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

TL;DR

This paper introduces an explicit isomorphism between van Daele $K$-theory and $KK$-theory for graded $C^*$-algebras with real structures using the Cayley transform, with applications to physics and geometry.

Contribution

It constructs a new explicit isomorphism via the Cayley transform connecting van Daele $K$-theory to $KK$-theory for graded $C^*$-algebras with real structures, compatible with index pairings.

Findings

Provides an explicit isomorphism between van Daele $K$-theory and $KK$-theory.

Shows compatibility with index pairings and Kasparov products.

Applies to real $K$-theory and topological phases of matter.

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 $C^*$-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.