论文标题
受控的KK理论和Milnor精确序列
Controlled KK-theory, and a Milnor exact sequence
论文作者
论文摘要
我们介绍了与一对$（a，b）$的可分离$ c^*$ - 代数相关的受控$ kk $ - 理论组。粗略地，这些由通常的$ k $ - 理论组的元素$ k_0（b）$组成，这些元素大约上下班，$ a $ a $。我们的主要结果表明，这些组与Milnor精确序列与Kasparov的$ kk $ group有关，以使Rørdam的$ kl $ group被确定为我们受控的$ kk $ groups的倒数限制。 如果涉及的$ C^*$ - 代数满足UCT，则我们的Milnor精确序列与Schochet的Milnor序列一致，尽管我们的结果独立于UCT。随后的工作将在UCT申请。
We introduce controlled $KK$-theory groups associated to a pair $(A,B)$ of separable $C^*$-algebras. Roughly, these consist of elements of the usual $K$-theory group $K_0(B)$ that approximately commute with elements of $A$. Our main results show that these groups are related to Kasparov's $KK$-groups by a Milnor exact sequence, in such a way that Rørdam's $KL$-group is identified with an inverse limit of our controlled $KK$-groups. In the case that the $C^*$-algebras involved satisfy the UCT, our Milnor exact sequence agrees with the Milnor sequence associated to a $KK$-filtration in the sense of Schochet, although our results are independent of the UCT. Applications to the UCT will be pursued in subsequent work.
