$KK$-theory as the $K$-theory of $C^*$-categories

Let complex $C^{*}$ algebras be endowed with a norm-continuous action of a fixed compact second countable group. From a separable $C^{*}$-algebra $A$ and a $\sigma $-unital $C^{*}$-algebra $B$, we construct a $C^{*}$-category $\mathrm{Rep} (A,B)$ and an isomorphism \[ \kappa :K^{i+1}(\mathrm{Rep} (A,B))\rightarrow KK^i(A,B),\;\;\;i\in \mathbb{Z}_2, \] where on the left-hand side are Karoubi's topological $K$-groups, and on the right-hand side are Kasparov’s equivariant bivariant $K$-groups.

Keywords

$K$-theory, $KK$-theory, $C^*$-category

2010 Mathematics Subject Classification

19J99, 19K35, 46L89, 46Mxx

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Published 1 January 2000