‖‖푣푥‖‖푤푥‖). 풮

Now, outside an 푠-ball around 푥, we have

푠2/2푡 −푚 ∫ (푝푡(푥,푦)(푣푥),푎푦푝푡(푥,푦)(푤푥)) 푑푦 = 푂(푒 푡 ‖푎‖‖푣푥‖‖푤푥‖), 풮 ∁퐵푠(푥) def 1/4 2 −푚/2 as 푡 → 0. Now set 푠 = 푡 and note that (ℎ푡(푥,푦)) = (2휋푡) ℎ2푡(푥,푦). The estimate of Lemma 3.3.12 on the integral over 퐵푠(푥) then completes the proof.

51 3.3. The metric space of localized states

In order to estimate the scaling of the truncated heat flow 푃훬푝푡(푣푥) with 훬, we will 2 relate 푝푡(푣푥) to the spectral resolution of 퐷 , as follows.

Definition 3.3.14. Let 푃휆 be the projection onto the 휆-eigenspace of the first-order ̃ elliptic differential operator 퐷 and let 퐸휆 be its integral kernel, so that for all sections 푣 of 풮 we have ̃ 푃휆(푣)(푦) = ∫ 퐸휆(푥,푦)(푣푥)푑푥. 푀 ̃ Then, 퐸휆 ∶ 풮 → 훤(풮) is the associated lifting 퐸휆(푣푥)∶ 푦 ↦ 퐸휆(푥,푦)(푣푥).

−푡휆2 2 In particular, we have 푝푡 = ∑휆 푒 퐸휆 weakly. To estimate the 퐿 -norm of 퐸휆(푣푥), we will need the following classical result by Hörmander[Hör68].

Theorem 3.3.15 ([Hör68, Theorem 4.4]). There exists a constant 퐶 such that

̃ dim푀−1 sup ∥퐸휆(푥,푦)∥ ≤ 퐶(1 + |휆|) 푥,푦∈푀 uniformly in 푥,푦,휆. In particular, there exists a constant 푐 such that

dim푀−1 ‖퐸휆(푣푥)‖ ≤ 푐(1 + |휆|) ‖푣푥‖ , 퐻 풮푥 for all 푣푥 ∈ 풮, all 푥 ∈ 푀 and all 휆 ∈ 휎(퐷) ⊂ 푅.

Due to the polynomial scaling of the fiberwise inner product, we can now show that the exponential dependence of (1 − 푃훬)푝푡(푣푥) on 훬 implies that we retain the asymptotic properties of the heat flow when we truncate such that 훬2푡 = 푐log훬 for sufficiently large 푐.

Lemma 3.3.16. We have for all 푎 ∈ 퐵(퐻),

2 ∣⟨푝 (푣 ),(푎 − 푃 푎푃 )푝 (푤 )⟩ ∣ = 푂(‖푣 ‖‖푤 ‖‖푎‖푡1−2푚푒−푡푧훬 ), 푡 푥 훬 훬 푡 푥 퐻 푥 푥 for all fixed 0 ≤ 푧 < 1, uniformly in 푣푥,푤푥 ∈ 풮푥,푥 ∈ 푀, as 훬 → ∞.

Proof. Recall that the integral transform associated to the kernel 푝푡(푥,푦) equals the −푡퐷2 −푡휆2 bounded linear operator 푤 ↦ 푒 푤 on 퐻, so that 푝푡 = ∑휆 푒 퐸휆 weakly. Thus, for −푡휆2 푤 ∈ 퐻, we have ⟨푃훬푝푡(푣푥),푤⟩ = ∑|휆|<훬∈휎(퐷) 푒 ⟨퐸휆(푣푥),푤⟩.

52 3.3. The metric space of localized states

The difference to be estimated then consists of the sum of the missing terms, which 2 2 −푡(휆1+휆2) equals ∑ 2 푒 ⟨퐸휆 (푣푥),푎퐸휆 (푤푥)⟩. 휆1,휆2∉[−훬,훬] 1 2 First note that Theorem 3.3.15 provides a global constant 푐 such that

푚−1 |⟨퐸 (푣 ),푎퐸 (푤 )⟩| ≤ ‖푣 ‖‖푤 ‖‖푎‖푐2 ((1 + 휆2)(1 + 휆2)) . 휆1 푥 휆2 푥 푥 푥 1 2

−푡휆2 2 푚−1 −(1−휖)푡훬2 Now, ∑|휆|>훬 푒 (1 + 휆 ) is, for 0 < 휖 ≤ 1, bounded by 푒 times the shifted −푡휖휆2 2 푚−1 −푡휆2 2 푚−1 sum ∑|휆|>훬 푒 (1 + 휆 ) . Moreover, the entire sum ∑휆 푒 (1 + 휆 ) is, by the 2 1 −푚 heat asymptotics for the Laplace-type operator 퐷 , bounded by 푂(푡 2 ).

2 1−2푚 −푡(1−휖)훬2 Thus, we obtain a bound of 푂(푐 ‖푣푥‖‖푤푥‖‖푎‖푡 푒 ).

Our localization map is thus given by a truncated, rescaled heat flow, as follows.

def −2 Definition 3.3.17. Let 푡훬 = 2푚훬 log훬. The map 퐹훬 ∶ 풮 → 푃훬퐻 is given by

2 푚/4 −푡훬휆 푣푥 ↦ (2휋푡훬) ∑ 푒 퐸휆(푣푥). |휆|≤훬

There exists finite 훬 such that 퐹훬 is injective, by Lemma 3.3.13 and injectivity of the heat flow 푝푡(푣푥).

Now we are in a position to connect 퐹훬 to the localization question of Proposition 3.3.8.

Proposition 3.3.18. Consider the map 훷훬 ∶ ℙ풮 → ℙ(퐻훬) given by

def 훷훬([푣푥]) = [퐹훬(푣푥)] ∈ ℙ(퐻훬), for 훬 sufficiently large that 퐹훬 is injective. Then, 훷훬([푣푥]) is localized near 푥 in the sense that 휂(휇 ) = 푂(훬̃ −2), 푊(휇 ,훿 )2 = 푂(훬̃ −2). 훷훬([푣푥]) 2 훷훬([푣푥]) 푥

2 1−2푚 −푡훬푧훬 ̃ −2 Proof. Note that for any 휖 > 0 we may pick 푧 such that 푡훬 푒 = 푂(훬 ). Thus, for 푎 ∈ 훤(End풮) we have 푚/2 2 ̃ −2 ⟨퐹훬(푣푥),푎퐹훬(푣푥)⟩ = (2휋푡) ⟨푝푡(푣푥),푎푝푡(푣푥)⟩ + ‖푣푥‖ ‖푎‖푂(훬 ) = (푣 ,푎 푣 ) + ‖푣 ‖2 푂(̃ (푘)(푎)훬−푘 + ‖푎‖훬−2) 푥 푥 푥 푥 Lip푥

53 3.3. The metric space of localized states

2 2 ̃ −2 so that in particular ‖퐹훬(푣푥)‖ = ‖푣푥‖ (1 + 푂(훬 )). 푓 (푦) def= 푑(푥,푦)2 ‖푣 ‖2 = 1 푊(휇 ,훿 )2 = 훷 ([푣 ])(푓 ) With 푥 and 푥 , we have 2 훷훬([푣푥]) 푥 훬 푥 푥 which ̃ −2 def ̃ −1 is 푂(훬 ), and with 푔(푦)푖 = 휙(푦)푖 − 휙(푥)푖 we see that 훷훬([푣푥])(푔푖) is 푂(훬 ). The dispersion of the associated measure is therefore 푂(훬̃ −2).

Proof of Proposition 3.3.8. Let 휇1,휇2 be the measures associated to 훷훬(푣푥) and 훷훬(푤푦) ̃ −2 respectively. Then, 휂(휇푖) = 푂(훬 ) by Proposition 3.3.18 and we have ̃ −1 ∣푑푀(푥,푦) − 푑(휇1,휇2)∣ ≤ 푊2(훿푥,휇1) + 푊2(휇2,훿푦) = 푂(훬 ).

def Let 푝 = 퐸휇 [휙]. Then, 푑ℝ푛 (휙(푥),푝) ≤ 푊2(휙(푥),휙∗휇1)+√휂(휇) and the first term is, 훷훬(푣푥) ̃ −1 by bi-Lipschitz equivalence, 푂(푊2(푥,휇1)) so that 푑ℝ푛 (휙(푥),푝) = 푂(훬 ). Therefore, ̃ −1 one has 푑(푥,푏(훷훬(푣푥))) = 푂(훬 ) as well.

Finally, for probability measures 휈, 푥 = 푏(휈) and 0 ≠ 푣 ∈ 풮푥, we have 푑(훷훬(푣),휈) ≤ ̃ −1 푊2(훿푥,휈)+푊2(훷훬(푣),훿푥), and Lemma 3.3.5 leads to bounds of 푂(√휂(휈)) and 푂(훬 ) (when combined with Proposition 3.3.18), respectively, on the latter.

The space ℙ(퐻훬) in terms of the truncation of a spectral triple ̃ −1 By Corollary 3.3.10, there exists 휖 = 푂(훬 ) such that the space ℙ(퐻훬)휖2 is 휖-close to 푀, when equipped with the Kantorovich-Rubinstein metric. By Kantorovich-Rubinstein duality, that metric can be computed by Connes’ formula (3.1):

푑훬(v,w) = sup {∣∫ 푓푑휇푣 − ∫ 푓푑휇푤∣ ∣ ∥[퐷,푓]∥ ≤ 1} 푓∈퐶∞(푀) 푀 푀

∞ Now, each element v of ℙ(퐻훬) induces a state 휔푣 of 퐶 (푀)훬, which corresponds to representatives 푣 ∈ 퐻훬 of v as 2 2 휔푣(푃훬푓푃훬) = ⟨푣,푃훬푓푃훬푣⟩/‖푣‖퐻 = ⟨푣,푓푣⟩/‖푣‖퐻

= ∫ 푓(푥)푑휇푣(푥). 푀

With this identification, we have

푑훬(v,w) = sup {∣휔푣(푃훬푓푃훬) − 휔푤(푃훬푓푃훬)∣ ∣ ∥[퐷,푓]∥ ≤ 1}. (3.3) 푓∈퐶∞(푀)

54 3.4. The PointForge algorithm: associating a finite metric space

It is an open question whether, in the limit 훬 → ∞, this metric can be approximated arbitrarily well in Gromov-Hausdorff distance by the functional on the truncation of the spectral triple given by

̃ 푑훬(v,w) = sup {∣휔푣(푓) − 휔푤(푓)∣ ∣ ∥[퐷훬,푓]∥ ≤ 1}. (3.4) ∞ 푓∈퐶훬(푀)

̃ Although we clearly have ∥[퐷훬,푃훬푓푃훬]∥ ≤ ∥[퐷,푓]∥ so that 푑훬 ≤ 푑훬, it is a highly non- trivial undertaking to obtain a bound in the opposite direction. See e.g. [DLM14b; Sui20] for further perspective on the problem. ̃ Importantly, the distance 푑훬 is directly amenable to algorithmic computation; see Section 3.4. Therefore, that is the metric we propose to use in the PointForge algorithm introduced below. The question of correctness of that algorithm, in the sense of Gromow-Hausdorff convergence of its result to the original compact Riemannian ̃ manifold, then at least partly hinges on the relation between the metrics 푑훬 and 푑훬. Attacking this difficult question is however beyond the scope of the present chapter.

3.4 The PointForge algorithm: associating a finite metric space

Once a set of localized vector states is found, the Connes (Kantorovich-Rubinstein) distance between them will serve as an estimate for the geodesic distance between the points in 푀 near which they are concentrated. Keeping in mind the discussion of Section 3.3, we will regard the truncated metric of Equation (3.4) as the natural metric on ℙ(퐻훬). Localized vector states can be found by minimizing the dispersion functional in 퐻. Apart from the comparison of the metrics (3.3) and (3.4), nonzero dispersion induces a distortion of estimated distances (see section 3.3, below). Therefore, the dispersion supplies a lower bound on the Gromov-Hausdorff distance between any graph of localized states and the manifold 푀. Computationally speaking, then, it would be desirable to minimize the number of states (and, hence, computational resources) required to approach this bound. The main other factor, besides localization, that influences the Gromov–Hausdorff distance is the density (in the Hausdorff sense) of our set of points inside 푀. Optimally, therefore, the states would be equidistributed on 푀.

55 3.4. The PointForge algorithm: associating a finite metric space

In order to construct a potential whose minima are both localized and roughly (that is, under the map 휙) equidistributed, we add an electrostatic repulsion term to the dispersion. Given a set 푉 of states, the next state is then generated as the minimum of the energy functional

−1 def −1 2 푒(푣;푉) = −휂(푣) + 푔푒 ∑(∑(⟨푣,휙푖푣⟩ − ⟨푤,휙푖푤⟩) ) . (3.5) 푤∈푉 푖

The value of the coupling constant 푔푒 should ideally be sufficiently large to overcome local variation in minimal dispersion but is otherwise not expected to influence the generated states much – this is consistent with our observations for 푀 = 푆1,푆2.

The PointForge algorithm Using the functional (3.5) we propose the following algorithm to construct states and thus a finite metric space 푀훬 that models the metric information about 푀 contained at cutoff 훬. As preparation, we must estimate the number 푁 of states to generate.

• Estimate vol푀훬 and dim푀훬, e.g. using the asymptotic formulas of [Ste19b]. 2 • Estimate the Euclidean dispersion 휂0 = 퐸휈 [‖푋‖ ] under the multivariate normal distribution 휈 of covariance matrix 2훬−2 log훬id on ℝdim푀훬 .

푁 = 푀 /( (퐵 )휂dim푀훬/2) 퐵 • Set vol 훬 vol dim푀훬 0 , where dim푀훬 is the Euclidean unit ball of dimension dim푀훬.

For cases where 휙 is a Riemannian embedding of 푀, any 푔푒 will suffice to lead to equidistributed states in 푀, while for 푔푒 = 0 the states generated numerically would mostly lie very close together. However in the cases where 휙 is far from Riemannian, we need to chose 푔푒 to be sufficiently large to overcome local variations in minimal 2 −1 2 −1 2 −1 dispersion, and assume this to mean that −훼 휂0 + 푔푒훼 휂0 ≥ −훽 휂0 , where 훼 and 훽 are the optimal local Lipschitz constants of 휙 and 휙−1, respectively. This ensures that states in regions of 푀 where the dispersion is over-reported (due to stretching by 휙) will be generated once the regions where the dispersion is under-reported are saturated with states, instead of being skipped. Then, simply generate 푁 states by minimizing the iterative energy functional and calculate the Connes distance between them:

56 3.4. The PointForge algorithm: associating a finite metric space

1: while |푉| ≤ 푁 do 2: Find a vector 푤 (locally) minimizing 푒(푤;푉). 3: Append 푤 to 푉. 4: for 푣 ∈ 푉, do 5: Set 푑(푣,푤) = max{|⟨푣,푎푣⟩ − ⟨푤,푎푤⟩|∶ ‖[퐷,푎]‖ ≤ 1}. 6: end for 7: end while The algorithm, including the distance calculation and the examples 푆1 and 푆2, has been implemented in Python and is publicly available at [SG19].

Implementation: calculating the metric on ℙ(퐻훬)

When 푣,푤 ∈ 퐻훬, the distance between the vector states ⟨푣| ⋅ |푣⟩ and ⟨푤| ⋅ |푤⟩ of the algebra 퐴 = 퐶∞(푀) equals

max{|⟨푣,푎푣⟩ − ⟨푤,푎푤⟩|∶ ‖[퐷훬,푎]‖ ≤ 1}, 푎∈퐴훬 as in discussed in section 3.3.

The functional 푎 ↦ ⟨푣|푎|푣⟩ − ⟨푤|푎|푤⟩ is linear and the space {푎 ∈ 퐴훬 ∣ ‖[퐷훬,푎]‖ ≤ 1} is convex, which ensures that computing the minimum is computationally feasible.

Indeed, if 푎0,…,푎푛 is a basis for (퐴훬)sa, we can reformulate the problem as: 푛+1 Problem 1. Minimize ∑푖 푐푖 (⟨푣,푎푖푣⟩ − ⟨푤,푎푖푤⟩) over 푐 ∈ ℝ , subject to the constraint

퐼 ∑푖 푐푖[퐷훬,푎푖] [ ∗ ] > 0 ∑푖 푐푖[퐷훬,푎푖] 퐼

With the constraints formulated as a linear matrix inequality, we have put the problem in a form directly amenable to techniques from semi-definite programming. A reasonably effective algorithm, given the scale of the problem, is then provided by the Splitting Cone Solver of [ODo+16].

∞ Complexity and the dimension of 퐶 (푀)훬 Step 2 of the PointForge algorithm amounts to finding a local minimum ofa quadratic function under quadratic constraints in a vector space of dimension dim퐻훬, 2 which can be done in 푂(dim퐻훬) , e.g. with the BFGS algorithm.

57 3.5. Example: 푆2

The problem in step 5 is convex, of dimension dim퐶훬(푀). This factor is what limits the computational feasibility of high 훬 in our experiments, so it would be informative to analyze the scaling of dim퐶훬(푀) with 훬. As a simple example, one can represent the generator 푒푖휃 of 퐶∞(푆1) as the shift operator on 퐻 = 푙2, with basis indexed by ℤ, where the corresponding Dirac operator acts ∞ 1 diagonally as 퐷푒푛 = 푛푒푛. It is then easy to see that the dimension of 퐶 (푆 )훬 is equal to dim퐻훬 = 2⌊훬⌋ + 1. 2 For 푀 = 푆 , if we choose an orthonormal basis 푒푙푚 of eigenvectors of 퐷 and introduce 푌 ⟨(푒 ⋅ 푒 ), 푌 ⟩ 2 the spherical harmonics 0 푙푚 then we can express 푙1푚1 푙2,푚2 0 푙3푚3 퐿 (푀) in terms of 3푗-symbols. In particular, these vanish unless ||푙1| − |푙2|| ≤ 푙3 ≤ |푙1| + |푙2|, which tells ∞ 2 us that 퐶 (푆 )훬 is spanned by (0푌푙푚)훬 for 푙 ≤ 2훬 and is thus of dimension bounded by (2훬 + 1)2. The general situation is not entirely clear. However, our Proposition 3.3.8, as noted dim푀 ∞ there, provides a lower bound of 훩(훬 ) on the scaling of dim퐶 (푀)훬 with 훬.

3.5 Example: 푆2

The simplest interesting example of a commutative spectral triple that allows for an isometric embedding in ℝ3 is probably the sphere 푆2. This section will cover the ∞ 2 2 2 application of the PointForge algorithm to truncations of (퐶 (푆 ),퐿 (푆 ,풮푆2 ),퐷푆2 ), and thereby illustrate (and test the optimality of) the analytic results of Section 3.3.

Implementation ∞ 2 The main ingredients are the vector space 퐶훬(푆 ), the spectrum of 퐷푆2 , the element 휙, 2 2 and their representation on 퐿 (푆 ,풮푆2 )훬. ∞ 2 The vector space 퐶훬(푆 ) is spanned by the spherical harmonic functions 푌푙푚 up to 푙 = 2훬, as in section 3.4. An eigenbasis 푒푙푚 of 퐷 can be expressed in terms of the spin- 1 weighted spherical harmonics 푠푌푙푚, with 푠 = ± 2 , as discussed e.g. in [GVF01], section ∞ 2 9.A. The matrix coefficients of the representation of 퐶훬(푆 ) can then be expressed in terms of triple integrals of spin-weighted spherical harmonics. Note that a brute- force approach of calculating the inner products ⟨푒푙푚,푌푙′푚′ 푒푙″푚″ ⟩ in order to obviate knowledge of the representation-theoretic machinery attached to 푆2 would have been possible, however it would have introduced the additional complexity of calculating 2 ∞ 2 (dim퐻훬) ⋅ dim퐶훬(푆 ) ⋅ rk풮 integrals numerically.

58 3.5. Example: 푆2

푧 푥 − 푖푦 The element 휙 is just the idempotent associated to the Bott projection ( ), 푥 + 푖푦 −푧 where 푥,푦,푧 are the standard coordinates on the embedding 푆2 ↪ ℝ3. Note that this embedding 휙 is isometric, although that is not necessary for the algorithm or the theory in Section 3.3 to work. The source code to this implementation is publicly available as part of the full Python implementation of the algorithm at [SG19].

The localized state densities

Because the measures associated to states in ℙ(퐻훬) are of the form (푣,푣)vol푀, with 푣 in the finite-dimensional vector space 푃훬퐻, one can easily plot the corresponding function (푣,푣) on 푀. This allows us to test them, by simply plotting the corresponding fiberwise inner product of the spinor spherical harmonics in the continuum. We can then compare these with the numerical states generated through the PointForge algorithm for different 훬. The expectation is that the numerical states will be comparable to the states obtained through 훷훬 but will be slightly less localized.

Figure 3.1 shows plots for 훷훬(푣푥), for fixed 푣푥 ∈ 풮 is fixed, and plots of numerical states for 훬 = 4,10. It is evident that the states are indeed peaked neatly near 푥, in both cases. We thus find that the states are well localized and become more localized the largerthe cutoff is. Other than this qualitative comparison we also have analytic control. Proposition 3.3.6 gives the functional form of the dispersion as a function of the cut-off 훬 as log훬/훬2. We can check this relation explicitly by plotting the size of the dispersion against the cutoff value, as done in Figure 3.2 for the cutoff upto 훬 = 16.

Distribution of states over the sphere Plotting several states simultaneously allows us to show how the repulsion term dis- tributes them over the sphere. Figure 3.3 shows 17 states for 훬 = 11. The distribution of states in Figure 3.3 has some inhomogeneities, some gaps between states are very large. This is because we only generated 17 states instead of the 110 we would expect to generate in the PointForge algorithm. Restricting the number of states reduced computation time, and allowed for a clearer visualization of the independent states. In the right hand Figure we see the states as densities on the sphere, while the left hand plot shows the densities in the 휃,휙 plane.

59 3.5. Example: 푆2

(a) Analytic state for 훬 = 4 (b) Analytic state for 훬 = 10

(c) Numerical state for 훬 = 4 (d) Numerical state for 훬 = 10 Figure 3.1: Plot of analytic and truncated localized states.

60 3.5. Example: 푆2

Figure 3.2: Plot of the dispersion of states versus the value of the cutoff for the states. The dashed line is a fit of the analytic result that the dispersion should scale like log훬/훬2.

To test how the repulsive potential acts we can generate states on the sphere and just plot the coordinates for their center of mass associated with the embedding maps 휙푖. We show this in Figure 3.4 for a maximal eigenvalue of 훬 = 10, it is clear that without potential all states generated cluster at one point, while even a weak repulsive potential leads to points that are evenly distributed.

Error analysis

2 def If a measure 휇 on 푆 is reasonably localized, so that 푥 = 퐸휇 [휙(푋)] ≠ 0, then it possesses a unique 휙-barycenter 푝 given by the projection of 푥 onto the sphere. The Euclidean distance between 푥 and 휙(푝) is then given by √1 − ‖푥‖2. Figure 3.5(a) shows how closely the geodesic distance between barycenters is approxi- ̃ mated by the truncated Connes distance 푑훬. The monotone scaling of the error reflects the fact that antipodal, imperfectly localized measures are significantly closer in Wasser- stein distance than their barycenters are, due to the presence of the cut locus. ̃ Interestingly, the error is strictly positive, so that 푑훬(휇1,휇2) turns out to be – for the

61 3.5. Example: 푆2

(a) States in 휃 − 휙 plane (b) States on the sphere Figure 3.3: Localized states on the sphere, the left hand image shows the states projected on the two dimensional plane using a sinusoidal projection, while the right hand image shows the states on the sphere.

states considered – a better approximation to 푑푀(푝1,푝2) than 푊1(휇1,휇2) = 푑훬(휇1,휇2) itself. In particular, as long as the error is positive, the convergence of 푑훬(휇1,휇2) to ̃ 푑푀(푝1,푝2) as the dispersions fall implies convergence of 푑훬 to 푑훬 as well. Whether this points to special behaviour of the (truncated) Connes distance between localized elements of ℙ(퐻훬) remains to be seen.

2 For measures 휇1,휇2 on 푆 , the analysis of Section 3.3 shows that the error |푑푀(푝1,푝2)− 2 푊1(휇1,휇2)| is bounded by 푊2(푝1,휇1)+푊2(푝2,휇2), and moreover that we have 푊2(푝,휇) 2 2 ≤ 훽 퐸휇 [∥휙(푝) − 휙(푋)∥ ], where 훽 = 휋/2 is the Lipschitz constant of 휙. Thus, we have a 휋2 2 2 bound of 4 (∑푖 휂(휇푖) + (1 − ‖푥푖‖ )) on the squared error |푑푀(푝1,푝2)−푊1(휇1,휇2)| . Since the other terms in this inequality can readily be calculated, it provides us with a the- oretical lower (upper) bound on 푊1(휇1,휇2). Figure 3.5(b) shows this lower bound, and ̃ the upper bound provided by 푑훬, for 푊1(휇1,휇2), with 푑푀(푝1,푝2) shown for reference.

62 3.5. Example: 푆2

(a) No repulsive potential (b) Repulsive potential with coupling 0.001

(c) Repulsive potential with coupling 0.1 (d) Repulsive potential with coupling 100 Figure 3.4: This shows how the states are distributed dependent on the repulsive poten- tial. We can see that even a weak repulsive potential suffices to lead to well distributed points. This figure shows the point distribution in the Sinusoidal projection, flattening the sphere onto the plane.

63 3.6. Embedding a distance graph in ℝ푛

(a) Absolute error (b) Bounds for 푑훬(휇1,휇2) Figure 3.5: Distance errors and bounds for pairs of localized states at 훬 = 5.

3.6 Embedding a distance graph in ℝ푛

Let 푀 be a compact Riemannian manifold, and assume that 푀 embeds isometrically 푛 into ℝ . Now take a finite set 푉 of points in 푀 and their geodesic distances 푑(⋅,⋅)|푉×푉, e.g. by generating states as above and calculating their distances using (3.1). Optimally, we would ask for a way to embed 푉 in ℝ푛 such that its image under this embedding equals its image under some Riemannian isometry 푀 → ℝ푛.

Of course, without knowledge of 푀 such a problem is unsolvable for any given 푉. Instead, we hope for our embedding procedure to satisfy such a property asymptotically, i.e. that for sequences of 푉 of increasing density their embeddings converge to an embedding of 푀, under some suitable notions of density and convergence. This is an open problem, and since our primary purpose at this point is one of visualization, we will only take it as a guiding principle.

Stress and local isometry of embeddings The field of optimal graph embedding is well-established and provides many approaches to questions similar to the above. A particular model of interest is metric multidimen- sional scaling5, where one looks for an embedding 푋 ∶ 푉 → ℝ푛 that minimizes the

5See e.g. [BG05]

64 3.6. Embedding a distance graph in ℝ푛

stress function,

2 ∑푝≠푞∈푉 푤(푝,푞)(푑(푝,푞) − ||푋(푝) − 푋(푞)||) 휎(푋) = , ∑푝≠푞∈푉 푤(푝,푞) where 푤 is a positive weight function: this is just a weighted version of the second Gromov-Wasserstein distance between 푉 ⊂ 푀 and 푋(푉) ⊂ ℝ푛. Because our 푀 is not assumed to be Euclidean, the usual choice 푤 = 1 would be quite unnatural here. In particular, an isometric embedding of 푀 in the Riemannian sense would not necessarily have minimal stress, because the model instead asks for isometry in the sense of maps of metric spaces, not Riemannian manifolds. Since all tangent space information is lost when discretizing like this, the Riemannian notion of isometry does not translate immediately and we must replace it using a measure of locality. By the smoothness of an isometric embedding 휙 of 푀, the relative defect |푑(푝,푞) − ||휙(푝) − 휙(푞)|||/푑(푝,푞) must converge to 0 as 푝 → 푞. That is to say, as long as we only worry about pairs of points that are close in 푀, the stress function above places the correct restriction on 푋 - the further they are apart, the less sense the corresponding contribution to 휎 makes. This motivates us to pick a positive weight function 푤(푝,푞) of 푑(푝,푞) that decays monotonically and sufficiently quickly to suppress those lengths that cannot be approximated well by an Euclidean embedding. For example, imagine two points connected by a shortest geodesic (a great circle arc) of length 푙 ≤ 휋 on the unit sphere and let that sphere be embedded isometrically in ℝ3. In ℝ3, the shortest geodesic connecting the points is a chord of length 푐(푙) = 2sin(푙/2). The defect for small geodesic distances 푙 is thus quite small, being 푂(푙3). It reaches its maximum when the points are antipodal, with a relative error of (휋 − 2)/휋. The weight function should suppress the contribution of the larger distances to the stress 휎, in order to still recognize when an embedding of the distance graph is locally isometric. 푛 Let 휙 ∶ 푀 → ℝ be isometric and let 푤푘 be a sequence of weight functions depending on the cardinality 푘 of 푉 ⊂ 푀. If 푤푘(푙) = 표(1) for fixed 푙 and the marginal defect, which is bounded by

2 푘 푤 (푝,푞)(푑(푝,푞) − ‖휙(푝) − 휙(푞)‖) sup푝,푞∈푀 푘 , inf|푉|=푘 ∑푝,푞∈푉 푤푘(푝,푞) is summable in 푘, we can at least be sure that the stress function 휎 converges to 0 for embeddings 푋 = 휙|푉.

65 3.6. Embedding a distance graph in ℝ푛

The optimal choice of 푤 then depends (at least somewhat) on the geometry of 푀 itself; the curvature inf휙∶푀→ℝ푛 sup{|푑(푝,푞) − ‖휙(푝) − 휙(푞)‖| ∣ 푝,푞 ∈ 푀,푑(푝,푞) ≤ 휖}, as function of 휖, together with the Hausdorff distance between 푉 and 푀, determines the optimal behaviour of 푤.

Implementation

For dim푀 = 2, we expect the length of the smallest edges to scale roughly as 푘−1/2. For

푤푘(푙) = exp(−√푘푙) the infimum in the denominator of the marginal defect, above, is roughly bounded from below by its value for an equidistributed 푉, which is of order 푘2 ∫휋 (푙)푤 (푙)푑푙 ∼ 푘 푘 → ∞ 푂(푘−3/2) 0 sin 푘 as . The supremum in its numerator is , so this sequence 푤푘 will do in the narrow sense that it will asymptotically detect when a 푛 sequence {휙푘 ∶ 푉푘 → ℝ } corresponds asymptotically to an isometric embedding of 푀, assuming the 푉푘 are roughly equidistributed. Given the choice of weights, minima of the resulting stress function can be found efficiently using the weighted SMACOF algorithm for stress majorization. Asimple Python implementation of the weighted SMACOF algorithm is part of [SG19], but for more intensive use we recommend the more efficient FORTRAN version with Python bindings [Ste19a].

The 퐷푐 operator on the sphere The results of Section 3.3 apply to any Dirac-type commutative spectral triple. In ∞ 2 particular, they apply to perturbations (퐶 (푀),퐿 (푀,풮),퐷풮 +퐵) of a Dirac spectral triple, as long as the perturbation 퐵 does not change the principal symbol of 퐷. We will apply the PointForge and embedding algorithms both to the sphere and to a perturbation thereof that will arise in Chapter 4. All spin manifolds of dimension ≤ 4 satisfy (the two-sided version of) the higher Heisen- berg equation introduced in [CCM14]. Chapter 4 explores the constraint that existence of solutions to the one-sided higher Heisenberg equation,

1 ⟨푌[푌,퐷]…[푌,퐷]⟩ = 훾, (3.6) 푛! ⏟⏟⏟ repeated 푛 times places on the truncation of a spectral triple. There we found that (3.6), with 푌 and 훾 obtained from the Dirac spectral triple of 푆2, is solved by a one-parameter class of

66 3.6. Embedding a distance graph in ℝ푛

(a) 퐷푆2 (b) 퐷푆2 + 푐퐵

Figure 3.6: Locally almost-isometric embeddings corresponding to 퐷푆2 and 퐷푆2 + 푐퐵, with shaded 푆2 for reference

operators {퐷푐 ∣ 푐 ∈ ℝ} ⊂ 풟, where

퐷푐 = 퐷푆2 + 푐퐵.

Here 퐵 is a bounded, self-adjoint operator 퐵 = sign(퐷)cos(휋퐷푆2 ). This class of solu- tions does not strictly describe spectral triples, since the pseudo differential operator 퐷푐 does not satisfy the first order condition. As discussed there, however, failure ofthis condition is not detectable by standard methods at the level of truncations of spectral triples.

Result The PointForge algorithm returns a metric graph, given an operator system spectral triple (퐴,퐻,퐷) and a designated element 휙 ∈ 퐴푛. We apply the locally isometric em- bedding above not only to the example from Section 3.5, but also (tentatively) to the ∞ 2 triple (퐶 (푆 )훬,퐻훬,퐷푐,훬) of [GS20], in order to investigate the metric properties of the latter. Here 훬 = 5, corresponding dim퐻훬 = 84, which leads to 35 states. This leads to the results in Figure 3.6. There we can see that the embedded points for 2 퐷푆2 , the left hand plot, lie outside the shaded 푆 that is included for reference, while on 2 the other hand the points for 퐷푆2 + 푐퐵, in the right hand plot, lie inside the shaded 푆 . The transparency of the dots increases with distance to the viewer. Both embeddings

67 3.7. Final remarks

show some deviation from the sphere: for 퐷푆2 , the radii of the embedded points lie in [1.06,1.12], with an average of 1.09; for 퐷푆2 + 푐퐵, in [0.94,0.98] averaging 0.96.

3.7 Final remarks

The PointForge algorithm we introduced in section 3.4 was designed to reconstruct metric spaces from truncations of commutative (Dirac) spectral triples. However, the ingredients of the algorithm need not originate as truncations of a commutative spectral triple at all; the steps apply verbatim to arbitrary operator system spectral triples, provided a special ’embedding’ element 휙 is given. Obtaining such 휙 could either be related to the higher Heisenberg equation of [CCM14], or, computational resources allowing, be disposed of entirely as discussed in section 3.3. This would provide one with the means to construct finite metric spaces associated to an arbitrary noncommutative spectral triple. It would be interesting to elaborate on this and relate it to quantization and e.g. fuzzy spaces, to get a geometric sense of the relation between a commutative spectral triple and its noncommutative deformations. For the case of the fuzzy sphere, which is not a truncation of a Dirac-type spectral triple but, unlike those, is a family of genuine spectral triples, it would be interesting to compare the localized states discussed here to the coherent states of [DLV13]. A proper generalization to the context of fuzzy spaces and other finite spectral triples could be particularly useful in connection to more physically inspired explorations thereof, such as [BG16]. The ensemble of finite, random spectral triples defined there has shown signs of a phase transition [BG16; Gla17] and canbe characterized through spectral dimension measures [BDG19], which can be taken as an indication of possibly emergent geometric properties. The PointForge algorithm might then be an interesting tool to further explore some exemplary spectral triples from this class to gather further insights. It would also be instructive to test how the PointForge algorithm works for spectral triples of different topologies, e.g. the non-commutative torus [PS06] or a fuzzy torus [BG19]. Another possible application in this direction would be to exploit the explicit scale dependence of the present formalism in order to obtain a better understanding of the gravitational properties of noncommutative approaches (such as [CC96]) to quantum field theory.

68 Chapter 4

Understanding truncated noncommutative geometries through computer simulations1

When aiming to apply mathematical results of non-commutative geometry to physical problems the question arises how they translate to a context in which only a part of the spectrum is known. In this chapter we aim to detect when a finite-dimensional triple is the truncation of the Dirac spectral triple of a spin manifold. To that end, we numerically investigate the restriction that the higher Heisenberg equation [CCM14] places on a truncated Dirac operator. We find a bounded perturbation of the Dirac operator on the Riemann sphere that induces the same Chern class.

4.1 Introduction

The spectral viewpoint is central to non-commutative geometry, which makes it a natu- ral framework to investigate the relation between energy and geometry. To understand low-energy (that is, physical) observations, we need to be able to distinguish commuta- tive spectral triples from classically meaningless configurations, using only low-energy data. Connes’ spectral reconstruction theorem [Con13] tells us when a spectral triple (퐴,퐻,퐷) is the Dirac triple of a spincmanifold. However, checking the conditions under which

1This work was written jointly with Lisa Glaser and published as [GS20].

69 4.1. Introduction

the theorem holds requires knowledge of all spectral information: they can not be applied when we only consider a finite part of the frequency (energy) spectrum. That is, the usual spectral expressions do not reveal much about the nature of the universe to an observer with access to only finite spectral information. This is highly relevant when applying non-commutative geometry to physical problems, since in realistic systems only approximate knowledge is available. It is also highly relevant when using spectral triples to discretize geometries through finite algebras and Hilbert spaces, and in most attempts to use numerical methods to explore spectral triples. In order to engage the issue, we explore whether it is possible to use the higher Heisen- berg equation [CCM14] to detect, at a finite frequency level, whether a given truncated spectral triple corresponds to a spincmanifold. The analysis starts with a computer sim- ulation of the higher Heisenberg constraint (introduced below) on the sphere, which leads to a new analytic solution of the corresponding equation. Lastly, the methods from Chapter 3 are applied to generate and visualise finite metric graphs that represent (what is argued to be) the metric space corresponding to the finite-scale geometries involved. The remainder of this introduction is structured as follows. Section 4.1 briefly intro- duces the relevant concepts from noncommutative geometry, such as spectral triples, the spectral action principle, and the relation of the latter to observations based on finite spectra. Section 4.1 expands on the notion of information contained in finite spectra and introduces the problem of detecting ‘commutativity’ at finite scale, whereafter Section 4.1 introduces the higher Heisenberg relation as a possible approach to that problem and gives a brief overview of the structure of the chapter itself.

Background: noncommutative geometry and the cutoff scale By Gelfand duality, a (compact Hausdorff) space 푋 may be entirely understood in terms of the algebra 퐶(푋) of continuous functions. Moreover, each commutative unital 퐶∗-algebra is of this form 퐶(푋) for some 푋. Noncommutative geometry starts by the observation that we can extend this duality to spincmanifolds: the spincmanifold 푀 (and, therefore, its metric) can be described uniquely in terms of the spectral triple (퐶∞(푀),퐻,퐷), where 퐷 is the associated Dirac operator and 퐻 is a Hilbert space of spinors. This description of spin geometry in terms of operators on Hilbert spaces then allows one to extend many spin-geometric notions to the study of more general geometric

70 4.1. Introduction

objects, the noncommutative spectral triples2 (퐴,퐻,퐷). In particular, the resulting flexibility allows one to use the same language to describe both ordinary spingeometry and the field theories common in particle physics [CCM07]. A very simple choice of algebra, together with the spectral action principle [CC97] (see below) leads to the standard model, minimally coupled to general relativity. One interesting feature of this latter formulation is that while the classical (metric) geometry is described through an infinite-dimensional algebra and Hilbert space, the particles of the standard model are encoded in a finite-dimensional non-commutative al- gebra. Fundamentally finite dimensional spectral triples allow for a description of spaces that are discretized, but still retain their original symmetry group. Examples of these, often called fuzzy, spaces are the fuzzy sphere [GP95], fuzzy projective spaces [Bal+02] or the fuzzy torus [DO03]. General finite spectral triples have been classified [Kra98; Ćać11; CC08] and parametrized [Bar15]. In the present chapter, however, we will be concerned with truncated, not fundamentally finite, spectral triples. By the assumption of diffeomorphism invariance, all observables in pure gravity – including the action – must be expressible in terms of global geometric invariants. The spectral action principle [CC97] in noncommutative geometry asserts that, moreover, the action should be formulated in terms of the spectrum of the Dirac operator 퐷 alone. The identification of such global invariants with zeta residues allows them to bewritten in terms of asymptotic traces of 퐷, and this induces the prescription

푆(퐷) = tr(휒(퐷/훬)) for the bare action, where 휒 should be a suitable smooth cutoff function. The scale parameter 훬 controls the relative contributions of Dirac eigenvalues. At finite cutoff scale 훬 we are then automatically invited to think of the corresponding system as described by a finite-rank, truncated Dirac operator. Recent work has started numerically exploring the path integral,

풵 = ∫d퐷푒−풮(퐷), (4.1) with 풮 a trace of powers of 퐷, over finite-rank Dirac operators, as a possible nonpertur- bative description for quantum gravitational phenomena [BG16; Gla17; BDG19].

2Here, 퐴 is a possibly noncommutative C∗-algebra, corresponding to the ‘topological’ aspect of the noncommutatige geometry, and 퐷 a possibly unbounded selfadjoint operator, corresponding to the ‘metric’ aspect thereof, both represented on a Hilbert space 퐻. See [GVF01] for an introduction.

71 4.1. Introduction

Geometry at finite scale

Spectral descriptions of continuum geometry involve infinite-dimensional algebras and Hilbert spaces. If these are to be applied to physics involving measurement at finite energies, to be captured in computer simulations or to be described approximately, we must understand how (much) information can be contained in partial spectra. This in- volves extending the tools that have been developed to understand infinite-dimensional non-commutative geometries to truncated spectral triples, as has been done for the residue functionals in [Ste19b].

A particular difficulty, which is central to the present chapter, relates to the recognition of (possibly almost-commutative) manifolds at the truncated level. In carrying out the path integral (4.1), for instance, one should in principle restrict to Dirac operators that actually correspond to (possibly almost-commutative) spincstructures for the given (fixed) manifold 푀, just like the path integral in Euclidean quantum gravity restricts the integration to fields that describe Riemannian metrics as opposed to being fully arbitrary. However, it is a priori unclear what this restriction means for the integration variable 퐷.

Although Connes’ reconstruction theorem [Con13] allows us to detect when a spectral triple (퐴,퐻,퐷) corresponds to the Dirac triple on a spin manifold, it is not clear how to implement those conditions as a constraint on an integral over operators 퐷. Moreover, it is not clear when a finite-rank Dirac operator 퐷 corresponds to a cutoff of such a spin geometry. This complicates the proposed identification of path integrals over finite-rank Dirac operators with finite-scale path integrals over spin geometries. The one-sided higher Heisenberg equation recalled below (and more generally, its two-sided version) offers a possible approach to constraining the domain of integration in(4.1).

The higher Heisenberg equation

In [CCM14] Chamseddine, Connes and Mukhanov introduce a non-commutative analogue to the Heisenberg relation of quantum mechanics. This ‘higher Heisenberg equation’ neatly captures the relation between the scalar fields (smooth functions) and the Dirac operator that is central to noncommutative geometry in a single algebraic equation. The one-sided version of this equation, applicable to (disjoint sums of) even-dimensional 푛-spheres, works as follows. For 푀 a 푛-dimensional manifold, there trivially exists a covering 휙 ∶ 푀 → 푆푛; let its components be denoted by 푌푖, 1 ≤ 푖 ≤ 푛. 푖 푛/2 2 Then, the section 푌 = 푌 훤푖 of the trivial Clifford bundle of rank 2 satisfies 푌 = 1

72 4.1. Introduction

and 푌∗ = 푌, and moreover 1 ⟨푌[푌,퐷]…[푌,퐷]⟩ = 훾, (4.2) 푛! ⏟⏟⏟ repeated 푛 times where 훾 is the grading on the spinor bundle and ⟨⋅⟩ denotes the 퐶∞(푀)-valued fiberwise trace on the Clifford algebra bundle. Ifa general Riemannian manifold 푀 admits such 푌, moreover, they must necessarily be of the form considered above, ensuring that 푀 ∞ 푛 is a disjoint sum of even-dimensional spheres. The 푌푖 then generate 퐶 (푆 ) and the spectral triple (퐶∞(푀),퐻,퐷) is unitarily equivalent to the direct sum of any splitting of (퐶∞(푆푛),퐻,퐷) into irreducible components. For more general (spin) 푀, the real structure on the spinor bundle induces a two- sided version of the equation above, corresponding to a map 휙 × 휙′ that induces a (not necessarily isometric) embedding 푀 → 푆푛 × 푆푛. We are presently concerned only with the one-sided equation as a first example. In this chapter we propose to use the higher Heisenberg relation to constrain general selfadjoint matrices 퐷, in order to induce them to correspond to truncated Dirac operators of reasonable Riemannian geometries on the underlying manifold. Computer simulations then allow us to explore numerically the effects of this constraint. A real spectral triple consists of (풜,ℋ,퐷) together with a real structure 퐽 and a chirality 훾 that satisfy a number of conditions. An introduction can be found e.g. in [GVF01]. One axiom that has special significance, is the first order condition

[[퐷,푎],퐽푏∗퐽−1] = 0 ∀푎,푏 ∈ 풜, which ensures that 퐷 acts as first-order differential operator in the commutative case, and is the second algebraic constraint (besides the one corresponding to the higher Heisenberg equation) appearing in Connes’ reconstruction theorem. To recover the metric on a spincmanifold from the corresponding spectral triple, one can define a metric on the space of states 휔1,휔2 ∈ 푆(풜),

푑(휔1,휔2) = sup{휔1(푎) − 휔2(푎)|||[퐷,푎]|| ≤ 1}. (4.3) 푎∈풜 In the commutative case, the pure states correspond to atomic measures, that is, points, on the underlying manifold. In Chapter 3 we used this definition of distance, together with a notion of locality, to associate finite metric spaces to truncated non-commutative geometries.

73 4.2. The Heisenberg relation in simulations

In section 4.2 we explain the truncation and our simulations methods and present results for the circle and the two-sphere. This section in particular discusses the reasoning behind our choice of truncation, how it is implemented and some possible problems in this choice. In section 4.3 we show that one of the Dirac operators found in the previous section is a better solution to the Heisenberg relation, while not strictly belonging to a spectral triple in the infinite size limit. In our conclusion, section 4.4, we summarize the results and collect some questions that are opened by our work.

4.2 The Heisenberg relation in simulations

In noncommutative geometry one describes a spin manifold in terms of the associated spectral triple (퐴,퐻,퐷). From a mathematical perspective, it is desirable to be able to describe such a spectral triple as a limit of finite-dimensional data of increasing precision, just like one can describe a Riemannian manifold as a Gromov-Hausdorff limit of finite metric spaces. From a physical perspective, the same desire results from the view that one should be able to gain at least some information about the geometry by probing it at finite energies.

One natural approach to such a ‘cutoff’ of the geometric data (퐴,퐻,퐷) is to pick a scale 훬, then define def 푃훬 = 휒[−훬,훬](퐷) to be the spectral projection onto the eigenspaces of 퐷 of eigenvalue |휆| ≤ 훬, and then take the finite-dimensional data

(푃훬퐴푃훬,푃훬퐻,푃훬퐷) (4.4) as our starting point. As we will notice again later, these truncated triples do not necce- sarily satisfy the conditions for a spectral triple using the truncated real structure 퐽훬. We accept this as one of the limitations of the current program, constructing finite spectral triples with a finite real structure would lead back towards the fuzzy spaces de- fined by Barrett [Bar15]. In this setting, Chapter 3 reconstructed (asymptotically) spin ∞ manifolds 푀 from the data (푃훬퐶 (푀)푃훬,푃훬퐻,푃훬퐷푀) associated to the commutative ∞ spectral triple (퐶 (푀),퐻,퐷푀). Some properties of the induced metric on the state spaces of 푃훬퐴푃훬 and 퐴 were previously investigated in [DLM14b], and for the sphere specifically in [DLV13].

74 4.2. The Heisenberg relation in simulations

The truncated higher Heisenberg equation All spin3 manifolds of dimension ≤ 4 satisfy (the two-sided of) the higher Heisenberg equation (4.2), whereas clearly not all spectral triples do. This suggests to use the equation to recognize many cases in which a spectral triple does not correspond to a spin manifold, without needing to check the rather elusive conditions of the spec- tral reconstruction theorem. We will extend this tool to the finite-dimensional data (푃훬퐴푃훬,푃훬퐻,푃훬퐷) introduced above, and explore what type of truncated triple solves the truncated higher Heisenberg relation.

Given a solution 푌,퐷 of equation (4.2) and the spectral projection 푃훬 = 휒[−훬,훬](퐷), the defect def 푛 훿(푌훬,퐷훬,훾훬) = ⟨푌훬[퐷훬,푌훬] ⟩ − 푛!푘훾훬 (4.5) strongly converges (superpolynomially) to zero as 훬 → ∞. Simple examples like the circle (see below) show, however, that we cannot expect the defect to converge to zero in any Schatten 푝-norm including 푝 = ∞. One wonders then which restrictions on the truncated triple are enforced by minimizing 훿(푌훬,퐷훬,훾훬). ′ The direct approach to this question starts by searching for an operator 퐷 on 푃훬퐻 that comes at least close to solving (4.2) in the sense of minimizing the constraint 2 2 ∥훿(푌 ,퐷′,훾 )∥ = ∥⟨푌 [퐷′ ,푌 ]푛⟩ − 푛!푘훾 ∥ , (4.6) 훬 훬 2 훬 훬 훬 훬 2 for fixed 훬. The Hilbert-Schmidt norm is a natural choice here; all Schatten norms are equivalent in finite dimensions and this is the least computationally expensive among them. This, then, is the constraint whose solutions we investigate numerically below:

• Fix a cutoff 훬,

• Take 푃훬,푌훬,훾훬 from the corresponding commutative spectral triple (that is, here, from the circle and the (spin) sphere), ′ • Look for the arguments 퐷훬 (matrices of dimension rank푃훬) that minimize (4.6).

The second step means that, for the sphere, the possible matrix size of the truncations will be restricted to the sums of multiplicities of eigenspaces. To have some more ′ freedom in the choice of matrix size for 퐷훬 one could, instead of 푃훬, use some other projection in its commutant. It seems, however, that in the cases of the circle and the sphere doing so would introduce a further defect in 훿(푌훬,퐷푆푛,훬,훾훬). 3Unlike equation (4.2), its more general two-sided version involves the spin structure. In the example 푀 = 푆2 considered here, the spin structure does not play a role.

75 4.2. The Heisenberg relation in simulations

Computation In order to numerically investigate the behaviour of (4.6) in practice, we use an annealing type algorithm. Simulated annealing algorithms find optima of a given function by running a random walk in its domain, with transition probability depending on the value of the optimized function and a global ‘temperature’ parameter 푇 that is decreased in time. The algorithm we use is called thermal annealing, and controls the temperature by postulating that the information theoretic and thermodynamic entropy densities must agree [VLH03]. This is a convenient choice for our problem since it has few free parameters, and we are only interested in the final result. The free parameters in question are a constant 푐 which governs the speed at which the temperature is lowered and the final temperature 푇푓. Any choice of 푐 that does not lead to freezing out of the system before equilibrium is reached is valid, while the final temperature governs how strongly the system is allowed to fluctuate around the final state. Weset 푇푓 = 0.001 and adjust 푐 to the simulations in question, testing several 푐 to ensure the results are equivalent. 4 The annealing algorithm runs until some 푇 < 푇푓 is reached , and then simulate the system at this low temperature for a while. The quantities of interest to us are then the configuration with the lowest value of the constraint, as well as an average overthe states at the final temperature.

The circle as a simple example A first example of an algebraic relation, analogous to (4.2), whose solution describes a spin manifold is as follows [Con00]. Assume that 푈 ∈ 퐵(퐻) is unitary and 퐷 is a self- adjoint unbounded operator on 퐻 such that 0 ∈ 휎(퐷) and 퐷−1 ∈ 퐿(1,∞)(퐻). Assume, moreover, that the pair 푈,퐷 is represented irreducibly. Then, if 푈 and 퐷 satisfy 푈∗[퐷,푈] = 1 , (4.7) the triple (퐴,퐻,퐷), where 퐴 is a dense subalgebra of the C∗algebra generated by 푈, ∞ 1 2 1 is unitarily isomorphic to the spectral triple (퐶 (푆 ),퐿 (푆 ),퐷푆1 ) that describes the circle. Under such an isomorphism 푈 is mapped to the generator 휃 ↦ 푒푖휃 of 퐶(푆1) (up to the obvious phase ambiguity in equation (4.7)).

Given the spectral projection 푃훬 as in section 4.2, the operator 푈훬 = 푃훬푈푃훬 is no longer unitary and is even nilpotent, so (4.7), with 푈 replaced by 푈훬, cannot be solved in 퐷. 4The nature of the algorithm means that we do not have perfect control of the finite temperature, however the exact finite temperature is not important in our case.

76 4.2. The Heisenberg relation in simulations

The corresponding version of (4.6) is

2 ∗ 2 ‖훿(푈훬,퐷훬)‖2 = ||푈훬[퐷훬,푈훬] − 1||2 . (4.8)

In order to counter the spurious symmetry 퐷 ↦ 퐷 + 푐퐼 of (4.7), we demand that 퐷훬 additionally satisfies 퐷퐽 = 퐽퐷, where 퐽 is the real structure corresponding to the point- wise complex conjugation map on 퐿2(푆1). This anticommutation with 퐽 ensures that the spectrum of 퐷훬 is symmetric around 0, and our computer simulations parametrize the space of such 퐷훬 as follows: Pick a eigenbasis of 푃훬퐷푆2 and find unitary 푉 in 푃훬 such that 푉∗퐽푉 is just complex conjugation of the coefficients in this eigenbasis. Then, we 퐷 푖푉∗퐻 푉 퐻 ∈ 푀 (ℝ) parametrize 훬 as 0 , where 0 rank푃훬 is an arbitrary real antisymmetric matrix. Using the constraint (4.8) as a weight for thermal annealing we collect two types of observations. On the one hand, we measure the Dirac operator that leads to the smallest value of the constraint. This is ideally going to be very close to the Dirac operator for the circle. To compensate for small numerical fluctuations, we also measure 500 times after the low final temperature is reached and average these measurements. In Figure 4.1 we see that the eigenvalues of the simulated Dirac operators turn out very close to those of the circle Dirac. They can not be distinguished in the upper plot, while the lower plot shows the difference from the analytic spectrum for the average andthe best eigenvalues. The small difference is an effect of the cutoff, which is also reinforced by the difference being larger for larger eigenvalues.

Examining thematrix entries 훿(푈훬,퐷훬)푖푗 of the constraint we find that the violations of this equation are minimal almost everywhere. The only sizable deviation is on the entry ∗ for 푖,푗 = 0 where the deviation is ∼ −1. Since the defect 푈훬[퐷푆1,훬,푈훬] − 푃훬 equals the projection onto that kernel, it is not surprising to find the maximum there. Hence our simulations find the truncated circle Dirac operator, which we know tobe the correct solution. This is a good test for the formalism, and encourages us to move on from the simple circle to the more complicated sphere.

푆2 simulations The version of equation (4.2) corresponding to the sphere 푆2 is

훿(푌훬,퐷훬,훾훬) = ⟨푌훬[퐷훬,푌훬][퐷훬,푌훬]⟩ − 훾훬. (4.9)

77 4.2. The Heisenberg relation in simulations

Figure 4.1: Comparing the eigenvalues of the Dirac operator with the smallest value of the constraint to that of the average over operators (with error indicating the statistical fluctutations) and the exact circle. The results are all so close together thatwecannot distinguish them in the upper plot, the lower plot shows only the difference between the simulation results and the exact numbers.

푧 푥 − 푖푦 Here, 푌 = ( ), with 푥,푦,푧 the standard coordinates on ℝ3, viewed as 푥 + 푖푦 −푧 functions on 푆2 through its standard embedding. That is, 푌 + 1 is twice the Bott projector. The angular brackets denote the 퐵(푃훬퐻)-valued trace on 푀2(퐵(푃훬)) and 2 2 훾훬 is the truncation of the usual grading on 퐿 (푆 ,푆). See the Appendix 4.A for the representation used in the numerical simulations.

For the sphere the Dirac operator has a few symmetries that the truncated operator should satisfy for the truncated operator to still interact correctly with the truncated chirality and real structure. This leads us to consider different parametrizations for the operator.

78 4.2. The Heisenberg relation in simulations

Parametrizing the Dirac operator In order to cancel the symmetry 퐷 ↦ 퐷 + 푐퐼 of (4.9) and to enforce symmetry of the spectrum of 퐷훬, we have tested two different additional constraints. The first, stronger constraint is that 퐷훬 correspond to the (truncation of the) same K-cycle as 퐷푆2 – that is, −1 −1 that 퐷훬|퐷훬| = 푃훬퐷푆2 |퐷푆2 | , so that the bounded transform of a hypothetical even ∞ 2 ∞ 2 spectral triple (퐶 (푆 ),퐻,퐷,훾) with 퐷훬 = 푃훬퐷 equals that of (퐶 (푆 ),퐻,퐷푆2 ,훾). The second, strictly weaker constraint is that 퐷훬 anticommute with 훾훬, which is necessary for ∞ 2 퐷훬 to possibly correspond to part of an even spectral triple (퐶 (푆 ),퐻,퐷,훾). These constraints lead to the parametrizations

−푃 0 푅 푆 퐷 = ( ) or 퐷 = ( ), 훬 0 푃 훬 −푆 −푅

∗ −1 −1 respectively, where 푃 is positive (ensuring that 퐷훬 = 퐷훬 and 퐷훬|퐷훬| = 푃훬퐷푆2 |퐷푆2 | ) ∗ and 푅,푖푆 selfadjoint (ensuring that 퐷훬 = 퐷훬 and 퐷훬훾훬 = −훾훬퐷훬). The former parametrization is faster than the latter since both eigendecompositions of 푃 and the search for optimal 퐷훬 occur in a vector space of half the dimension. The geometries parametrized through 푃 are strictly a subclass of those parametrized through 푅,푆, hence we know that solutions arising in the first ensemble also exist in the second. Our simulations however show that to find the same optimal solutions in the 푅,푆 parametrization requires longer runtimes and much lower temperatures. This is because the larger configuration space takes longer to explore and lowers the relative fraction of the most optimal solutions. We have tested that the 푅,푆 simulations do not allow for additional, more optimal solutions than the 푃 parametrization, hence the results shown will all use the 푃 parametrization.

Results Tovisualize the results of our simulations we will again look both at averages over roughly 150 measurements near the minimum as well as at the actual numerical minimum of equation (4.9) that was encountered. If we look at the operators as heatmaps, with each pixel in heatmap representing one entry of the matrix, as in see Figure 4.2, we see that the average Dirac operator in the −푃 ⊕ 푃 parametrization commutes (up to numerical error) with 퐷푆2 . This simple structure of the simulated Dirac operators implies they are well described, quantitatively, by their spectrum. In Figure 4.3, we compare the measured eigenvalues

79 4.2. The Heisenberg relation in simulations

Figure 4.2: The average Dirac operator is almost entirely real, and completely diagonal.

Figure 4.3: Comparing the average eigenvalues, and the best case eigenvalues of the simulations with those of the sphere. We can see that the results differ considerably between odd and even 훬, but that neither agrees with the sphere.

80 4.2. The Heisenberg relation in simulations

with those of the sphere. The Figure shows results for spectral cutoffs of 훬 = 5,6, which showcases a clear difference between odd and even cutoffs.

The simulated Dirac operators are (up to numerical error) diagonal in an eigenbasis of 1 퐷푆2 , but the simulated eigenvalues are shifted up or down by roughly 2 . The direction of the shift appears dependent on the parity of the eigenvalue and of the cutoff 훬. That is to say, it seems we are be dealing with a bounded perturbation of 퐷푆2 with particularly simple structure.

In particular, the localized zeta function asymptotics (which measure at least volume and dimension) must agree for this perturbation and the sphere. When we have identified the numerical solutions analytically, in Section 4.3 below, we will show in Figure 4.7 how this fact is reflected by the finite parts of the spectrum obtained.

Results for the Heisenberg equation

The operators in Figure 4.2 arise from minimization of the Heisenberg constraint given 2 by the squared Hilbert-Schmidt norm ∥훿(푌 ,훾 ,퐷)∥ , so it is interesting to see whether 훬 훬 2 patterns arise in the corresponding matrix entries of 훿(푌훬,훾훬,퐷); we show these in Figure 4.4. Clearly, the simulations come close to fully letting 훿(푌훬,훾훬,퐷) vanish.

For the operator 퐷푆2,훬, however, the defect 훿(푌훬,퐷푆2,훬,훾훬) does not vanish and equals

(1 + 휆)(1 + 4휆) 훿(푌 ,퐷 2 ,훾 ) = − (퐸 + 퐸 )훾 (4.10) 훬 푆 ,훬 훬 2(1 + 2휆)2 휆 −휆

′ ′ where 퐸휆 projects onto the eigenspace corresponding to 휆 = max{휆 ∈ 휎(퐷) ∣ |휆 | ≤ 훬}; 1 1/푝 this is of norm ∼ 2 and of divergent (푂(훬 )) 푝-Schatten norm for 푝 < ∞ as 훬 → ∞.

For each 훬 considered, we found a 퐷훬 with ‖훿(푌훬,훾훬,퐷훬)‖2 ≈ 0 and in particular the constraint satisfied ‖훿(푌훬,훾훬,퐷)‖2 ≪ ‖훿(푌훬,훾훬,퐷푆2,훬)‖2. Additionally these optimal 퐷훬 seem to be quite simple and symmetric, and shows a remarkable consistency across different sizes, as shown in Figure 4.5. Since the matrix size (therankof 푃훬) grows as 푂(훬2) it is hard to obtain reliable results for larger 훬, however the results we obtained suggest that there might be a similar type of solution for all sizes, i.e. a compatible ′ ′ chain of finite size Dirac operators that might arise as 푃훬퐷 푃훬 for some 퐷 that solves (4.2) exactly. It is thus useful to supplement the numerical results with some analytic explorations.

81 4.2. The Heisenberg relation in simulations

(a) The exact sphere

(b) Average operator Figure 4.4: Heatmap plot of the Heisenberg relation for the operator parametrized through 푃 and the sphere for 훬 = 6. The uppermost plot shows the finite size defects in the sphere, while the lower plots show the defect generated by an averaged Dirac operator. 82 4.3. An alternative analytic solution to the Heisenberg relation

Figure 4.5: Average eigenvalues for the 4 smallest truncations of the sphere.

4.3 An alternative analytic solution to the Heisenberg relation

The simulations above suggest that, for finite 훬, there might be a class of operators 퐷 ∈ 퐵(푃 퐻) that lead to lower values of the constraint ∥훿(푌 ,훾 ,퐷)∥ than the truncations 훬 훬 훬 2 of 퐷푆2 do. Since the 퐷 that show up commute with 퐷푆2 and seem to be compatible across alternating choices of 훬 (see Figure 4.5), we are led to look analytically for a corresponding general solution of 훿(푌,훾,퐷) = 0 inside the commutant of 퐷푆2 . Let us denote by 풟 the space of selfadjoint operators with discrete spectrum that commute with 퐷푆2 and anticommute with 훾, that is, those of the form 푓(퐷) for some antisymmetric 푓 ∈ 퐶(ℝ,ℝ). Is there an analytic solution 퐷 ∈ 풟 to equation (4.9)? The Appendix 4.A exhibits the coefficients of the representation of 푌,훾,퐷 on 퐻 in the basis chosen for the simulations. Since the generators 푌푖 are laddering, i.e. band, matrices in this basis, the resulting version of equation (4.9) is easy to solve analytically.

It leads to the following recursion for the sequence 휇푙 of positive eigenvalues of 퐷, 1 3 labeled by the spinor momenta 푙 = 2 , 2 ,…,

2 2 2 2 2 휇푙 − 2푎푙휇푙휇푙−1 + 2푏푙휇푙+1휇푙 = 푎푙휇푙−1 + 푏푙휇푙+1 + 16푙 (1 + 푙)

2 2 where 푎푙 = (1+푙) (2푙−1), 푏푙 = 푙 (3+2푙) The corresponding recursion equation, with

83 4.3. An alternative analytic solution to the Heisenberg relation

휇−1/2 = 0, has the unique one-parameter solution 1 휇 = (푙 + ) + 푐sin(휋푙). 푙 2

That is to say, the unique one-parameter solution {퐷푐 ∣ 푐 ∈ ℝ} ⊂ 풟 to equation (4.9) is

퐷푐 = 퐷푆2 + 푐퐵,

−1 where the bounded and selfadjoint operator 퐵 equals cos(휋퐷푆2 )(퐷푆2 |퐷푆2 | ). Looking at this we see that it agrees with the Dirac operator we found in our simulations using the parametrization with 푃. In particular for 푐 = ±1/2 this agrees with the simulations with an even/ odd maximal eigenvalue, as shown in Figures 4.3 and 4.5.

Spectral triple axioms

For nonzero 푐, the full operator 퐷푆2 +푐퐵 does not satisfy the first-order axiom (condition 2 in the reconstruction theorem of [Con13]) because [[퐵,푌푖],푌푗] is not zero for all 푖,푗; 퐵, although pseudodifferential of order zero, is not an endomorphism of the spinor (1,∞) bundle. The defect [[퐵,푌푖],푌푗], however, is compact (it is in fact in 퐿 (퐻)). As we will see in the next subsection the boundary effects caused by the truncation to finite matrix sizes mask this difference and lead to violations of the first order axiomfor 퐷푆2 alone that are of the same order of magnitude as the violation for 퐵.

Boundary defects As mentioned in the introduction, replacing a solution 푌,훾,퐷 of the one-sided higher Heisenberg equation (4.2) by 푌훬,훾훬,퐷훬 leads to a nontrivial defect.

For operators in 풟 this introduces an additional term in 훿(푌훬,훾훬,퐷) that is not present in 훿(푌,훾,퐷). This term is a multiple of the 훾 operator projected onto the highest eigenspace of |퐷|, where the coefficient equals (1 − 2푙) 푐 휇2 + 휇 (휇 + 2휇 ) − 1, 푙 푙 16푙2 푙−1 푙−1 푙

1+9푙2+6푙3 with 푐푙 = 16푙2(푙+1)2 . In terms of the parameter 푐, above, this means that additionally to solving 훿(푌,훾,퐷푐) = 0 we can solve the finite-cutoff equation 훿(푌훬,훾훬,퐷푐) = 0 (uniquely) by 푐 = 푠(훬)/2, where the sign 푠(훬) equals the parity cos(휋휆max) of the highest eigenvalue 1 휆max of |퐷푆2 | below 훬 (so that the corresponding eigenvalue of 푐퐵 is + 2 ): see Figure 4.6.

84 4.3. An alternative analytic solution to the Heisenberg relation

Figure 4.6: Spectra of 퐷푆2 and 퐷푆2 + 푐퐵 for even/odd 휆max.

The finite-rank operators 퐷푐,훬, for any 푐 ∈ ℝ, never satisfy the first-order condition that [[퐷훬,푌훬],푌훬] should vanish. For 퐷 = 퐷푆2 , for which the defect vanishes in the strong limit 훬 → ∞, there is a boundary defect of asymptotically constant norm and of unbounded trace norm as 훬 → ∞ – that is, ‖[[퐷푆2,훬,푌훬],푌훬‖ ∼ 1 and ‖[[퐷푆2,훬,푌훬],푌훬‖1 = 푂(훬).

As mentioned above, the defect [[퐵훬,푌훬],푌훬] does not vanish in the strong limit 훬 → ∞. However, precisely when 푐 = 푠(훬)/2 as above, the highest-order terms of [[푐퐵훬,푌훬],푌훬] and [[퐷푆2,훬,푌훬],푌훬] cancel each other. As a result, the defect [[퐷푐,훬,푌훬],푌훬] is of norm 푂(훬−1) and trace norm 푂(1). In this sense it is hard to computationally detect the fact that 퐷푆2,훬 comes from a spectral triple while (for nonzero 푐) 퐷푐,훬 does not.

Visualisation: a locally isometric graph embedding

The operator 퐷푆2 + 푐퐵 seems, at least on 푃훬퐻 for finite 훬, to come closer to satisfying the higher Heisenberg equation (4.2) than the original solution 퐷푆2 does, and neither its spectral asymptotics nor the first-order equation allow us to discern at the finite

85 4.3. An alternative analytic solution to the Heisenberg relation

(a) Dimension estimate (b) Volume estimate

Figure 4.7: Finite-rank estimates of the spectral asymptotics of 퐷푆2 and 퐷푆2 +푐퐵, where 1 푐 is ± 2 for 훬 even/ odd. level that it does not form a commutative spectral triple with 퐶∞(푆2) and 퐿2(푆2,푆). This suggests to pretend it does arise from a spin geometry and to compare at least the resulting metric on 푆2 to the standard one. First of all, since the difference 퐵 is bounded, the Weyl asymptotics agree in the sense that the first zeta residues must be equal in both value and argument. This isalready detectable at the truncated level, e.g. using the finite-rank zeta approximations from [Ste19b]: see Figure 4.7. One interesting feature of these figures is that the dimension and volume estimators converge faster for the 퐷푆2 + 푐퐵 operator than for the truncated sphere. The asymptotics corresponding to total scalar curvature, however, are completely dif- 2⋅4휋 −4휋 ferent for 퐷푆2 + 푐퐵 (the corresponding residue is not 6⋅4휋 but rather 6⋅4휋 ) because it is 2 the 푂(푡−푛/2+1) term in the asymptotics of tr푒−푡퐷 and is therefore highly sensitive to bounded shifts when the dimension equals 2. Chapter 3 developed a method to associate a finite metric space to ‘operator system ∞ spectral triples’ (푃훬퐶 (푀)푃훬,푃훬퐻,푃훬퐷). The method, briefly, is as follows.

def 2 2 • The embedding 푌 is used to define the dispersion 훿(푣) = ∑푖⟨푣,푌푖 푣⟩ − ⟨푣,푌푖푣⟩ of a vector 푣 ∈ 퐻, which measures the degree to which the corresponding vector state is localized. In the commutative case, this corresponds to the statistical variance of the position variable 푌 under the measure induced by 푣.

86 4.4. Conclusions

• One iteratively constructs a reasonably dense (finite) set of localized states by minimizing the dispersion, combined with an electrostatic repulsion to avoid repetition. Up to the distortion induced by imperfect localization, this results in the commutative case in generating a set of roughly equidistributed points on the underlying manifold.

• The Connes distance formula (4.3) is used to calculate the distance between the generated states, in order to obtain a metric graph. In the commutative case, those distances correspond to the Kantorovich-Wasserstein distance between the measures induced by the localized states, which reduces to the geodesic distance in the limit of perfect localization.

• The SMACOF algorithm is utilized to embed the obtained metric graph in ℝ푛 in an asymptotically locally isometric way. This means that, asymptotically as 훬 → ∞, the embedding is pressured to be Riemannian.

For 퐷푆2 and 퐷푆2 + 푐퐵, this procedure yields the images displayed in Figure 4.8. There we can see that the embedded points for 퐷푆2 , the left hand plot, lie outside the shaded 2 푆 that is included for reference, while on the other hand the points for 퐷푆2 +푐퐵, in the right hand plot, lie inside the shaded 푆2. The transparency of the dots increases with distance to the viewer. Both embeddings show some deviation from the sphere: for 퐷푆2 , the radii of the embedded points lie in [1.06,1.12], with an average of 1.09; for 퐷푆2 + 푐퐵, in [0.94,0.98] averaging 0.96.

4.4 Conclusions

In this chapter we explored the behaviour of the truncated one-sided higher Heisenberg relation in dimensions 1 and 2. In the one-dimensional case the simulations yielded the expected result, showing that the truncation of the Dirac operator on the circle is closest to solving the corresponding truncated relation. The two-dimensional version of the truncated Heisenberg relation, however, lead to a new minimum that differs from (but commutes with) the truncated Dirac operator on the sphere. We found 1 analytically that this numerical minimum corresponds to the truncation at 푐 = ± 2 of a new one-parameter family 퐷푐 = 퐷푆2 + 푐퐵 of exact solutions to the non-truncated higher Heisenberg equation. While these bounded perturbations 퐷푐 of 퐷푆2 satisfy most conditions of the reconstruction theorem, they fail to satisfy the first-order condition. Unlike many other geometric properties, however, this defect turns out to not be detectable at the truncated level.

87 4.4. Conclusions

(a) 퐷푆2 (b) 퐷푆2 + 푐퐵

Figure 4.8: Locally almost-isometric embeddings corresponding to 퐷푆2 and 퐷푆2 + 푐퐵, with shaded 푆2 for reference

An interesting comparison here is the case of the four-dimensional version of the higher Heisenberg relation. That relation is solved not only by the four-sphere, but also by an additional, genuinely non-commutative, spectral triple, the Connes-Landi sphere [CL01]. This similarity invites the question whether the Heisenberg relation might invite more freedom the larger the dimension becomes.

There are many interesting extensions of this work waiting to be explored. In particular the Heisenberg relation needs to be understood in more detail. It is unclear how its one-sided version behaves in higher dimensions and, just as importantly, when more freedom is allowed for the parameter 푌. Our results, seen in context with the Connes- Landi sphere, suggest that more conditions are required to ensure that we deal with truncations of genuine Dirac spectral triples. In addition, it would be interesting to explore the two-sided equation of the Heisenberg relation. In that context, allowing the embedding maps 푌 to vary as well as the Dirac operator enlarges the resulting ensemble to contain all spin manifolds of the dimensions considered. With additional conditions, this would be a solid basis for a spectral version of random geometry, which could be compared to and begin a dialogue with results in quantum gravity, such as those of dynamical triangulations [Lol98] and spinfoams [Per13].

88 4.A. Representation of 푌,훾,퐷푆2

4.A Representation of 푌,훾,퐷푆2

2 Let 푆 be the standard spinor bundle over 푆 , with Dirac operator 퐷푆2 , and let 푥,푦,푧 be the standard coordinate functions on 푆2 ⊂ ℝ3. Then, the Dirac-type spectral triple ∞ 2 2 2 given by (퐶 (푆 ),퐿 (푆 ;푆),퐷푆2 ) can be represented as follows. Let {|푙,푚⟩± ∣ 푙 ∈ ℤ≥0 + 1 2 ,−푙 ≤ 푚 ≤ 푙} be an orthonormal basis of the Hilbert space 퐻. Then, we represent the generators 푎 = 푥 − 푖푦 and 푏 = 푧 of the algebra 퐶∞(푆2), the grading 훾 of 푆 and the Dirac operator 퐷푆2 as follows:

√(푙 + 푚 + 1)(푙 − 푚) 푎|푙,푚⟩ = − |푙,푚 + 1⟩ ± 2푙(푙 + 1) ∓ √(푙 + 푚 + 1)(푙 + 푚 + 2) + |푙 + 1,푚 + 1⟩ 2(푙 + 1) ± √(푙 − 푚)(푙 − 푚 − 1) − |푙 − 1,푚⟩ , 2푙 ± 푚 푏|푙,푚⟩ = |푙,푚⟩ ± 2푙(푙 + 1) ∓ √(푙 − 푚 + 1)(푙 + 푚 + 1) + |푙 + 1,푚⟩ 2(푙 + 1) ± √(푙 − 푚)(푙 + 푚) + |푙 − 1,푚⟩ , 2푙 ±

훾|푙,푚⟩± =|푙,푚⟩∓ , 1 퐷 2 |푙,푚⟩ = ± (푙 + )|푙,푚⟩ . 푆 ± 2 ± This representation was chosen to align well with that of [Dab+05]. We then write the 푏 푎 matrix 푌 as ( ). 푎∗ −푏

89

Part II

Schatten classes for Hilbert modules over commutative 퐶∗-algebras

Chapter 5

Introduction

Words can never trace out all the fibers that knit us to the old.

George Eliot, Letter to Charles Bray, Christmas Day, 1858

The trace is a fundamental and highly versatile invariant of operators on Hilbert spaces. In many applications, however, one is rather concerned with continuous families of such operators. From the perspective of Gelfand duality, the natural framework for such continuous families is that of Hilbert C∗-modules over an abelian base. The present study provides a systematic construction of trace and Schatten classes in this setting.

The finite-rank trace The ∗-algebra 푀(퐴) of finite matrices over a C∗-algebra 퐴 comes naturally equipped with a positive linear map

tr∶ 푀(퐴) → 퐴, (푎푖푗)푖푗 → ∑푎푖푖, 푖 which clearly commutes with the entrywise lift of linear maps between C∗-algebras, and is cyclic if and only if the algebra 퐴 is commutative.

93 5. Introduction

If 퐸 is a finitely generated projective Hilbert 퐴-module, any compact adjointable en- domorphism of 퐸 can be represented as an element of 푀(퐴) by a choice of isomor- phism between 퐸 and a complemented submodule of 퐴푛. Whenever 퐴 is commutative, cyclicity implies that the trace of the resulting matrix is invariant under the choice of isomorphism and so constitutes the 퐴-valued (Hattori-Stallings) trace on End퐴(퐸). Chapter 6, below, introduces a robust framework that generalizes the construction to countably generated Hilbert modules.

Continuous families of Schatten-class operators Given a finite-rank Hermitian vector bundle 푉 over a locally compact Hausdorff space 푋, Gelfand duality and the Serre-Swan theorm imply that the finite-rank trace is just the fiberwise trace tr∶ 훤(End푉) → 퐶0(푋) of continuous sections of the endomorphism bundle. If 푉 → 푋 is instead a continuous field of separable Hilbert spaces, then there is stilla trace map on the fiberwise trace classes. The challenge is then to unify these fiberwise trace classes in such a way as to yield a 퐶0(푋)-valued fiberwise trace that retains the fundamental properties of the trace class on a fixed Hilbert space 퐻. As a fundamental example, consider the trivial bundle 퐻 × 푋 → 푋 with fiber a fixed separable Hilbert space 퐻. The C∗-algebra of its adjointable endomorphisms consists of all ∗-stronglycontinuous, bounded families of bounded operators on 퐻. One wonders str whether this algebra, denoted 퐶b (푋,퐵(퐻)), contains two-sided ideals of “continuous Schatten-class operators” such that some or most of the usual theory of Schatten classes on Hilbert spaces is preserved. In order to ensure continuity of the trace, the least one should demand of such an ideal is that the pointwise Schatten norms lie in 퐶0(푋). On the other hand, the strongest reasonable condition at hand is that the families are themselves Schatten-norm con- 푝 tinuous, that is, that they lie in the Banach space 퐶0(푋,ℒ (퐻)). Through careful control over the relation between the Schatten classes on the standard Hilbert space 2 푙 (ℂ) and the complex matrix algebras 푀푛(ℂ), we are able in Theorem 6.2.11 to show str that these conditions do in fact coincide on 퐶b (푋,퐵(퐻)) and yield a two-sided ideal that is contained in the compact operators and is closed under its Banach norm. Kasparov’s stabilization theorem and unitary invariance of the Schatten norms allow us to easily generalize the trivial bundle example to all continuous fields of separable Hilbert spaces in Theorem 6.3.1. A further upshot of this approach is that much of

94 5. Introduction

the pointwise Schatten-class theory, including the Hölder-von Neumann inequality, carries over easily to the general case, cf. Theorem 6.3.4.

Frames and the fiberwise trace The theory of Schatten classes on Hilbert spaces is often mediated through the language of orthonormal bases and diagonalization. The approach of Section 6.2 shows that one may very well work with (standard normalized) frames [FL02; RT03] instead. This allows for straightforward generalization of the familiar formulas to the theory of Hilbert modules, and indeed, the result is what one would hope for: the fiberwise trace turns out to be the norm-convergent sum over the diagonal in a given frame, cf. Theorem 6.1.5. In the context of frames, it was earlier remarked in [DH94],[FL02, Proposition 4.8] that the obvious notion of Hilbert-Schmidt inner product is invariant both under the choice of frame and under the adjoint. That observation is supplied with the necessary context as a special case of our Schatten-class operators in Section 6.4.

Applications By the same principles as for the finite-rank trace, [Alm73] introduced, in the context of K-theory, the Fredholm determinant of endomorphisms of finitely generated modules over unital commutative algebras. As the Fredholm determinant is interesting in its own right, Section 7.1 uses the result on the Schatten classes to extend its definition and basic properties to the setting of countably generated modules over unital commutative C∗-algebras. A straightforward generalization of [Alm74] remains however elusive, due to the conceptual problems in generalizing the relevant category. Spectral geometry is the study of Riemannian manifolds 푀 via the spectra of differ- ential operators, such as spinc-Dirac operators 퐷, on 푀. An important example of 2 spectral invariant is the localized heat trace 푡 ↦ tr푓푒−푡퐷 ,푓 ∈ 퐶(푀). It determines the volume and total scalar curvature of 푀, is strongly related to the Atiyah-Singer Index Theorem [Gil95], and is able to describe classical field theories on 푀 through the spectral action principle [CM08, Chapter 11]. The first step in generalizing the above to unbounded Kasparov cycles, that is, certain (퐶0(푀),퐶0(푁))-Hilbert bimodules carrying a selfadjoint, regular, 퐶0(푁)-invariant unbounded operator 푆, is to make sense −푡푆2 −푧 of the expressions tr푓푒 and tr푓|푆| as elements of 퐶0(푁). Sections 7.2,7.3 embark on the necessary theory. Open questions for further research remain, particularly in the direction of zeta residues and compatibility with the interior product in unbounded KK-theory.

95 5.1. Preliminaries

The noncommutative case As briefly discussed in Chapter 8, the case of noncommutative C∗-algebras 퐴 is markedly different. We adapt a basic example from [FL02] to show that one is induced tocon- template not a trace ideal in the adjointable operators, but a trace ideal inside some smaller ∗-algebra dependent on a choice of Hilbert bimodule structure. In the non- commutative case, it is more natural not to work with an 퐴-valued “trace” but instead to consider generalizing the induced map tr∗ ∶ 퐻∗(푀푁(퐴),푀푁(푀)) → 퐻∗(퐴,푀) in Hochschild (or cyclic) homology, as initiated in [Nis91]. Although this cleanly ad- dresses the limitation of noncyclicity, it introduces additional complications to the issue of convergence. This in turn further reinforces the idea that one should investigate trace classes on bimodules instead. This direction of research, however, remains wide-open.

5.1 Preliminaries

We start by recalling the notion of frames on Hilbert 퐶∗-modules over 퐶∗-algebras. For basic definitions on Hilbert 퐶∗-modules, adjointable maps, tensor products, et cetera we refer to e.g. [Weg93]. We also recall the definition of unbounded Kasparov cycles [BJ83]. Keep in mind that we will, in later sections, specialize to the case of abelian 퐶∗-algebras, that is, those of the form 퐶0(푋) for 푋 a locally compact Hausdorff space. Hilbert 퐶∗-modules over such 퐶∗-algebras are given by the sections of continuous fields of Hilbert spaces; cf. [Tak79; DD63].

Frames on Hilbert C∗-modules We start this section by recalling two well-known results on Hilbert C∗-modules. For completeness we include their (short) proofs.

∗ Proposition 5.1.1. Let 퐴 be a C -algebra and let 퐸퐴 be a Hilbert 퐴-module. Then 퐸퐴퐴 is dense in 퐸, and the map 푢∶ 푣 ⊗퐴 푎 ↦ 푣푎, 퐸퐴 ⊗퐴 퐴 → 퐸퐴, is unitary.

Proof. Let {푒휆} be an approximate unit of 퐴. For 푣 ∈ 퐸퐴 one has ⟨푣 − 푣푒휆,푣 − 푣푒휆⟩ = ⟨푣,푣⟩ − 푒휆 ⟨푣,푣⟩ − ⟨푣,푣⟩푒휆 + 푒휆 ⟨푣,푣⟩푒휆, which converges to 0; thus, 푣 is the norm limit of the sequence 푣푒휆 ∈ 퐸퐴퐴. Clearly 푢 is isometric, so that its range is closed. As the range is dense, it must be surjective and, therefore, unitary.

96 5.1. Preliminaries

∗ Proposition 5.1.2. Let 퐴 and 퐵 be C -algebras and let 퐸퐴 be a Hilbert 퐴-module. If 휙 ∶ 퐴 → 퐵 is a ∗-homomorphism, then 퐵 is a left 퐴-module with the action 푎⋅푏 def= 휙(푎)푏. Then, there is an adjointable map

휙∗ ∶ 퐸퐴 → 퐸퐴 ⊗퐴 퐵, such that ⟨휙 푣,휙 푤⟩ = 휙(⟨푣,푤⟩ ). ∗ ∗ 퐵 퐴

Moreover, if 푇 ∈ ℒ(퐸퐴) then 휙∗푇 ∶= 푇⊗1 is an adjointable endomorphism of 퐸퐴 ⊗퐴 퐵, i.e. there is an induced map 휙∗ ∶ ℒ(퐸퐴) → ℒ(퐸퐴 ⊗퐴 퐵).

Proof. Recall that the map id∗ ∶ 푣 ⋅ 푎 ↦ 푣 ⊗퐴 푎, 퐸퐴 → 퐸퐴 ⊗퐴 퐴, is an isomorphism. We set 휙 def= (id⊗ 휙) ∘ id and find that for 푣 ∈ 퐸 ,푎 ∈ 퐴 one has ⟨휙 푣,휙 푣⟩ = ∗ 퐴 ∗ 퐴 ∗ ∗ 퐵 ⟨휙(푎),⟨푣,푣⟩ ⋅ 휙(푎)⟩ = 휙(푎)∗휙(⟨푣,푣⟩)휙(푎) = 휙(⟨푣 ⋅ 푎,푣 ⋅ 푎⟩).

A convenient basic fact about separable Hilbert spaces 퐻 is that they possess countable orthonormal bases {푒푖}. For one thing, this allows one to explicitly relate the compact operators 퐵0(퐻) to the direct limit 푀(ℂ) of matrix algebras over the base field ℂ and 1 in particular to treat the trace on 퐿 (퐻) using the series expression tr푇 = ∑푖 ⟨푒푖,푇푒푖⟩. The situation for Hilbert C∗-modules is slightly less straightforward: we will introduce the analogous but strictly weaker concept of a frame. In spite of the increased generality, we will see that frames provide sufficient flexibility to mimic standard treatments of trace-class operators on Hilbert spaces in the setting of Hilbert 퐶0(푋)-modules. ∗ ∗ Definition 5.1.3. Let 퐸퐴 be a countably generated Hilbert C -module over a C - algebra 퐴.A frame 푒 of 퐸퐴 is a sequence 푒푖 of elements of 퐸퐴, such that

∞ ⟨푣,푤⟩ = ∑⟨푣,푒푖⟩⟨푒푖,푤⟩, 푖=1 in norm, for all 푣,푤 ∈ 퐸퐴.

Such objects 푒 were called ‘standard normalized frames’ in [FL02]. Note that the subsequent treatment in [RT03], which is very similar to the definition used here, is ∗ different for non-unital C -algebras: we require the 푒푖 to be in 퐸퐴, not in the ‘multiplier module’ ℒ(퐴퐴,퐸퐴). This choice will later imply, for instance, that we do not consider the identity on the 퐶0(푋)-module 퐶0(푋), for noncompact spaces 푋, to be in the trace class.

97 5.1. Preliminaries

Example 5.1.4. Let 퐻 be a separable Hilbert space. Let 푃 ∈ 퐵(퐻) be a projection and 퐾 = 푃퐻 ⊂ 퐻. Then, if {푒푖} is an orthonormal basis of 퐻, we have ∞ ∞ ⟨푣,푤⟩ = ∑⟨푃푣,푒푖⟩⟨푒푖,푃푤⟩ = ∑⟨푣,푃푒푖⟩⟨푃푒푖,푤⟩, 푖=1 푖=1 for all 푣,푤 ∈ 퐾. That is, 푒 = {푃푒푖} is a frame of 퐾. Note that 푒 is not an orthonormal basis, because the 푒푖 might be neither orthogonal nor of norm 1.

Now, in the context of trace-class operators on a separable Hilbert space 퐻, frames ‘are as good as orthonormal bases’, in the sense of Corollary 5.1.6 below. Lemma 5.1.5. Let 푒,푓 be frames of a separable Hilbert space 퐻 and let 푇 be a bounded ∞ ∗ ∗ endomorphism of 퐻. Then, the series ∑푖=1 ⟨푇 푓푖,푇 푓푖⟩ converges if and only if ∞ ∑푖=1 ⟨푇푒푖,푇푒푖⟩ converges, and the limits agree.

∞ Proof. Assume that ∑푖=1 ⟨푇푒푖,푇푒푖⟩ < ∞. Then for finite subsets 퐹 ⊂ ℕ, ∞ ∞ ∗ ∗ ∗ ∗ ! ∗ ∗ ∑⟨푇 푓푖,푇 푓푖⟩ = ∑∑⟨푇 푓,푒⟩⟨푒,푇 푓⟩ = ∑∑⟨푒,푇 푓⟩⟨푇 푓,푒⟩ 푖∈퐹 푖∈퐹 푗=1 푖∈퐹 푗=1 ∞ ≤ ∑⟨푇푒,푇푒⟩. 푗=1

∞ ∗ ∗ Being bounded and monotone, the series ∑푖=1 ⟨푇 푓푖,푇 푓푖⟩ must converge. If we now switch 푇 and 푇∗, 푒 and 푓 and repeat the calculation, we see that the limits must in fact agree.

Corollary 5.1.6. Let 푒 be a frame of a separable Hilbert space 퐻. Then, for bounded 2 ∞ endomorphisms 푇 of 퐻, we have 푇 ∈ ℒ (퐻) whenever ∑푖=1 ⟨푇푒푖,푇푒푖⟩ < ∞. Moreover, 1 ∞ for 푇 ∈ ℒ (퐻), one has tr푇 = ∑푖=1 ⟨푒푖,푇푒푖⟩.

∞ Proof. For the first part, let 푓 be an orthonormal basis and note that ∑푖=1 ⟨푇푒푖,푇푒푖⟩ ∞ ∗ ∗ converges whenever ∑푖=1 ⟨푇 푓푖,푇 푓푖⟩ does by Lemma 5.1.5. That, then, is equivalent to 푇∗ ∈ ℒ2 (퐻), which in turn is equivalent to 푇 ∈ ℒ2 (퐻). For the second part, note that ℒ1 (퐻) = ℒ2 (퐻)ℒ2 (퐻). It is thus sufficient to con- sider an element 푇 = |푆|2 ∈ ℒ1 (퐻) with 푆 ∈ ℒ2 (퐻). Then, tr|푆|2 = tr|푆∗|2 = ∞ ∗ ∗ ∞ ∑푖=1 ⟨푆 푓푖,푆 푓푖⟩, which equals ∑푖=1 ⟨푆푒푖,푆푒푖⟩ by Lemma 5.1.5.

98 5.1. Preliminaries

We will see later that the Example 5.1.4 is a very good prototype for the general situation for Hilbert C∗-modules as well.

∗ 2 2 Example 5.1.7. Let 퐴 be a unital C -algebra and let 푙 (퐴) = 푙 ⊗ℂ 퐴퐴 be its standard 2 def module. Let {푒푖} be the standard orthonormal basis of 푙 and define {푒푖 = 푒푖 ⊗ 1퐴} in 푙2(퐴). Then clearly

∞ 2 ⟨푣,푤⟩ = ∑⟨푣,푒푖⟩⟨푒푖,푤⟩; (푣,푤 ∈ 푙 (퐴)). 푖=1

2 ∗ If 푃 ∈ ℒ(퐸퐴) is a projection (i.e. 푃 = 푃 = 푃) and 퐹퐴 = 푃(퐸퐴), then

∞ ∞ ⟨푃푣,푃푤⟩ = ∑⟨푃푣,푒푖⟩⟨푒푖,푃푤⟩ = ∑⟨푣,푃푒푖⟩⟨푃푒푖,푤⟩, 푖=1 푖=1 so {푃푒푖} is a frame of 퐹퐴.

∗ 2 Each frame of 퐸퐴 gives rise to a unitary 휃푒 ∶ 퐸퐴 → (휃푒휃푒 )푙 (퐴), as follows.

2 Proposition 5.1.8. Let 푒 be a frame of 퐸퐴. The frame transform 휃푒 ∶ 퐸퐴 → 푙 (퐴), given by

휃푒(푣) ∶= (⟨푒푖,푣⟩)푖 ∗ is adjointable, and its adjoint satisfies 휃푒 (푒푘 ⊗ 푎) = 푒푘 ⋅ 푎. Moreover, 푒 is a frame if and ∗ only if 휃푒 휃푒 = id퐸.

For the proof we refer to [RT03, Theorem 3.5].

Remark 5.1.9. Note that, unless 퐴 is unital, the converse does not hold: not every 2 isometry sending 퐸퐴 to a complemented submodule of 푙 (퐴) is induced by a frame. ∗ The frame elements 푒푖 would be given by 휃푒 (훿푖푗1퐴)푗, but the latter is not an element of 푙2(퐴) unless 퐴 is unital. This is where our treatment differs from that of [RT03], which works with frames (in the present sense) of the multiplier module ℒ(퐸퐴,퐴) instead.

Frames are compatible with ∗-homomorphisms. In particular, this means that char- acters of a commutative C∗-algebra map frames of Hilbert 퐶∗-modules to frames of Hilbert spaces.

99 5.1. Preliminaries

∗ Proposition 5.1.10. Let 퐴, 퐵 be C -algebras, 퐸퐴 a Hilbert 퐴-module and 휙 ∶ 퐴 → 퐵 a ∗-homomorphism. If 푒 is a frame of 퐸퐴 and if 휙 is surjective, then 휙∗(푒) is a frame of 퐸퐴 ⊗퐴 퐵.

def Proof. Consider 푓 = 휙∗(푒) = {휙∗(푒푖)}푖 ∈ 퐸퐴 ⊗퐴 퐵. Note that 휃푓(휙∗푣) = 휙∗휃푒(푣) inside 2 푙 (퐵), so that ⟨휃푓(휙∗푣),휃푓(휙∗푤)⟩ = 휙(⟨휃푒(푣),휃푒(푤)⟩) = 휙(⟨푣,푤⟩) = ⟨휙∗푣,휙∗푤⟩. Thus, with Proposition 5.1.8 it follows that 푓 is a frame.

Existence of frames Kasparov’s stabilization Theorem [Kas80] shows that example 5.1.7 describes the gen- eral unital (and, as we will see, the non-unital) case very well.

∗ Theorem 5.1.11. Let 퐴 be a unital C -algebra and let 퐸퐴 be a countably generated Hilbert 퐴-module. Then there exists a projection 푃2 = 푃 = 푃∗ in ℒ(푙2(퐴)) such that 2 퐸퐴 ≃ 푃(푙 (퐴)). In particular, 퐸퐴 possesses a frame.

For the proof we refer to [Kas80] (see also [Lan95, Theorem 6.2]). The non-unital case requires more effort, but the end result is the same. We referto [Kaa17, Section 2] for a proof.

Proposition 5.1.12 ([Kaa17, Proposition 2.6]). Let 퐴 be a C∗-algebra. Then, all countably generated Hilbert 퐴-modules possess a frame.

The standard module over abelian C∗-algebras For the rest of this subsection, let 푋 be a locally compact Hausdorff space. The C∗- ∗ algebra 퐶0(푋) is abelian – and, by Gelfand duality, all abelian C -algebras are of this type.

We will investigate the Hilbert 퐶0(푋)-module 퐶0(푋,퐻), for 퐻 a separable Hilbert space, which will later provide a useful tool in investigating the Schatten classes of operators on more general Hilbert 퐶0(푋)-modules. Westart with some basic definitions and results, whose proof we leave to the reader.

Definition 5.1.13. Let 푓 be a map from a locally compact topological space 푋 to a normed space. We say that 푓 vanishes at infinity whenever for all 휖 > 0, the set {푥 ∈ 푋∶ ∥푓(푥)∥ ≥ 휖} is compact.

100 5.1. Preliminaries

Definition 5.1.14. Let 푌 be a Banach space, equipped with its norm topology. The space 퐶0(푋,푌) consists of the continuous functions from 푋 to 푌 that vanish at infinity.

Proposition 5.1.15. Let 푌 be a Banach space and 푋 be a locally compact topologi- def cal space. Then, 퐶0(푋,푌) is a Banach space when equipped with the norm ∥푓∥ = ∥푓(푥)∥ 푓 ∈ 퐶 (푋,푌) 푥 ↦ ∥푓(푥)∥ 퐶 (푋) sup푥∈푋 . Moreover, for 0 , the map lies in 0 . Proposition 5.1.16. Let 퐻 be a separable Hilbert space. Then, the Banach space ∗ 퐶0(푋,퐻) has the structure of a Hilbert C -module when equipped with the 퐶0(푋)- def valued inner product ⟨푣,푤⟩(푥) = ⟨푣(푥),푤(푥)⟩퐻. ∗ Proposition 5.1.17. The Hilbert 퐶 -module 퐶0(푋,퐻) is unitarily equivalent to the ∗ tensor product 퐻 ⊗ℂ 퐶0(푋) of Hilbert 퐶 -modules.

Proof. Let {푒푖}푖 be an orthonormal basis of 퐻, and for 푣 ∈ 퐶0(푋,퐻) write 푣푖 ∶ 푥 ↦ ∞ ∗ ⟨푣(푥),푒푖⟩; this defines a sequence of functions in 퐶0(푋). Because ⟨푣,푣⟩ = ∑푖=1 푣푖 푣푖 converges pointwise, is positive and lies in 퐶0(푋), it converges in the norm of 퐶0(푋) by Dini’s theorem.

Consider then the 퐶0(푋)-linear map 휃∶ 퐶0(푋,퐻) → 퐻 ⊗ℂ 퐶0(푋),푣 ↦ ∑푖 푒푖 ⊗ 푣푖. This series converges in 퐻 ⊗ℂ 퐶0(푋) because

∥∑푒푖 ⊗ 푣푖∥ = ∥∑ ⟨푣푖,⟨푒푖,푒푗⟩푣푗⟩∥ = ∥∑⟨푣푖,푣푖⟩∥. 푖∈퐹 푖,푗∈퐹 푖∈퐹

Moreover, taking limits on both sides we see that 휃 is isometric.

Now, the map 푚∶ 퐻 ⊗ℂ 퐶0(푋) → 퐶0(푋,퐻),ℎ ⊗ 푎 ↦ ℎ푎, is isometric because

∗ ⟨∑ℎ푖 ⊗ 푎푖,∑ℎ푖 ⊗ 푎푖⟩ = ∑⟨ℎ푖,ℎ푗⟩푎푖 푎푗 = ⟨∑푚(ℎ푖 ⊗ 푎푖),∑푚(ℎ푖 ⊗ 푎푖)⟩. 푖 푖 푖푗 푖 푖

As 푚 inverts 휃 by orthonormality of the {푒푖}푖, we conclude that 휃 is a surjective 퐶0(푋)- linear isometry, so that it is adjointable (with adjoint 푚) and moreover unitary. There- fore, 퐶0(푋,퐻) is unitarily equivalent to 퐻 ⊗ℂ 퐶0(푋).

∗ Proposition 5.1.18. The C -algebra ℒ(퐶0(푋,퐻)) of adjointable endomorphisms ∗ of the Hilbert 퐶0(푋)-module 퐶0(푋,퐻) is isomorphic to the C -algebra of bounded, str ∗-strongly continuous maps from 푋 to 퐵(퐻), denoted 퐶b (푋,퐵(퐻)).

101 5.1. Preliminaries

Proof. Consider a point 푥 in 푋 as a character 휒 ∶ 퐶0(푋) → ℂ given by evaluation in 푥. 휒 ∶ 퐶 (푋,퐻) → 퐶 (푋,퐻)⊗ ℂ By Proposition 5.1.2 there is an adjointable map ∗ 0 0 퐶0(푋,퐻) whose image 휒∗(퐶0(푋,퐻)) can be canonically identified with 퐻. Accordingly, in terms of the map 휒∗ ∶ ℒ(퐶0(푋,퐻)) → ℒ(퐶0(푋,퐻)) given by 푇 ↦ 푇 ⊗ 1, we have canonically 휒∗ℒ(퐶0(푋,퐻)) ≃ 퐵(퐻).

Now take 푇 ∈ ℒ(퐶0(푋,퐻)) and a convergent sequence 푥푖 → 푥 in 푋. Let ℎ ∈ 퐻 and let 푣0 ∈ 퐶0(푋,퐻) be constant in a neighborhood of 푥 with 푣0(푥) = ℎ. Since 푇 ∗ is an adjointable endomorphism, both 푇푣0 and 푇 푣0 lie in 퐶0(푋,퐻), so that 푇푣0(푥푖) 푇푣 (푥) ∥(휒 푇)ℎ − (휒 푇)ℎ∥ → 0 converges to 0 in norm. That is to say, 푖∗ ∗ (and similarly for 푇∗). As 푥 and ℎ were arbitrary, we conclude that 푇 ∈ 퐶str(푋,퐵(퐻)). Moreover, if ‖푇(푥)‖ > 퐶 for some 푥 ∈ 푋 and 퐶 > 0, there is 푣0 as above with ‖푣0‖ ≤ 1 and ‖푇푣0‖ > 퐶 − 휖 for all 휖 > 0, so that we conclude that ‖푇‖ ≥ 퐶. Thus, if 푇 preserves 퐶0(푋,퐻) it str must lie in 퐶b (푋,퐵(퐻)). str ∗ Conversely, for all 푇 ∈ 퐶b (푋,퐵(퐻)) we have that 푥 ↦ 푇(푥)푣(푥) and 푥 ↦ 푇 (푥)푣(푥) lie ∗ in 퐶0(푋,퐻) for all 푣 ∈ 퐶0(푋,퐻). As ⟨푇푣,푤⟩(푥) = ⟨푇(푥)푣(푥),푤(푥)⟩ = ⟨푣(푥),푇 (푥)푤(푥)⟩ we conclude that the pointwise adjoint provides an adjoint of 푇 as an operator of 퐶0(푋,퐻). That is, all such 푇 are adjointable operators on 퐶0(푋,퐻).

General Hilbert C∗-modules over a commutative base

We now apply the above results to the case of general Hilbert 퐶0(푋)-modules. For a deeper topological understanding of such modules, see [Tak79; DD63].

Proposition 5.1.19. Let 퐸 be a Hilbert 퐶0(푋)-module. Then there exists a ∗-strongly str continuous projection 푃 ∈ 퐶b (푋,퐵(퐻)) such that 퐸 is isomorphic to the subset 훤0(푋,푃) ⊂ 퐶0(푋,퐻) of those elements ℎ of 퐶0(푋,퐻) satisfying ℎ(푥) ∈ ran푃푥. More- over, under this identification ℒ(퐸) is isomorphic to the set of those elements 푇 ∈ str 퐶b (푋,퐵(퐻)) that satisfy 푃푇 = 푇푃 = 푇.

Proof. If 푋 is compact, this is a direct consequence of Kasparov’s Stabilization Theorem (cf. Theorem 5.1.11) and Propositions 5.1.17 and 5.1.18. If not, consider 퐸 as an + 퐶0(푋) -module and note that the endomorphism 푃 from Theorem 5.1.11 lies in str + + 퐶b (푋 ,퐵(퐻)) in terms of the one-point compactification 푋 of 푋. But since (푒,푒) ∈ 퐶0(푋) for 푒 ∈ 퐸, the map 푃 must project into a subspace of 퐶0(푋,퐻). str For the last statement, let 푇 ∈ 퐶b (푋,퐵(퐻)) such that 푃푇 = 푇푃 = 푇. Then 푃푇푃 = 푇 so that 푇 preserves 훤0(푋,푃). Conversely, let 푆 ∈ ℒ(훤0(푋,푃)). Then, the map 푇 = 푆푃 is

102 5.1. Preliminaries

a composition of adjointable operators and is therefore an element of ℒ(퐶0(푋,퐻)) ≃ str 퐶b (푋,퐵(퐻)) using Proposition 5.1.18. Weclearly have 푃푇 = 푇푃 = 푇 so that the claim follows.

Note that the fibers 푃푥퐻 of 퐸 may vary quite wildly with 푥 ∈ 푋, as the following example shows. Example 5.1.20. Let 푈 ⊂ 푋 be open and let 푃 be the orthogonal projection onto a closed subspace 푉 ⊂ 퐻. Then, 퐸 = 퐶0(푈,푉) ⊂ 퐶0(푋,푉) is (in particular) a Hilbert 퐶0(푋)-module because the action of 퐶0(푋) on 퐶0(푋,푉) by pointwise multiplication preserves the subspace 퐶0(푈,푉). The fibers of 퐸 are 푉 for 푥 ∈ 푈 and {0}, for 푥 ∉ 푈.

The following example illustrates how the projections associated to such bundles behave.

Example 5.1.21. Let 푈 ⊂ 푋 be open and let 푉 = span푣0 ⊂ 퐻, with ‖푣0‖퐻 = 1. We will investigate the projection associated to the Hilbert 퐶0(푋)-module 퐸 = 퐶0(푈,푉) by Proposition 5.1.19. 2 Let {휂푖}푖 be a compactly supported partition of unity on 푈, so that ∑푖 휂푖 (푥) = 1푈(푥) 2 def for 푥 ∈ 푋. Then, 푣 = ∑푖 푣0휂푖 ⟨푣,푣0⟩ for all 푣 ∈ 퐸, so that {푒푖 = 푣0휂푖}푖 is a frame of 퐶0(푈,푉). In fact, any frame 푓 is of this form: we have 휂푖 = ⟨푣0,푓푖⟩.

Now, let {푤푖}푖 be an orthonormal basis of 퐻. Then, 휃푒, the frame transform of 푒, maps 푤 ∈ 퐶0(푈,푉) to ∑푖 푤푖 ⟨푒푖,푤⟩. ∗ Although the image 훤0(푋,휃푒휃푒 ) of 휃푒 (consisting of those elements 푤 of 퐶0(푋,퐻) for which ⟨푤푖,푤⟩ has support contained in that of 푒푖) is isomorphic to 퐶0(푈,푉) through ∗ the map 휃푒 , it looks decidedly different from the isomorphic subspace 퐶0(푈,푉) of ∗ 퐶0(푋,퐻) we started with. The associated projection 푃 = 휃푒휃푒 maps elements 푤 to

∑푖 푤푖휂푖 ∑푗 휂푗 ⟨푤,푤푗⟩.

In the previous Example, note that ‖푃(푥)‖ = 1 for 푥 ∈ 푈 and ‖푃(푥)‖ = 0 for 푥 ∉ 푈. str Thus, 푃 ∈ 퐶b (푋,퐵(퐻)) lies in 퐶푏(푋,퐵(퐻)) if and only if 푈 is clopen, that is, if and only if the bundle {(푥,ℎ) ∣ ℎ = 1푈(푥)ℎ} is locally trivial. This illustrates a general criterion for local triviality: str Remark 5.1.22. If 푃 ∈ 퐶푏(푋,퐵(퐻)) ⊂ 퐶b (푋,퐵(퐻)) is a projection, then each 푥 ∈ 푋 has a neighbourhood on which ∥푃(푦) − 푃(푥)∥ < 1, so that there exists a continuous ∗ map 푦 ↦ 푢푦 with 푃(푦) = 푢푦푃(푥)푢푦 by [Weg93, Proposition 5.2.6]. We conclude that the bundle {(푥,ℎ) ∣ ℎ ∈ 푃(푥)퐻} is locally trivial.

103 5.1. Preliminaries

Conversely, if the bundle is locally trivial, at least when the fibers are constant, we can choose 푃 to be norm continuous. More precisely, if we let 푝 ∶ 퐹 → 푋 be a locally trivial bundle of Hilbert spaces with separable, infinite-dimensional fiber 퐻. By [RW98, Corollary 4.79] 퐹 is isomorphic to 푋 × 퐻, and 훤0(퐹) is isomorphic to 퐶0(푋,퐻). We may thus choose 푃 = id퐻 ∈ 퐶푏(푋,퐵(퐻)) in Proposition 5.1.19. If instead 퐻 is finite-dimensional, then by the Serre-Swan theorem [GVF01, Theo- 푛 rem 2.10] there exists a projection 푝 ∈ 푀푛(퐴) with 푝퐴 ≃ 훤0(퐹). Thus, in Proposi- tion 5.1.19 we may choose 퐻 = ℂ푛 and 푃 = 푝.

104 Chapter 6

Schatten classes for Hilbert 퐶0(푋)-modules

We introduce a new definition of Schatten classes of endomorphisms on Hilbert mod- ∗ ules over separable abelian C -algebras 퐴 = 퐶0(푋), such that the resulting trace class is equipped with an 퐴-valued trace. A detailed investigation of the case of the standard module 퐻 ⊗ℂ 퐴 ≃ 퐶0(푋,퐻) leads us to establish several desirable properties that are familiar from the Hilbert space case.

6.1 Introduction and definition

When 퐴 is abelian, i.e. 퐴 ≃ 퐶0(푋) for some locally compact Hausdorff space 푋, each adjointable operator 푇 on a Hilbert 퐴-module 퐸퐴 can be localized by the pure states 휒 of 퐴 to yield a family 휒∗푇 of operators on the Hilbert spaces 휒∗퐸퐴. We will unify the 푝 푝 pointwise Schatten classes ℒ (휒∗퐸퐴) into a two-sided ideal ℒ (퐸퐴) ⊂ ℒ(퐸퐴) and 1 define an 퐴-valued trace on ℒ (퐸퐴). Assumption 6.1.1. We will require all of our Hilbert 퐴-modules to be countably generated in order to ensure access to frames using Proposition 5.1.12. We will denote the character space of a commutative 퐶∗-algebra 퐴 equipped with the ∗ ̂ ̂ ̂ weak topology by 퐴, so that 퐴 ≃ 퐶0(퐴) with 퐴 a locally compact Hausdorff space.

The least we should demand of ‘Schatten-class operators’ 푇 on 퐸퐴 is that their pointwise

105 6.1. Introduction and definition

푝 ̂ Schatten norm, i.e. the trace of |휒∗푇| , varies continuously with 휒 ∈ 퐴. In fact, this is the way to ensure that the ‘trace-class operators’ have traces with values in 퐴 and that the other Schatten classes respect this property in their pairing. The most we could reasonably demand, in contrast, is that the operators 휒∗푇 are continuous in Schatten norm with respect to some trivialization of 퐸퐴 (see Definition 6.2.1, below). It will turn out that these requirements, properly understood, are equivalent and yield a well-behaved Schatten class.

푝 Definition 6.1.2. The 푝-th Schatten class ℒ (퐸퐴) for 1 ≤ 푝 < ∞ is the space of all endo- 푝 ̂ 푝 morphisms 푇 ∈ ℒ(퐸퐴) for which the function tr|푇| ∶ 퐴 → ℝ ∪ {∞},휒 ↦ tr|휒∗푇| lies in 퐴.

The following proposition, familiar from the Hilbert space case, is immediate from the definition:

Proposition 6.1.3. Let 1 ≤ 푝 < ∞ and let 푇 ∈ ℒ(퐸퐴). Then

푝 푝 푝 1 푇 ∈ ℒ (퐸퐴) ⟺ |푇| ∈ ℒ (퐸퐴) ⟺ |푇| ∈ ℒ (퐸퐴).

Remark 6.1.4. Recall that Dini’s theorem, translated to the abelian C∗-algebraic con- text, states the following: if 푎푖 is a sequence of positive elements in 퐴, then ∑푖 푎푖 con- verges in norm if and only if the function 푥 ↦ ∑푖 푥(푎푖) is an element of 퐶0(푋) ≃ 퐴. This theorem plays a crucial role throughout, because it allows us to relate the fiberwise Schatten norms on bundles of Hilbert spaces to various expressions for the element 푝 푝 tr|푇| ∈ 퐴, for 푇 ∈ ℒ (퐸퐴).

푝 We will use the existence of frames (cf. Proposition 5.1.12), to relate ℒ (퐸퐴) to the Schatten classes ℒ푝 (푙2(퐴)) on the standard module 푙2(퐴) and to relate the trace tr|푇|푝 to a series expression in terms of (arbitrary) frames.

푝 ∗ 푝 2 Theorem 6.1.5. Let 푇 ∈ ℒ(퐸퐴). Then 푇 ∈ ℒ (퐸퐴) if and only if 휃푒푇휃푒 ∈ ℒ (푙 (퐴)) 푝 for any frame 푒 of 퐸퐴 with frame transform 휃푒. Equivalently, 푇 ∈ ℒ (퐸퐴) if and only ∞ 푝 푝 푝 ∗ if the series ∑푖=1 ⟨푒푖,|푇| 푒푖⟩ converges in norm; the limit equals tr|푇| = tr휃푒|푇| 휃푒 .

Proof. We start with the second part. As 휒∗(푒) is a frame of the Hilbert space 휒∗퐸퐴, 푝 ∞ 푝 ∞ 푝 one has tr|휒∗푇| = ∑푖=1 ⟨휒∗(푒푖),|휒∗푇| 휒∗(푒푖)⟩ = ∑푖=1 휒(⟨푒푖,|푇| 푒푖⟩). Hence if 푇 ∈ 푝 ∞ 푝 푝 ℒ (퐸퐴), the positive series ∑푖=1 ⟨푒푖,|푇| 푒푖⟩ converges in norm to an element tr|푇| of 퐴 by Dini’s theorem.

106 6.1. Introduction and definition

∞ 푝 Conversely, if the series ∑푖=1 ⟨푒푖,|푇| 푒푖⟩ converges in norm, then the limit provides an 푝 푝 푝 element tr|푇| ∈ 퐴 such that 휒(tr|푇| ) = tr|휒∗푇| for all characters 휒 of 퐴, so that 푝 푇 ∈ ℒ (퐸퐴). 휃∗휃 = |휃 푇휃∗|푝 = 휃 |푇|푝휃∗ For the first part, because 푒 푒 id퐸퐴 , we have 푒 푒 푒 푒 . Furthermore, note def ∗ 2 def that {푓푖 = 휃푒(푒푖)}푖 is a frame of 휃푒휃푒 푙 (퐴) ≃ 퐸퐴 so that the elements {ℎ푖 = 휒∗푓푖}푖 form ∗ 2 a frame of 휒∗휃푒휃푒 푙 (퐴) by Proposition 5.1.10. Now, by definition of the ℎ푖 we have 푝 ∗ 푝 푝 ∗ 푝 휒∗(휃푒|푇| 휃푒 )(ℎ푖) = 휒∗(휃푒|푇| 푒푖) and so ⟨ℎ푖,휒∗(휃푒|푇| 휃푒 )(ℎ푖)⟩ = 휒(⟨휃푒푒푖,휃푒|푇| 푒푖⟩) 푝 = 휒(⟨푒푖,|푇| 푒푖⟩). That is,

∞ ∞ 푝 ∗ 푝 ∗ 푝 tr휒∗휃푒|푇| 휃푒 = ∑⟨ℎ푖,휒∗(휃푒|푇| 휃푒 )(ℎ푖)⟩ = ∑휒(⟨푒푖,|푇| 푒푖⟩). 푖=1 푖=1

∞ 푝 푝 ∗ 푝 2 Thus, if ∑푖=1 ⟨푒푖,|푇| 푒푖⟩ converges to an element of 퐴, we have 휃푒|푇| 휃푒 ∈ ℒ (푙 (퐴)). 푝 ∗ 푝 2 ∞ 푝 Conversely, if 휃푒|푇| 휃푒 ∈ ℒ (푙 (퐴)) then the function 휒 ↦ ∑푖=1 휒(⟨푒푖,|푇| 푒푖⟩) for ̂ 휒 ∈ 퐴 lies in 퐶0(푋) ≃ 퐴. The series must then converge in norm by Dini’s theorem.

푝 푝 푝 푝 Corollary 6.1.6. Let 푆 ∈ ℒ(퐸퐴) and 푇 ∈ ℒ (퐸퐴). If |푆| ≤ |푇| , then 푆 ∈ ℒ (퐸퐴) and in particular tr|푆|푝 ≤ tr|푇|푝.

푝 푝 Proof. Let 푒 be a frame of 퐸퐴. Then ∑푖∈퐹 ⟨푒푖,|푆| 푒푖⟩ ≤ ∑푖∈퐹 ⟨푒푖,|푇| 푒푖⟩ for all finite 퐹 ⊂ ℕ; in particular, the left-hand side is Cauchy whenever the right-hand side is. By Theorem 6.1.5, this will suffice.

Remark 6.1.7. The above Corollary is weaker than the Hilbert space version, cf. Lemma 6.2.7(3) below. Instead, Corollary 6.2.16 below gives a stronger result but an additional assumption on 푆 is required. Note that this is the only point in the treatment of this chapter where such a difference between the Hilbert module and Hilbert space Schatten classes appears.

푝 The most straightforward road to analyzing the structure and properties of ℒ (퐸퐴) now lies open: we will investigate ℒ푝 (푙2(퐴)) and use the pullback by the frame trans- 푝 푝 2 forms to transfer its properties to ℒ (퐸퐴). It will turn out that ℒ (푙 (퐴)) is indeed very well-behaved, so that this allows us to recover many of the familiar properties of the Schatten classes of operators on Hilbert spaces.

107 6.2. The Schatten class on the standard module

6.2 The Schatten class on the standard module

∗ Let 퐻 be a separable Hilbert space and let 퐴 = 퐶0(푋) be an abelian 퐶 -algebra. Re- call from Section 5.1 that the Hilbert 퐴-module 퐻 ⊗ℂ 퐴 is isomorphic to 퐶0(푋,퐻) ̂ through the canonical isomorphism 휒∗(퐻 ⊗ℂ 퐴) ≃ 퐻 (with a character 휒 ∈ 퐴 always corresponding via the Gelfand transform to a point 푥 ∈ 푋). Recall, moreover, that its str endomorphism space is given by 퐶b (푋,퐻) with the fiberwise action given simply by 푇(ℎ)(푥) def= 푇(푥)(ℎ(푥)) for 푥 ∈ 푋. Because all fibers are identified canonically with 퐻, we may canonically compare the localizations of an operator 푇 ∈ ℒ(퐻 ⊗ℂ 퐴) between different fibers. This technically useful difference between 퐻 ⊗ℂ 퐴 and other Hilbert 퐴-modules will allow us to use topologies on 퐵(퐻) to define particular subsets of ℒ(퐻 ⊗ℂ 퐴). Most importantly,

Definition 6.2.1. The space of continuous Schatten-class operators on 퐶0(푋,퐻) ≃ 퐻⊗ℂ 푝 str 퐴 is the subspace 퐶0(푋,ℒ (퐻)) of 퐶b (푋,퐵(퐻)).

Note that where the requirement that tr|푇|푝 ∈ 퐴 is the least restrictive among reason- able criteria for a ‘Schatten-class operator’ 푇, as discussed above Definition 6.1.2, the condition of Definition 6.2.1 is arguably the most restrictive. However, we will prove that the demands are, in fact, equivalent, so that the 푝-th Schat- 푝 ten class ℒ (퐻 ⊗ℂ 퐴) on the standard module can be identified with the Banach space 푝 퐶0(푋,ℒ (퐻)) of continuous Schatten-class operators. This will later —specifically in Theorem 6.3.1—allow us to combine the properties that follow straightforwardly from either of the two definitions.

푝 푝 Remark 6.2.2. Clearly, one has 퐶0(푋,ℒ (퐻)) ⊂ ℒ (퐶0(푋,퐻)) because, for contin- uous maps 푇 from 푋 to ℒ푝 (퐻) and 푥,푦 ∈ 푋, we have ∥푇(푥) − 푇(푦)∥ ≥ |‖푇(푥)‖ − 푝 푝 ∥푇(푦)∥ | so that 푥 ↦ ‖푇(푥)‖ ∈ 퐴. 푝 푝

The ostensibly more restrictive definition of the continuous Schatten class has some ad- 푝 vantages to that of ℒ (퐶0(푋,퐻)). For instance, it is immediate from the definition that 푝 푝 퐶0(푋,ℒ (퐻)) is closed under addition, as we have not yet shown for ℒ (퐶0(푋,퐻)). Moreover, we can easily obtain a continuous version of the Hölder–von Neumann inequality:

1 1 1 푝 Proposition 6.2.3. Let 푝,푞,푟 ≥ 1 such that 푝 + 푞 = 푟 , and let 푆 ∈ 퐶0(푋,ℒ (퐻)) and 푞 푟 푇 ∈ 퐶0(푋,ℒ (퐻)). Then 푆푇 ∈ 퐶0(푋,ℒ (퐻)) and ‖푆푇‖푟 ≤ ‖푆‖푝 ‖푇‖푞.

108 6.2. The Schatten class on the standard module

Proof. For any 푥,푦 ∈ 푋 we have ∥푆푇(푥) − 푆푇(푦)∥ ≤ ∥푆(푥) − 푆(푦)∥ ‖푇(푥)‖ + ∥푆(푦)∥ ∥푇(푥) − 푇(푦)∥ , 푟 푝 푞 푝 푞 by the Hölder–von Neumann inequality [Sim05, Theorem 2.8]. Weconclude that 푆푇 is 푟 continuous as a map from 푋 to ℒ (퐻). Moreover, since ‖푆푇(푥)‖푟 ≤ ‖푆(푥)‖푝 ‖푇(푥)‖푞 푟 for all 푥, the statement on the norms follows as well as the claim that 푆푇 ∈ 퐶0(푋,ℒ (퐻)).

푝 푝 We now identify a subset of 퐶0(푋,ℒ (퐻)) ⊆ ℒ (퐻 ⊗ℂ 퐴) (cf. Remark 6.2.2 for 푝 the latter inclusion) whose completion in the Banach norm of 퐶0(푋,ℒ (퐻)) is all 푝 푝 of ℒ (퐻 ⊗ℂ 퐴). This will show that ℒ (퐻 ⊗ℂ 퐴) coincides with the Banach space 푝 퐶0(푋,ℒ (퐻)). Moreover, the fact that this common subset consists of finite-rank 푝 operators, in the Hilbert module sense, allows us to show that ℒ (퐸퐴) ⊂ 풦(퐸퐴) in Theorem 6.3.1. ∗ Proposition 6.2.4. The finite-rank operators on 퐶0(푋,퐻) (in the Hilbert C -module 푝 sense) lie in 퐶0(푋,ℒ (퐻)).

Proof. Let 푇 = |푣⟩⟨푤| with 푣,푤 ∈ 퐶0(푋,퐻). Then, for 푥,푦 ∈ 푋, we have

∥(푇푥 − 푇푦)∥ ≤ ∥|푣푥 − 푣푦⟩⟨푤푦|∥ + ∥|푣푦⟩⟨푤푥 − 푤푦|∥ . 푝 푝 푝 Pointwise in 퐻, however, we have ∥|휉⟩⟨휂|∥ = ‖휉‖∥휂∥, so that norm continuity of 푣 푝 and 푤 finish the proof.

Lemma 6.2.5. Let 푉 ⊂ 퐻 be finite-dimensional and consider 퐶0(푋,퐵(푉)) as a subspace str ⟂ of 퐶b (푋,퐵(퐻)) by the map 푇 ↦ 푇⊕0 ∈ 퐵(푉)⊕퐵(푉 ) ⊂ 퐵(퐻). Then, all elements of 퐶0(푋,퐵(푉)) are finite rank operators on the Hilbert modules 퐶0(푋,퐻). In particular, 푝 we have 퐶0(푋,퐵(푉)) ⊂ 퐶0(푋,ℒ (퐻)) for all 1 ≤ 푝 < ∞.

1 2 푛 Proof. Let 푇 ∈ 퐶0(푋,퐵(푉)) and decompose so that 푇 = 푆|푇| . Then let {푒푖}푖=1 be an 1 2 orthonormal basis of 푉. Because ∑푖 |푒푖⟩⟨푒푖| = id푉, we see that 푇 = 푆∑푖 |푒푖⟩⟨푒푖||푇| = 1 2 ∑푖 |푆푒푖⟩⟨|푇| 푒푖|, where we denote for 푅 ∈ 퐶0(푋,퐵(푉)) by 푅푒푖 the element 푥 ↦ 푅(푥)푒푖 of 퐶0(푋,푉). We conclude that 푇 is of finite rank. Remark 6.2.6. By a theorem of Fell [Fel61, Theorem 4.1], compact operators of bounded rank on 퐶0(푋,퐻) automatically have continuous trace. See remark 6.3.2, below, for the link to the study of continuous-trace C∗-algebras.

109 6.2. The Schatten class on the standard module

Some properties of the Schatten classes on Hilbert spaces We assemble here some more or less well-known properties of the ordinary Schatten classes on 퐵(퐻). The purpose is to show that one can use the series that defines the trace of |푇|푝 to control the rate at which certain finite-rank approximations of 푇 will converge to 푇 in Schatten norm. Lemma 6.2.7. Let 푇 ∈ ℒ푝 (퐻).

1. For 1 ≤ 푝 < 2 one has ‖푇‖푝 = inf ∑∞ ‖푇푒 ‖푝, where the infimum is taken 푝 {푒푖}푖 푖=1 푖 over orthonormal bases {푒푖}푖 of 퐻. 푝 ∞ 푝 2. For 푝 ≥ 2, one has ‖푇‖ = sup ∑ ‖푇푒푖‖ , where the supremum is over 푝 {푒푖}푖 푖=1 orthonormal bases {푒푖}푖 of 퐻. 3. Let 푝 ≥ 2. For any bounded endomorphism 푆 ∈ ℒ(퐻), if |푆|2 ≤ 푇 for some 푝/2 푝 푝 푝/2 푇 ∈ ℒ (퐻), then 푆 ∈ ℒ (퐻) and ‖푆‖푝 ≤ ‖푇‖푝/2.

∗ Proof. Let {푒푖}푖 be an orthonormal eigenbasis of (the compact, normal operator) 푇 푇, ordered by decreasing of the corresponding eigenvalues {휆푖}. 푝 ∞ 푝/2 First note that tr|푇| = ∑푖=1 ⟨푇푒푖,푇푒푖⟩ . Any other orthonormal basis {푓푖 = 푈푒푖} of 퐻 is related to {푒푖} by some unitary operator 푈 ∈ 퐵(퐻). 푝/2 For 1 ≤ 푝 < 2, the function 푥 ↦ 푥 is concave on ℝ+. Thus, since

∗ 2 ⟨푇푓푖,푇푓푖⟩ = ∑⟨푓푖,푇 푇푒푗⟩⟨푒푗,푓푖⟩ = ∑휆푗 ∣⟨푒푗,푓푖⟩∣ , 푗 푗

푝 푝/2 2 푝 we find that ∥푇푓푖∥ ≥ ∑푗 휆푗 ∣⟨푒푗,푓푖⟩∣ = ⟨푓푖,|푇| 푓푖⟩. We conclude that

∞ ∞ 푝 푝 ∗ 푝 푝 ∑∥푇푓푖∥ ≥ ∑⟨푓푖,|푇| 푓푖⟩ = tr푈 |푇| 푈 = tr|푇| . 푖=1 푖=1

푝/2 For 푝 ≥ 2 the function 푥 ↦ 푥 is convex on ℝ+ and we find, mutatis mutandis in the argument as above that now

∞ ∞ 푝 푝 ∗ 푝 푝 ∑∥푇푓푖∥ ≤ ∑⟨푓푖,|푇| 푓푖⟩ = tr푈 |푇| 푈 = tr|푇| . 푖=1 푖=1

110 6.2. The Schatten class on the standard module

2 ∗ 푝 푝 푝 푝 For the final claim, if |푆| ≤ 푅 푅, one has ‖푆푒푖‖ ≤ ‖푅푒푖‖ so that ‖푆‖푝 ≤ ‖푅‖푝 = ∗ 푝/2 ‖푅 푅‖푝/2.

We will need the following Corollary in the proof of Lemma 6.2.9. Corollary 6.2.8. Let 푇 ∈ 퐵(퐻) and let 푒 be a finite-rank projection. For any 푝 ≥ 2 we have 푇푒 ∈ ℒ푝 (퐻) and, in fact, 푝 푝 ‖푇푒‖푝 ≤ tr푒|푇| 푒.

∗ Proof. As in Lemma 6.2.7, let 푒푇 푇푒 have eigenbasis {푔푖} with eigenvalues {휆푖}. Then, 2 2 for any 푣 ∈ 퐻, we have ⟨푇푒푣,푇푒푣⟩ = ⟨푒|푇| 푒푣,푣⟩ = ∑푗 휆푗|⟨푣,푔푗⟩| . In particular,

2 푝/2 푝/2 푝 |⟨푣,푔푗⟩| ⟨푇푒푣,푇푒푣⟩ = ‖푣‖ (∑휆푗 ) . 푗 ⟨푣,푣⟩

푝/2 By convexity of 푥 ↦ 푥 on ℝ+ for 푝 ≥ 2 we have

푝/2 푝−2 푝/2 2 푝−2 푝 ⟨푇푒푣,푇푒푣⟩ ≤ ‖푣‖ ∑휆푗 |⟨푣,푔푗⟩| = ‖푣‖ ⟨푣,푒|푇| 푒푣⟩. 푗 But from the Lemma it then follows that 푝 푝 푝−2 푝 푝 ‖푇푒‖푝 = sup∑∥푇푒푓푖∥ ≤ sup∑∥푓푖∥ ⟨푓푖,푒|푇| 푒푓푖⟩ ≤ tr푒|푇| 푒. {푓푖} 푖 {푓푖} 푖

With respect to a choice of orthonormal basis on 퐻, we can view ℒ푝 (퐻) as a completion of the direct limit Matℂ of finite matrix algebras Mat푛(ℂ) in the Schatten norm. This can be done ‘uniformly’, where the convergence of the limit is controlled by the trace, as we will show in Proposition 6.2.10 below. Given 푇 ∈ ℒ(퐻) and a sequence of increasing finite-dimensional subspaces 푃퐻 ⊂ 퐻, the operator 푃푇푃 converges to 푇 ∗-strongly as 푃 → id퐻. The following Lemma allows us to control the 푝-norm of the difference when increasing the rank of 푃 by one, such that 푃푇푃 → 푇 in 푝-norm precisely when 푇 ∈ ℒ푝 (퐻). Lemma 6.2.9. Let 푝 ∈ [1,∞) and let 푇 ∈ ℒ푝 (퐻). Let 푒 be a finite-rank projection in 퐻, let 푃 be a finite-rank projection with 푃푒 = 푒푃 = 0 and let 푄 = 푃 + 푒. Then,

1/2푝 1/2푝 ‖푄푇푄 − 푃푇푃‖ ≤ ‖푇‖1/2 (( 푒|푇∗|푝푒) + ( 푒|푇|푝푒) ) 푝 푝 tr tr

111 6.2. The Schatten class on the standard module

Proof. One has 푄푇푄−푃푇푃 = 푃푇푒+푒푇푒+푒푇푃 = 푒푇푄+푃푇푒, so that ‖푄푇푄−푃푇푃‖푝 1 2 2 ≤ ‖푒푇푄‖푝 + ‖푃푇푒‖푝. Now compose 푇 as 푇 = 푆|푇| with |푆| = |푇|. Then, by the 1/2 Hölder-von Neumann inequality, ‖푒푇푄‖푝 ≤ ‖푒푆‖2푝‖|푇| 푄‖2푝 and similarly ‖푃푇푒‖푝 1/2 1/2 1/2 1/2 ≤ ‖푃푆‖2푝‖|푇| 푒‖2푝. Moreover, ‖푃푆‖2푝 ≤ ‖푇‖푝 and ‖|푇| 푄‖2푝 ≤ ‖푇‖푝 , and as ∗ ∗ 1/2 ‖푒푆‖2푝 = ‖푆 푒‖2푝, we may now apply Corollary 6.2.8 directly to ‖푆 푒‖2푝 and ‖|푇| 푒‖2푝 in order to finish the proof.

푝 ̂ 푝 Identification of ℒ (퐻 ⊗ℂ 퐴) with 퐶0(퐴,ℒ (퐻)) It is a well-known fact that the finite-rank operators are dense in ℒ푝 (퐻) for any 푝, as can easily be seen from the spectral theorem for compact self-adjoint operators. This argu- 푝 ment, however, does not extend uniformly to self-adjoint 푇 ∈ 퐶0(푋,ℒ (퐻)) unless 푇 is continuously diagonalizable. The explicit, albeit apparently somewhat clumsy, result of Lemma 6.2.9 presents a solution to this problem. Namely, orthogonal projections str 푃 ∈ 퐵(퐻) can be lifted to constant projections 퐶b (푋,퐵(퐻)). This allows the Lemma str to be applied uniformly to all of 퐶b (푋,퐵(퐻)), as we do in Proposition 6.2.10 below. This will then provide the main ingredient of main result of this section, to wit, the 푝 푝 identification of ℒ (퐻 ⊗ℂ 퐴) with 퐶0(푋,ℒ (퐻)) (Theorem 6.2.11). str Proposition 6.2.10. Let 푇 ∈ 퐶b (푋,퐵(퐻)), let 푝 ∈ [1,∞) and assume that the function 푝 푝 tr|푇| ∶ 푥 ∈ 푋 ↦ tr|푇(푥)| is defined everywhere and lies in 퐴 = 퐶0(푋). def 푛 Let {푒푖}푖 be an orthonormal basis of 퐻 and let, for any 푛 ≥ 0, 푃푛 = ∑푖=1 |푒푖⟩⟨푒푖| def be the corresponding spectral projections. Then the operators 푇푛 = 푃푛푇푃푛 (as in 푝 Lemma 6.2.9) are elements in 퐶0(푋,ℒ (퐻)) and, for 푚 ≥ 푛, 푚 2푝 2푝−1 푝 푝 ∗ 푝 ‖푇푚 − 푇푛‖푝 ≤ 2 suptr|푇(푥)| ∑ ⟨푒푖,(|푇(푥)| + |푇 (푥)| )푒푖⟩ 푥∈푋 푖=푛+1

푝 Proof. Note that 푇푛 is a finite-rank operator and in particular 푇푛 ∈ 퐶0(푋,ℒ (퐻)), by Lemma 6.2.5. The result follows from Lemma 6.2.9 applied to the projections 푃 = 푃푛, 2푝 2푝−1 2푝 2푝 푄 = 푃푚 and 푒 = 푃푚 − 푃푛 and by the simple equality (푥 + 푦) ≤ 2 (푥 + 푦 ). str 푝 Theorem 6.2.11. Let 푇 ∈ 퐶b (푋,퐵(퐻)). Then 푇 ∈ 퐶0(푋,ℒ (퐻)) if and only if 푝 푇 ∈ ℒ (퐶0(푋,퐻)).

Proof. The implication ⇒ was already established in Remark 6.2.2. For the converse, str 푝 assume that 푇 ∈ 퐶b (푋,퐵(퐻)) and that 푥 ↦ tr|푇(푥)| lies in 퐶0(푋). Then, pick an

112 6.2. The Schatten class on the standard module

orthonormal basis {푒푖}푖 of 퐻 and write again 푇푛 = 푃푛푇푃푛 as in Proposition 6.2.10. 푝 ∗ 푝 ∞ 푝 Since 푥 ↦ tr|푇(푥)| = tr|푇 (푥)| is in 퐶0(푋), the series ∑푖=1 ⟨푒푖,|푇(푥)| 푒푖⟩ as well ∞ ∗ 푝 as the series ∑푖=1 ⟨푒푖,|푇 (푥)| 푒푖⟩ must converge uniformly on compact subsets of 푋 by Dini’s theorem. That, in turn, implies that

푛+푘 1/2푝 2푝−1 푝 푝 ∗ 푝 sup‖푇푛+푘(푥) − 푇푛(푥)‖푝 < [2 suptr|푇(푥)| ∑ ⟨푒푖,(|푇(푥)| + |푇 (푥)| )푒푖⟩] 푥∈푋 푥∈푋 푖=푛+1

푝 goes to zero for large 푛. Since 퐶0(푋,ℒ (퐻)) is a Banach space, the sequence 푇푛 thus 푝 converges (to 푇) in Schatten 푝-norm, so that 푇 ∈ 퐶0(푋,ℒ (퐻)). Remark 6.2.12. In terms of tensor products of Banach spaces, Theorem 6.2.11 trans- 푝 푝 lates to the statement that ℒ (퐻 ⊗ℂ 퐴) ≃ ℒ (퐻) ⊗휖 퐴, the injective tensor product. 푝 Corollary 6.2.13. The continuous Schatten class 퐶0(푋,ℒ (퐻)) forms a two-sided str ideal in 퐶b (푋,퐵(퐻)).

푝 ′ str Proof. Let 푇 ∈ 퐶0(푋,ℒ (퐻)) and let 푇 ∈ 퐶b (푋,퐵(퐻)). Then, for any basis {푒푖}푖 1 def def 2 of 퐻 the operators 푇푛 = 푃푛푇푃푛 (푝 ≥ 2) or 푇푛 = 푃푛푆푃푛|푇| 푃푛 (푝 < 2), with 푃푛 and 푆 as in Proposition 6.2.10, converge to 푇 in the continuous Schatten 푝-norm. Now, note that ∥푇′(푇 − 푇 )∥ ≤ ∥푇′∥‖푇 − 푇 ‖ and ∥(푇 − 푇 )푇′∥ ≤ ∥푇′∥‖푇 − 푇 ‖ ; thus, 푚 푛 푝 푚 푛 푝 푚 푛 푝 푚 푛 푝 ′ ′ ′ ′ 푝 푇 푇푛 and 푇푛푇 converge to 푇 푇 and 푇푇 , respectively, in the norm of 퐶0(푋,ℒ (퐻)).

Remark 6.2.14. This result does not follow directly from the fact that ℒ푝 (퐻) is an ideal of 퐵(퐻) equipped with a Banach norm such that the inclusion into 퐵(퐻) is continuous: each such ideal induces a two-sided ideal 퐶푏(푋,퐼) ⊂ 퐶푏(푋,퐵(퐻)), but str that is not necessarily an ideal of 퐶b (푋,퐵(퐻)). An easy counterexample is given by 퐼 = 퐵(퐻) itself.

푝 Corollary 6.2.15. The continuous Schatten class 퐶0(푋,ℒ (퐻)) is contained in the ∗ compact operators on the Hilbert 퐶 -module 퐶0(푋,퐻).

Proof. The operators 푇푛 of Proposition 6.2.10 are of finite rank in the Hilbert module sense, because they are contained in 퐶0(푋,퐵(푉)) for some finite-dimensional 푉 ⊂ 퐻 푝 so that we can apply Lemma 6.2.5. As 푇푛 → 푇 for 푇 ∈ 퐶0(푋,ℒ (퐻)) in the Schatten 푝-norm, 푇푛 → 푇 in operator norm as well. We conclude that 푇 is compact in the Hilbert module sense.

113 6.3. Properties of the Schatten classes on Hilbert modules

푝 The following slight strengthening of Corollary 6.1.6 to 퐶0(푋,ℒ (퐻)) now translates 푝 to ℒ (퐶0(푋,퐻)). str 푝 Corollary 6.2.16. If 0 ≤ 푆 ≤ 푇 ∈ 퐶b (푋,퐵(퐻)) and 푇 ∈ ℒ (퐶0(푋,퐻)) and addition- norm 푝 푝 푝 ally we have 푆 ∈ 퐶푏 (푋,퐵(퐻)), then 푆 ∈ ℒ (퐶0(푋,퐻)) and tr|푆| ≤ tr|푇| .

Proof. The operators 푆,푇 are pointwise compact, norm continuous (this is where we use the additional assumption on 푆) and positive, so that their individual eigenvalues 휆푘 (ordered decreasingly) are continuous ([Kat95, p. IV.3.5], see also Lemma 7.2.2 below).

Then, by the min-max theorem, the 푘’th singular value of 휒∗푆 is bounded by the 푘’th singular value of 휒∗푇, so that the same holds for their 푝’th powers. As the Schatten norm of 휒∗푇 is the sum of those 푝’th powers, which converges to a continuous function, 푝 the convergence must be uniform by Dini’s theorem. Thus, the series ∑푘 휆푘(푇) of 푝 elements of 퐴 is Cauchy, so that the series ∑푘 휆푘(푆) must be Cauchy as well. We conclude that 푥 ↦ ∥휒 푆∥ lies in 퐴. ∗ 푝 Remark 6.2.17. The additional assumption in Corollary 6.2.16 is necessary because the positive compact operators on a Hilbert 퐶∗-module, in contrast to those on a Hilbert space, may not necessarily form an order ideal: there is an additional continuity requirement on the (pointwise compact) localizations. In contrast, as in Corollary 6.1.6, the positive trace-class operators on a Hilbert 퐶∗-module do form an order ideal.

6.3 Properties of the Schatten classes on Hilbert modules

We now return to the general setup of countably generated Hilbert 퐶∗-modules over ∗ 2 commutative 퐶 -algebras. For the case of the standard module 푙 (퐴) with 퐴 = 퐶0(푋) Theorem 6.2.11 shows that ℒ푝 (푙2(퐴)) is a Banach space and a two-sided ideal of ℒ(푙2(퐴)) that is moreover contained in 풦(푙2(퐴)). These are very desirable properties for general, countably generated Hilbert 퐴-modules. Fortuitously, the existence of frames (Proposition 5.1.12) and the pull-back criterion of Theorem 6.1.5 allows us to 푝 easily establish equivalent properties of ℒ (퐸퐴) for all countably generated Hilbert 퐴-modules 퐸퐴. 푝 Theorem 6.3.1. The space ℒ (퐸퐴) is a two-sided ideal of ℒ(퐸퐴) that is contained in 풦(퐸퐴).

Proof. Choose a frame 푒 of 퐸퐴 and let 휙푒 be the ∗-homomorphism induced by the 2 ∗ 푝 frame transform: 휙푒 ∶ ℒ(퐸퐴) → ℒ(푙 (퐴)),푇 ↦ 휃푒푇휃푒 . Recall that 푇 ∈ ℒ (퐸퐴) if

114 6.3. Properties of the Schatten classes on Hilbert modules

푝 2 푝 and only if 휙푒(푇) ∈ ℒ (푙 (퐴)) by Theorem 6.1.5. By Theorem 6.2.11, then, ℒ (퐸퐴) 푝 is closed under finite linear combinations. Moreover, for 푆 ∈ ℒ(퐸퐴),푇 ∈ ℒ (퐸퐴), we 푝 2 푝 2 have 휙푒(푆푇) ∈ ℒ (푙 (퐴)) and 휙푒(푇푆) ∈ ℒ (푙 (퐴)) by Corollary 6.2.13. We conclude 푝 that ℒ (퐸퐴) is a two-sided ideal. 푝 ∗ 푝 2 2 Moreover, as in the previous paragraph, 푇 ∈ ℒ (퐸퐴) iff 휃푒푇휃푒 ∈ ℒ (푙 (퐴)) ⊂ 풦(푙 (퐴)) ∗ ∗ ∗ by Corollary 6.2.15. But if 휃푒푇휃푒 is compact, then the operator 푇 = 휃푒 휃푒푇휃푒 휃푒 is com- ∗ pact as well (essentially because 휃푒 (|푒푖⟩⟨푒푗|)휃푒 = |푒푖⟩⟨푒푗| for any 푖,푗 ∈ ℕ). Remark 6.3.2. Dixmier’s definition of continuous-trace C∗-algebras [Dix77, Chapter 4.5] applies to 풦(퐸퐴). By Theorem 6.3.1, the corresponding trace and Hilbert-Schmidt 1 2 classes agree with our ℒ (퐸퐴) and ℒ (퐸퐴) respectively. The projections satisfying Fell’s criterion are then the compact finite-rank ones, that is, those that correspond to finitely generated projective modules. Remark 6.3.3. In the light of Theorem 6.3.1, we have obtained an a fortiori method of determine whether a positive operator 푇 on a Hilbert C∗-module over an abelian 퐶∗-algebra is compact: it suffices that tr푇푝 lies in 퐴 for some 푝 ≥ 1. Compare e.g. the proof of [KS17, Proposition 7] and [ibid., Remark 6] to see that showing such compactness directly can be a nontrivial undertaking.

푝 Using Theorems 6.1.5 and 6.2.11, we can pull back the Banach norm on 퐶0(푋,ℒ (퐻)) 푝 to ℒ (퐸퐴), and this turns out rather well: 1/푝 Theorem 6.3.4. The function ‖⋅‖ ∶ 푇 ↦ ∥tr|푇|푝∥ is a norm that turns ℒ푝 (퐸 ) 푝 퐴 퐴 푝 into a normed vector space. Moreover, for all 푇 ∈ ℒ (퐸퐴),

1. ‖푇‖ = sup ∥휒 푇∥ 푝 휒∈퐴̂ ∗ 푝 ∗ 2. ‖푇 ‖푝 = ‖푇‖푝

3. ‖푆푇‖푝 ≤ ‖푆‖‖푇‖푝 for all 푆 ∈ ℒ(퐸퐴)

4. ‖푇‖ ≤ ‖푇‖푝

푞 푝 1 1 5. For 푝,푞,푟 ≥ 1, if 푆 ∈ ℒ (퐸퐴) and 푇 ∈ ℒ (퐸퐴) with 푝 + 푞 = 1/푟 then 푆푇 ∈ 푟 ℒ (퐸퐴) and ‖푆푇‖푟 ≤ ‖푆‖푞 ‖푇‖푝.

푝 Moreover, ℒ (퐸퐴) is a Banach space.

115 6.4. The Hilbert module of Hilbert–Schmidt operators

def ∗ 2 Proof. Let 푒 be a frame of 퐸퐴 and consider the projection 푃푒 = 휃푒휃푒 in ℒ(푙 (퐴)). ∗ 푝 Recall that the invertible ∗-homomorphism 휙푒 ∶ 푇 ↦ 휃푒푇휃푒 maps ℒ (퐸퐴) into the 푝 2 subspace 푃푒ℒ (푙 (퐴))푃푒 by Theorem 6.1.5. Moreover, elementary calculation shows 휙 ∣ 푝 that 푒 ℒ (퐸퐴) is an isomorphism of normed spaces. 푝 For the properties of the norm, recall that 휒(tr|푇|푝) = tr|휒 푇|푝 = ∥휒 푇∥ . Thus, the ∗ ∗ 푝 1/푝 1/푝 norm satisfies ∥tr|푇|푝∥ = sup ∥휒 푇∥ . Therefore, by the analogous proper- 퐴 휒∈퐴̂ ∗ 푝 ℒ푝 (퐻) ‖푇∗‖ = ‖푇‖ ‖푇‖ ≥ ‖푇‖ = ∥휒 푇∥ ‖푆푇‖ ≤ ties of , we see that 푝 푝, 푝 sup휒∈퐴̂ ∗ and 푝 sup ∥휒 (푆)∥∥휒 푇∥ which is bounded by ‖푆‖‖푇‖ . 휒∈퐴̂ ∗ ∗ 푝 푝 ∗ ∗ ∗ 푟 2 For property 5, recall that 휃푒푆푇휃푒 = 휃푒푆휃푒 휃푒푇휃푒 ∈ ℒ (푙 (퐴)) by Proposition 6.2.3, 푟 and so 푆푇 ∈ ℒ (퐸퐴) by Theorem 6.1.5. The norm inequality then follows from the Hölder–von Neumann inequality for operators on Hilbert spaces. 푝 Finally, to establish completeness of ℒ (퐸퐴) it is enough to prove that its pullback 2 푝 2 푝 2 to 푙 (퐴), which equals 푃푒ℒ (푙 (퐴))푃푒, is a closed subspace of ℒ (푙 (퐴)). But if 푝 2 푃푒푇푛푃푒 → 푇 in ℒ (푙 (퐴)) then 2 ‖푃푒푇푛푃푒 − 푃푒푇푃푒‖푝 = ‖푃푒(푃푒푇푛푃푒 − 푇)푃푒‖푝 ≤ ‖푃푒‖ ‖(푃푒푇푛푃푒 − 푇)‖푝 → 0 as 푛 → ∞, in virtue of the just-proved inequality 3. Hence, 푃푒푇푛푃푒 converges to 푃푒푇푃푒 ∈ 푝 2 푃푒ℒ (푙 (퐴))푃푒 as desired.

6.4 The Hilbert module of Hilbert–Schmidt operators

2 The Hilbert–Schmidt class ℒ (퐸퐴) is a somewhat special case among the Schatten classes, because the map 푇 ↦ tr푇∗푇 is a positive definite quadratic form. That is, it induces an inner product as we will now explore. 2 2 Definition 6.4.1. The pairing ⟨⋅,⋅⟩2 ∶ ℒ (퐸퐴) × ℒ (퐸퐴) → 퐴 is given by 1 ⟨푆,푇⟩ def= ∑ 푖푘 tr|푇 + 푖푘푆|2 2 4 푘∈ℤ/4ℤ

When viewed fiberwise, this is just the ordinary Hilbert–Schmidt inner product: 2 Proposition 6.4.2. For 푆,푇 ∈ ℒ (퐸퐴) and a character 휒 of 퐴 the pairing ⟨푆,푇⟩2 ∗ ∞ satisfies 휒(⟨푆,푇⟩2) = tr((휒∗푆) 휒∗푇). Moreover, the series ∑푖=1 ⟨푆푒푖,푇푒푖⟩ converges in norm to ⟨푆,푇⟩2 for any frame 푒 of 퐸퐴.

116 6.4. The Hilbert module of Hilbert–Schmidt operators

Proof. Since 휒∗ is a homomorphism, the first part follows from the fact that tr휒∗(|푇 + 푖푘푆|2) = 휒(tr|푇+푖푘푆|2) by the polarization identity for the fiberwise Hilbert–Schmidt inner product. Since 1 ⟨푆푒 ,푇푒 ⟩ = ⟨푒 ,푆∗푇푒 ⟩ = ∑ 푖푘 ⟨푒 ,|푇 + 푖푘푆|2푒 ⟩, 푖 푖 푖 푖 4 푖 푖 푘∈ℤ/4ℤ the second part follows from Theorem 6.1.5. 2 Corollary 6.4.3. The pairing ⟨⋅,⋅⟩2 on ℒ (퐸퐴) is non-degenerate and sesquilinear.

Next, because 퐴 is commutative 퐸퐴 is automatically an 퐴-bimodule. In fact, there is a def ∗-homomorphism 휌∶ 퐴 → ℒ(퐸퐴) given by 휌(푎)(푣) = 푣 ⋅ 푎. This makes ℒ(퐸퐴) an 퐴-bimodule with 푎 ⋅ 푇 ⋅ 푏 = 휌(푎) ∘ 푇 ∘ 휌(푏) for all 푎,푏 ∈ 퐴 and 푇 ∈ ℒ(퐸퐴). 푝 Proposition 6.4.4. The 퐴-bimodule structure of ℒ(퐸퐴) restricts to ℒ (퐸퐴) for all 1 ≤ 푝 < ∞ and satisfies ‖푇 ⋅ 푎‖푝 ≤ ‖푎‖‖푇‖푝 as well as ‖푎 ⋅ 푇‖푝 ≤ ‖푎‖‖푇‖푝.

Proof. Since any ∗-homomorphism between 퐶∗-algebras is norm decreasing, this fol- lows from Theorem 6.3.1 since ∥푇휌(푎)∥ ≤ ∥휌(푎)∥‖푇‖ ≤ ‖푎‖‖푇‖ . 푝 푝 푝

All this leads to the following result for the case that 푝 = 2: 2 Proposition 6.4.5. With the above right 퐴-action and the inner product ⟨⋅,⋅⟩2, ℒ (퐸퐴) becomes a Hilbert 퐴-module.

Proof. Note that 휒∗(푇∘휌(푎)) = 휒(푎)휒∗푇 for all 푎 ∈ 퐴 and characters 휒 of 퐴. Thus, with Proposition 6.4.2, the inner product is 퐴-sesquilinear. All that is left to show, therefore, 2 is that ℒ (퐸퐴) is complete. That, however, was proven already in Theorem 6.3.1. 2 Proposition 6.4.6. Let 퐻 be a separable Hilbert space. Then, ℒ (퐻 ⊗ℂ 퐴) is isomor- 2 phic, as a Hilbert 퐴-module, to ℒ (퐻) ⊗ℂ 퐴.

2 Proof. Under the isomorphism 퐻 ⊗ℂ 퐴 ≃ 퐶0(푋,퐻), ℒ (퐻 ⊗ℂ 퐴) is mapped isomet- 2 rically onto 퐶0(푋,ℒ (퐻)) by Theorem 6.2.11. Under this identification, the Hilbert ∗ 2 ∗ 퐶 -module structure of ℒ (퐻 ⊗ℂ 퐴) coincides with the canonical Hilbert 퐶 -module 2 2 structure on 퐶0(푋,ℒ (퐻)) induced by the inner product on the Hilbert space ℒ (퐻). 2 2 ∗ Thus, we have ℒ (퐻 ⊗ℂ 퐴) ≃ 퐶0(푋,ℒ (퐻)) as Hilbert 퐶 -modules. Now, invoke 2 2 once again the isomorphism ℒ (퐻) ⊗ℂ 퐴 ≃ 퐶0(푋,ℒ (퐻)), this time for the Hilbert space ℒ2 (퐻), to complete the proof.

117 6.5. The trace class and the trace

6.5 The trace class and the trace

1 We will refer to the ideal ℒ (퐸퐴) ⊂ ℒ(퐸퐴) consisting of those operators 푇 for which tr|푇| is given by an element of 퐴, as the trace class. In the case of Schatten classes of Hilbert spaces, i.e. 퐴 = ℂ and 퐸퐴 = 퐻, it is customary to identify the trace class as the ideal generated by squares of elements of the Hilbert–Schmidt class, in order to relate the Hilbert–Schmidt inner product to a linear function, the trace, on ℒ1 (퐻). The situation here is completely analogous:

2 2 2 2 Proposition 6.5.1. Let ℒ (퐸퐴)ℒ (퐸퐴) = {푅푆 ∣ 푅 ∈ ℒ (퐸퐴),푆 ∈ ℒ (퐸퐴)} as a sub- 2 2 1 set of ℒ(퐸퐴). Then, ℒ (퐸퐴)ℒ (퐸퐴) = ℒ (퐸퐴).

2 2 1 Proof. The inclusion ℒ (퐸퐴)ℒ (퐸퐴) ⊂ ℒ (퐸퐴) is a direct consequence of the Hölder– 1 von Neumann inequality in Theorem 6.3.4(5)Conversely, let 푇 ∈ ℒ (퐸퐴). Then, 1 1 푇 = 푆|푇| 2 in the usual weak polar decomposition, with |푆| = |푇| 2 . By Proposi- 1 2 2 tion 6.1.3, 푆 and |푇| lie in ℒ (퐸퐴).

2 This furnishes us with a way to turn the bilinear map ⟨⋅,⋅⟩2 on ℒ (퐸퐴) into a linear 1 map tr on ℒ (퐸퐴) called the trace:

1 1 ∗ 2 Definition 6.5.2. The trace on ℒ (퐸퐴) is the map tr∶ 푇 ↦ ⟨푆 ,|푇| ⟩2, where 푇 = 1 푆|푇| 2 is the weak polar decomposition.

Proposition 6.5.3. The trace is well-defined. Moreover, let 푒 be a frame. Then, the series ∑푖 ⟨푒푖,푇푒푖⟩ converges in norm to tr푇.

2 ∗ Proof. Assume 푇 = 푅푆 with 푅,푆 ∈ ℒ (퐸퐴) and let 푒 be a frame. Then ⟨푅 ,푆⟩2 by ∗ definition equals ∑푖 ⟨푅 푒푖,푆푒푖⟩, which converges in norm by Proposition 6.4.2. As ∗ ⟨푅 푒푖,푆푒푖⟩ = ⟨푒푖,푇푒푖⟩, we have two expressions for tr푇: one independent of the de- composition 푇 = 푅푆 and one independent of the choice 푒 of frame. The proposition follows.

1 Corollary 6.5.4. Let 휒 be a character of 퐴 and let 푇 ∈ ℒ (퐸퐴). Then, tr휒∗푇 = 휒(tr푇).

Proof. Note that 휒(⟨푒푖,푇푒푖⟩) = ⟨휒∗푒푖,휒∗푇휒∗푒푖⟩. Since 휒∗푒푖 is a frame of 휒∗퐸퐴 the result then follows from Corollary 5.1.6.

118 6.5. The trace class and the trace

1 Corollary 6.5.5. For 푇 ∈ ℒ (퐸퐴), |tr푇| ≤ tr|푇|, as elements of 퐴. In particular, if 퐴 is unital |tr푇| ≤ tr|푇| ≤ ‖푇‖1 1퐴.

̂ Proof. For all 휒 ∈ 퐴 we have 휒(|tr푇|) = |tr휒∗푇| ≤ tr|휒∗푇| = 휒(tr|푇|) by the inequal- ity |tr푆| ≤ tr|푆| on ℒ1 (퐻) for Hilbert spaces 퐻. As the characters separate 퐴, the first statement follows immediately. The last statement follows from the inequality ∗ 푎 ≤ ‖푎‖1퐴 for positive elements of any unital C -algebra.

Now, we can finally show that the trace is cyclic. The standard approach is as follows:

1 1 Proposition 6.5.6. If 푆,푇 ∈ ℒ(퐸퐴) are such that 푆푇 ∈ ℒ (퐸퐴) and 푇푆 ∈ ℒ (퐸퐴), then tr푆푇 = tr푇푆.

Proof. Consider the value of a character 휒 ∈ 퐴̂on the difference tr푆푇 − tr푇푆 ∈ 퐴 and use Corollary 6.5.4 above:

휒(tr푆푇 − tr푇푆) = 휒(tr(푆푇)) − 휒(tr(푇푆))

= tr(휒∗(푆푇)) − tr(휒∗(푇푆))

= tr(휒∗(푆)휒∗(푇)) − tr(휒∗(푇)휒∗(푆))

1 We may now use the tracial property of the trace on ℒ (휒∗퐸퐴) (cf. [Sim05, Corollary 3.8]) and the fact that 휒 separates 퐴 to conclude the proof.

119

Chapter 7

Applications of Schatten classes

We investigate several applications of the just-developed theory of Schatten classes, to wit: the Fredholm determinant, the operator zeta functions, and a new definition of summability of unbounded Kasparov (퐴,퐵)-cycles when 퐵 is commutative.

7.1 The Fredholm determinant

As a first application of the above theory of Schatten classes for Hilbert modules over unital abelian 퐶∗-algebras, we consider the Fredholm determinant. Let 퐴 be a commu- ∗ tative C -algebra and let 퐸퐴 be a countably generated 퐴-module.

Definition 7.1.1. Let 퐺(퐸퐴) ⊂ ℒ(퐸퐴) be the set of bounded, invertible endomor- 퐸 +푇 푇 ∈ ℒ1 (퐸 ) phisms of 퐴 of the form id퐸퐴 , where 퐴 .

Note that 퐺(퐸퐴) is a group under the multiplication of ℒ(퐸퐴) because we have

(id+푇)−1 = id−푇(id+푇)−1,

1 and ℒ (퐸퐴) is an ideal. We will define the Fredholm determinant, first on the standard module, and then by pullback by a frame transform on general countably generated Hilbert 퐴-modules. The following definition of the Fredholm determinant on a Hilbert space 퐻 is well- known. See e.g. [Sim05, Chapter 3] for a brief discussion in the context of Lidskii’s theorem.

121 7.1. The Fredholm determinant

1 Definition 7.1.2. Let 푇 ∈ ℒ (퐻ℂ). Then, the Fredholm determinant of id+푇 is ∞ 푘 det(id+푇) def= ∑tr⋀ 푇. 푘=0

푘 푘 Recall that the series converges by the estimate ‖⋀ 푇‖1 ≤ ‖푇‖1/푘!.

Remark 7.1.3. Wehave |det(id+푇1)−det(id+푇2)| ≤ ‖푇1 − 푇2‖1 exp(‖푇1‖1 +‖푇2‖1 + 1), cf. [Sim05, Theorem 3.4]. Thus, 푇 ↦ det(id+푇) is a continuous function on ℒ1 (퐻).

The Fredholm determinant of id+푇 is invariant under conjugation of 푇 by partial isometries that commute with 푇: Lemma 7.1.4. If 푢∶ 퐻 → 퐾 is a partial isometry of Hilbert spaces, and 푇 ∈ ℒ1 (퐻) ∗ ∗ ∗ ∗ is such that 푢 푢푇 = 푇푢 푢 = 푇, then id+푢푇푢 ∈ 퐺(퐾ℂ) and in fact det(id퐾 +푢푇푢 ) = det(id퐻 +푇).

Proof. Note that we have

(id+푢푇푢∗)(id−푢푇(id+푇)−1푢∗) = (id+푢푇푢∗) − 푢(id+푇)푇(id+푇)−1푢∗ = id,

∗ so that id+푢푇푢 ∈ 퐺(퐾ℂ). Next, note that tr⋀푘 푢푇푢∗ = tr⋀푘 푢∗푢푇 = tr⋀푘 푇 so that indeed

∞ ∞ 푘 푘 ∗ ∗ det(id퐻 +푇) = ∑tr⋀ 푇 = ∑tr⋀ 푢푇푢 = det(id퐾 +푢푇푢 ). 푘=0 푘=0

Proposition 7.1.5. If 퐾 is a separable Hilbert space equipped with a frame 푒 and 1 ∗ 푇 ∈ ℒ (퐾), then det(1 + 휃푒푇휃푒 ) = det(1 + 푇) in terms of the corresponding frame 2 transform 휃푒 ∶ 퐾 → 푙 .

∗ Proof. Since 휃푒 휃푒 = id퐾 commutes with 푇 we can apply Lemma 7.1.4. ∗ Definition 7.1.6. Let 퐸퐴 be a countably generated Hilbert 퐶 -module over a unital ∗ 1 and abelian C -algebra 퐴. For 푇 ∈ ℒ (퐸퐴), the Fredholm determinant det(id+푇) of ̂ id+푇 is the function on 퐴 given by 휒 ↦ det(휒∗(id+푇)).

122 7.1. The Fredholm determinant

1 Proposition 7.1.7. Let 퐴 be unital and abelian as above. For 푇 ∈ ℒ (퐸퐴), the Fred- ̂ holm determinant lies in 퐴 ≡ 퐶(퐴) and as such we have 휒(det(id+푇)) = det(휒∗(id+푇)).

∗ Proof. Let 푒 be a frame of 퐸퐴. Note that det(휒∗(id+푇)) = det(휒∗(id+휃푒푇휃푒 )) for ̂ ∗ ∗ all 휒 ∈ 퐴 by Proposition 7.1.5. That is, det(id+푇) = det(id+휃푒푇휃푒 ). Now, 휃푒푇휃푒 ∈ 1 2 ℒ (푙 (퐴)) by Theorem 6.1.5. With Remark 7.1.3 we see that 휒 ↦ det(id+휒∗푆) is continuous whenever 푆 ∈ ℒ1 (푙2(퐴)). Thus, since 퐴 is unital (and thus 퐴̂compact) we find that det(id+푇) ∈ 퐶(퐴)̂ = 퐴.

Proposition 7.1.8. Let 푇 ∈ 퐶(푋,ℒ1 (퐻)) with 푋 = 퐴̂. Then we have

푘 푘 ⋀ (푇) ∈ 퐶(푋,ℒ1 (⋀ 퐻)).

푘 푘 In particular, one has det(id+푧푇) = ∑푘≥0 푧 tr⋀ (푇), and 푧 ↦ det(id+푧푇) is entire (as an 퐴-valued function on ℂ).

Proof. Let 퐴,퐵 ∈ ℒ1 (퐻) and note that ⋀푘+1(퐴) − ⋀푘+1(퐵) = (⋀푘(퐴) − ⋀푘(퐵)) ∧ 퐴 + ⋀푘(퐵) ∧ (퐴 − 퐵), which can be iterated to yield

푘−1 푘 푘 푚 푘−1−푚 ∥⋀ (퐴) − ⋀ (퐵)∥ ≤ ‖퐴 − 퐵‖1 ∑‖퐴‖1 ‖퐵‖1 . 1 푚=0

As a consequence, we see that ⋀푘(푇) ∈ 퐶(푋,ℒ1 (⋀푘 퐻)) whenever 푇 ∈ 퐶(푋,ℒ1 (퐻)). Moreover, we have the pointwise series expression

푘 푘 det(휒∗(id+푧푇)) = det(id+푧 휒∗푇) = ∑푧 tr⋀ (휒∗푇) 푘≥0

푘 as in [Sim05, Lemma 3.3]. Since tr⋀푘(휒 푇) ≤ ∥휒 푇∥ /푘! the series ∑ 푧푘 tr⋀푘(푇) ∗ ∗ 1 푘≥0 in 퐴 converges absolutely for all 푧 ∈ ℂ. This implies in particular that 푧 ↦ det(id+푧푇) is entire.

Remark 7.1.9. If 푓 ∈ 퐴 is invertible, then det(id−푓−1푇) = 0 in 퐴 if and only if 휒(푓) = ̂ 푓(푥) is a (nonzero) eigenvalue of 휒∗푇 for all 휒 ∈ 퐴 (corresponding to the point 푥 ∈ 푋). Proposition 7.1.10. The Fredholm determinant is multiplicative in the sense that

det(id+푇)(id+푆) = det(id+푇)det(id+푆).

123 7.1. The Fredholm determinant

Proof. This follows simply from the analogous property of the Fredholm determinant on Hilbert spaces, since

휒(det((id+푇)(id+푆))) = det(휒∗((id+푇)(id+푆)))

= det(휒∗(id+푇))det(휒∗(id+푆))) = 휒(det(id+푇)휒(det(id+푆)),

̂ for all id+푇, id+푆 in 퐺(퐸퐴) and all 휒 ∈ 퐴.

−1 Proposition 7.1.11. For 0 ≤ |푧| < ‖푇‖1 , the Fredholm determinant satisfies

∞ (−1)푛+1 det(id+푧푇) = exp(∑ 푧푛 tr푇푛), 푛=1 푛 and the series converges absolutely.

푛 푛 푛 Proof. Absolute convergence follows from the fact that ‖tr푇 ‖ ≤ ‖푇 ‖1 ≤ ‖푇‖1, so that the sum of the norms of the summands is bounded by log(1 + 푧‖푇‖1).

As the characters separate 퐴, it will suffice to prove the formula for the case 퐸퐴 = 퐻ℂ, where it is well-known. For |푧|‖푇‖1 < 1, by absolute convergence of the trace and Lidskii’s theorem,

∞ (−1)푛+1 ∞ 휆 (푇)푛 ∑ 푧푛 tr푇푛 = ∑(−1)푛+1푧푛 ∑ 푘 푛 푛 푛=1 푛=1 푘

= ∑log(1 + 푧휆푘(푇)), 푘 so that the exponential of the right-hand side equals det(id+푧푇) by Definition 7.1.2.

1 Proposition 7.1.12. Let 푇 ∈ ℒ (퐸퐴). Then id+푇 is invertible (that is, id+푇 ∈ 퐺(퐸퐴)) if and only if det(id+푇) ∈ 퐴 is invertible.

Proof. By Proposition 7.1.10, det(id+푇) is invertible whenever id+푇 is. For the converse statement, in view of Proposition 7.1.5 we may assume without loss of 2 generality that 퐸퐴 = 푙 (퐴) ≃ 퐶(푋,퐻) - where 푋 is compact by the assumption that 퐴 be unital. So, if det(id+푇) is invertible then it follows that id+휒∗푇 is invertible for all

124 7.2. The zeta function

휒 by [Sim05, Theorem 3.5b)]. Moreover, as for 푆 ,푆 ∈ 퐺(퐻) we have ∥푆−1 − 푆−1∥ ≤ 1 2 1 2 1 −1 −1 −1 ∥푆1 ∥∥푆2 ∥‖푆1 − 푆2‖1, the 퐵(퐻)-valued map 휒 ↦ (id+휒∗푇) lies in 퐶푏(푋,퐵(퐻))) whenever it is bounded. −1 −1 Now, for all eigenvalues 휆 ∈ 휎(휒∗푇) we have det(id−휆 휒∗푇) = 0 so that det(id−휆 푇) is not invertible in 퐴. Thus, if there exist a sequence 휆((휒푖)∗푇) ∈ 휎((휒푖)∗푇) converging −1 (in ℂ) to −1, the element det(id+푇) = lim푖 det(id−휆(푥푖) 푇) is contained in the closed set consisting of the non-invertible elements of 퐴, which contradicts the assumpion. ̂ We conclude that there exists 휇 > 0 with inf푛 |1 + 휆푛(휒∗푇)| > 휇 uniformly for 휒 ∈ 퐴. ∥( +휒 푇)−1∥ = |1 + 휆 (휒 푇)|−1 < 1/휇 푥 In particular, id ∗ sup푛 푛 ∗ for all . We conclude that −1 휒 ↦ (id+휒∗푇) is bounded and therefore continuous.

Corollary 7.1.13. The Fredholm determinant is a homomorphism from the group 퐺(퐸퐴) to the group of invertible elements of 퐴.

alg In particular, the map det extends the (matrix) determinant homomorphism 퐾1 (퐴) = 2 퐺퐿∞(퐴)/[퐺퐿∞(퐴),퐺퐿∞(퐴)] → 퐴 to all of 퐺(푙 (퐴)). Remark 7.1.14. In [Alm73; Alm74] it was shown that the Fredholm determinant is the unique additive invariant of endomorphisms 푇 ∶ 퐸 → 퐸 on finitely generated projective 퐴-modules. It is an interesting open question to see under which additional conditions this result extends to the countably generated Hilbert module context. Clearly, the Fredholm determinant discussed above gives rise to a additive map from a countably generated Hilbert 퐴-module 퐸 equipped with a trace-class operator 푇 to analytic 퐴- valued functions.

7.2 The zeta function

As a second application we consider zeta functions associated to positive Schatten class operators on a Hilbert module. Again, 퐴 is a commutative 퐶∗-algebra and we identify ̂ 퐴 = 푋 so that 퐴 ≅ 퐶0(푋).

푝 푧 Definition 7.2.1. Let 0 ≤ 푇 ≤ 1 ∈ ℒ (퐸퐴), for 푝 ≥ 1. For 푧 ∈ ℂ with ℜ푧 > 푝, define 푇 using the continuous functional calculus in ℒ(퐸퐴). Then, the associated zeta function is the function on the complex half-plane ℜ푧 > 푝 given by

휁(푧,푇) def= tr푇푧.

125 7.2. The zeta function

We will show that the zeta function is in fact holomorphic (in the sense of Banach space- valued holomorphic functions, see e.g. [Rud91, Definition 3.30]) on the defining half-plane. First, let us recall that the individual (fiberwise) eigenvalues of positive compact opera- tors are themselves continuous.

̂ Lemma 7.2.2. Let 0 ≤ 푇 ∈ 풦(퐸퐴). For 푘 ≥ 0 and 휒 ∈ 퐴 be the character corresponding to the point 푥 ∈ 푋, let {휆푘(푥)}푘 be the eigenvalues of 휒∗푇, in decreasing order with multiplicity. Then, the map 푥 ↦ 휆푘(푥) is an element of 퐴.

Proof. A proof can be found in [MT05, Theorem 6.4.2] (see also op.cit. Section 6.6).

Remark 7.2.3. Note that it is important to know that both 푇 is pointwise compact and that 푇 is norm continuous. Dropping the latter assumption is fatal (see e.g. Exam- ple 5.1.21). By self-adjointness and norm continuity, the full spectrum of 푇 is continuous in norm topology, cf. [Kat95, Remark IV.3.3]. However, the spectral projections (or even the eigenvectors) can in general not be continuously extended over any open neighbour- hood, even in the case where the module is finitely generated: see e.g. [Kad84]. A fortiori, continuous diagonalizability (‘diagonability’) is entirely out of the question in general.

The classical treatment of zeta functions of operators on Hilbert spaces is in terms 푧 푧 of the Dirichlet series tr푇 = ∑푘 휆푘. The Jensen–Cahen theorem (see e.g. [HR64]) shows that these series converge uniformly on angular regions contained in the defining half-plane, so that the limit is in fact holomorphic. The theorem translates very well to the Hilbert module setting.

푝 Lemma 7.2.4. Suppose that 0 ≤ 푇 ≤ 1 ∈ ℒ (퐸퐴). For 0 < 훼 < 휋/2, denote the angular region {푧 ∈ ℂ ∣ |Arg(푧 − 푝)| ≤ 훼} by 퐶훼. Then, for all 휖 > 0 there exists 푚0 ≥ 0 such that, for all 푛 ≥ 푚 ≥ 푚0 and all 푧 ∈ 퐶훼,

푛 푧 ∥∑휆푘∥ < 휖. 푘=푚 퐴

126 7.2. The zeta function

푛 푝 푧−푝 Proof. The proof is based on [HR64, Theorem 2]. Consider the series ∑푘=푚 휆푘휆푘 . def 푞 푝 def 푧 푧 Write 퐴(푝,푞) = ∑푘=푝 휆푘 and 훥푧,푘 = 휆푘+1 − 휆푘. Then, by Abel’s lemma on partial summation [Abe26], we have

푛 푛−1 푧 푧 ∑휆푘 = ∑퐴(푚,푘)훥푧,푘 + 퐴(푚,푛)휆푛 푘=푚 푘=푚

∞ 푝 푝 Now, because the series ∑푘=0 휆푘(푥) converges pointwise to tr푇 , Dini’s theorem ∞ 푝 shows that ∑푘=0 휆푘 converges in norm. In particular, for all 휖 > 0 there exists 푚0 such that ∥퐴(푚,푞)∥ < 휖cos훼 for all 푞 ≥ 푚 ≥ 푚0.

By [HR64, Lemma 2], we have 훥푧,푘 ≤ |푧|/푝훥푝,푘. Thus, as |푧|/푝 ≤ sec훼 throughout 퐶훼, we have 푛 푛−1 푧 푝 푝 푝 ∥∑휆푘∥ < 휖(∑훥푝,푘 + 휆푛) = 휖휆푛 < 휖∥푇 ∥. 푘=푚 푘=푚

As in the classical case, the Jensen–Cahen theorem paves the way for a holomorphic zeta function.

푝 Theorem 7.2.5. Let 0 ≤ 푇 ≤ 1 ∈ ℒ (퐸퐴). Then the map 푧 ↦ 휁(푧,푇) is holomorphic on the half-plane ℂℜ푧>푝 = {푧 ∣ ℜ푧 > 푝}. Moreover, for all compact subsets 퐾 ⊂ ℂℜ푧>푝, all 푥 ∈ 푋 and all 휖 > 0, there is a neighbourhood 푈 of 푥 on which

sup|휁(푧,푇)(푦) − 휁(푧,푇)(푥)| < 휖 푧∈퐾 for all 푦 ∈ 푈.

푛 푧 Proof. For the first statement we consider the 퐴-valued function 휁푛(⋅,푇)∶ 푧 ↦ ∑푘=0 휆푘 on ℂℜ푧>푝. Recall that all bounded functionals 퐴 → ℂ decompose in four positive linear functionals. By the Riesz representation theorem, all such positive linear functionals are given by positive, finite, regular Borel measures 휇 on 퐴̂under the identification def 푧 푧 휙휇(푓) = ∫푓푑휇. In particular, we have 휙휇(휆푘) = ∫휆푘(푥) 푑휇(푥). Now, if 퐶 is a contour 푧 ℂ ∮ 휆 (푥)푧 푥 ∈ 푋 around 0 in ℜ푧>푝, the contour integral 퐶 푘 vanishes for all . Note that |휆푧| = 휆ℜ푧 ∮ ∫휆 (푥)푧푑휇(푥) = ∫(∮ 휆 (푥)푧)푑휇(푥) = 0 푘 푘 and so, by Fubini’s theorem, 퐶 푘 퐶 푘 . By Morera’s theorem, we conclude that 휙휇(휁푛) is holomorphic on ℂℜ푧>푝. Moreover, as 휁푛(⋅,푇) → 휁(⋅,푇) uniformly on compact subsets of ℂℜ푧>푝 by Lemma 7.2.4, the

127 7.2. The zeta function

function 휙휇(휁(⋅,푇)) is holomorphic on ℂℜ푧>푝 as well. By [Rud91, Theorem 3.31], we may conclude that 휁(⋅,푇) is holomorphic, as an 퐴-valued function, on the half-plane ℜ푧 > 푝. For the second statement, note that by Lemma 7.2.4, for all 휖 > 0 there is, for all compact subsets 퐾 ⊂ ℂℜ푧>푝, some 푚0 with ‖휁푚(푧,푇) − 휁(푧,푇)‖퐴 < 휖 for all 푧 ∈ 퐾 and all 푚 ≥ 푚0.

Assume without loss of generality that 휆푘(푥) > 0 and pick a neighbourhood 푈 of 푥 on which 휆푘(푦) > 0, for all 푘 = 1,…,푚. For any 휖0 > 0 let 푉 ⊂ 푈 be such that |ln휆푘(푥) − 푧 푧ln휆푘(푦) 푧 ln휆푘(푦)| < 휖0 for all 푥,푦 ∈ 푉. Then, 휆푘(푦) = 푒 for all 푧 ∈ 퐾, so that |휆푘(푥) − 푝 푧 푧푠 푧푠 휖1 휆푘(푦) | ≤ |1−푒 ||휆푘(푥)| for some 푠 ∈ ℂ with |푠| < 휖0. Moreover, |1−푒 | ≤ |1−푒 | ≤ 휖 푒휖1 휖 = 휖 |푧| 1 , where 1 0 sup푧∈퐾 . 푝 휖1 If we now pick 휖0 such that 휖1푒 |휆푘(푥)| < 휖/푚, we conclude that for all 푦 ∈ 푉 and 푧 푧 all 푧 ∈ 퐾 we have |휆푘(푥) − 휆푘(푦) | < 휖/푚. Consequently, we find that ‖휁푚(⋅,푇)(푥) − 휁푚(⋅,푇)(푦)‖ < 휖 as desired.

Remark 7.2.6. It would be desirable to extend the previous Lemma and Theorem to 푧 the functions tr푎푇 , for 푎 ∈ ℒ(퐸퐴). However, since the spectral projections of 푇 are in general not even weakly continuous, this would be a nontrivial extension. We expect that a possibility for such an extension would be to investigate the functions

푥 ↦ tr푝푘(푥)푎(푥)푝푘(푥)/rank(푝푘(푥)), where 푝푘(푥) is the spectral projection on the eigenspace belonging to the eigenvalue 휆푘(푥) of 푇(푥), and then use these expressions as coefficients in the continuous Dirichlet series.

The next step in the classical case would be to show that certain operators have zeta functions that can be continued meromorphically to all of ℂ, with a discrete set of poles. The residues at these poles then yields interesting information about the operator 푇. For instance, if 푇 is the bounded transform (1 + 퐷2)−1/2 of a pseudodifferential operator these residues give geometric information about the pertinent background manifold (for more details, cf. [BGV04]). For this reason, it would be very desirable to have a reasonable criterion under which our zeta function of operators on Hilbert C∗-modules can be continued meromorphically to all of ℂ, but further research in that direction is beyond the scope of the present work.

128 7.3. Summability of unbounded Kasparov cycles

7.3 Summability of unbounded Kasparov cycles

To motivate investigating summability of unbounded KK-cycles, we will recall some foundational results that motivate the notion of summability of (the special case of) spectral triples.

Summability and spectral triples We will aim to generalize the following property [GVF01, Definition 10.8] of certain spectral triples in noncommutative geometry. Definition 7.3.1. A spectral triple (퐴,퐻,퐷) is said to be finitely summable if there 2 −푝/2 1 exists 푝 ∈ ℝ≥0 such that 푎(1 + 퐷 ) is in ℒ (퐻) for all 푎 ∈ 퐴.

By the Jensen-Cahen theorem, for such spectral triples the zeta function 휁(푧,푎) = tr푎(1 + 퐷2)−푧/2 is holomorphic on the half-plane ℜ푧 > 푝, for any 푎 ∈ 퐴. Example 7.3.2. Let 푀 be a compact Riemannian manifold of dimension 푚 and 퐷 be a Dirac-type operator on a smooth hermitian bundle 풮 over 푀. Then, by the asymptotic 2 expansion [BGV04, Theorem 2.30] of the heat kernel 푘푡(푥,푦) of 퐷 , there exists smooth families 훹푖 of morphisms of 풮 such that, for 푦 sufficiently close to 푥, ∞ −푚/2 −푑(푥,푦)2/4푡 푖 푘푡(푥,푦) ∼ (4휋푡) 푒 ∑푡 훹푖(푥,푦), 푖=0 in the sense that the latter is an asymptotic expansion of the former as 푡 → 0. In particular, for endomorphisms ℎ of 풮 this provides us with the expansion

∞ 2 ℎ푒−푡퐷 ∼ (4휋푡)−푚/2 ∑푡푖 ∫ (ℎ훹 (푥,푥))푑 (푥). tr tr풮푥 푖 vol 푖=0 푀 Under a Mellin transform, this proves that (퐴,퐻,퐷) is in fact 푚+-summable, and that moreover 휁(푎,⋅) extends holomorphically to ℂ ⧵ {푚 − 2푗 ∣ 푗 ≥ 0}. Moreover,

휁(푧,푎)훤(푧) = ∫ 푎(푥) (ℎ훹 (푥,푥))푑 (푥), res푧=푗−푚 tr풮푥 푗 vol 푀 훹(푥,푥) = 풮 where in particular tr풮푥 rk so that

res푧=−푚 휁(푧,푎)훤(푧) = ∫ 푎(푥)푑vol(푥) 푀

129 7.3. Summability of unbounded Kasparov cycles

and (4휋)푚/2 tr 훹 (푥,푥) = 푠(푥)/6 − 1 ∫ 푠(푦)푑vol(푦), where 푠 denotes the scalar cur- 풮푥 1 4 푀 vature of 푀.

The previous example shows at the very least that the residues of the zeta function (or, in analytically less tractable situations, the Dixmier traces) associated to 퐷 are of geometric interest (in the sense of metric geometry), and that moreover the infinimal degree 푝 of summability itself already contains some geometric information. A further application of the relevance of these residues to metric geometry in the truly noncommutative setting can be seen in (the literature based on) [CC97]. An important motivation for the development of noncommutative geometry comes from the fact that it is a natural framework in which to consider vast generalizations of the Atiyah-Singer index theorem. The topic of zeta residues (or Dixmier traces) and summability interlocks with that story from the very beginning, so it may be informative to dwell upon it for a while.

The Chern character and zeta residues in differential topology Let (퐴,퐻,퐹,훾) be an even Fredholm module that is (푛+1)-summable, in the sense that [퐹,푎] ∈ ℒ푛+1 (퐻) for all 푎 in (a dense subalgebra1 풜 of) 퐴. To simplify the exposition, we assumed that the module is even. Then, the Chern character 훤( 푛 + 1) Ch (퐹)∶ (푎 ,…,푎 ) ↦ 2 tr(훾퐹[퐹,푎 ]⋯[퐹,푎 ]) ∗ 0 푛 2푛! 0 푛 of the module defines a class in the periodic cyclic cohomology of 풜, as in [Con94, Definition 4.1.훽.3]. Under Connes’ pairing [Con85, Corollary II.2.17] between cyclic cohomology and K-theory, this Chern character implements Atiyah’s index map on the K-theory of 풜 [Con94, Proposition 4.1.훾.4]. This, however, is far from sufficient to encapsulate the statement of the Atiyah-Singer index theorem. On a smooth vector bundle 퐸 → 푀, if 퐹 is an elliptic psuedodifferential operator of order 0, it is not at all clear a priori how to interpret the trace above in terms of local quantities associated to 퐹. Now, the definition of a spectral triple (퐴,퐻,퐷) is precisely tailored to ensure that its def 2 −1/2 bounded transform (퐴,퐻,퐹퐷 = 퐷(1 + 퐷 ) ) is a Fredholm module, and moreover 푝 we have [퐹퐷,푎] ∈ ℒ (퐻) for all 푎 ∈ 퐴 whenever (퐴,퐻,퐷) is 푝-summable.

1Which is assumed to be closed under the holomorphic functional calculus, so that the K-theory of 풜 agrees with that of 퐴 [Con94, Proposition 4.1.훽.7]

130 7.3. Summability of unbounded Kasparov cycles

Atiyah and Bott showed by a simple calculation that, for a graded spectral triple 퐷 = 퐷− ⊕ 퐷+, 2 −푧 index퐷+ = res푧=0 훤(푧)tr훾(1 + 퐷 ) . In fact, the invariant on the left-hand side can be vastly refined. In this language, Connes and Moscovici showed [CM95] that there exist universal constants 푐푝푞 such that the cycle

푧−푝−2∑ 푘 def (푘1) (푘푝) 푗 푗 훷퐷(푎0,…,푎푝) = ∑ 푐푝푘 res푧=0 tr(훾푎0[퐷,푎0] ⋯[퐷,푎푝] |퐷| ) 푘푗≥0

(푘) is cohomologous in (periodic) cyclic cohomology to Ch∗(퐹퐷). Here 푥 denotes the 푘th iterated commutator [퐷2,…,[퐷2,푥]] with 퐷2. All this goes to show that there is an important and natural role to play for zeta functions in the context of spectral triples, even if they are only viewed topologically, that is, as unbounded representatives of the associated Fredholm modules. As (unbounded) KK- cycles are the natural bivariant generalization of (spectral tripes) Fredholm modules, it is to be expected that traces and zeta residues will be instrumental in further developing KK-theory in the framework of noncommutative geometry.

Summability of right-commutative unbounded Kasparov cycles We now apply the theory of Schatten classes on Hilbert modules to arrive at a notion of summability for unbounded Kasparov cycles over a commutative 퐶∗-algebra. This no- tion is supposed to generalize summability for spectral triples (as unbounded Kasparov cycles over ℂ) as just defined. We refer to [BJ83; Mes14; KL12; MR16] for all relevant notions of unbounded Kasparov cycles, external and internal Kasparov product, and to [KS18; SV19] for the specific application to Riemannian submersions and immersions to be discussed below. ∗ In order to set the notation, for 퐴,퐵 two 퐶 -algebras we let (퐴퐸퐵,푆) be an unbounded Kasparov 퐴 − 퐵 cycle, consisting of

• 퐸퐵 is a (graded) Hilbert 퐵-module

• 퐴 is represented on 퐸퐵 by adjointable (even) operators.

• 푆 is a regular, self-adjoint (and odd) operator, densely defined on dom(푆) ⊂ 퐸퐴.

2 −1/2 • 푎(1 + 푆 ) is in 풦(퐸퐵) for all 푎 ∈ 퐴.

131 7.3. Summability of unbounded Kasparov cycles

• There exists a dense subalgebra 풜 ⊂ 퐴 that preserves dom(푆) and is such that for all 푎 ∈ 풜, the commutator [푆,푎] extends to an adjointable operator on 퐸퐵. ∗ Definition 7.3.3. Let 퐵 be a unital, commutative C -algebra and let (퐴퐸퐵,푆) be an unbounded Kasparov 퐴 − 퐵 cycle. We say that (퐴퐸퐵,푆) is 푝-summable if

1 2 − 2 푞 (1 + 푆 ) ∈ ℒ (퐸퐵).

For simplicity, we restrict to the case where 퐴 is unital. See [Car+14, Section 2] for an understanding of the nonunital case. Example 7.3.4. If (퐴,퐻,퐷) is a 푝-summable spectral triple (cf. [GVF01, Definition 10.8]), then it is an unbounded 푝-summable Kasparov (퐴,ℂ) cycle.

By the very definition of the Schatten classes, the localization of an unbounded 푝- summable Kasparov (퐴,퐵)-cycle along a character of 퐵 yields a 푝-summable spectral triple. In the other direction, it is not true that if all such localized spectral triples are 푝-summable, then the original unbounded KK-cycle is 푝-summable: one needs the fiberwise ℒ푝-norms of the resolvents to be continuous over the base. There is, however, a reasonably simple criterion that is sufficient for a pointwise 푝- summable spectral triple to be 푝-summable. All our examples will be of this type. ∗ Proposition 7.3.5. Let 퐵 be a commutative C -algebra and let (퐴퐸퐵,퐷) be an un- bounded Kasparov (퐴,퐵)-cycle that is boundedly pointwise 푝-summable, in the sense that 2 −1/2 sup‖푥∗(1 + 퐷 ) ‖푝 < ∞. 푥∈퐵̂

Assume moreover that 푥 ↦ ‖푥∗푎‖ lies in 퐵 (as opposed to 푀(퐵)) for all 푎 ∈ 퐴. Then, if there exists for each 푥 ∈ 퐵̂ a neighbourhood 푈 and for each 푦 ∈ 푈 a (not def ∗ necessarily surjective) isometry 휏푦 ∶ 퐸푦 → 퐸푥 such that, with 퐴푦 = 퐷푥 − 휏푦퐷푦휏푦,

• dom퐴푦 is dense in 퐸푥

−1 • 퐴푦(퐷푥 + 푖) extends to a bounded operator on 퐸푥,

−1 • ∥퐴푦(퐷푥 + 푖) ∥ → 0 as 푦 → 푥, then (퐴퐸퐵,퐷) is 푝-summable.

132 7.3. Summability of unbounded Kasparov cycles

∗ ∗ Proof. Write 푇 = 휏푥퐷푥휏푥 and 푆 = 휏푦퐷푦휏푦. As dom푇 ∩ dom퐴푦 = dom퐴푦 is dense in 퐻, the fact that 퐴푦 is 푇-bounded implies that the domain of the closure of 퐴 contains dom푇. Under the last assumption, let 0 < 휖 < 1. There exists a neighbourhood 푉 ⊂ 푈 ̂ −1 −1 of 푥 ∈ 퐵 such that ∥퐴푦(푇 + 푖) ∥ < 휖, so that 1+퐴푦(푇+푖) has an inverse that satisfies −1 −1 −1 ∥(1 + 퐴푦(푇 + 푖) ) ∥ ≤ (1 − 휖) . Thus, the operator 푆 + 푖 = 푇 + 퐴푦 + 푖 is (selfad- −1 −1 −1 −1 joint and) invertible, and its inverse satisfies (푆 + 푖) = (푇 + 푖) (1 + 퐴푦(푇 + 푖) ) , so that in particular ∥(푆 + 푖)−1∥ ≤ (1 − 휖)−1 ∥(푇 + 푖)−1∥ and ∥(푇 + 푖)−1∥ ≤ (1 + 푞 푞 푞 −1 ∗ −1 2 ∗ −1/2 휖)∥(푆 + 푖) ∥ . As ∥(휏푦퐷푦휏푦 + 푖) ∥ = ∥(1 + 휏푦퐷푦휏푦) ∥ for all 푦, the continu- 푞 푞 푞 ity of the trace follows from the pointwise existence. That the trace of 푎(1 + 퐷2)−푞/2 2 −푞/2 then vanishes at infinity for 푎 ∈ 퐴 follows from the fact that tr푥∗푎(1 + 퐷 ) ≤ 2 −1/2 ‖푥∗푎‖∥푥∗(1 + 퐷 ) ∥ . 푞

Some remarks are in order here.

• The maps 휏푏 are to be thought of as a slightly weaker version of parallel transport: we need not assume that the bundle 퐸퐵 be locally trivial.

2 −푞/2 2 −푞/2 • We have assumed here that 푥∗(1 + 퐷 ) , instead of 푥∗푎(1 + 퐷 ) , is in the trace class; this corresponds to the ‘compact fiber’ assumption in the case of Riemannian submersions, below.

Example 7.3.6 (Differentiable frames). Let 푒 be a frame such that 푒푖 ∈ dom(퐷) for all ∗ ̂ 푖, so that dom(휃푒퐷휃푒 ) contains all finite sequences in 퐵. Then, for all 푦 ∈ 퐵, 푦(푒) is a ∗ ∗ frame of 퐸푦 so that 휃푥(푒)휃푦(푒) ∶ 퐸푦 → 퐸푥 is an isometry such that 퐴푦 = 퐷푥 − 휏푦퐷푦휏푦 is densely defined and symmetric. Now, the condition in the proposition amounts to 2 −1/2 norm 퐴푦(1 + 퐷푥) ∈ 퐶푏 (푈,퐵(퐻)). That is, in this setup the question is whether the relative perturbations of 퐷푥 are norm continuous.

Compatibility with the unbounded Kasparov product We expect the summability of (a reasonably large class of) unbounded KK-cycles to correspond to some fiber dimension and, therefore, be additive under the internal and external unbounded Kasparov product. Apart from the examples presented below, a very rough general analysis goes as follows.

Let (퐴퐸퐵,퐷) be an unbounded KK-cycle. Consider the localizations 휌푦 ∶ 퐴 → ℒ(퐸푦)

133 7.3. Summability of unbounded Kasparov cycles

of the representation of 퐴 by elements 푦 ∈ 퐵̂. These will in general not be faithful, even if the representation 휌∶ 퐴 → ℒ(퐸) is. Instead, each 휌푦 will define a closed ideal def ̂ 퐼푦 = ker휌푦 of 퐴, so that there is a closed subset 퐹푦 of 퐴 with 퐼푦 = {푓 ∈ 퐴∶ 푓(푥) = 0 for all 푥 ∈ 퐹푦} and so 휌푦 is in fact a faithful representation of 퐶0(퐹푦) ≃ 퐴/퐼푦.

If we assume that the 퐶0(퐹푦)-spectral triples (푦∗퐸,푦∗퐷) are given by first-order elliptic operators on some vector bundle over 퐹푦, then by the analysis of e.g. [See67], they are dim퐹푦-summable. That is to say, under this assumption of local ellipticity and under for instance one of the assumptions of Proposition 7.3.5, the summability of (퐸,퐷) is 퐹 equal to sup푦∈퐵̂ dim 푦. As for additivity of the fiber dimension under the unbounded Kasparov product: ′ under the interior product of modules 퐴퐸퐵, 퐵퐸퐶, the subspaces 퐹푦 dual to ker휌푦 and ′ ′ ′ 퐹 휌 (휌 ⊗ 휌 ) = ∩ ′ 휌 퐴̂ 푧 dual to ker 푧 satisfy ker 퐵 푧 푦∈∁퐹푧 ker 푦 so that the closed subset of corresponding to this ideal is ⋃ ′ 퐹푦, which has Hausdorff dimension bounded from 푦∈퐹푧 퐹′ + 퐹 above by dim 푧 sup푦 dim 푦.

Remark 7.3.7. The rather strong assumption of ellipticity of the localizations made here fails very much in general, but it may fail in a way that does not impact the rest of the story. For instance, if we consider the general construction of (bounded representatives) of the shriek map in [CS84, Definition 2.1], then we see that we might end upwith a vertically elliptic operator over the neighbourhood {푥 ∣ 푑(푓(푥),푦) < 휖} of the ‘true’ fiber 푓−1(푦). Due to this vertical ellipticity, however, the principal symbols constructed there still correspond (to the bounded transforms of) dim푓−1(푦)-summable operators.

There is a large class of nontrivial examples associated to pseudodifferential operators, which deserves a somewhat more detailed treatment.

Example: Riemannian submersions

In [KS18] the factorization of the Dirac operator 퐷푌 on 푌 in terms of a vertical operator 푆 and the Dirac operator 퐷푋 on 푋 was studied for a Riemannian submersion 푌 → 푋 of compact spin푐 manifolds (more general proper Riemannian submersions were considered in [KS17; Dun18; Dun20]). 2 2 We let 퐿 (풮푋) and 퐿 (풮푌) denote the Hilbert space completions of the spinor mod- ∞ ∞ ∞ ules 훤 (풮푋) and 훤 (풮푌), respectively. Based on a certain 퐶 (푌)-module of smooth ∗ sections of the vertical spinor bundle 풮푉 one then defines a Hilbert 퐶 -module 퐸퐶(푋)

134 7.3. Summability of unbounded Kasparov cycles

between 퐶(푌) and 퐶(푋), together with a self-adjoint and regular unbounded operator 퐷푉 on 퐸, such that 2 2 퐿 (풮푌) ≅ 퐸⊗̂퐶(푋)퐿 (풮푋) and in such a way that the operator 퐷푌 corresponds to the tensor sum 퐷푉 ⊗ 훾퐸 + 1 ⊗∇ 퐷푋 for some metric connection ∇ on 퐸퐶(푋) and the grading operator 훾퐸 on 퐸 (up to an explicit error term related to the curvature).

Let us analyze here the summability aspects of the operator 퐷푉 ∶ dom(퐷푉) → 퐸. The main property that we will use below is that 퐷푉 is the closure of a so-called vertically ∞ ∞ ∞ elliptic operator 풟 ∶ 훤 (풮푉) → 훤 (풮푉). This means that for all 푓 ∈ 퐶 (푌), [풟,푓] is ∞ an endomorphism of 훤 (풮푉) that is invertible at all points where 푑푓|ker푑휋 is nonzero (see [KS18; KS17] for more details). In fact, this allows one to prove [KS17, Theorem 3] that the pair (퐸,퐷푉) is an unbounded Kasparov 퐶(푌) − 퐶(푋) cycle. ∗ As far as summability is concerned, note that the restrictions 휒 풟 = 풟푥 of 풟 to the fibers of 푥 ∈ 푋 (for the character 휒 ∶ 퐶(푋) → ℂ) are elliptic Dirac type operators. Since the dimension of the fibers is constant and equal to dim퐹 for the typical fiber 퐹, one 2 −1/2 has ∥(1 + 퐷푥) ∥ < ∞ for all 푝 > dim퐹 [Con96] (cf. [GVF01, Theorem 11.1]). The 푝 question, then, is whether this pointwise trace is continuous.

푝 2 −1/2 Proposition 7.3.8. The ℒ -norm of (1 + 풟푥) defines (as 푥 varies over 푋) a con- tinuous function on 푋 for any 푝 > dim퐹. Consequently, the unbounded Kasparov 퐶(푌) − 퐶(푋) cycle (퐸,퐷푉) is 푝-summable for all 푝 > dim퐹.

Proof (based on [KS17, Section 2.3]). For simplicity of exposition, we will assume that the bundle 풮푌 → 푌 is locally trivial over 푋; that is, we assume that around each point −1 푥0 ∈ 푋 there exists a neighbourhood 푈 ⊂ 푋 such that 1) 휋 (푈) is diffeomorphic to 푈 × 퐹, and 2) the bundle 풮푌 → 푌 can be smoothly and unitarily trivialized over 푈. If we pull sections of 풮푌 over 푈 back through this trivialization, we obtain a family 푘 of Dirac-type differential operators {풟푥} on the trivial bundle ℂ over the compact Riemannian manifold 퐹. In particular, these operators can be written as

dim퐹 휕 풟푥 = ∑ 퐴푗(푥,푧) + 퐵(푥,푧) 푗=1 휕푧푗 with 퐴푗,퐵 symmetric matrix-valued smooth functions on 푈×퐹 and 퐴푗 invertible. Thus, ′ −1 ′ there are smooth families 푍푥,푥′ (푧) = 퐴푗(푥 ,푧)퐴푗 (푥,푧), 푊푥,푥′ (푧) = 퐵(푥 ,푧)−푍푥,푥′ 퐵(푥,푧)

135 7.3. Summability of unbounded Kasparov cycles

of matrices such that 풟푥′ = 푍푥,푥′ 풟푥 + 푊푥,푥′ , with in particular lim푥′→푥(id−푍푥,푥′ ) = 0 = lim푥′→푥 푊푥,푥′ .

Now denote the closure of 풟푥 by 퐷푥. Note that 퐷푥 is selfadjoint by compactness of 퐹. Moreover, because 풟푥 is an elliptic differential operator of order 1, the resolvent −1 푝 2 푘 (퐷푥 + 푖) lies in ℒ (퐿 (퐹,ℂ )) for all 푝 > dim퐹. −1 −1 Then, 퐷푥(퐷푥 + 푖) being bounded, we see that lim푥′→푥 ‖(퐷푥′ − 퐷푥)(퐷푥 + 푖) ‖ = 0. Now, apply Proposition 7.3.5: that is, by the resolvent identity, we conclude that

−1 −1 lim ‖(퐷 + 푖) − (퐷 ′ + 푖) ‖ = 0 푥′→푥 푥 푥 푝 for all 푝 > dim퐹, and so (1 + 퐷2)−1/2 ∈ ℒ푝 (퐸) if and only if 푝 > dim퐹.

Note that when we combine this result with the factorization result [KS18] of 퐷푌 in 푐 terms of 퐷푉 and 퐷푋 we obtain for Riemannian submersions of spin manifolds the desired additivity of summability for the unbounded interior Kasparov product.

Example: Embedding spheres in Euclidean space We consider a special class of immersions, given by the embedding of spheres 푆푛 in Euclidean space ℝ푛+1. This is based on [CS84; SV19; Ver19]. As in [CS84] the em- bedding 푆푛 → ℝ푛+1 gives rise to an immersion class in KK-theory. For spheres, the 푛 unbounded representative is given by the module 퐶0(푆 × (−휖,휖)) based on a normal neighborhood of 푆푛 ⊂ ℝ푛+1, equipped with the regular self-adjoint operator 푆 given by the multiplication operator with a suitable function 푓 ∶ (−휖,휖) → ℝ. For convenience, we will take 푆 to be multiplication by the function 휋 휋푠 푓(푠) = tan( ); (푠 ∈ (−휖,휖)). 2휖 2휖 −1 푛 Since (푖 + 푓) is clearly a 퐶0-function on 푆 × (−휖,휖), we find as in [SV19, Lemma −1 푛 2.3] that (푖 + 푆) is a compact operator on the Hilbert module 퐶0(푆 × (−휖,휖)) and so 푛 푛+1 forms an unbounded Kasparov 퐶(푆 ) − 퐶0(ℝ ) cycle. 2 −1/2 푝 푛 Proposition 7.3.9. We have (1 + 푆 ) ∈ ℒ (퐶0(푆 × (−휖,휖))) for any 푝 > 0. Hence 푛 푛+1 the unbounded Kasparov 퐶(푆 ) − 퐶0(ℝ ) is 푝-summable for all 푝 > 0.

Proof. For any locally compact Hausdorff space 푋, the pointwise localizations of the 푝 Hilbert 퐶0(푋)-module 퐶0(푋) are one-dimensional, so that the pointwise ℒ -norm

136 7.3. Summability of unbounded Kasparov cycles

of any 푔 ∈ ℒ(퐶0(푋)) = 퐶푏(푋) is given by pointwise evaluation of |푔|. Hence, tr(1 + 2 −푝/2 2 −푝/2 푆 ) = (1 + 푓 ) , which lies in 퐶0 for all 푝 > 0.

Again, this is a confirmation of additivity for summability under the unbounded interior Kasparov product. Indeed, in [SV19] it was shown that 퐷푆푛 can be related to the immersion class defined by 푆 as above and 퐷ℝ푛+1 in the following way. Namely, the unbounded interior product of 푆 and 퐷ℝ푛+1 is equal to the unbounded interior product of 퐷푆푛 with a so-called index cycle 푇. The latter represents the identity at the bounded level but is in fact a 푝-summable Kasparov cycle for all 푝 > 1. Proposition 7.3.10. The selfadjoint closure 푇 of the operator

0 −푖휕푠 − 푖푓(푠) 푇0 = ( ) 푖휕푠 + 푖푓(푠) 0

∞ 2 on 퐶푐 ((−휖,휖),ℂ ) is 푝-summable for all 푝 > 1.

휋 2 2 Proof. As in [SV19, Lemma 2.11], for |휆| > 2휖 we have 훥휖 + 1 < 푇 + 휆 + 1, where ∞ ⊕2 훥휖 is the closure of the Dirichlet Laplacian on 퐶푐 ((−휖,휖)) . On the other hand, if ∞ ⊕2 2 훥휖/2 is the closure of the Dirichlet Laplacian on 퐶푐 ((−휖/2,휖/2)) and 푐 = 푓 (휖/2) + ′ 2 |푓 (휖/2)|, then ∥푇 − 훥휖/2|퐿2((−휖/2,휖/2))⊕2 ∥ = 푐 < ∞ so that, by the min-max principle, 2 the singular values 휎푛(푇 ) are bounded from above by 휎푛(훥휖/2) + 푐. Thus, one has 2 −1 2 2 −1 −1 −1 푝 ‖(훥휖/2 + 휆 + 푐 + 1) ‖푝 ≤ ‖(푇 + 휆 + 1) ‖푝 ≤ ‖(훥 + 1) ‖푝 and so (푇 ± 휆푖) ∈ ℒ if and only if 푝 > 1.

+ As such, the summability of 퐷ℝ푛+1 plus that of the immersion cycle (i.e. 0 ) indeed coincides with the summability of 퐷푆푛 plus that of the index cycle.

Example: Actions of ℤ 2 1 Consider the standard 퐶(푋)-module 퐸 = 퐿 (푆 ) ⊗ℂ 퐶(푋) equipped with the ‘Dirac’ operator 퐷 = 퐷푆1 ⊗ 1. −푝 −푝 −1 Then tr휒∗|퐷| = tr|퐷푆1 | < ∞ for all characters 휒 and 푝 > 1 so that we have |퐷| ∈ ℒ푝 (퐸) for 푝 > 1. As an example of an unbounded Kasparov cycle, consider a homeomorphism on 푋 and def 푛 consider the action 푛 ⋅ 푓 = 푓 ∘ 휙 of ℤ on 퐶(푋). Let 퐶(푋) ⋊휙 ℤ be the corresponding

137 7.3. Summability of unbounded Kasparov cycles

full crossed product C∗-algebra. Consider the unitary 푈 ∈ ℒ(퐸) given by 푈 = 푆 ⊗ 1, where 푆 is multiplication by 휃 ↦ 푒푖휋휃 on 퐿2(푆1). Then, the map 휌 defined on finite sums in 퐶(푋) ⋊휙 ℤ by 푘 휌∶ ∑푓푘푢푘 ↦ ∑푓푘푈 , 푘 푘 where the left action of 퐶(푋) is just given by pointwise multiplication, extends by univer- 푘 푘 sality to a representation of 퐶(푋)⋊휙 ℤ. Moreover, [퐷,∑푘 푓푘푈 ] = ∑푘 푘푓푘푈 because 푘 [퐷푆1 ,푆 ] = 푘푆, so that there is a dense subalgebra of 퐶(푋) ⋊휙 ℤ with [퐷,푎] bounded. Weconclude that (퐸퐶(푋),퐷) is an unbounded 푝-summable Kasparov 퐶(푋)⋊휙 ℤ−퐶(푋) cycle for all 푝 > 1. In particular, one has

푘 −푧 푘 −푧 휁퐷(푓푢푘,푧)(푥) = tr휒∗(푓푈 퐷 ) = 푓(푥)tr퐿2(푆1) 푆 |퐷푆1 | , so that 휁퐷(∑푘 푓푘푢푘,푧) extends meromorphically to ℂ ⧵ {1} and in fact

res푧=1 휁퐷(∑푓푘푢푘,푧) ∝ 푓0. 푘

Summability and the exterior Kasparov product One of the key results in [BJ83] was an explicit and linear formula for the external Kasparov product. More precisely, they showed that two unbounded Kasparov cycles (restricting to the even-odd case for simplicity) (퐸퐵,훾,푆) and (퐹퐶,푇) can be combined into an external product unbounded KK-cycle over the minimal tensor product 퐵 ⊗ ℂ:

((퐸 ⊗ 퐹)퐵⊗퐶,푆 ⊗ 1 + 훾 ⊗ 푇). For spectral triples this can be understood as the direct product of the corresponding (noncommutative) spaces. In any case, it is desirable to have an additive property of summability for this external product in the case of a commutative base. Lemma 7.3.11. If 푎,푏 are positive, (resolvent) commuting, regular operators on a Hilbert C∗-module, then for 푝,푞 > 0 one has (1 + 푎 + 푏)−푝−푞 ≤ (1 + 푎)−푝/2(1 + 푏)−푞(1 + 푎)−푝/2

Proof. By positivity of 푎,푏, we have (푏 + 1)−1 ≥ (푎 + 푏 + 1)−1 ≤ (푎 + 1)−1, and by com- mutativity of the C∗-algebra generated by the resolvents of 푎,푏, we have (푎+푏+1)−푝−푞 ≤ (푎 + 1)−푝/2(푎 + 푏 + 1)−푞(푎 + 1)−푝/2 ≤ (푎 + 1)−푝/2(푏 + 1)−푞(푎 + 1)−푝/2.

138 7.3. Summability of unbounded Kasparov cycles

Corollary 7.3.12. The summability of unbounded Kasparov modules (over commu- tative 퐶∗-algebras) is additive under the exterior product.

Proof. The corresponding selfadjoint operators (푆 ⊗ 1) and (훾 ⊗ 푇) on 퐸 ⊗ 퐹 anti- commute, and the actions of 퐵,퐶 commute. Thus, we have |푆 ⊗ 1 + 훾 ⊗ 푇|2 = |푆 ⊗ 2 2 1| +|훾⊗푇| , whose summands commute. Moreover, the exterior product {푒푖 ⊗푓푗}푖푗 of frames is a frame. We conclude, with the Lemma, that |푆 ⊗ 1 + 1 ⊗ 푇 + 푖|−푝−푞 ≤ |푆 ⊗ 1 + 푖|−푝|1 ⊗ 푇 + 푖|−푞, and the 퐵 ⊗ 퐶-valued trace of the latter is just the tensor product of the traces of |푆 + 푖|−푝 and |푇 + 푖|−푞.

Summability and the interior Kasparov product Of course, the real challenge is to establish the compatibility of summability with the internal unbounded Kasparov product. Clearly, Lemma 7.3.11 is then not sufficient. We leave its (challenging) extension to further research, but provide a very first step if only to exhibit the easier part. The following Lemma would appear to be relevant. The operators 푠 and 푡 in its state- ment should correspond to the 푝th and 푞th (absolute) powers of the respective resol- vents. ∗ Lemma 7.3.13. Let 퐵,퐶 be separable, commutative C -algebras and let 퐸퐵, 퐹퐶 be ∗ countably generated Hilbert C -modules, and let 푒 be any frame of 퐸퐵. Then, for all 2 1 푠 ∈ ℒ (퐸퐵) and all 0 ≤ 푡 ∈ ℒ (퐹퐶), one has ∗ 1 (푠 ⊗ 1)(1 ⊗푒 푡)(푠 ⊗ 1) ∈ ℒ (퐸퐵 ⊗퐵 퐹퐶), where ⊗푒 denotes the product along the Grassman connection associated to 푒,

def (1 ⊗푒 푡)(푥 ⊗퐵 푦) = ∑푒푖 ⊗퐵 ⟨푒푖,푥⟩푡푦. 푖

def Proof. Choose a frame 푓푗 of 퐹퐶, so that 휂푖푗 = 푒푖 ⊗퐵 푓푗 is a frame of 퐸퐵 ⊗퐵 퐹퐶. Because

⟨푥 ⊗ 푦,(1 ⊗푒 푡)푥 ⊗ 푦⟩ = ∑⟨푦,⟨푥,푒푖⟩푡⟨푒푖,푥⟩푦⟩ ≥ 0, 푖 the product operator is positive so that the statement of the Lemma is equivalent to the convergence of its trace along a frame of 퐸퐵 ⊗퐵 퐹퐶. Now,

∗ ⟨휂푖푗,(푠 ⊗ 1)(1 ⊗∇ 푡)(푠 ⊗ 1)휂푖푗⟩ = ∑⟨푓푗,⟨푒푖,푠푒푘⟩푡⟨푠푒푘,푒푖⟩푓푗⟩. 푘

139 7.3. Summability of unbounded Kasparov cycles

def 2 ∗ 1 Write 푠푖푘 = ⟨푒푖,푠푒푘⟩. For finite subsets 퐹 ⊂ ℕ , one has ∑푖푘∈퐹 푠푖푘푡푠푖푘 ∈ ℒ (퐹퐶), and then ∥tr∑ 푠∗ 푡푠 ∥ = ∥tr푡∑ 푠∗ 푠 ∥ ≤ ‖푡‖ ∥∑ 푠∗ 푠 ∥ . As the series ∑ 푠∗ 푠 and 푖푘∈퐹 푖푘 푖푘 푖푘∈퐹 푖푘 푖푘 1 푖푘∈퐹 푖푘 푖푘 퐵 푖푘 푖푘 푖푘 ∗ ∗ ∑푘 푠푖푘푠푖푘 are Cauchy (with limits tr푠 푠 ∈ 퐵 and ⟨푠푒푖,푠푒푖⟩ ∈ 퐵, respectively), we conclude ∗ 1 that ∑푖푘 푠푖푘푡푠푖푘 converges in ℒ (퐹퐶), so that the statement follows.

The rather extreme assumptions of the following Lemma, which is itself a simple corollary of Lemma 7.3.11, allow us to compare the 푝 + 푞-th power of the resolvent of the unbounded Kasparov product to the expression in the statement of Lemma 7.3.13.

Lemma 7.3.14. Let ∇ be any 푇-connection on 퐸퐵. Then, if either

def • (퐸퐵,푆) is graded, {푆 ⊗ 1,훾 ⊗∇ 푇} = 0 and 푆 ×∇ 푇 = 푆 ⊗ 1 + 훾 ⊗∇ 푇, or

def 0 푆 ⊗ 1 + 푖 ⊗∇ 푇 • [푆 ⊗ 1,1 ⊗∇ 푇] = 0 and 푆 ×∇ 푇 = ( ), 푆 ⊗ 1 − 푖 ⊗∇ 푇 0

−푝−푞 −푝/2 −푞 −푝/2 then |푆 ×∇ 푇 + 푖| ≤ |푆 ⊗ 1 + 푖| |1 ⊗ 푇 + 푖| |푆 ⊗ 1 + 푖| .

As the resolvent of 푆 ⊗ 1 is just that of 푆 tensored with 1, all that we need now in order to apply Lemma 7.3.13 is an estimate on the 푞-th absolute power of the resolvent of 1 ⊗∇ 푇 or 훾 ⊗∇ 푇. In order to treat this resolvent when ∇ is a Grassman connection, we will work in the larger module 퐻 ⊗ℂ 퐹. To this end, let 푢 be the unitary isomorphism between the Hilbert 퐶-modules 퐻 ⊗ℂ 퐹 and 퐻 ⊗ℂ 퐵 ⊗퐵 퐹. Then, for any frame 푒 of 퐸퐵, let def ∗ ∗ ̃ def 푄푒 = 푢 (휃푒 휃푒 ⊗1)푢, so that we have 푄푒(퐻⊗ℂ 퐹) ≃ 퐸⊗퐵 퐹. Let 푇 = 1⊗푇∶ 퐻⊗ℂ 퐹 → 퐻 ⊗ℂ 퐹. Now, the conditions of the following Lemma are satisfied rather generally, cf. the analysis of [KL13, Lemma 5.3] and the differentiable frames constructed in [Kaa17]. ̃ Lemma 7.3.15. If there exists a frame 푒 of 퐸퐵 such that [푇,푄푒] extends to a bounded, ̃ adjointable operator on 퐻 ⊗ℂ 퐹, then there exists bounded, adjointable 퐴 = 푄푇(1 − 푄) + (1 − 푄)푇푄̃ ∈ ℒ(퐻 ⊗ 퐹) 1 ⊗ 푇 = 푄푇푄̃ ℂ such that the operator ∇0 satisfies

1 ⊗ 푇 = 푄(푇̃ − 퐴) = (푇̃ − 퐴)푄. ∇0

|1 ⊗ 푇 + 휇|−2 ≤ ‖퐴‖2 푄|푇̃ + 휇|−2푄 = ‖퐴‖2 1 ⊗ |푇 + 휇|−2 In particular, ∇0 ∇0 .

140 7.3. Summability of unbounded Kasparov cycles

As in [KL13, Proposition 6.6]. Note that 퐴 is bounded and adjointable by the identity 퐴 = [푄,푇](1̃ − 푄) + (1 − 푄)[푇,푄]̃ . Now, we have 푄푇푄̃ = 푄(푄푇푄̃ + (1 − 푄)푇(1̃ − 푄)) = 푄(푇−퐴)̃ and 푄푇푄̃ = (푄푇푄+(1−푄)̃ 푇(1−푄))푄̃ = (푇−퐴)푄̃ . The expression for the resolvent now follows from the resolvent identity.

We conclude that, under the joint assumptions of Lemma 7.3.15 and Lemma 7.3.13, the unbounded Kasparov product is 푝 + 푞-summable whenever 푞 ≤ 2.

Remark 7.3.16. There is a ℒ(퐹)-valued, unbounded, ‘partial trace’

휏푒 ∶ dom휏푒 ⊂ ℒ(퐸 ⊗퐵 퐹) → ℒ(퐹), given by 휏푒(푥)(푓) = ∑푖 ⟨푒푖|푥|푒푖 ⊗ 푓⟩, where ⟨푒푖| is seen as an 퐹-valued map on 퐸⊗퐵 퐹. 1 Clearly we have dom휏푒 ⊃ ℒ (퐸 ⊗퐵 퐹). Note that tr휏푒(푥) = tr푥 so that for positive 푥 1 1 we have 푥 ∈ ℒ (퐸 ⊗퐵 퐹) ⊂ dom휏푒 if and only if 휏푒(푥) ∈ ℒ (퐹). 2 As seen in Lemma 7.3.13, all operators of the form 푠(1 ⊗푒 푡)푠 with 푠 ∈ ℒ (퐸) are 1 1 contained in dom휏푒, and moreover 휏푒(푠(1 ⊗푒 푡)푠) ∈ ℒ (퐹) whenever 푡 ∈ ℒ (퐹). 푞 푞 푞 1 It follows trivially that when 0 ≤ 푥 ≤ 1⊗푒 푥0, we have 휏푒(푠푥 푠) ∈ ℒ (퐹) whenever 푥0 ∈ 푞 푞 ℒ (퐹). However, we have no such guarantee whenever 0 ≤ 푥 ≤ 1⊗푒 푥0 with 푥0 ∈ ℒ (퐹) ∗ instead: unfortunately, the range of the partial trace 휏푒 is the noncommutative C - algebra ℒ(퐹), so that the techniques for traces with values in a commutative C∗-algebra developed here (such as Corollary 6.2.16 in particular) do not apply. See Chapter 8.

141

Chapter 8

Outlook: the noncommutative case

The approach to the Schatten classes outlined in Chapter 6 relies heavily on the com- mutativity of the algebra 퐴. Not just the common proof technique, which boils down 푝 to combining the properties of ℒ (퐻ℂ) with the uniformity argument of Theorem 6.2.11, fails in the noncommutative case: key results no longer hold. For instance, we can build on [FL02, Example 1.1] to show that the set {푇 ∈ ℒ(푙2(퐴)) ∣ tr|푇|푝 ∈ 퐴} is no longer a two-sided ideal in ℒ(푙2(퐴)) when 퐴 = 퐵(퐻) in the following example. Moreover, the following example shows that the finite matrices 푀(퐵(퐻)) cannot be contained in a two-sided ideal in ℒ(푙2(퐵(퐻))) on which the trace is even weakly convergent. Example 8.0.1. Let 퐴 be a unital C∗-algebra that contains a copy of the Cuntz algebra 푛 ∗ ∗ 풪푛: a family {푢푖}푖=1 of isometries such that 푢푖 푢푖 = 1 and ∑푗 푢푗푢푗 = 1. Let 푇푛 ∈ Mat푛(퐴) be given by

푢1 0 … 0 푇푛 = [ ⋮ ⋮ ⋮ ] 푢푛 0 … 0 so that in particular ‖푇푛‖ = 1 and Mat푛(퐴)

1 0 … 1 0 … ∗ ∗ 푇푛푇푛 = [0 0 ⋱] 푇푛푇푛 = [0 ⋱ ⋱]. ⋮ ⋱ ⋱ ⋮ ⋱ ⋱

143 8.1. Ideals inside a smaller endomorphism algebra

∗ Consider the “trace” tr on Mat푛(퐴) given by 푇 ↦ ∑푖 푇푖푖. Then, the element 푆 = 푇1푇1 ∈ ∗ Mat1(퐴) satisfies tr푇푛푆푇푛 = 푛.

In the language of frames, note that 푒 = (푢1,…,푢푛) is a frame of the Hilbert 퐴-module 퐴 ∗ ∗ ∗ because ∑푖 푎 푢푖 푢푖푏 = 푎 푏 for 푎,푏 ∈ 퐴. The frame transform 휃푒 now sends the identity ∗ ∗ 1퐴 to 푇푛푇푛, so that tr휃푒1퐴휃푒 = 푛.

In particular, if 퐴 contains a copy of 풪∞ (as does 퐵(퐻) in particular), we must conclude that no trace ideal in ℒ(푙2(퐴)) can contain Mat(퐴).

In the general case, then, there is no hope for an extension of Theorem 6.3.1 to provide for a two-sided trace ideal in ℒ(푙2(퐴)) that at least contains Mat퐴, as it should in order to really extend the “trace” on Mat푛 퐴.

8.1 Ideals inside a smaller endomorphism algebra

As a counterpoint to Example 8.0.1, we can assign a subalgebra of ℒ(푙2(퐴)) with an associated trace ideal in the general case, as the following example shows.

Example 8.1.1 (The Schatten classes of[Nis91]). For 푎 ∈ 퐴 and 푡 ∈ 퐵(퐻), we have 2 2 2 푡 ⊗ 푎 ∈ ℒ(푙 (퐴)) by the inclusion ℒ(푙 ) ⊗ ℒ(퐴) ↪ ℒ(푙 ⊗ℂ 퐴). Moreover, as the projective tensor norm on 퐵(퐻)⊗퐴 dominates the operator norm of ℒ(푙2(퐴)), there 2 is an inclusion 퐵(퐻)⊗휋 퐴 ↪ ℒ(푙 (퐴)) of Banach algebras. This implies the inclusion 푝 2 ℒ (퐻) ⊗휋 퐴 ⊂ 퐵(퐻) ⊗휋 퐴 ⊂ ℒ(푙 (퐴)). 푝 For 푇 ∈ ℒ (퐻) ⊗휋 퐴 there exist (cf. [Rya02, Chapter 2.1]) bounded sequences 푝 1 {푎푖}푖,{푡푖}푖 in the unit ball of 퐴 and ℒ (퐻), respectively, and {휆푖}푖 ∈ 푙 with ∑푖 휆푖푡푖 ⊗ 푎푖 = 푇, and similar for 푆 ∈ 퐵(퐻) ⊗휋 퐴. Thus, ∑푖푗 휆푖휇푗푠푖푡푗 ⊗ 푎푖푏푗 = 푆푇 converges 푝 in ℒ (퐻) ⊗휋 퐴 because ∥푠푖푡푗∥ ≤ ‖푠푖‖∥푡푗∥ . Mutatis mutandis, we also see that 푝 푝 푝 푝 푇푆 ∈ ℒ (퐻) ⊗휋 퐴 and therefore conclude that ℒ (퐻) ⊗휋 퐴 is a two-sided ideal in 퐵(퐻) ⊗휋 퐴. 푝 By projectivity, the map trℒ푝(퐻) ⊗1퐴 extends continuously to ℒ (퐻) ⊗휋 퐴. With the algebraic identification Mat퐴 = Matℂ⊗퐴, this projective trace extends that on Mat퐴.

The crucial difference between Example 8.0.1 and Example 8.1.1, is that the Schatten 2 classes of the latter are only two-sided ideals inside 퐵(퐻) ⊗휋 퐴, not in all of ℒ(푙 (퐴)).

144 8.1. Ideals inside a smaller endomorphism algebra

Example 8.1.1, however, does not solve the problem fully. First of all, we now treat the commutative and noncommutative case very differently1, without a clear bridge between the two cases. Second, there are likely to be many examples where the algebra 퐵(퐻) ⊗휋 퐴 is insufficiently large to accomodate a given representation, even whenthe 퐴-valued “trace” may still be perfectly well-defined on a subset thereof that does forma two-sided ideal – as is of course the case when 퐴 is commutative, or even almost so. In the light of this example, we should perhaps not require the domain of an 퐴-valued “trace” to be an ideal inside all of ℒ(푙2(퐴)), but rather inside some subalgebra. That would be particularly natural in the language of KK-cycles: if one has a KK-cycle (퐴퐸퐵,퐹), there is already a designated subalgebra 퐴 ⊂ ℒ(퐸퐵) and one might like to say 푝 the cycle is 푝-summable if {푎 ∈ 퐴∶ [퐹,푎] ∈ ℒ (퐴퐸퐵)} is dense inside 퐴, or something 푝 to that effect. We would then only need ℒ (퐴퐸퐵), whatever its definition will be, to be a two-sided ideal in (or, at least, be closed under multiplication by) 퐴 ⊂ ℒ(퐸퐵), not in all of ℒ(퐸퐵).

Example 8.1.2. Note that the functional 푇 ↦ √‖tr푇∗푇‖ on the matrix algebras is subadditive, bounded from below by the operator norm. Only from the left, it is even unitarily invariant and submultiplicative in the usual sense. This, given a normed subalgebra 퐶 ⊂ ℒ(푙2(퐴)), might lead one to define a two-sided Hilbert-Schmidt ideal 2 ∗ ∗ “relative to 퐶” as the completion of 푀(퐴) in the norm ‖푇‖2;퐶 = sup{‖tr푆 푇 푇푆‖ ∣ 푆 ∈ 퐶,‖푆‖퐶 ≤ 1}.

Remark 8.1.3. Building on the previous example, a rather different extension on the lines of the Hattori-Stallings trace would be to first compose with the quotient map

푞∶ 퐴 → 퐴/[퐴,퐴], i.e. to define the trace with values in the latter as ∑푖 푞(⟨푒푖,푇푒푖⟩) for any given frame. The resulting space of Hilbert-Schmidt operators (i.e. those for which the trace of the square converges in the norm on the abelianization) is then a two-sided ideal in the adjointable endomorphisms of a countably generated Hilbert 퐴-module. The square of this Hilbert-Schmidt ideal then gives a two-sided trace class ideal. Unlike for the Hattori-Stallings trace it is not clear, however, that this construction possesses even a non-canonical lift to 퐴 itself.

1 str Note that 퐵(퐻) ⊗휋 퐶0(푋) ⊂ 퐶0(푋,퐵(퐻)) ⊊ 퐶 (푋,퐵(퐻)) unless 퐻 is finite-dimensional or 푋 is b 푝 푝 finite (assuming of course it is Hausdorff). Moreover, even though ℒ (퐻) ⊗휋 퐶0(푋) ⊂ 퐶0(푋,ℒ (퐻)) 1 2 always, the spaces need not agree: the diagonal elements of ℒ (푙 ) ⊗휋 퐶0(푋) consists of the absolutely 1 2 summable sequences in 퐶0(푋), whereas the diagonal elements of 퐶0(푋,ℒ (푙 )) consists of the uncondi- tionally summable sequences, cf. [Rya02].

145 8.2. The bivariant Chern character

8.2 The bivariant Chern character

Because the trace on Mat(퐴) is not cyclic for noncommutative 퐴, it is more natural to regard it as a map (the Dennis trace) between Hochschild or cyclic complexes instead. That brings us to a highly desired – but elusive – application: the bivariant Chern character. This introduces two additional complications, however: that of working with more general locally convex algebras 풜 ⊂ 퐴 and that of the trace on Mat(풜)⊗푛̂ with values in the tensor product 풜⊗푛̂ .

If (퐴퐻,훾,퐹) is a finitely summable (even) Fredholm module (with respect to the dense subalgebra 풜 ⊂ 퐴), Connes’ Chern character

훤(푛 + 1) Ch (퐹)∶ (푎 ,…,푎 ) ↦ tr(훾퐹[퐹,푎 ]⋯[퐹,푎 ]) ∗ 0 2푛 2푛! 0 2푛

(see also pp. 130) and the Chern character of an idempotent 푒 ∈ 푀푘(풜),

∗(푒) = (푛!)−1 ∑ 푒 ⊗ ⋯ ⊗ 푒 , Ch 푖0푖1 푖2푚푖0 0≤푖푗≤푘

2푛 produce classes in HC (풜) and HC2푛(풜) respectively that are independent of the choices of representative (퐴퐻,퐹) and 푒 of their respective classes in K-theory and K- ∗ homology. Moreover, their pairing ⟨Ch (푒),Ch∗(퐹)⟩ equals the index of F on the K-theory class [푒], cf. [Con94, Chapter 4.1].

If (퐵퐸퐴,퐹) is a (bounded) Kasparov (ℬ,풜)-module and 푒 ∈ 푀푘(ℬ) a projection 푖 representing the classes [퐸,퐹] ∈ KK (ℬ,풜) and [푒] ∈ 퐾0(ℬ) respectively, then their 푖 Kasparov product [푒] ⊗퐵 [퐸,퐹] is an element of 퐾 (풜). The bivariant index problem, as posed by Connes in [Con83] and confronted by Nistor in [Nis93], is to determine a map Ch∶ KK푖(ℬ,풜) → HC2푛+푖(ℬ,풜) such that

∗ ∗ ∗ ⟨Ch ([푒] ⊗퐵 [퐸,퐹]),휙⟩ = ⟨Ch [푒],Ch ([퐸,퐹])(휙)⟩ for all 휙 ∈ HC2푛+푖(풜). For a discussion of the precise definition of HC2푛+푖(ℬ,풜) in this context and the way in which to regard its elements as maps from HC•(풜) to HC•(ℬ), see [Nis93]. Now, the connection between the topic of the Connes-Chern character and the present discussion of the trace is that the construction of the Chern character proposed in [Nis91; Nis93] factors through a generalization of the (Dennis trace) map tr푛. That is,

146 8.2. The bivariant Chern character

consider the map

tr푛 ∶ 푀(풜)⊗(푛+1) → 풜⊗(푛+1), tr푛(푇0 ⊗ ⋯ ⊗ 푇푛) = ∑ 푇0 ⊗ ⋯ ⊗ 푇푛 . 푖0,푖1 푖푛,푖0 푖0,⋯,푖푛 Given a choice ⊗̂ of tensor product appropriate to the computation of HC(풜), a 푝 proper generalization of tr푛 would then be constructed from a pair (ℒ ,ℒ∞ ) of 풜,⊗̂ 풜,⊗̂ subalgebras of ℒ(퐻 ⊗ℂ 퐴) with at least the following properties:

푛 푝 • The domain domtr•, which is spanned by all tensor products ⨂̂ ℒ 푖 such 푖=1 풜,⊗̂ 푘 −1 that the reciprocals of the 푝푖 sum to ∑푖=1 푝푖 = 1, is closed under contractions, and 푛 • ⊗(푛+1)̂ • the series defining tr converges to a continuous map domtr → ⨁푛 풜 .

푝 Nistor’s original construction takes ℒ def= ℒ푝 (퐻)⊗ 풜 and ℒ∞ def= 퐵(퐻)⊗ 풜, as 풜,⊗̂ 휋 풜,⊗̂ 휋 in Example 8.1.1. It is important to note that the resulting definition of a ‘1-summable bounded Kasparov module’ (퐻⊗ℂ 퐴,퐹) requires both that the representation of 풜 ⊂ 퐴 1 land in 퐵(퐻) ⊗휋 풜 and that the commutators [퐹,푎] land in ℒ (퐻) ⊗휋 풜, again as in Example 8.1.1. This is more restrictive than one might desire, as we will see below.

The Chern character of the Toeplitz extension An important case for the Chern character is that of the boundary map in the Pimsner- Voiculescu sequence. That is to say, consider the Toeplitz extension [Pim97]

2 0 → 풦(푙 (퐴)) → 풯(퐴,훼) → 퐴 ⋊훼 ℤ → 0.

1 Its class in KK (퐴 ⋊훼 ℤ,퐴) is represented by the unbounded cycle (퐸,퐷) presented in Example 8.2.1 below, cf. [GMR18, Theorem 1] and [AM19, Section 4]. Example 8.2.1. Let 퐴 be a unital C∗-algebra equipped with an automorphism 훼∶ 퐴 → ∗ 퐴. Then, consider the crossed product C -algebra 퐴 ⋊훼 ℤ generated by 퐴 and a unitary 푢 satisfying 훼푛(푎) = 푢푛푎푢∗푛. 2 1 Equip the Hilbert 퐴-module 퐸 = 퐿 (푆 ) ⊗ℂ 퐴 with the Dirac operator 퐷 = 퐷푆1 ⊗ 1 and consider the unitary 푈 = 푆 ⊗ 1 ∈ ℒ(퐸), where 푆 is multiplication by 휃 ↦ 푒푖휋휃 on 2 1 푘 푘 퐿 (푆 ). Then, we may represent 퐴 ⋊훼 ℤ on 퐸 by the map 휌∶ ∑푘 푎푘푢 ↦ ∑푘 푎푘푈 .

147 8.2. The bivariant Chern character

Note that there is a dense subalgebra of 퐴⋊훼 ℤ with [퐷,푏] bounded because [퐷푆1 ,푆] = 1, so that (퐸,퐷) is an unbounded Kasparov (퐴 ⋊훼 ℤ,퐴)-module.

Taking the interior Kasparov product with this cycle, in turn, implements the boundary map 휕 in the Pimsner-Voiculescu six-term exact sequence [PV80]

−1 1−훼∗ 휄∗ K0(퐴) K0(퐴) K0(퐴 ⋊훼 ℤ)

휕 휕 . K (퐴 ⋊ ℤ) K (퐴) K (퐴) 1 훼 휄∗ 1 −1 1 1−훼∗

In [Nes88], Nest constructed a – notoriously complicated – map

• •+1 #∶ HC (풜) → HC (풜 ⋊훼 ℤ), compatible with the boundary map 휕 discussed above, that yields the compatible dia- gram HCev(풜) HCev(풜) HCev(풜 ⋊ ℤ) −1 휄∗ 훼 1−훼∗ # # . −1 odd 휄∗ odd 1−훼∗ odd HC (풜 ⋊훼 ℤ) HC (풜) HC (풜) It would be a beautiful result to realize both the boundary map 휕 in KK-theory and the corresponding map # in cyclic cohomology as resulting from the same, simple unbounded Kasparov cycle (퐸,퐷). The former is already known to be its class in KK-theory, and the latter should arise as its Chern character. Nistor’s theory of bivariant Chern characters, however, does not apply here in the 푝 desired generality. The problem is that the spaces 퐵(퐻) ⊗휋 풜 and ℒ (퐻) ⊗휋 풜 may be too small, as the following example shows. Example 8.2.2. Let 푀 be a compact smooth manifold that is equipped with a diffeo- morphism 휙∶ 푀 → 푀 and consider the unbounded Kasparov (퐶(푀) ⋊휙 ℤ,퐶(푀))- cycle (퐸,퐷) as in Example 8.2.1 (note: this is precisely the 1+-summable cycle of Sec- tion 7.3). ∞ If 휙 is topologically 2-transitive and 푓 ∈ 퐶 (푀) takes at least two different values 푧1,푧2, str 2 1 then the element 푓푢0 ∈ 퐶b (푀,ℒ(퐿 (푆 ))) is not norm continuous. That is to say, in that case the image of 푓푢0 in ℒ(퐸) does not lie in (the obvious representation of) 2 1 ℒ(퐿 (푆 )) ⊗휋 퐶(푀).

148 8.2. The bivariant Chern character

Even though we have constructed well-behaved Schatten classes in the case where 퐴 is a commutative C∗-algebra, that does not suffice to treat this example. We are likely to wish to consider the cyclic cohomology of some locally convex subalgebra 풜 ⊂ 퐴, say, of smooth functions, where Dini’s theorem no longer applies. Moreover, even if we really desire to work with 퐴 itself, the relation (if any) between convergence of the tensor trace tr푛 and that of tr depends on the choice of topological tensor product. Given the fickle nature of these complications, it seems advisable for further treatment of the subject to embark with very specific examples in mind.

149

Bibliography

[AB84] A. Andrianov and L. Bonora. “Finite-Mode Regularization of the Fermion Functional Integral”. In: Nuclear Physics B 233.2 (1984), pp. 232–246. [Abe26] N. H. Abel. “Untersuchungen Über Die Reihe: 1+(m/1) x+ m⋅(m- 1)/(1⋅ 2)⋅ x2+ m⋅(m-1)⋅(m-2)/(1⋅ 2⋅ 3)⋅ x3+…u.s.w.” In: Journal Fur Die Reine Und Angewandte Mathematik 1 (1826), pp. 311–339. [Alm73] G. Almkvist. “Endomorphisms of Finitely Generated Projective Mod- ules over a Commutative Ring”. In: Arkiv for Matematik 11 (1973), pp. 263–301. [Alm74] G. Almkvist. “The Grothendieck Ring of the Category of Endomor- phisms”. In: Journal of Algebra 28 (1974), pp. 375–388. [AM19] F. Arici and B. Mesland. Toeplitz Extensions in Noncommutative Topol- ogy and Mathematical Physics. Nov. 2019. arXiv: 1911 . 05823 [math.OA]. [Ara96] J. Aramaki. “An Extension of the Ikehara Tauberian Theorem and Its Application”. In: Acta Mathematica Hungarica 71.4 (1996), pp. 297– 326. [Bal+02] A. P. Balachandran, B. P. Dolan, J. Lee, X. Martin, and D. O’Connor. “Fuzzy Complex Projective Spaces and Their Star-Products”. In: Journal of Geometry and Physics 43.2-3 (2002), pp. 184–204. [Bar07] J. W. Barrett. “Lorentzian Version of the Noncommutative Geometry of the Standard Model of Particle Physics”. In: Journal of Mathematics and Physics 48.1 (2007), pp. 012303, 7.

151 Bibliography

[Bar15] J. W. Barrett. “Matrix Geometries and Fuzzy Spaces as Finite Spec- tral Triples”. In: Journal of Mathematics and Physics 56.8 (2015), pp. 082301, 25. [BDG19] J. W. Barrett, P. Druce, and L. Glaser. “Spectral Estimators for Finite Non-Commutative Geometries”. In: Journal of Physics A 52.27 (2019), pp. 275203, 33. [Ber03] M. Berger. A Panoramic View of Riemannian Geometry. Springer- Verlag, Berlin, 2003, pp. xxiv+824. isbn: 3-540-65317-1. [Ber19] T. Berendschot. “Truncated Geometry: Spectral Approximation of the Torus”. Radboud University Nijmegen, June 2019. [BG05] I. Borg and P. J. F. Groenen. Modern Multidimensional Scaling. 2nd ed. Springer Series in Statistics. Springer, New York, 2005, pp. xxii+614. isbn: 978-0-387-25150-9 0-387-25150-2. [BG16] J. W. Barrett and L. Glaser. “Monte Carlo Simulations of Random Non-Commutative Geometries”. In: Journal of Physics A 49.24 (2016), pp. 245001, 27. [BG19] J. W. Barrett and J. Gaunt. Finite Spectral Triples for the Fuzzy Torus. 2019. arXiv: 1908.06796 [math.QA]. [BGV04] N. Berline, E. Getzler, and M. Vergne. Heat Kernels and Dirac Opera- tors. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004. x+363. isbn: 3-540-20062-2. [BJ83] S. Baaj and P. Julg. “Théorie Bivariante de Kasparov et Opérateurs Non Bornés Dans Les C -Modules Hilbertiens”. In: C. R. Acad. Sci. Paris Sér. I Math. 296.21 (1983), pp. 875–878. [BLS94] G. Bimonte, F. Lizzi, and G. Sparano. “Distances on a Lattice from Non-Commutative Geometry”. In: Physics Letters B 341.2 (1994), pp. 139–146. [BN02] M. Belkin and P. Niyogi. “Laplacian Eigenmaps and Spectral Tech- niques for Embedding and Clustering”. In: Advances in Neural Infor- mation Processing Systems 14. Ed. by T. G. Dietterich, S. Becker, and Z. Ghahramani. MIT Press, 2002, pp. 585–591. [BN07] M. Belkin and P. Niyogi. “Convergence of Laplacian Eigenmaps”. In: Advances in Neural Information Processing Systems. 2007, pp. 129–136.

152 Bibliography

[Ćać11] B. Ćaćić. “Moduli Spaces of Dirac Operators for Finite Spectral Triples”. In: Quantum Groups and Noncommutative Spaces. Aspects Math., E41. Vieweg + Teubner, Wiesbaden, 2011, pp. 9–68. [Car+07] A. L. Carey, A. Rennie, A. Sedaev, and F. Sukochev. “The Dixmier Trace and Asymptotics of Zeta Functions”. In: Journal of Functional Analysis 249.2 (2007), pp. 253–283. [Car+14] A. L. Carey, V. Gayral, A. Rennie, and F. A. Sukochev. “Index Theory for Locally Compact Noncommutative Geometries”. In: Memoirs of the American Mathematical Society 231.1085 (2014), pp. vi+130. [CC08] A. H. Chamseddine and A. Connes. “Why the Standard Model”. In: Journal of Geometry and Physics 58.1 (2008), pp. 38–47. [CC96] A. H. Chamseddine and A. Connes. “Universal Formula for Noncom- mutative Geometry Actions: Unification of Gravity and the Standard Model”. In: Physical Review Letters 77.24 (1996), pp. 4868–4871. [CC97] A. H. Chamseddine and A. Connes. “The Spectral Action Principle”. In: Communications in Mathematical Physics 186.3 (1997), pp. 731– 750. [CCM07] A. H. Chamseddine, A. Connes, and M. Marcolli. “Gravity and the Standard Model with Neutrino Mixing”. In: Advances in Theoretical and Mathematical Physics 11.6 (2007), pp. 991–1089. [CCM14] A. H. Chamseddine, A. Connes, and V. Mukhanov. “Geometry and the Quantum: Basics”. In: Journal of High Energy Physics 12 (2014), 098, front matter+24. [CCM15] A. H. Chamseddine, A. Connes, and V. Mukhanov. “Quanta of Ge- ometry: Noncommutative Aspects”. In: Physical Review Letters 114.9 (2015), pp. 091302, 5. [CL01] A. Connes and G. Landi. “Noncommutative Manifolds, the Instan- ton Algebra and Isospectral Deformations”. In: Communications in Mathematical Physics 221.1 (2001), pp. 141–159. [CL90] A. Connes and J. Lott. “Particle Models and Noncommutative Geome- try”. In: Nuclear Physics B. Vol. 18B. 1990, 29–47 (1991).

153 Bibliography

[CM08] A. Connes and M. Marcolli. Noncommutative Geometry, Quantum Fields and Motives. Vol. 55. American Mathematical Society Collo- quium Publications. American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2008, pp. xxii+785. isbn: 978-0-8218-4210-2. [CM95] A. Connes and H. Moscovici. “The Local Index Formula in Noncom- mutative Geometry”. In: Geometric and Functional Analysis 5.2 (1995), pp. 174–243. [Col87] Y. Colin de Verdière. “Construction de Laplaciens Dont Une Partie Finie Du Spectre Est Donnée”. In: Annales Scientifiques de l’École Normale Supérieure 20.4 (1987), pp. 599–615. [Con00] A. Connes. “A Short Survey of Noncommutative Geometry”. In: Journal of Mathematics and Physics 41.6 (2000), pp. 3832–3866. [Con13] A. Connes. “On the Spectral Characterization of Manifolds”. In: Journal of Noncommutative Geometry 7.1 (2013), pp. 1–82. [Con83] A. Connes. “Cohomologie Cyclique et Foncteurs Extn”. In: Comptes Rendus de l’Academie des Sciences de Paris. I 296.23 (1983), pp. 953– 958. [Con85] A. Connes. “Noncommutative Differential Geometry”. In: Inst. Hautes Études Sci. Publ. Math. 62 (1985), pp. 257–360. [Con90] J. B. Conway. A Course in Functional Analysis. 2nd ed. Vol. 96. Gradu- ate Texts in Mathematics. Springer-Verlag, New York, 1990, pp. xvi+399. isbn: 0-387-97245-5. [Con94] A. Connes. Noncommutative Geometry. Academic Press, Inc., San Diego, CA, 1994. xiv+661. isbn: 0-12-185860-X. [Con96] A. Connes. “Gravity Coupled with Matter and the Foundation of Non-Commutative Geometry”. In: Communications in Mathematical Physics 182.1 (1996), pp. 155–176. [CS12] D. Chelkak and S. Smirnov. “Universality in the 2D Ising Model and Conformal Invariance of Fermionic Observables”. In: Inventiones Mathematicae 189.3 (2012), pp. 515–580. [CS20] A. Connes and W. D. van Suijlekom. “Spectral Truncations in Noncom- mutative Geometry and Operator Systems”. In: Communications in Mathematical Physics (2020).

154 Bibliography

[CS84] A. Connes and G. Skandalis. “The Longitudinal Index Theorem for Foliations”. In: Publications of the Research Institute for Mathematical Sciences 20.6 (1984), pp. 1139–1183. [Dab+05] L. Dabrowski, G. Landi, M. Paschke, and A. Sitarz. “The Spectral Geometry of the Equatorial Podleś Sphere”. In: Comptes Rendus Mathe- matique. Academie des Sciences. Paris 340.11 (2005), pp. 819–822. [DAn16] F. D’Andrea. “Pythagoras Theorem in Noncommutative Geometry”. In: Noncommutative Geometry and Optimal Transport. Vol. 676. Con- temp. Math. Amer. Math. Soc., Providence, RI, 2016, pp. 175–210. [DD63] J. Dixmier and A. Douady. “Champs Continus d’espaces Hilbertiens et de C*-Algèbres”. In: Bulletin de la Societe Mathematique de France 91 (1963), pp. 227–284. [DG75] J. J. Duistermaat and V. W. Guillemin. “The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics”. In: Inventiones Mathematicae 29.1 (1975), pp. 39–79. [DH94] Y. Denizeau and J.-F. Havet. “Correspondances d’indice Fini. I. Indice d’un Vecteur”. In: Journal of Operator Theory 32.1 (1994), pp. 111– 156. [Dix77] J. Dixmier. C*-Algebras. North-Holland Publishing Co., Amsterdam- New York-Oxford, 1977, pp. xiii+492. isbn: 0-7204-0762-1. [DLM14a] F. D’Andrea, F. Lizzi, and P. Martinetti. “Deformations of the Canon- ical Commutation Relations and Metric Structures”. In: SIGMA. Symmetry, Integrability and Geometry. Methods and Applications 10 (2014), Paper 062, 14. [DLM14b] F. D’Andrea, F. Lizzi, and P. Martinetti. “Spectral Geometry with a Cut-off: Topological and Metric Aspects”. In: Journal of Geometry and Physics 82 (2014), pp. 18–45. [DLV13] F. D’Andrea, F. Lizzi, and J. C. Várilly. “Metric Properties of the Fuzzy Sphere”. In: Letters in Mathematical Physics 103.2 (2013), pp. 183– 205. [DM99] A. Dimakis and F. Müller-Hoissen. “Some Aspects of Noncommutative Geometry and Physics”. In: Novelties in String Theory (Göteborg, 1998). World Sci. Publ., River Edge, NJ, 1999, pp. 327–347. [DO03] B. P. Dolan and D. O’Connor. “A Fuzzy Three Sphere and Fuzzy Tori”. In: Journal of High Energy Physics 10 (2003), pp. 060, 16.

155 Bibliography

[Dun18] K. van den Dungen. The Kasparov Product on Submersions of Open Manifolds. Nov. 2018. arXiv: 1811.07824 [math.KT]. [Dun20] K. van den Dungen. Localisations of Half-Closed Modules and the Unbounded Kasparov Product. June 2020. arXiv: 2006 . 10616 [math.KT]. [Fel61] J. M. G. Fell. “The Structure of Algebras of Operator Fields”. In: Acta Mathematica 106 (1961), pp. 233–280. [FGD95] P. Flajolet, X. Gourdon, and P. Dumas. “Mellin Transforms and Asymp- totics: Harmonic Sums”. In: Theoretical Computer Science. Vol. 144. 1-2. 1995, pp. 3–58. [FL02] M. Frank and D. R. Larson. “Frames in Hilbert C*-Modules and C*- Algebras”. In: J. Operator Theory 48.2 (2002), pp. 273–314. [Gil95] P. B. Gilkey. Invariance Theory, the Heat Equation, and the Atiyah- Singer Index Theorem. 2nd ed. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995, pp. x+516. isbn: 0-8493-7874-4. [Gla17] L. Glaser. “Scaling Behaviour in Random Non-Commutative Geome- tries”. In: Journal of Physics A 50.27 (2017), pp. 275201, 18. [GMR18] M. Goffeng, B. Mesland, and A. Rennie. “Shift-Tail Equivalence and an Unbounded Representative of the Cuntz-Pimsner Extension”. In: Ergodic Theory and Dynamical Systems 38.4 (2018), pp. 1389–1421. [GP95] H. Grosse and P. Prešnajder. “The Dirac Operator on the Fuzzy Sphere”. In: Letters in Mathematical Physics 33.2 (1995), pp. 171– 181. [GS20] L. Glaser and A. B. Stern. “Understanding Truncated Non-Commutative Geometries through Computer Simulations”. In: Journal of Mathemat- ical Physics 61.3 (2020), pp. 033507, 11. [GS21] L. Glaser and A. B. Stern. “Reconstructing Manifolds from Truncations of Spectral Triples”. In: Journal of Geometry and Physics 159 (2021), p. 103921. [GVF01] J. M. Gracia-Bondía, J. C. Várilly, and H. Figueroa. Elements of Non- commutative Geometry. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Boston, Inc., Boston, MA, 2001. xviii+685. isbn: 0-8176-4124-6.

156 Bibliography

[Hör68] L. Hörmander. “The Spectral Function of an Elliptic Operator”. In: Acta Mathematica 121 (1968), pp. 193–218. [HR64] G. H. Hardy and M. Riesz. The General Theory of Dirichlet’s Series. Cambridge Tracts in Mathematics and Mathematical Physics, No. 18. Stechert-Hafner, Inc., New York, 1964. vii+78. [IKM01] B. Iochum, T. Krajewski, and P. Martinetti. “Distances in Finite Spaces from Noncommutative Geometry”. In: Journal of Geometry and Physics 37.1-2 (2001), pp. 100–125. [JL13] S. Jordan and R. Loll. “Causal Dynamical Triangulations without Preferred Foliation”. In: Physics Letters B 724.1 (2013), pp. 155–159. [Kaa17] J. Kaad. “Differentiable Absorption of Hilbert C*-Modules, Connec- tions, and Lifts of Unbounded Operators”. In: Journal of Noncommuta- tive Geometry 11.3 (2017), pp. 1037–1068. [Kad84] R. V. Kadison. “Diagonalizing Matrices”. In: American Journal of Mathematics 106.6 (1984), pp. 1451–1468. [Kas80] G. G. Kasparov. “Hilbert C*-Modules: Theorems of Stinespring and Voiculescu”. In: J. Operator Theory 4.1 (1980), pp. 133–150. [Kat95] T. Kato. Perturbation Theory for Linear Operators. Classics in Mathe- matics. Springer-Verlag, Berlin, 1995. xxii+619. isbn: 3-540-58661-X. [KL12] J. Kaad and M. Lesch. “A Local Global Principle for Regular Operators in Hilbert C*-Modules”. In: Journal of Functional Analysis 262.10 (2012), pp. 4540–4569. [KL13] J. Kaad and M. Lesch. “Spectral Flow and the Unbounded Kasparov Product”. In: Advances in Mathematics 248 (2013), pp. 495–530. [Kra98] T. Krajewski. “Classification of Finite Spectral Triples”. In: Journal of Geometry and Physics 28.1-2 (1998), pp. 1–30. [KS17] J. Kaad and W. D. van Suijlekom. Factorization of Dirac Operators on Almost-Regular Fibrations of Spinc Manifolds. 2017. arXiv: 1710. 03182 [math.FA]. [KS18] J. Kaad and W. D. van Suijlekom. “Riemannian Submersions and Factorization of Dirac Operators”. In: J. Noncommut. Geom. 12.3 (2018), pp. 1133–1159.

157 Bibliography

[Lan95] E. C. Lance. Hilbert C*-Modules. Vol. 210. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1995. x+130. isbn: 0-521-47910-X. [Lev06] B. Levy. “Laplace-Beltrami Eigenfunctions towards an Algorithm That ”Understands” Geometry”. In: IEEE International Conference on Shape Modeling and Applications 2006 (SMI’06). 2006, pp. 13–13. [Loh96] J. Lohkamp. “Discontinuity of Geometric Expansions”. In: Commen- tarii Mathematici Helvetici 71.2 (1996), pp. 213–228. [Lol98] R. Loll. “Discrete Approaches to Quantum Gravity in Four Dimen- sions”. In: Living Reviews in Relativity 1 (1998), pp. 1998–13, 35. [LS17] T. A. Loring and H. Schulz-Baldes. “Finite Volume Calculation of K-Theory Invariants”. In: New York Journal of Mathematics 23 (2017), pp. 1111–1140. [Mar06] P. Martinetti. “Carnot-Carathéodory Metric and Gauge Fluctuation in Noncommutative Geometry”. In: Communications in Mathematical Physics 265.3 (2006), pp. 585–616. [Mes14] B. Mesland. “Unbounded Bivariant K and Correspondences in Non- commutative Geometry”. In: Journal Fur Die Reine Und Angewandte Mathematik 691 (2014), pp. 101–172. [MP49] S. Minakshisundaram and Å. Pleijel. “Some Properties of the Eigen- functions of the Laplace-Operator on Riemannian Manifolds”. In: Canadian Journal of Mathematics 1 (1949), pp. 242–256. [MR16] B. Mesland and A. Rennie. “Nonunital Spectral Triples and Metric Completeness in Unbounded KK-Theory”. In: Journal of Functional Analysis 271.9 (2016), pp. 2460–2538. [Mrč05] J. Mrčun. “On Isomorphisms of Algebras of Smooth Functions”. In: Proceedings of the American Mathematical Society 133.10 (2005), pp. 3109–3113. [MT05] V. M. Manuilov and E. V. Troitsky. Hilbert C*-Modules. Vol. 226. Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 2005, pp. viii+202. isbn: 0-8218-3810-5. [Nes88] R. Nest. “Cyclic Cohomology of Crossed Products with Z”. In: Journal of Functional Analysis 80.2 (1988), pp. 235–283.

158 Bibliography

[Nis91] V. Nistor. “A Bivariant Chern Character for p-Summable Quasihomo- morphisms”. In: K-Theory 5.3 (1991), pp. 193–211. [Nis93] V. Nistor. “A Bivariant Chern-Connes Character”. In: Annals of Mathe- matics. Second Series 138.3 (1993), pp. 555–590. [ODo+16] B. O’Donoghue, E. Chu, N. Parikh, and S. Boyd. “Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding”. In: Journal of Optimization Theory and Applications 169.3 (2016), pp. 1042–1068. [Per13] A. Perez. “The Spin-Foam Approach to Quantum Gravity”. In: Living Reviews in Relativity 16.1 (Feb. 2013), p. 3. [Pim97] M. V. Pimsner. “A Class of C*-Algebras Generalizing Both Cuntz- Krieger Algebras and Crossed Products by Z”. In: Free Probability Theory (Waterloo,ON, 1995). Vol. 12. Fields Inst. Commun. Amer. Math. Soc., Providence, RI, 1997, pp. 189–212. [Ply86] R. J. Plymen. “Strong Morita Equivalence, Spinors and Symplectic Spinors”. In: Journal of Operator Theory 16.2 (1986), pp. 305–324. [PS06] M. Paschke and A. Sitarz. “On Spin Structures and Dirac Operators on the Noncommutative Torus”. In: Letters in Mathematical Physics 77.3 (2006), pp. 317–327. [PV80] M. Pimsner and D. Voiculescu. “Exact Sequences for K-Groups and Ext-Groups of Certain Cross-Product C -Algebras”. In: Journal of Operator Theory 4.1 (1980), pp. 93–118. [Rie04] M. A. Rieffel. Gromov-Hausdorff Distance for Quantum Metric Spaces. Matrix Algebras Converge to the Sphere for Quantum Gromov- Hausdorff Distance. American Mathematical Society, Providence, RI, 2004, pp. viii+91. [Rie99] M. A. Rieffel. “Metrics on State Spaces”. In: Documenta Mathematica 4 (1999), pp. 559–600. [Ros97] S. Rosenberg. The Laplacian on a Riemannian Manifold. Vol. 31. London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997, pp. x+172. isbn: 0-521-46300-9 0-521-46831- 0. [RT03] I. Raeburn and S. J. Thompson. “Countably Generated Hilbert Mod- ules, the Kasparov Stabilisation Theorem, and Frames with Hilbert Modules”. In: Proc. Amer. Math. Soc. 131.5 (2003), pp. 1557–1564.

159 Bibliography

[Rud91] W. Rudin. Functional Analysis. Second. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. xviii+424. isbn: 0-07-054236-8. [RV06] A. Rennie and J. C. Varilly. Reconstruction of Manifolds in Noncommu- tative Geometry. 2006. arXiv: math/0610418. [RW98] I. Raeburn and D. P. Williams. Morita Equivalence and Continuous- Trace C*-Algebras. Vol. 60. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998. xiv+327. isbn: 0-8218-0860-5. [Rya02] R. A. Ryan. Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2002. xiv+225. isbn: 1-85233-437-1. [See67] R. T. Seeley. “Complex Powers of an Elliptic Operator”. In: Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966). Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. [SG19] A. Stern and L. Glaser. A Python Implementation of the PointForge Algorithm, with the Sphere as an Example. 2019. url: https:// github.com/abstern/pointforge. [Sim05] B. Simon. Trace Ideals and Their Applications. Second. Vol. 120. Math- ematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. viii+150. isbn: 0-8218-3581-5. [SS16] L. Schneiderbauer and H. C. Steinacker. “Measuring Finite Quantum Geometries via Quasi-Coherent States”. In: Journal of Physics A 49.28 (2016), pp. 285301, 44. [Ste19a] A. Stern. A Simple Implementation of the SMACOF Algorithm in FORTRAN. 2019. url: https://github.com/abstern/py- SMACOF-weighted. [Ste19b] A. B. Stern. “Finite-Rank Approximations of Spectral Zeta Residues”. In: Letters in Mathematical Physics 109.3 (2019), pp. 565–577. [Sui15] W. D. van Suijlekom. Noncommutative Geometry and Particle Physics. Mathematical Physics Studies. Springer, Dordrecht, 2015, pp. xvi+237. isbn: 978-94-017-9161-8 978-94-017-9162-5. [Sui20] W. D. van Suijlekom. Gromov-Hausdorff Convergence of State Spaces for Spectral Truncations. 2020. arXiv: 2005.08544 [math.QA].

160 Bibliography

[SV19] W. D. van Suijlekom and L. S. Verhoeven. Immersions and the Un- bounded Kasparov Product: Embedding Spheres into Euclidean Space. 2019. arXiv: 1911.06044 [math.QA]. [Tak79] A. Takahashi. “A Duality between Hilbert Modules and Fields of Hilbert Spaces”. In: Rev. Colombiana Mat. 13.2 (1979), pp. 93–120. [Ver19] L. Verhoeven. “Embedding the Circle into the Plane in Unbounded KK-Theory”. Master’s thesis. Radboud University Nijmegen, 2019. [Vil03] C. Villani. Topics in Optimal Transportation. Vol. 58. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003, pp. xvi+370. isbn: 0-8218-3312-X. [Vil09] C. Villani. Optimal Transport. Vol. 338. Grundlehren Der Mathema- tischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009, pp. xxii+973. isbn: 978-3-540- 71049-3. [VLH03] J. de Vicente, J. Lanchares, and R. Hermida. “Placement by Thermody- namic Simulated Annealing”. In: Physics Letters A 317.5 (Oct. 2003), pp. 415–423. [Wal12] J.-C. Wallet. “Connes Distance by Examples: Homothetic Spectral Metric Spaces”. In: Reviews in Mathematical Physics 24.9 (2012), pp. 1250027, 26. [Weg93] N. E. Wegge-Olsen. K-Theory and C*-Algebras. Oxford Science Publica- tions. The Clarendon Press, Oxford University Press, New York, 1993. xii+370. isbn: 0-19-859694-4. [Wie88] N. Wiener. The Fourier Integral and Certain of Its Applications. Cam- bridge Mathematical Library. Cambridge University Press, Cambridge, 1988, pp. xviii+201. isbn: 0-521-35884-1. [Wod87] M. Wodzicki. “Noncommutative Residue. I. Fundamentals”. In: K- Theory, Arithmetic and Geometry (Moscow, 1984–1986). Vol. 1289. Lecture Notes in Math. Springer, Berlin, 1987, pp. 320–399. [ŻS01] K. Życzkowski and W. Słomczyński. “The Monge Metric on the Sphere and Geometry of Quantum States”. In: Journal of Physics A 34.34 (2001), pp. 6689–6722.

161

Samenvatting

Dit proefschrift verkent twee facetten van de spectrale meetkunde: dat van eindige benadering en dat van continue verandering. In deze samenvatting, bedoeld voor de geïnteresseerde van buiten het vakgebied, probeer ik een idee te geven van wat de drie schuingedrukte termen uit die vorige zin inhouden in de context van dit proefschrift.

Spectrale meetkunde Welk geluid een muziekinstrument maakt, wordt bepaald door de frequenties waarmee het kan trillen (als reactie op de handelingen van de muzikant). De hoogte van die frequenties is vervolgens afhankelijk van de vorm van het instrument. Zo kan je aan bijvoorbeeld de lengte van een pianosnaar of de straal van een trommel al zien hoe hoog die zal klinken. Wiskundig gezien zouden we zeggen: de meetkunde (de vorm, in dit geval van het instrument) zegt iets over het spectrum (de frequenties) van bepaalde operatoren (zoals de wiskundige bewerking die – in bovenstaand voorbeeld – de voortstuwing van geluidsgolven door het instrument beschrijft). De spectrale meetkunde gaat over de omgekeerde kwestie: als we alleen dat spectrum (de frequenties dus) weten – of het nou van een instrument, een tafel, of een waterstofatoom is – wat kunnen we daarmee zeggen over de vorm?

Eindige benadering Omdat we het in de praktijk moeten doen met een beperkt waarnemingsvermogen en een beperkte rekencapaciteit, introduceert dit proefschrift een methode om vormen te beschrijven aan de hand van een klein deel van de mogelijke resonantiefrequenties2.

2Dat wil hier zeggen: van stationaire geluidsgolven of, algemener, van het spectrum van een operator van het Laplace-type.

163 Samenvatting

Het blijkt zo te zijn dat steeds hogere frequenties steeds meer detail toevoegen, maar juist steeds minder van doen hebben met de grote lijnen (zie figuur 1). Daarom kunnen we de lagere frequenties gebruiken om een soort schets te maken.

Figuur 1: Een klein deel van de frequenties overheerst het geluid: ook zonder de hoge frequenties (onder) blijven de grote lijnen behouden.

De hoofdstukken 2 en 3 gaan dus over de vraag: hoe weet je welke vorm een object heeft, als je alleen maar deels weet hoe het klinkt? Eerst wordt in hoofdstuk 2 een techniek ontwikkeld om – door het gedrag van hittestroom3 op heel korte termijn te bekijken en in verband te brengen met het gedrag van stationaire hittegolven van steeds hogere frequentie – de meetkundige rol van de laagfrequente resonanties te isoleren. Vervolgens legt hoofdstuk 3 een verband tussen lokalisatie – in de zin van op-één-plek- zijn – en de ingrediënten uit hoofdstuk 2. Door combinaties van laagfrequente golven te maken die zoveel mogelijk zijn gelokaliseerd, krijgen we een (schetsmatig) beeld van de kleinste stukjes waar de vorm uit bestaat (zie figuur 2). Die stukjes worden vervolgens aan elkaar gekoppeld door een formule van Connes, die de afstand tussen punten relateert aan de frequenties van golven die ertussen bewegen.

Figuur 2: Optimaal gelokaliseerde, Figuur 3: Een reconstructie door laagfrequente golven op de bol. PointForge.

Op basis van deze wiskundige constructie schreven we vervolgens het computerprogramma

3Hittestroom is gerelateerd aan het gedrag van geluidsgolven: het gaat allebei om het verspreiden van trillingen door het materiaal.

164 Samenvatting

PointForge, dat een reeks resonantiefrequenties kan vertalen in een afbeelding van het resonerende object (zie figuur 3). De vraag die centraal staat in hoofdstuk 4 is: voor welke patronen van resonantie- frequenties bestaat er daadwerkelijk een vorm die zo klinkt? In dit hoofdstuk zetten we de eerste stappen in die richting, door een natuurkundig concept te gebruiken om zulke patronen te detecteren en dat concept aan te passen aan de situatie waar alleen kennis van de lage frequenties beschikbaar is. In een computerverificatie van deze methode – een experiment, zogezegd – vonden we een nieuwe meetkundige structuur die verrassend genoeg op lage frequenties beter de eigenschappen van een bol vertoont dan de bol dat zelf doet. Dat wil zeggen, als we niet al van tevoren hadden geweten welke van de getalletjes in onze computer daadwerkelijk een bol voorstelden, hadden we de verkeerde gekozen. Als de vraag uit dit hoofdstuk in verder onderzoek wordt beantwoord, kunnen we de wereld van alle mogelijke vormen verkennen in termen van de wereld van alle ‘correcte’ patronen van resonantiefrequenties: we kunnen dan ‘op gehoor’ vormen construeren. De beoogde toepassing van dit hele onderzoek naar de gedeeltelijke resonantiespectra ligt in de quantumzwaartekracht, omdat de daarvoor cruciale interactie tussen meetkunde en energie veel tastbaarder is in termen van resonantiefrequenties. Verder is er een sterk verband met de computerwetenschap, omdat in machine learning en computer graphics een zoektocht gaande is naar nieuwe methodes om grip te krijgen op de meetkunde van zowel abstracte vormen (zoals datasets) als concrete vormen (voor beeldmanipulatie en animatie).

Continue verandering Het spoor is een fundamenteel en veelzijdig wiskundig gereedschap in de functionaal- analyse, en in het bijzonder in de spectrale meetkunde; bijvoorbeeld alle meetkundige informatie uit hoofdstuk 2 wordt uitgedrukt in termen van sporen. Kort en inadequaat gezegd is het spoor een manier om het spectrum van een operator in één getal samen te vatten, zoals je bijvoorbeeld een gemiddelde zou gebruiken om een hoop verschillende getallen te representeren. We komen zometeen op deze sporen terug. In de spectrale meetkunde wordt het concept van verandering (van vorm) beschreven door een verder verfijnd concept van de vormen zelf – we zouden zeggen dat devorm langs iets, zoals bijvoorbeeld langs een tijdslijn, verandert, en beschouwen vervolgens dat hele veranderende proces in één keer, zoals in figuur 4.

165 Samenvatting

Figuur 4: Twee cirkels veranderen in Figuur 5: Een regenboog verliest zijn één cirkel: een verandering van vorm hoogfrequente kleuren. is zelf ook een vorm.

Wiskundig gezien wordt zo’n veranderende vorm beschreven met zogenaamde Hilbert- modules over abelse C∗-algebras. In deze context, echter, bestond zoiets als een spoor nog niet, en dat belemmerde ons om onze spoorformules uit de spectrale meetkunde toe te passen in deze bredere sfeer en zo een beter begrip te krijgen van de relatie tussen verandering en eindpunt. Het laatste deel van dit proefschrift introduceert dan ook een systematische theorie van sporen in de context van zulke Hilbert-modules.

166 Acknowledgements

…et haec olim meminisse iuvabit.

Virgil, Aeneid 1.203

I extend my deepest gratitude to my doctoral advisor, Walter van Suijlekom. With your tireless scientific enthusiasm, generous availability and steadfast consideration ofmy personal well-being, you brought a true centre to what would otherwise have easily been an unmoored experience as a PhD student mostly–in absence. I thank Francesca Arici, John Barrett, Klaas Landsman, Hessel Posthuma and Raimar Wulkenhaar for taking place in the manuscript committee and for taking the time to consider the final draft. Most of Part I of this thesis was written in a two-and-a-half-year–long collaboration with Lisa Glaser. Dear Lisa, thank you so much for your work, your companionship and your advice. This thesis would likely not have been written without you, and certainly not with my sanity intact. I truly appreciate Jens Kaad for his hospitality and advice during and following a month- long visit to Odense. His perspective on the noncommutative Chern character problem was essential for the decision to focus on the abelian case, even though that leaves the original question wide-open. Furthermore, his close reading and valuable comments made for a significant improvement of Chapters 5–7. Thanks to my comrades in Nijmegen! Thank you, Ruben, for your munificence as a walking encyclopedia and for uncountably many though-provoking exchanges. Peter, thank you for appreciating the yin of PhD studies. Julius, thank you for your com- panionship in the office and your talent at the ping–pong table. Henrique, thankyou

167 Acknowledgements

for bringing a unique philosophical perspective and truly animating our discourse far beyond your stay. Sohail, fellow traveler from the west coast, thank you for making me feel I was among colleagues even during the many days I spent alone in the local library. Bert, thank you for welcoming me to Nijmegen with truly heartening joie de vivre and for demonstrating the way of amiable audacity. Thank you, Johan, for being a role model and for bringing reassuring advice when most needed. Even if I do not share your faith, you have instilled in me a new respect for it. Many thanks to Chris Ripken, who proposed we bring the lost tribe of mathematicians into the fold of workplace representation, and for our efforts together with Jordy Davelaar to help PhD students prosper. I salute Florian Zeiser and Veronica Fabiani for picking up the thread where we left it and turning it into so much more. The greatest risk to this thesis arrived very close to its completion. Some people in particular were instrumental in turning that moment around. First of all, Harry Linders, without your phone call I would have abandoned this thesis. I am deeply grateful for the way things turned out: thank you! Larissa Wilhelm-Moody, you have my sincere gratitude for patiently guiding me to take responsibility for my personal future. Gabriel Lord, thank you for playing the role as guardian of the finish line with valor. My sincere thanks to Renate Loll, for both insightful advice and scientific inspiration. Gert Heckman, Erik Koelink, Francisca Maalcke-Luesken, and Monique Slegers: thank you for taking the time to show me the way. I deeply appreciate Felicia Flamm and Laura Sitniakowsky for their encouragement, good advice, and invaluable perspective. I am grateful to those who played a very important role in inspiring me to pursue a PhD in the first place. Thanks to Petr Dunin-Barkowski, who showed me the joyof collaboration and the freedom of scientific experimentation. Thanks to Özgür Ceyhan, for inestimable guidance and support and for introducing me to the complexity of mathematical physics. Thanks again to Hessel Posthuma, for being a patient and inspiring thesis advisor in what was for me a very difficult time, and for introducing me to nommutative geometry. I am indebted to Hugo Bouma for originally persuading me towards the exact sciences – it was my great appreciation for you that swayed me then, and it is with great appreciation that I now thank you for playing this pivotal role.

168 Acknowledgements

Lieve Tera, het is met trots en vreugd dat ik dit proefschrift aan jou opdraag. De dingen komen zelden alleen, maar komen allemaal door en na en om elkaar. En al die dingen zoeken ruimte om in te zijn, en zo worden we vol van hart, steeds groter en groter. En het hart is een goed thuis voor al die dingen, en ook voor ons. In het mijne ben jij, en dat doet mij heel veel goed. Lieve familie, lieve Mas, Roel, Victor, Trienke en Aldo, dankjulliewel voor zoveel steun en betrokkenheid! Lars, vriend, ik zou niet weten waar ik moet beginnen. Dankjewel voor zevenentwintig jaar broederschap. Lieve Jesse en lieve Nina, ik heb dit proefschrift geschreven in jullie allereerste levensjaren. Jullie zijn, en dus was die tijd, mij onnoemmelijk dierbaar. Dit werk heeft ook jullie wat gekost, toen ik moe, afgeleid, ongeduldig of chagrijnig was terwijl jullie beter verdienden. Jullie hebben er ook aan bijgedragen, door me de beste motivatie te geven om wat te maken van mezelf. Ik hoop dat het het waard was, en dank jullie van harte voor jullie geduld. Lieve Laura, met jou komen voor mij de meest fundamentele verlangens in vervulling: me thuis te voelen in de wereld en dit leven betekenis te geven. Ik had niet durven ho- pen dat het zo zou kunnen zijn. Wat dit proefschrift betreft: dankjewel voor vijf jaar aanmoediging, geruststelling, steun, afleiding, opbeuring, inspiratie, begrip, gehoor en geduld. Het is gezien; het is niet onopgemerkt gebleven.

169 Acknowledgements

Dit werk maakt deel uit van het onderzoekprogramma van de Stichting voor Fundamenteel Onderzoek der Materie (FOM), die deel uitmaakt van de Neder- landse Organisatie voor Wetenschappelijk Onderzoek (NWO).