48 research outputs found
On the Unbounded Picture of -Theory
In the founding paper on unbounded -theory it was established by Baaj and
Julg that the bounded transform, which associates a class in -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 -theory,
which becomes isomorphic to -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
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
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
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.Comment: 42 pages, minor changes, acknowledgements adde
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.Comment: 52 page
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.Comment: 32 page
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.Comment: 45 page
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.Comment: 28 pages. Proof of local index theorem is include
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.Comment: v2: minor changes; to appear in Ergodic Theory and Dynamical System
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.Comment: 36 page