The universal property of graded $KK^G$-theory
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913006619721728 |
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| author | Burgstaller, Bernhard |
| author_facet | Burgstaller, Bernhard |
| contents | A universal category-theoretical characterization of groupoid equivariant $KK^G$-theory for ${\mathbb{Z}}_2$-graded $C^*$-algebras is established, by observing the ``$KK$-axiom'' that for each $[s,{\cal E} \oplus B, \mathbb{F}] \in KK^G(A,B)$, the `corner-embedding' $*$-homomorphism ${\bf j}: B \rightarrow {\sf cl} \big({\cal K}_B({\cal E} \oplus B) + s(A) + \mathbb{F} \cdot s(A) \big)$ is invertible in $KK^G$. This $KK$-axiom and homotopy-invariance characterize graded $KK^G$-theory universally and completely, thus directly extending the well-known characterization of $KK$-theory for ungraded $C^*$-algebras via stability, homotopy invariance and splitexactness by Higson. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23157 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The universal property of graded $KK^G$-theory Burgstaller, Bernhard K-Theory and Homology Operator Algebras 19K35, 46L80, 20L05, 20M18 A universal category-theoretical characterization of groupoid equivariant $KK^G$-theory for ${\mathbb{Z}}_2$-graded $C^*$-algebras is established, by observing the ``$KK$-axiom'' that for each $[s,{\cal E} \oplus B, \mathbb{F}] \in KK^G(A,B)$, the `corner-embedding' $*$-homomorphism ${\bf j}: B \rightarrow {\sf cl} \big({\cal K}_B({\cal E} \oplus B) + s(A) + \mathbb{F} \cdot s(A) \big)$ is invertible in $KK^G$. This $KK$-axiom and homotopy-invariance characterize graded $KK^G$-theory universally and completely, thus directly extending the well-known characterization of $KK$-theory for ungraded $C^*$-algebras via stability, homotopy invariance and splitexactness by Higson. |
| title | The universal property of graded $KK^G$-theory |
| topic | K-Theory and Homology Operator Algebras 19K35, 46L80, 20L05, 20M18 |
| url | https://arxiv.org/abs/2603.23157 |