The operator $K$-functor and extensions of $C^*$-algebras

G. G. Kasparov

Izvestiya. Mathematics , Tome 16 (1981) no. 3, pp. 513-572

Résumé

In this paper a general operator $K$-functor $K_*K^*(A,B)$ is constructed, depending on a pair $A$, $B$ of $C^*$-algebras. Special cases of this functor are the ordinary cohomological $K$-functor $K^*(B)$ and the homological $K$-functor $K_*(A)$. The results (homotopy invariance, Bott periodicity, exact sequences, etc.) permit one to compute $K_*K^*(A,B)$ effectively in concrete examples. The main result, concerning extensions of $C^*$-algebras, consists in a description of a "stable type" of extensions of the most general form: $0\to B\to D\to A\to0$. It is shown that the sum of such an extension with a fixed decomposable extension of the form $0\to\mathscr K\otimes B\to D_0\to A\to0$ is uniquely determined by an element of the group $KK^1(A,B)$. Bibliography: 25 titles.