Jens Kaad

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

Abstract Hilbert C ⁎ -modules are the analogues of Hilbert spaces where a C ⁎ -algebra plays the role of the scalar field. With the advent of Kasparovʼs 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.

SPECTRAL FLOW AND THE UNBOUNDED KASPAROV PRODUCT

Abstract 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 σ-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–Schrodinger operators can be nicely explained in terms of the interior Kasparov product.

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.

The Podleś sphere as a spectral metric space

Abstract We study the spectral metric aspects of the standard Podleś sphere, which is a homogeneous space for quantum S U ( 2 ) . The point of departure is the real equivariant spectral triple investigated by Dąbrowski and Sitarz. The Dirac operator of this spectral triple interprets the standard Podleś 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 Podleś sphere with the structure of a compact quantum metric space in the sense of Rieffel.

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.

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

K

H_0 = \alpha \cdot (-i \nabla )

Factorization of Dirac operators on toric noncommutative manifolds

H0=α·(-i∇) for all space dimensions