arXiv subjects

Jens Kaad

Publications and source records attributed to Jens Kaad.

At least 37 records · Page 2Linked to original sources

arXiv2020

Index theory on the Miščenko bundle

We consider the assembly map for principal bundles with fiber a countable discrete group. We obtain an index-theoretic interpretation of this homomorphism by providing a tensor-product presentation for the module of sections associated to the Miščenko line bundle. In addition, we give a proof of Atiyah's $L^2$-index theorem in the general context of principal bundles over compact Hausdorff spaces. We thereby also reestablish that the surjectivity of the Baum-Connes assembly map implies the Kadison-Kaplansky idempotent conjecture in the torsion-free case. Our approach does not rely on geometric $K$-homology but rather on an explicit construction of Alexander-Spanier cohomology classes coming from a Chern character for tracial function algebras.

math.KT↗

arXiv2019

The unbounded Kasparov product by a differentiable module

In this paper we investigate the unbounded Kasparov product between a differentiable module and an unbounded cycle of a very general kind that includes all unbounded Kasparov modules and hence also all spectral triples. Our assumptions on the differentiable module are weak and we do in particular not require that it satisfies any kind of smooth projectivity conditions. The algebras that we work with are furthermore not required to possess a smooth approximate identity. The lack of an adequate projectivity condition on our differentiable module entails that the usual class of unbounded Kasparov modules is not flexible enough to accommodate the unbounded Kasparov product and it becomes necessary to twist the commutator condition by an automorphism. We show that the unbounded Kasparov product makes sense in this twisted setting and that it recovers the usual interior Kasparov product after taking bounded transforms. Since our unbounded cycles are twisted (or modular) we are not able to apply the work of Kucerovsky for recognizing unbounded representatives for the bounded Kasparov product, instead we rely directly on the connection criterion developed by Connes and Skandalis. In fact, since we do not impose any twisted Lipschitz regularity conditions on our unbounded cycles, even the passage from an unbounded cycle to a bounded Kasparov module requires a substantial amount of extra care.

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arXiv2018

Factorization of Dirac operators on toric noncommutative manifolds

We factorize the Dirac operator on the Connes-Landi 4-sphere in unbounded KK-theory. We show that a family of Dirac operators along the orbits of the torus action defines an unbounded Kasparov module, while the Dirac operator on the principal orbit space -an open quadrant in the 2-sphere- defines a half-closed chain. We show that the tensor sum of these two operators coincides up to unitary equivalence with the Dirac operator on the Connes-Landi sphere and prove that this tensor sum is an unbounded representative of the interior Kasparov product in bivariant K-theory. We also generalize our results to Dirac operators on all toric noncommutative manifolds subject to a condition on the principal stratum. We find that there is a curvature term that arises as an obstruction for having a tensor sum decomposition in unbounded KK-theory. This curvature term can however not be detected at the level of bounded KK-theory.

math.OA↗

arXiv2017

On the Global Limiting Absorption Principle for Massless Dirac Operators

We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators $H_0 = α\cdot (-i \nabla)$ for all space dimensions $n \in \mathbb{N}$, $n \geq 2$. This is a new result for all dimensions other than three, in particular, it applies to the two-dimensional case which is known to be of some relevance in applications to graphene. We also prove an essential self-adjointness result for first-order matrix-valued differential operators with Lipschitz coefficients.

math.SP↗

arXiv2017

Factorization of Dirac operators on almost-regular fibrations of spin$^c$ manifolds

We establish the factorization of the Dirac operator on an almost-regular fibration of spin$^c$ manifolds in unbounded KK-theory. As a first intermediate result we establish that any vertically elliptic and symmetric first-order differential operator on a proper submersion defines an unbounded Kasparov module, and thus represents a class in KK-theory. Then, we generalize our previous results on factorizations of Dirac operators to proper Riemannian submersions of spin$^c$ manifolds. This allows us to show that the Dirac operator on the total space of an almost-regular fibration can be written as the tensor sum of a vertically elliptic family of Dirac operators with the horizontal Dirac operator, up to an explicit `obstructing' curvature term. We conclude by showing that the tensor sum factorization represents the interior Kasparov product in bivariant K-theory.

math.FA↗

arXiv2017

On a theorem of Kucerovsky for half-closed chains

Kucerovsky's theorem provides a method for recognizing the interior Kasparov product of selfadjoint unbounded cycles. In this paper we extend Kucerovsky's theorem to the non-selfadjoint setting by replacing unbounded Kasparov modules with Hilsum's half-closed chains. On our way we show that any half-closed chain gives rise to a multitude of twisted selfadjoint unbounded cycles via a localization procedure. These unbounded modular cycles allow us to provide verifiable criteria avoiding any reference to domains of adjoints of symmetric unbounded operators.

math.OA↗

arXiv2017

Operator *-correspondences in analysis and geometry

An operator *-algebra is a non-selfadjoint operator algebra with completely isometric involution. We show that any operator *-algebra admits a faithful representation on a Hilbert space in such a way that the involution coincides with the operator adjoint up to conjugation by a symmetry. We introduce operator *-correspondences as a general class of inner product modules over operator *-algebras and prove a similar representation theorem for them. From this we derive the existence of linking operator *-algebras for operator *-correspondences. We illustrate the relevance of this class of inner product modules by providing numerous examples arising from noncommutative geometry.

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arXiv2016

Morita invariance of unbounded bivariant K-theory

We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator *-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator *-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov's bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a refinement of the usual interior Kasparov product. We illustrate our results by proving C^1-versions of well-known C^*-algebraic Morita equivalences in the context of hereditary subalgebras, conformal equivalences and crossed products by discrete groups.

math.KT↗

arXiv2016

A local global principle for regular operators in Hilbert C*-modules

Hilbert C*-modules are the analogues of Hilbert spaces where a C*-algebra plays the role of the scalar field. With the advent of Kasparov's celebrated KK-theory they became a standard tool in the theory of operator algebras. While the elementary properties of Hilbert C*-modules can be derived basically in parallel to Hilbert space theory the lack of an analogue of the Projection Theorem soon leads to serious obstructions and difficulties. In particular the theory of unbounded operators is notoriously more complicated due to the additional axiom of regularity which is not easy to check. In this paper we present a new criterion for regularity in terms of the Hilbert space localizations of an unbounded operator. We discuss several examples which show that the criterion can easily be checked and that it leads to nontrivial regularity results.

math.OA↗

arXiv2016

Riemannian submersions and factorization of Dirac operators

We establish the factorization of Dirac operators on Riemannian submersions of compact spin$^c$ manifolds in unbounded KK-theory. More precisely, we show that the Dirac operator on the total space of such a submersion is unitarily equivalent to the tensor sum of a family of Dirac operators with the Dirac operator on the base space, up to an explicit bounded curvature term. Thus, the latter is an obstruction to having a factorization in unbounded KK-theory. We show that our tensor sum represents the bounded KK-product of the corresponding KK-cycles and connect to the early work of Connes and Skandalis.

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arXiv2016

Algebraic $K$-theory and a semi-finite Fuglede-Kadison determinant

In this paper we apply algebraic $K$-theory techniques to construct a Fuglede-Kadison type determinant for a semi-finite von Neumann algebra equipped with a fixed trace. Our construction is based on the approach to determinants for Banach algebras developed by Skandalis and de la Harpe. This approach can be extended to the semi-finite case since the first topological $K$-group of the trace ideal in a semi-finite von Neumann algebra is trivial. On our way we also improve the methods of Skandalis and de la Harpe by considering relative $K$-groups with respect to an ideal instead of the usual absolute $K$-groups. Our construction recovers the determinant homomorphism introduced by Brown, but all the relevant algebraic properties are automatic due to the algebraic $K$-theory framework.

math.OA↗

arXiv2015

Pimsner algebras and Gysin sequences from principal circle actions

A self Morita equivalence over an algebra B, given by a B-bimodule E, is thought of as a line bundle over B. The corresponding Pimsner algebra O_E is then the total space algebra of a noncommutative principal circle bundle over B. A natural Gysin-like sequence relates the KK-theories of O_E and of B. Interesting examples come from O_E a quantum lens space over B a quantum weighted projective line (with arbitrary weights). The KK-theory of these spaces is explicitly computed and natural generators are exhibited.

math.QA↗

arXiv2015

On a spectral flow formula for the homological index

Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators {A(t)} parametrized by the real line. Then under certain conditions \cite{RS95} that include the assumption that the operators {D(t)= D+A(t)} all have discrete spectrum then the spectral flow along the path { D(t)} can be shown to be equal to the index of d/dt+D(t) when the latter is an unbounded Fredholm operator on L^2(R, H). In \cite{GLMST11} an investigation of the index=spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general. In \cite{CGPST14a} the question of what happens when the Fredholm condition is dropped altogether was investigated. In these circumstances the Fredholm index is replaced by the Witten index. In this paper we take the investigation begun in \cite{CGPST14a} much further. We show how to generalise a formula known from the setting of the L^2 index theorem to the non-Fredholm setting. Our main theorem gives a trace formula relating the homological index of \cite{CaKa:TIH} to an integral formula that is known, for a path of selfadjoint Fredholms with compact resolvent and with unitarily equivalent endpoints, to compute spectral flow. Our formula however, applies to paths of selfadjoint non-Fredholm operators. We interpret this as indicating there is a generalisation of spectral flow to the non-Fredholm setting.

math.FA↗

arXiv2014

Tate tame symbol and the joint torsion of commuting operators

We investigate determinants of Koszul complexes of holomorphic functions of a commuting tuple of bounded operators acting on a Hilbert space. Our main result shows that the analytic joint torsion, which compares two such determinants, can be computed by a local formula which involves a tame symbol of the involved holomorphic functions. As an application we are able to extend the classical tame symbol of meromorphic functions on a Riemann surface to the more involved setting of transversal functions on a complex analytic curve. This follows by spelling out our main result in the case of Toeplitz operators acting on the Hardy space over the polydisc.

math.KT↗

arXiv2014

Differentiable absorption of Hilbert C*-modules, connections, and lifts of unbounded operators

The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert C*-module is isomorphic to a direct summand in the standard module of square summable sequences in the base C*-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base C*-algebra. This leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The differentiable absorption theorem is then applied to construct densely defined connections (or correspondences) on Hilbert C*-modules. These connections can in turn be used to define selfadjoint and regular "lifts" of unbounded operators which act on an auxiliary Hilbert C*-module.

math.OA↗

arXiv2014

Anomalies of Dirac type operators on Euclidean space

We develop by example a type of index theory for non-Fredholm operators. A general framework using cyclic homology for this notion of index was introduced in a separate article [CaKa13] where it may be seen to generalise earlier ideas of Carey-Pincus and Gesztesy-Simon on this problem. Motivated by an example in two dimensions in [BGG+87] we introduce in this paper a class of examples of Dirac type operators on R^{2n} that provide non-trivial examples of our homological approach. Our examples may be seen as extending old ideas about the notion of anomaly introduced by physicists to handle topological terms in quantum action principles with an important difference, namely we are dealing with purely geometric data that can be seen to arise from the continuous spectrum of our Dirac type operators.

math.FA↗

arXiv2014

Canonical holomorphic sections of determinant line bundles

We investigate the analytic properties of torsion isomorphisms (determinants) of mapping cone triangles of Fredholm complexes. Our main tool is a generalization to Fredholm complexes of the perturbation isomorphisms constructed by R. Carey and J. Pincus for Fredholm operators. A perturbation isomorphism is a canonical isomorphism of determinants of homology groups associated to a finite rank perturbation of Fredholm complexes. The perturbation isomorphisms allow us to establish the invariance properties of the torsion isomorphisms under finite rank perturbations. We then show that the perturbation isomorphisms provide a holomorphic structure on the determinant lines over the space of Fredholm complexes. Finally, we establish that the torsion isomorphisms and the perturbation isomorphisms provide holomorphic sections of certain determinant line bundles.

math.KT↗

arXiv2014

Topological invariance of the homological index

R. W. Carey and J. Pincus in [CaPi86] proposed and index theory for non-Fredholm bounded operators T on a separable Hilbert space H such that TT* - T*T is in the trace class. We showed in [CGK13] using Dirac-type operators acting on sections of bundles over R^{2n} that we could construct bounded operators T satisfying the more general condition that (1-TT*)^n - (1-T*T)^n is trace class. We proposed there a "homological" index for these Dirac-type operators given by Tr( (1-TT*)^n - (1-T*T)^n ). In this paper we show that the index introduced in [CGK13] represents the result of a pairing between a cyclic homology theory for the algebra generated by T and T* and its dual cohomology theory. This leads us to establish homotopy invariance of our homological index (in the sense of cyclic theory). We are then able to define in a very general fashion a homological index for certain unbounded operators and prove invariance of this index under a class of unbounded perturbations.

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