Controlled KK-theory, and a Milnor exact sequence
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866914921596321792 |
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| author | Willett, Rufus Yu, Guoliang |
| author_facet | Willett, Rufus Yu, Guoliang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_10906 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Controlled KK-theory, and a Milnor exact sequence Willett, Rufus Yu, Guoliang Operator Algebras K-Theory and Homology 19K35, 46L80, 46L85 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. |
| title | Controlled KK-theory, and a Milnor exact sequence |
| topic | Operator Algebras K-Theory and Homology 19K35, 46L80, 46L85 |
| url | https://arxiv.org/abs/2011.10906 |