The UCT, the Milnor sequence, and a canonical decomposition of the Kasparov groups
The author considers the smallest full subcategory, \({\mathcal N}\), of separable nuclear \(C^*\)-algebras containing all separable type I \(C^*\)-algebras and being closed under strong Morita equivalence, direct limits, and crossed products by \(\mathbb{Z}\) and \(\mathbb{R}\). Moreover, for an exact sequence \(0 \to {\mathcal I} \to {\mathcal A} \to {\mathcal A}/{\mathcal I} \to 0\) let \({\mathcal N}\) contain all of the three \(C^*\)-algebras if it contains two of them (in particular, \({\mathcal N}\) is closed under extensions). It is shown that each \({\mathcal A} \in {\mathcal N}\) allows a \(KK\)-filtration, i.e., there is a sequence of commutative \(C^*\)-algebras \[ {\mathcal A}_0 \hookrightarrow {\mathcal A}_1 \hookrightarrow {\mathcal A}_2 \hookrightarrow \cdots \] with \({\mathcal A}^+_i \cong C (X_i)\) for some finite \(CW\)-complex \(X_i\), \(K_* ({\mathcal A}_i) \to K_* ({\mathcal A}_{i + 1})\) an inclusion, and \(\varinjlim {\mathcal A}_i\) \(KK\)-equivalent to \({\mathcal A}\). Then \(K_* ({\mathcal A}) \cong \varinjlim K_* ({\mathcal A}_i)\), and \({\mathcal A}\) satisfies the universal coefficient theorem for all \({\mathcal B}\), i.e., there is a natural short exact sequence \[ 0 \to \text{Ext}^1_\mathbb{Z} \bigl( K_* ({\mathcal A}), K_* ({\mathcal B}) \bigr) \to KK_* ({\mathcal A}, {\mathcal B}) \to \Hom_\mathbb{Z} \bigl( K_* ({\mathcal A}), K_* ({\mathcal B}) \bigr) \to 0 \] which splits unnaturally. Moreover, one has the Milnor \(\varprojlim^1\) sequence \[ 0 \to \varprojlim{}^1 KK_* ({\mathcal A}_i, {\mathcal B}) \to KK_* ({\mathcal A}, {\mathcal B}) \to \varprojlim{}^1 KK_* ({\mathcal A}_i, {\mathcal B}) \to 0 \] which also splits unnaturally. The main theorem states that the ingredients of both sequences and hence the \(KK\)-groups \(KK_* ({\mathcal A}, {\mathcal B})\) can be computed by taking the limits of the respective groups with \({\mathcal A}\) replaced by \({\mathcal A}_i\) independent of the specific \(KK\)-filtration.
scientific article; zbMATH DE number 847544