The uniqueness theorem for Kasparov theory
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914333725818880 |
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| author | Szabó, Gábor |
| author_facet | Szabó, Gábor |
| contents | Answering a question of Carrión et al in their recent landmark paper on C*-algebra classification, we prove a general uniqueness theorem for $KK$-theory. Given arbitrary separable C*-algebras $A$ and $B$ and a Cuntz pair consisting of two absorbing representations $φ,ψ: A\to\mathcal{M}(B\otimes\mathcal{K})$, the induced element of $KK(A,B)$ vanishes if and only if $φ$ and $ψ$ are strongly asymptotically unitarily equivalent. This improves upon the Lin-Dadarlat-Eilers stable uniqueness theorem. The conclusion is deduced by first showing the $K_1$-injectivity of an auxiliary C*-algebra associated to the C*-pair $(A,B)$, which is sometimes called the Paschke dual algebra in the literature. Most of the article is concerned with the treatment of an umbrella theorem, which yields such a uniqueness theorem for other variants of $KK$-theory. This encompasses nuclear $KK$-theory, ideal-related $KK$-theory, equivariant $KK$-theory, or any combinations thereof. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_23029 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The uniqueness theorem for Kasparov theory Szabó, Gábor Operator Algebras K-Theory and Homology 19K35, 46L55, 46L35 Answering a question of Carrión et al in their recent landmark paper on C*-algebra classification, we prove a general uniqueness theorem for $KK$-theory. Given arbitrary separable C*-algebras $A$ and $B$ and a Cuntz pair consisting of two absorbing representations $φ,ψ: A\to\mathcal{M}(B\otimes\mathcal{K})$, the induced element of $KK(A,B)$ vanishes if and only if $φ$ and $ψ$ are strongly asymptotically unitarily equivalent. This improves upon the Lin-Dadarlat-Eilers stable uniqueness theorem. The conclusion is deduced by first showing the $K_1$-injectivity of an auxiliary C*-algebra associated to the C*-pair $(A,B)$, which is sometimes called the Paschke dual algebra in the literature. Most of the article is concerned with the treatment of an umbrella theorem, which yields such a uniqueness theorem for other variants of $KK$-theory. This encompasses nuclear $KK$-theory, ideal-related $KK$-theory, equivariant $KK$-theory, or any combinations thereof. |
| title | The uniqueness theorem for Kasparov theory |
| topic | Operator Algebras K-Theory and Homology 19K35, 46L55, 46L35 |
| url | https://arxiv.org/abs/2601.23029 |