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