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Jens Kaad

Publications and source records attributed to Jens Kaad.

At least 19 recordsLinked to original sources
arXiv2026

Quantum deformations of the arc length metric

We investigate a q-deformation of the arc length metric on the unit circle. This q-deformation arises naturally from the Dirac operator by replacing the standard integers by their q-deformed analogues. Nonetheless, we show that the corresponding metric structure only makes sense at the level of quantum metric spaces as introduced by Marc Rieffel. This means that the quantum metric we obtain on the continuous functions on the circle does not arise from a classical metric on the circle. In the special case where q equals one we recover the usual arc length metric and we show that our family of quantum metric spaces depend continuously on the deformation parameter with respect to David Kerr's complete Gromov-Hausdorff distance.

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arXiv2025

Spectral localizers in KK-theory

We study the index homomorphism of even K-groups arising from a class in even KK-theory via the Kasparov product. Due to the seminal work of Baaj and Julg, under mild conditions on the C^*-algebras in question such a class in KK-theory can always be represented by an unbounded Kasparov module. We then describe the corresponding index homomorphism of even K-groups in terms of spectral localizers. This means that our explicit formula for the index homomorphism does not depend on the full spectrum of the abstract Dirac operator D, but rather on the intersection between this spectrum and a compact interval. The size of this compact interval does however reflect the interplay between the K-theoretic input and the abstract Dirac operator. Since the spectral projections for D are not available in the general context of Hilbert $C^*$-modules we instead rely on certain continuous compactly supported functions applied to D to construct the spectral localizer. In the special case where even KK-theory coincides with even K-homology, our work recovers the pioneering work of Loring and Schulz-Baldes on the index pairing.

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arXiv2025

Spectral metrics on quantum projective spaces

We show that the noncommutative differential geometry of quantum projective spaces is compatible with Rieffel's theory of compact quantum metric spaces. This amounts to a detailed investigation of the Connes metric coming from the unital spectral triple introduced by D'Andrea and Dabrowski. In particular, we establish that the Connes metric metrizes the weak-* topology on the state space of quantum projective space. This generalizes previous work by the second author and Aguilar regarding spectral metrics on the standard Podles spheres.

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arXiv2025

Noncommutative metric geometry of quantum circle bundles

In this paper we investigate quantum circle bundles from the point of view of compact quantum metric spaces. The raw input data is a circle action on a unital $C^*$-algebra together with a quantum metric of spectral geometric origin on the fixed point algebra. Under a few extra conditions on the spectral subspaces we show that the spectral geometric data on the base algebra can be lifted to the total algebra. Notably, the lifted spectral geometry is independent of the choice of frames and is permitted to interact with the total algebra via a twisted derivation. Under these conditions, it is explained how to assemble our data into a quantum metric on the total algebra in a way which unifies and generalizes a couple of results in the literature relating to crossed products by the integers and to quantum $SU(2)$. We apply our ideas to the higher Vaksman-Soibelman quantum spheres and endow them with quantum metrics arising from $q$-geometric data. In this context, the twist is enforced by the structure of the Drinfeld-Jimbo deformation arising from the Lie algebra of the special unitary group.

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arXiv2025

The quantum metric structure of quantum SU(2)

We introduce a two parameter family of Dirac operators on quantum SU(2) and analyse their properties from the point of view of non-commutative metric geometry. It is shown that these Dirac operators give rise to compact quantum metric structures, and that the corresponding two parameter family of compact quantum metric spaces varies continuously in Rieffel's quantum Gromov-Hausdorff distance. This continuity result includes the classical case where we recover the round 3-sphere up to a global scaling factor on the metric. Our main technical tool is a quantum SU(2) analogue of the Berezin transform, together with its associated fuzzy approximations, the analysis of which also leads to a systematic way of approximating Lipschitz operators by means of polynomial expressions in the generators.

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arXiv2024

Quantum metrics on crossed products with groups of polynomial growth

We show how to equip the crossed product between a group of polynomial growth and a compact quantum metric space with a compact quantum metric space structure. When the quantum metric on the base space arises from a spectral triple, which is compatible with the action of the group, we furthermore show that the crossed product becomes a spectral metric space. Lastly, we analyse the spectral triple at the crossed product level from the point of view of unbounded $KK$-theory and show that it arises as an internal Kasparov product of unbounded Kasparov modules.

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arXiv2023

External products of spectral metric spaces

In this paper, we present a characterization of compact quantum metric spaces in terms of finite dimensional approximations. This characterization naturally leads to the introduction of a matrix analogue of a compact quantum metric space. As an application, we show that matrix compact quantum metric spaces are stable under minimal tensor products and more specifically that matrix spectral metric spaces are stable under the external product operation on unital spectral triples. We present several noncommutative examples of matrix compact quantum metric spaces.

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arXiv2023

The Haar state on the Vaksman-Soibelman quantum spheres

In this note we present explicit formulae for the Haar state on the Vaksman-Soibelman quantum spheres. Our formulae correct various statements appearing in the literature and our proof is straightforward relying simply on properties of the modular automorphism group for the Haar state.

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arXiv2023

A comparison of two quantum distances

We show that Rieffel's quantum Gromov-Hausdorff distance between two compact quantum metric spaces is not equivalent to the ordinary Gromov-Hausdorff distance applied to the associated state spaces.

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arXiv2022

The Podles spheres converge to the sphere

We prove that the Podles spheres $S_q^2$ converge in quantum Gromov-Hausdorff distance to the classical 2-sphere as the deformation parameter $q$ tends to 1. Moreover, we construct a $q$-deformed analogue of the fuzzy spheres, and prove that they converge to $S_q^2$ as their linear dimension tends to infinity, thus providing a quantum counterpart to a classical result of Rieffel.

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arXiv2021

A higher Kac-Moody extension for two-dimensional gauge groups

Let $Γ$ be a finite dimensional Lie group and consider the smooth double loop group, i.e. the Fréchet Lie group of smooth maps from the 2-torus to $Γ$. For a finite dimensional Hilbert space V, let H denote the Hilbert space of vector valued $L^2$-functions on the 2-torus. The purpose of this paper is to construct a higher central extension of the smooth double loop group from the representation of the smooth double loop group on H induced by a smooth action of $Γ$ on V. This higher central extension comes from an action of the smooth double loop group on a 2-category and yields a group cohomology class of degree 3 on the smooth double loop group. We show by a concrete computation that this group cohomology class is non-trivial in general. We relate our higher central extension to the Kac-Moody extension of the smooth single loop group as a higher dimensional analogue of the latter. More generally, given a group G acting on a bipolarised Hilbert space, we apply higher category theory to construct a group cohomology class of degree 3 on G. As a second motivating example, we use these ideas to introduce a higher central extension of the group of invertible smooth functions on the noncommutative 2-torus.

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arXiv2021

Gromov-Hausdorff convergence of quantised intervals

The Podles quantum sphere S^2_q admits a natural commutative C*-subalgebra I_q with spectrum {0} \cup {q^{2k}: k = 0,1,2,...}, which may therefore be considered as a quantised version of a classical interval. We study here the compact quantum metric space structure on I_q inherited from the corresponding structure on S^2_q, and provide an explicit formula for the metric induced on the spectrum. Moreover, we show that the resulting metric spaces vary continuously in the deformation parameter q with respect to the Gromov-Hausdorff distance, and that they converge to a classical interval of length pi as q tends to 1.

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arXiv2021

Kernels of Hilbert Module Maps: A Counterexample

Answering a long standing question, we give an example of a Hilbert module and a nonzero bounded right linear map having a kernel with trivial orthogonal complement. In particular, this kernel is different from its own double orthogonal complement.

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arXiv2020

Gysin sequences and SU(2)-symmetries of C*-algebras

Motivated by the study of symmetries of C*-algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz-Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K-theory. In particular, starting from an irreducible representation of SU(2), we show that the corresponding Toeplitz algebra is equivariantly KK-equivalent to the algebra of complex numbers. In this way, we obtain a six term exact sequence of K-groups containing a noncommutative analogue of the Euler class.

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arXiv2020

On the Unbounded Picture of $KK$-Theory

In the founding paper on unbounded $KK$-theory it was established by Baaj and Julg that the bounded transform, which associates a class in $KK$-theory to any unbounded Kasparov module, is a surjective homomorphism (under a separability assumption). In this paper, we provide an equivalence relation on unbounded Kasparov modules and we thereby describe the kernel of the bounded transform. This allows us to introduce a notion of topological unbounded $KK$-theory, which becomes isomorphic to $KK$-theory via the bounded transform. The equivalence relation is formulated entirely at the level of unbounded Kasparov modules and consists of homotopies together with an extra degeneracy condition. Our degenerate unbounded Kasparov modules are called spectrally decomposable since they admit a decomposition into a part with positive spectrum and a part with negative spectrum.

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arXiv2020

Dynamics of compact quantum metric spaces

We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. We moreover provide a flexible set of assumptions ensuring that a continuous family of *-automorphisms of a compact quantum metric space, yields a field of crossed product algebras which varies continuously in Rieffel's quantum Gromov-Hausdorff distance. Lastly we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney C^1-topology.

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arXiv2020

The Podles sphere as a spectral metric space

We study the spectral metric aspects of the standard Podles sphere, which is a homogeneous space for quantum SU(2). The point of departure is the real equivariant spectral triple investigated by Dabrowski and Sitarz. The Dirac operator of this spectral triple interprets the standard Podles sphere as a 0-dimensional space and is therefore not isospectral to the Dirac operator on the 2-sphere. We show that the seminorm coming from commutators with this Dirac operator provides the Podles sphere with the structure of a compact quantum metric space in the sense of Rieffel.

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