受控KK-理论，以及一个Milnor正合序列
Controlled KK-theory, and a Milnor exact sequence
AI总结：
本文引入与可分$C^*$-代数对相关的受控$KK$-理论群，证明其通过Milnor正合序列与Kasparov的$KK$-群相关联，并将Rørdam的$KL$-群等同于受控群的逆极限。
摘要：
我们引入了与一对可分$C^*$-代数$(A,B)$相关的受控$KK$-理论群。粗略地说，它们由通常的$K$-理论群$K_0(B)$中与$A$的元素近似交换的元素组成。我们的主要结果表明，这些群通过一个Milnor正合序列与Kasparov的$KK$-群相关联，使得Rørdam的$KL$-群被等同于我们受控$KK$-群的逆极限。在涉及的$C^*$-代数满足UCT的情况下，我们的Milnor正合序列与Schochet意义上与$KK$-过滤相关的Milnor序列一致，尽管我们的结果独立于UCT。关于UCT的应用将在后续工作中探讨。
英文摘要：
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.
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