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\(KK\)-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
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