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Jens Kaad

Publications and source records attributed to Jens Kaad.

46 records · Page 3Linked to original sources
arXiv2013

A Serre-Swan theorem for bundles of bounded geometry

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. In this context it states that the category of Hilbert bundles of bounded geometry is equivalent to the category of operator *-modules over the operator *-algebra of continuously differentiable functions which vanish at infinity. Operator *-modules are generalizations of Hilbert C*-modules where C*-algebras have been replaced by a more flexible class of involutive algebras of bounded operators: Operator *-algebras. They play an important role in the study of the unbounded Kasparov product.

math.OA↗
arXiv2012

A transformation rule for the index of commuting operators

In the setting of several commuting operators on a Hilbert space one defines the notions of invertibility and Fredholmness in terms of the associated Koszul complex. The index problem then consists of computing the Euler characteristic of such a special type of Fredholm complex. In this paper we investigate transformation rules for the index under the holomorphic functional calculus. We distinguish between two different types of index results: 1) A global index theorem which expresses the index in terms of the degree function of the "symbol" and the locally constant index function of the "coordinates". 2) A local index theorem which computes the Euler characteristic of a localized Koszul complex near a common zero of the "symbol". Our results apply to the example of Toeplitz operators acting on both Bergman spaces over pseudoconvex domains and the Hardy space over the polydisc. The local index theorem is fundamental for future investigations of determinants and torsion of Koszul complexes.

math.KT↗
arXiv2011

On modular semifinite index theory

We propose a definition of a modular spectral triple which covers existing examples arising from KMS-states, Podles sphere and quantum SU(2). The definition also incorporates the notion of twisted commutators appearing in recent work of Connes and Moscovici. We show how a finitely summable modular spectral triple admits a twisted index pairing with unitaries satisfying a modular condition. The twist means that the dimensions of kernels and cokernels are measured with respect to two different but intimately related traces. The twisted index pairing can be expressed by pairing Chern characters in reduced versions of twisted cyclic theories. We end the paper by giving a local formula for the reduced Chern character in the case of quantum SU(2). It appears as a twisted coboundary of the Haar-state. In particular we present an explicit computation of the twisted index pairing arising from the sequence of corepresentation unitaries. As an important tool we construct a family of derived integration spaces associated with a weight and a trace on a semifinite von Neumann algebra.

math.OA↗
arXiv2011

Spectral flow and the unbounded Kasparov product

We present a fairly general construction of unbounded representatives for the interior Kasparov product. As a main tool we develop a theory of C^1-connections on operator * modules; we do not require any smoothness assumptions; our sigma-unitality assumptions are minimal. Furthermore, we use work of Kucerovsky and our recent Local Global Principle for regular operators in Hilbert C*-modules. As an application we show that the Spectral Flow Theorem and more generally the index theory of Dirac-Schrödinger operators can be nicely explained in terms of the interior Kasparov product.

math.OA↗
arXiv2011

A twisted spectral triple for quantum SU(2)

We initiate the study of a q-deformed geometry for quantum SU(2). In contrast with the usual properties of a spectral triple, we get that only twisted commutators between algebra elements and our Dirac operator are bounded. Furthermore, the resolvent only becomes compact when measured with respect to a trace on a semifinite von Neumann algebra which does not contain the quantum group. We show that the zeta function at the identity has a meromorphic continuation to the whole complex plane and that a large family of local Hochschild cocycles associated with our twisted spectral triple are twisted coboundaries.

math.QA↗
arXiv2010

Joint torsion of several commuting operators

We introduce the notion of joint torsion for several commuting operators satisfying a Fredholm condition. This new secondary invariant takes values in the group of invertibles of a field. It is constructed by comparing determinants associated with different filtrations of a Koszul complex. Our notion of joint torsion generalizes the Carey-Pincus joint torsion of a pair of commuting Fredholm operators. As an example, under more restrictive invertibility assumptions, we show that the joint torsion recovers the multiplicative Lefschetz numbers. Furthermore, in the case of Toeplitz operators over the polydisc we provide a link between the joint torsion and the Cauchy integral formula. We will also consider the algebraic properties of the joint torsion. They include a cocycle property, a symmetry property, a triviality property and a multiplicativity property. The proof of these results relies on a quite general comparison theorem for vertical and horizontal torsion isomorphisms associated with certain diagrams of chain complexes.

math.KT↗
arXiv2010

Invariance results for pairings with algebraic K-theory

To each algebra over the complex numbers we associate a sequence of abelian groups in a contravariant functorial way. In degree (m-1) we have the m-summable Fredholm modules over the algebra modulo stable m-summable perturbations. These new finitely summable K-homology groups pair with cyclic homology and algebraic K-theory. In the case of cyclic homology the pairing is induced by the Chern-Connes character. The pairing between algebraic K-theory and finitely summable K-homology is induced by the Connes-Karoubi multiplicative character.

math.KT↗
arXiv2009

A calculation of the multiplicative character

We give a formula, in terms of products of commutators, for the application of the odd multiplicative character to higher Loday symbols. On our way we construct a product on the relative K-groups and investigate the multiplicative properties of the relative Chern character.

math.KT↗
arXiv2007

KK-theory and Spectral Flow in von Neumann Algebras

We present a definition of spectral flow relative to any norm closed ideal J in any von Neumann algebra N. Given a path D(t) of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in K_0(J). In the case when N is semifinite, the numerical spectral flow of the path coincides with the value of trace on the associated K-class. Given a semifinite spectral triple (A,H,D) relative to a semifinite von Neumann algebra N, we construct a class [D] in KK^1(A,N') such that, for a unitary u in A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product of [u] and [D].

math.OA↗