The Cayley transform in complex, real and graded K -theory
2020 World Scientific Publishing Company. 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-algebras with a real structure. Isomorphisms between KK-theory and complex or real K-theory for ungraded C-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.