## $$KK$$-theory as the $$K$$-theory of $$C^*$$-categories.(English)Zbl 0972.19002

For a separable $$C^*$$-algebra $$A$$ and a $$\sigma$$-unital $$C^*$$-algebra $$B$$ the author considers an additive $$C^*$$-category $$\text{Rep}(A,B)$$ of $$A,B$$-bimodules with morphisms being $$B$$-homomorphisms that commute with the action of $$A$$ modulo compacts. He then defines the category $$\text{Rep}(A,B)$$ as the universal pseudoabelian $$C^*$$-category of $$\text{Rep}(A,B)$$ and proves that Karoubi’s topological $$K$$-group $$K^*(\text{Rep}(A,B))$$ is isomorphic to $$KK^{*+1}(A,B)$$.

### MSC:

 19K35 Kasparov theory ($$KK$$-theory) 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory 46M99 Methods of category theory in functional analysis

### Keywords:

$$C^*$$-category; $$KK$$-theory; topological $$K$$-theory
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