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arXiv · 2403.20081

Spanier-Whitehead K-Duality and Duality of Extensions of $C^*$-algebras

Ulrich Pennig·Taro Sogabe
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Abstract

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.

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Ulrich Pennig, Taro Sogabe. 2024-03-29. Spanier-Whitehead K-Duality and Duality of Extensions of $C^*$-algebras. https://doi.org/10.1093/imrn%2Frnae242

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