Unbounded bivariant $K$-theory and correspondences in noncommutative geometry

Abstract

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 $C^{\ast }$-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.

67 pages. Final version. Accepted for publication

Cited by in corpus (38)

• The $K$-theoretic bulk-edge correspondence for topological insulators
• Spectral flow and the unbounded Kasparov product
• Riemannian manifolds in noncommutative geometry
• A noncommutative framework for topological insulators
• Nonunital spectral triples and metric completeness in unbounded KK-theory
• The bulk-edge correspondence for the quantum Hall effect in Kasparov theory
• Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases
• Shift tail equivalence and an unbounded representative of the Cuntz-Pimsner extension
• Sums of regular selfadjoint operators in Hilbert-C*-modules
• Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators
• Gauge networks in noncommutative geometry
• Modular Theory, Non-Commutative Geometry and Quantum Gravity
• On globally non-trivial almost-commutative manifolds
• Gauge theory on noncommutative Riemannian principal bundles
• Wieler solenoids, Cuntz-Pimsner algebras and K-theory
• A reconstruction theorem for Connes-Landi deformations of commutative spectral triples
• Real structures on almost-commutative spectral triples
• Spectral Triples and Generalized Crossed Products
• Operator *-correspondences in analysis and geometry
• Locally bounded perturbations and (odd) unbounded KK-theory
• On the Unbounded Picture of $KK$-Theory
• Categorical Non-commutative Geometry
• Homotopy equivalence in unbounded KK-theory
• Spectral C*-categories and Fell bundles with path-lifting
• Geometric foundations for classical $\mathrm{U}(1)$-gauge theory on noncommutative manifolds
• Transversals, duality, and irrational rotation
• Equivariant $KK$-theory for non-Hausdorff groupoids
• Localisations of half-closed modules and the unbounded Kasparov product
• Coverings of Spectral Triples
• The Kasparov product on submersions of open manifolds
• An infinite-dimensional index theorem and the Higson-Kasparov-Trout algebra
• An equivariant orbifold index for proper actions
• Riemannian embeddings in codimension one as unbounded $KK$-cycles
• Matrix Geometries Emergent from a Point
• The Noncommutative Topology of Anti-Self-Dual Gauge Fields
• Singularity structures for noncommutative spaces
• Isometric Spectral Subtriples
• Noncommutative Geometry of Quantized Coverings