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Finite Approximation and Continuous Change in Spectral Geometry

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Finite Approximation and Continuous Change in Spectral Geometry 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
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