Spanier–Whitehead K-Duality and Duality of Extensions of <i>C</i>*-Algebras

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.