Spanier-Whitehead K-Duality and Duality of Extensions of $C^*$-algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909983492276224 |
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| author | Pennig, Ulrich Sogabe, Taro |
| author_facet | Pennig, Ulrich Sogabe, Taro |
| contents | KK-theory is a bivariant and homotopy-invariant functor on $C^*$-algebras that combines K-theory and K-homology. KK-groups form the morphisms in a triangulated category. Spanier-Whitehead K-Duality intertwines the homological with the cohomological side of KK-theory. Any extension of a unital $C^*$-algebra by the compacts has two natural exact triangles associated to it (the extension sequence itself and a mapping cone sequence). We find a duality (based on Spanier-Whitehead K-duality) that interchanges the roles of these two triangles together with their six-term exact sequences. This allows us to give a categorical picture for the duality of Cuntz-Krieger-Toeplitz extensions discovered by K. Matsumoto. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_20081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spanier-Whitehead K-Duality and Duality of Extensions of $C^*$-algebras Pennig, Ulrich Sogabe, Taro Operator Algebras K-Theory and Homology 19K35, 55P25, 46M15 KK-theory is a bivariant and homotopy-invariant functor on $C^*$-algebras that combines K-theory and K-homology. KK-groups form the morphisms in a triangulated category. Spanier-Whitehead K-Duality intertwines the homological with the cohomological side of KK-theory. Any extension of a unital $C^*$-algebra by the compacts has two natural exact triangles associated to it (the extension sequence itself and a mapping cone sequence). We find a duality (based on Spanier-Whitehead K-duality) that interchanges the roles of these two triangles together with their six-term exact sequences. This allows us to give a categorical picture for the duality of Cuntz-Krieger-Toeplitz extensions discovered by K. Matsumoto. |
| title | Spanier-Whitehead K-Duality and Duality of Extensions of $C^*$-algebras |
| topic | Operator Algebras K-Theory and Homology 19K35, 55P25, 46M15 |
| url | https://arxiv.org/abs/2403.20081 |