Commutator estimates on contact manifolds and applications

This article studies sharp norm estimates for the commutator of pseudo-differential operators with multiplication operators on closed Heisenberg manifolds. In particular, we obtain a Calderon commutator estimate: If $D$ is a first-order operator in the Heisenberg calculus and $f$ is Lipschitz in the Carnot-Caratheodory metric, then $[D,f]$ extends to an $L^2$-bounded operator. Using interpolation, it implies sharp weak--Schatten class properties for the commutator between zeroth order operators and H\"older continuous functions. We present applications to sub-Riemannian spectral triples on Heisenberg manifolds as well as to the regularization of a functional studied by Englis-Guo-Zhang.Comment: 31 pages, improved presentation and additional reference

Index formulas and charge deficiencies on the Landau levels

The notion of charge deficiency from Avron, Seiler, Simon (Charge deficiency, charge transport and comparison of dimensions, Comm. Math. Phys. 159) is studied from the view of $KK$-theory and is applied to the Landau levels in \C^n. We calculate the charge deficiencies of the higher Landau levels in \C^n by means of an Atiyah-Singer type index theorem.Comment: 18 page

A remark on twists and the notion of torsion-free discrete quantum groups

In this paper twists of reduced locally compact quantum groups are studied. Twists of the dual coaction on a reduced crossed product are introduced and the twisted dual coactions are proved to satisfy a type of Takesaki-Takai duality. The twisted Takesaki-Takai duality implies that twists of discrete, torsion-free quantum groups are torsion-free. Cocycle twists of duals of semisimple, compact Lie are studied leading to a locally compact quantum group contained in the Drinfeld-Jimbo algebra which gives a dual notion of Woronowicz deformations for semisimple, compact Lie groups. These cocycle twists are proven to be torsion-free whenever the Lie group is simply connected.Comment: 17 pages, to appear in Algebras and Representation Theory, http://www.springerlink.com/content/u33rr720672598r3

The Pimsner-Voiculescu sequence for coactions of compact Lie groups

The Pimsner-Voiculescu sequence is generalized to a Pimsner-Voiculescu tower describing the $KK$-category equivariant with respect to coactions of a compact Lie group satisfying the Hodgkin condition. A dual Pimsner-Voiculescu tower is used to show that coactions of a compact Hodgkin-Lie group satisfy the Baum-Connes property.Comment: 19 pages, to appear in Mathematica Scandinavic

Equivariant extensions of *-algebras

A bivariant functor is defined on a category of *-algebras and a category of operator ideals, both with actions of a second countable group $G$, into the category of abelian monoids. The element of the bivariant functor will be $G$-equivariant extensions of a *-algebra by an operator ideal under a suitable equivalence relation. The functor is related with the ordinary $Ext$-functor for $C^*$-algebras defined by Brown-Douglas-Fillmore. Invertibility in this monoid is studied and characterized in terms of Toeplitz operators with abstract symbol.Comment: 12 page

Spectral triples and finite summability on Cuntz-Krieger algebras

We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra of functions on a non-commutative space" coming from a sub shift of finite type. We show that any odd $K$-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are $\theta$-summable although their bounded transform, when constructed using the sign-function, will already on the level of analytic $K$-cycles be finitely summable bounded Fredholm modules. Using the unbounded Kasparov product, we exhibit a family of generalized spectral triples, possessing mildly unbounded commutators, whilst still giving well defined $K$-homology classes.Comment: 67 pages, minor changes in Section 5.1 and 6.