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

@article{Bourne2019TheCT, title={The Cayley transform in complex, real and graded \$K\$-theory}, author={Chris Bourne and J. Kellendonk and Adam T M Rennie}, journal={arXiv: K-Theory and Homology}, year={2019} }

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… CONTINUE READING

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