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.

Sums of regular selfadjoint operators in Hilbert-C*-modules

We introduce a notion of weak anticommutativity for a pair (S,T) of self-adjoint regular operators in a Hilbert-C*-module E. We prove that the sum $S+T$ of such pairs is self-adjoint and regular on the intersection of their domains. A similar result then holds for the sum S^2+T^2 of the squares. We show that our definition is closely related to the Connes-Skandalis positivity criterion in $KK$-theory. As such we weaken a sufficient condition of Kucerovsky for representing the Kasparov product. Our proofs indicate that our conditions are close to optimal.Comment: Final version. Minor editorial change

Shift tail equivalence and an unbounded representative of the Cuntz-Pimsner extension

We show how the fine structure in shift-tail equivalence, appearing in the noncommutative geometry of Cuntz-Krieger algebras developed by the first two authors, has an analogue in a wide range of other Cuntz-Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz-Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz- and Cuntz-Krieger algebras and for Cuntz-Pimsner algebras associated to vector bundles twisted by equicontinuous $*$-automorphisms.Comment: 30 page

Unbounded bivariant KK-theory and correspondences in noncommutative geometry

By introducing a notion of smooth connection for unbounded $KK$-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable $C^{*}$-module. The theory of operator spaces provides the required tools. Finally, the above mentioned $KK$-cycles with connection can be viewed as the morphisms in a category whose objects are spectral triples.Comment: 67 pages. Final version. Accepted for publicatio

The bordism group of unbounded KK-cycles

We consider Hilsum's notion of bordism as an equivalence relation on unbounded $KK$-cycles and study the equivalence classes. Upon fixing two $C^*$-algebras, and a $*$-subalgebra dense in the first $C^*$-algebra, a $\mathbb{Z}/2\mathbb{Z}$-graded abelian group is obtained; it maps to the Kasparov $KK$-group of the two $C^*$-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first $C^*$-algebra is the complex numbers (i.e., for $K$-theory) and is a split surjection if the first $C^*$-algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense $*$-subalgebra.Comment: 38 page

Gauge Theory for Spectral Triples and the Unbounded Kasparov Product

We explore factorizations of noncommutative Riemannian spin geometries over commutative base manifolds in unbounded KK-theory. After setting up the general formalism of unbounded KK-theory and improving upon the construction of internal products, we arrive at a natural bundle-theoretic formulation of gauge theories arising from spectral triples. We find that the unitary group of a given noncommutative spectral triple arises as the group of endomorphisms of a certain Hilbert bundle; the inner fluctuations split in terms of connections on, and endomorphisms of, this Hilbert bundle. Moreover, we introduce an extended gauge group of unitary endomorphisms and a corresponding notion of gauge fields. We work out several examples in full detail, to wit Yang--Mills theory, the noncommutative torus and the $\theta$-deformed Hopf fibration over the two-sphere.Comment: 50 pages. Accepted version. Section 2 has been rewritten. Results in sections 3-6 are unchange