The universal property of graded $KK^G$-theory

Abstract

A universal category-theoretical characterization of groupoid equivariant $KK^G$-theory for ${\mathbb{Z}}_2$-graded $C^*$-algebras is established, by observing the ``$KK$-axiom'' that for each $[s,{\cal E} \oplus B, \mathbb{F}] \in KK^G(A,B)$, the `corner-embedding' $*$-homomorphism ${\bf j}: B \rightarrow {\sf cl} \big({\cal K}_B({\cal E} \oplus B) + s(A) + \mathbb{F} \cdot s(A) \big)$ is invertible in $KK^G$. This $KK$-axiom and homotopy-invariance characterize graded $KK^G$-theory universally and completely, thus directly extending the well-known characterization of $KK$-theory for ungraded $C^*$-algebras via stability, homotopy invariance and splitexactness by Higson.