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Introduction

Additional affiliations

January 2015 - January 2016

October 2013 - August 2014

October 2012 - September 2013

## Publications

Publications (49)

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...

We prove that the Podleś spheres Sq2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_q^2$$\end{document} converge in quantum Gromov–Hausdorff distance to the classical...

Abstract 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 p...

We prove an approximation result for Lipschitz functions on the quantum sphere $S_q^2$, from which we deduce that the two natural quantum metric structures on $S_q^2$ have quantum Gromov-Hausdorff distance zero.

The Podleś quantum sphere Sq2 admits a natural commutative C⁎-subalgebra Iq with spectrum {0}∪{q2k:k∈N0}, which may therefore be considered as a quantised version of a classical interval. We study here the compact quantum metric space structure on Iq inherited from the corresponding structure on Sq2, and provide an explicit formula for the metric i...

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 t...

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.

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...

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...

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. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of $\ast$ -automorphisms of a com...

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 qu...

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 which describes the kerne...

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 which describes the kerne...

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 L²-index theorem in the general context of p...

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 Mi\v{s}\v{c}enko line bundle. In addition, we give a proof of Atiyah's $L^2$-index theorem in the general c...

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...

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 isospe...

We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators $H_0 = \alpha \cdot (-i \nabla)$ 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...

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-theo...

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...

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...

We clarify that one of the main results from "A local global principle for regular operators in Hilbert C*-modules" is already contained in an earlier paper by François Pierrot. Moreover, in the paper by Pierrot this result is proved in a stronger version that was only stated as a conjecture in our local global principle paper. We also present a sh...

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 *-al...

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 explici...

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 th...

In recent years operator space theory has made remarkable appearances in some areas of noncommutative geometry, notably in the study of $C^{*}$-algebras of real reductive groups and the unbounded picture of Kasaprov theory. In both these developments, a central r\^{o}le is played by operator modules and the Haagerup tensor product. This workshop br...

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 as minimal as possible and we do in particular not require that it satisfies any...

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 introd...

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 {...

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 len...

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 applic...

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 di...

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...

R. W. Carey and J. Pincus in [Geometric measure theory and the
calculus of variations (Arcata 1984), Proc. Sympos. Pure Math. 44, American Mathematical Society, Providence (1986), 149–161]
proposed an 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 [Comm. Math. P...

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 intr...

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...

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...

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...

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...

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 whic...

In the context of 2-summable Fredholm modules, we prove that the Connes-Karoubi multiplicative character coincides with Brown's determinant invariant on algebraic K-theory.

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 o...

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 jo...

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 h...

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.

In this paper we prove that the multiplicative character of A. Connes and M.
Karoubi and the determinant invariant of L. G. Brown, J. W. Helton and R. E.
Howe agree up to a canonical homomorphism.

We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko
(J).Given a semifinite spectral triple (A, H, D) relative to (N, τ) with A separable, we construct a class [D] ∈ KK1
(A, K(N)). For a unit...