The Cayley transform in complex, real and graded $K$-theory

Abstract

We use the Cayley transform to provide an explicit isomorphism at the level of cycles from van Daele $K$-theory to $KK$-theory for graded $C^{\ast }$-algebras with a real structure. Isomorphisms between $KK$-theory and complex or real $K$-theory for ungraded $C^{\ast }$-algebras are a special case of this map. In all cases our map is compatible with the computational techniques required in physical and geometrical applications, in particular index pairings and Kasparov products. We provide applications to real $K$-theory and topological phases of matter.

arXiv commentv2: 43 pages, minor revisions. To appear in Internat. J. Math

References in corpus (17)

• Periodic table for topological insulators and superconductors
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• Twisted equivariant matter
• Controlled topological phases and bulk-edge correspondence
• On the K-theoretic classification of topological phases of matter
• The $K$-theoretic bulk-edge correspondence for topological insulators
• A local global principle for regular operators in Hilbert C*-modules
• Boundary maps for $C^{\ast }$-crossed products with R with an application to the quantum Hall effect
• Bulk-boundary correspondence for disordered free-fermion topological phases
• Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases
• K-Theory for Real C*-algebras via Unitary Elements with Symmetries
• KK-theory and Spectral Flow in von Neumann Algebras
• Index theory and topological phases of aperiodic lattices
• Cyclic cohomology for graded $C^{\ast ,r}$-algebras and its pairings with van Daele $K$-theory
• Graded C*-algebras, graded K-theory, and twisted P-graph C*-algebras
• Non-Commutative Vector Bundles for Non-Unital Algebras
• On Graded K-theory, Elliptic Operators and the Functional Calculus

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• Interfaces of discrete systems - spectral and index properties
• Robustness of topological phases on aperiodic lattices