\(KK\)-theory as the \(K\)-theory of \(C^*\)-categories
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 \(KK^{*+1}(A,B)\).
scientific article; zbMATH DE number 1550789
scientific article