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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2602.22708 |
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| _version_ | 1866914352587603968 |
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| author | Inoué, Takao |
| author_facet | Inoué, Takao |
| contents | We formulate a Mayer--Vietoris/Čech viewpoint on $K$-theory for crossed products by discrete quantum groups, emphasizing how local-to-global gluing data interacts with quantum-group coefficients. Starting from a $G$-equivariant ideal cover and the associated Mayer--Vietoris six-term exact sequence, we package the resulting $K$-theoretic computation into a Čech-type spectral sequence whose $E^1$-page is explicitly described by iterated intersections. We then introduce a minimal ``twisted'' gluing mechanism controlled by a $\mathbb Z/2$-valued Čech $2$-cocycle and an involutive automorphism of the coefficient algebra. Under a Kirchberg--UCT hypothesis on the quantum-group crossed-product coefficient, the twist produces a nontrivial differential $d_2$ identified as $φ_*-\mathrm{id}$ on coefficient $K$-theory. In a concrete regime where the coefficient $K$-groups are cyclic (e.g.\ order $3$), the differential becomes an isomorphism and forces a $K$-theoretic obstruction to Morita triviality. This yields a conceptual mechanism for producing non-Morita-trivial twisted C$^*$-algebras with quantum-group crossed-product fibers, detected purely by Mayer--Vietoris/Čech data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22708 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mayer--Vietoris and Twisted Čech Spectral Sequences for C$^*$-Algebras with Free Quantum Group Coefficients Inoué, Takao Operator Algebras K-Theory and Homology Primary 46L80, Secondary 46L55, 46L85, 58B34, 20G42 We formulate a Mayer--Vietoris/Čech viewpoint on $K$-theory for crossed products by discrete quantum groups, emphasizing how local-to-global gluing data interacts with quantum-group coefficients. Starting from a $G$-equivariant ideal cover and the associated Mayer--Vietoris six-term exact sequence, we package the resulting $K$-theoretic computation into a Čech-type spectral sequence whose $E^1$-page is explicitly described by iterated intersections. We then introduce a minimal ``twisted'' gluing mechanism controlled by a $\mathbb Z/2$-valued Čech $2$-cocycle and an involutive automorphism of the coefficient algebra. Under a Kirchberg--UCT hypothesis on the quantum-group crossed-product coefficient, the twist produces a nontrivial differential $d_2$ identified as $φ_*-\mathrm{id}$ on coefficient $K$-theory. In a concrete regime where the coefficient $K$-groups are cyclic (e.g.\ order $3$), the differential becomes an isomorphism and forces a $K$-theoretic obstruction to Morita triviality. This yields a conceptual mechanism for producing non-Morita-trivial twisted C$^*$-algebras with quantum-group crossed-product fibers, detected purely by Mayer--Vietoris/Čech data. |
| title | Mayer--Vietoris and Twisted Čech Spectral Sequences for C$^*$-Algebras with Free Quantum Group Coefficients |
| topic | Operator Algebras K-Theory and Homology Primary 46L80, Secondary 46L55, 46L85, 58B34, 20G42 |
| url | https://arxiv.org/abs/2602.22708 |