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Koen van den Dungen

Publications and source records attributed to Koen van den Dungen.

At least 19 recordsLinked to original sources
arXiv2026

Analytic index theory and spectral flow in real Hilbert $C^*$-modules

We consider the analytic index and spectral flow of Fredholm operators on Hilbert $C^*$-modules. Our spaces and algebras are equipped with a real structure, so the analytic index and spectral flow takes value in the real $K$-theory group of a $σ$-unital $C^*$-algebra. We use Van Daele $K$-theory, which allows us to treat the eight real $K$-theory groups and the two complex groups on an equal footing. We provide a general definition of the analytic index for Clifford anti-linear and skew-adjoint Fredholm operators as well as self-adjoint and odd Fredholm operators. Our definition of spectral flow and its basic properties are valid for Wahl-continuous paths of Fredholm operators on a real Hilbert $C^*$-module. We also provide an analytic approach to the spectral flow as a decomposition into a finite sum of relative indices. Furthermore, we prove a real version of the Robbin-Salamon theorem, relating the spectral flow to a Fredholm index. Our description of the index and spectral flow relies on various isomorphisms between Kasparov's $KKR$-theory and Van Daele $K$-theory, which we systematically describe in the Appendix.

math.OA↗
arXiv2026

A K-theoretic note on the spectral localiser

We review the construction of the spectral localiser (due to Loring and Schulz-Baldes) from a K-theoretic perspective. We first give a K-theoretic argument providing a spectral flow expression for the even or odd index pairing in terms of the "infinite volume" spectral localiser. Our approach towards this first step is more direct, treats the even and odd cases on an equal footing, and has the advantage that the construction of the spectral localiser becomes immediately apparent from the computation of the index pairing via a Kasparov product. In a second step of "spectral truncation", we then describe how this spectral flow expression can be computed in terms of the signature of the "finite volume" spectral localiser. Throughout, we do not require invertibility of the operator representing the K-homology class, and the even index pairing then obtains an additional contribution coming from the Fredholm index.

math.KT↗
arXiv2025

Dirac-Schrödinger operators, index theory, and spectral flow

In this article we study generalised Dirac-Schrödinger operators in arbitrary signatures (with or without gradings), providing a general KK-theoretic framework for the study of index pairings and spectral flow. We provide a general Callias Theorem, which shows that the index (or the spectral flow, or abstractly the K-theory class) of Dirac-Schrödinger operators can be computed on a suitable compact hypersurface. Furthermore, if the zero eigenvalue is isolated in the spectrum of the Dirac operator, we relate the index (or spectral flow) of Dirac--Schrödinger operators to the index (or spectral flow) of corresponding Toeplitz operators. Combining both results, we obtain an index (or spectral flow) equality relating Toeplitz operators on the noncompact manifold to Toeplitz operators on the compact hypersurface. Our results generalise various known results from the literature, while presenting these results in a common unified framework.

math.KT↗
arXiv2025

Fredholm complexes of Hilbert C*-modules

We investigate complexes of Hilbert C*-modules, which are cochain complexes with (unbounded) regular operators between Hilbert C*-modules as differential maps. In particular, we provide various equivalent characterizations of the Fredholm property for such complexes of Hilbert C*-modules, and we define the Fredholm index taking values in the K-theory group of the C*-algebra. Among other properties of this index, we prove the stability under small or relatively compact perturbations, and we obtain alternative expressions for the index under the existence of a (weak or strong) Hodge decomposition.

math.OA↗
arXiv2024

Generalised Dirac-Schrödinger operators and the Callias Theorem

We consider generalised Dirac--Schrödinger operators, consisting of a self-adjoint elliptic first-order differential operator D with a skew-adjoint 'potential' given by a (suitable) family of unbounded operators. The index of such an operator represents the pairing (Kasparov product) of the K-theory class of the potential with the K-homology class of D. Our main result in this paper is a generalisation of the Callias Theorem: the index of the Dirac--Schrödinger operator can be computed on a suitable compact hypersurface. Our theorem simultaneously generalises (and is inspired by) the well-known result that the spectral flow of a path of relatively compact perturbations depends only on the endpoints.

math.KT↗
arXiv2022

Localisations of half-closed modules and the unbounded Kasparov product

In the context of the Kasparov product in unbounded KK-theory, a well-known theorem by Kucerovsky provides sufficient conditions for an unbounded Kasparov module to represent the (internal) Kasparov product of two other unbounded Kasparov modules. In this article, we discuss several improved and generalised variants of Kucerovsky's theorem. First, we provide a generalisation which relaxes the positivity condition, by replacing the lower bound by a relative lower bound. Second, we also discuss Kucerovsky's theorem in the context of half-closed modules, which generalise unbounded Kasparov modules to symmetric (rather than self-adjoint) operators. In order to deal with the positivity condition for such non-self-adjoint operators, we introduce a fairly general localisation procedure, which (using a suitable approximate unit) provides a 'localised representative' for the KK-class of a half-closed module. Using this localisation procedure, we then prove several variants of Kucerovsky's theorem for half-closed modules. A distinct advantage of the localised approach, also in the special case of self-adjoint operators (i.e., for unbounded Kasparov modules), is that the (global) positivity condition in Kucerovsky's original theorem is replaced by a (less restrictive) 'local' positivity condition, which is closer in spirit to the well-known Connes-Skandalis theorem in the bounded picture of KK-theory.

math.KT↗
arXiv2020

The Kasparov product on submersions of open manifolds

We study the Kasparov product on (possibly non-compact and incomplete) Riemannian manifolds. Specifically, we show on a submersion of Riemannian manifolds that the tensor sum of a regular vertically elliptic operator on the total space and an elliptic operator on the base space represents the Kasparov product of the corresponding classes in KK-theory. This construction works in general for symmetric operators (i.e. without assuming self-adjointness), and extends known results for submersions with compact fibres. The assumption of regularity for the vertically elliptic operator is not always satisfied, but depends on the topology and geometry of the submersion, and we give explicit examples of non-regular operators. We apply our main result to obtain a factorisation in unbounded KK-theory of the fundamental class of a Riemannian submersion, as a Kasparov product of the shriek map of the submersion and the fundamental class of the base manifold.

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arXiv2019

Index Theory for Globally Hyperbolic Spacetimes

An equality between the spectral flow of a family $A$ of self-adjoint Fredholm operators and the index of the associated differential operator $\frac{d}{dt}-iA$ with Atiyah-Patodi-Singer-style boundary conditions is shown. This generalizes a result by C.Bär and A. Strohmaier used to show their Lorentzian version of the Atiyah-Patodi-Singer Index Theorem.

math.SP↗
arXiv2019

Addendum to: Indefinite Kasparov modules and pseudo-Riemannian manifolds

We improve our previous results on indefinite Kasparov modules, which provide a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. In particular, we can weaken the assumptions that are imposed on indefinite Kasparov modules. Using a new theorem by Lesch and Mesland on the self-adjointness and regularity of the sum of two weakly anticommuting operators, we show that we still have an equivalence between indefinite Kasparov modules and pairs of Kasparov modules. Importantly, the weakened version of indefinite Kasparov modules now includes the main motivating example of the Dirac operator on a pseudo-Riemannian manifold. The appendix contains a construction of an approximate identity for weakly commuting operators, which is due to Lesch and Mesland.

math.KT↗
arXiv2019

Homotopy equivalence in unbounded KK-theory

We propose a new notion of unbounded $K\!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $σ$-unital $C^{*}$-algebras, we can then associate a semigroup $\overline{U\!K\!K}(A,B)$ of homotopy equivalence classes of unbounded cycles, and we prove that this semigroup is in fact an abelian group. In case $A$ is separable, our group $\overline{U\!K\!K}(A,B)$ is isomorphic to Kasparov's $K\!K$-theory group $K\!K(A,B)$ via the bounded transform. We also discuss various notions of degenerate cycles, and we prove that the homotopy relation on unbounded cycles coincides with the relation generated by operator-homotopies and addition of degenerate cycles.

math.KT↗
arXiv2018

The index of generalised Dirac-Schrödinger operators

We study the relation between spectral flow and index theory within the framework of (unbounded) KK-theory. In particular, we consider a generalised notion of 'Dirac-Schrödinger operators', consisting of a self-adjoint elliptic first-order differential operator D with a skew-adjoint 'potential' given by a (suitable) family of unbounded operators on an auxiliary Hilbert module. We show that such Dirac-Schrödinger operators are Fredholm, and we prove a relative index theorem for these operators (which allows cutting and pasting of the underlying manifolds). Furthermore, we show that the index of a Dirac-Schrödinger operator represents the pairing (Kasparov product) of the K-theory class of the potential with the K-homology class of D. We prove this result without assuming that the potential is differentiable; instead, we assume that the 'variation' of the potential is sufficiently small near infinity. In the special case of the real line, we recover the well-known equality of the index with the spectral flow of the potential.

math.KT↗
arXiv2018

Families of spectral triples and foliations of space(time)

We study a noncommutative analogue of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime can be reconstructed from the family of Dirac operators on the hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples. In the case of Riemannian signature, we prove that the construction yields in fact a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, we correspondingly obtain a 'Lorentzian spectral triple', which can also be viewed as the 'reverse Wick rotation' of a product spectral triple. This construction of 'Lorentzian spectral triples' fits well into the Krein space approach to noncommutative Lorentzian geometry.

math-ph↗
arXiv2018

Locally bounded perturbations and (odd) unbounded KK-theory

A regular symmetric operator on a Hilbert module is self-adjoint whenever there exists a suitable approximate identity. We say an operator is 'locally bounded' if the composition of the operator with each element in the approximate identity is bounded. We prove that the perturbation of a regular self-adjoint operator by a locally bounded symmetric operator is again regular and self-adjoint. We use this result to show that the Kasparov class represented by an unbounded Kasparov module is stable under locally bounded perturbations. As an application, we show that we obtain a converse to the 'doubling up' procedure of odd unbounded Kasparov modules. Finally, we discuss perturbations of unbounded Kasparov modules by unbounded multipliers. In particular, we explicitly construct an unbounded multiplier such that (after doubling up the module) the perturbed operator has compact resolvent.

math.OA↗
arXiv2016

Krein spectral triples and the fermionic action

Motivated by the space of spinors on a Lorentzian manifold, we define Krein spectral triples, which generalise spectral triples from Hilbert spaces to Krein spaces. This Krein space approach allows for an improved formulation of the fermionic action for almost-commutative manifolds. We show by explicit calculation that this action functional recovers the correct Lagrangians for the cases of electrodynamics, the electro-weak theory, and the Standard Model. The description of these examples does not require a real structure, unless one includes Majorana masses, in which case the internal spaces also exhibit a Krein space structure.

math-ph↗
arXiv2016

Indefinite Kasparov modules and pseudo-Riemannian manifolds

We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reversible. We present three examples of our framework: the Dirac operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an indefinite metric), the harmonic oscillator, and the construction via the Kasparov product of an indefinite spectral triple from a family of spectral triples. This last construction corresponds to a foliation of a globally hyperbolic spacetime by spacelike hypersurfaces.

math.KT↗
arXiv2014

On globally non-trivial almost-commutative manifolds

Within the framework of Connes' noncommutative geometry, we define and study globally non-trivial (or topologically non-trivial) almost-commutative manifolds. In particular, we focus on those almost-commutative manifolds that lead to a description of a (classical) gauge theory on the underlying base manifold. Such an almost-commutative manifold is described in terms of a 'principal module', which we build from a principal fibre bundle and a finite spectral triple. We also define the purely algebraic notion of 'gauge modules', and show that this yields a proper subclass of the principal modules. We describe how a principal module leads to the description of a gauge theory, and we provide two basic yet illustrative examples.

math-ph↗
arXiv2013

Pseudo-Riemannian spectral triples and the harmonic oscillator

We define pseudo-Riemannian spectral triples, an analytic context broad enough to encompass a spectral description of a wide class of pseudo-Riemannian manifolds, as well as their noncommutative generalisations. Our main theorem shows that to each pseudo-Riemannian spectral triple we can associate a genuine spectral triple, and so a K-homology class. With some additional assumptions we can then apply the local index theorem. We give a range of examples and some applications. The example of the harmonic oscillator in particular shows that our main theorem applies to much more than just classical pseudo-Riemannian manifolds.

math.OA↗