The stable homotopy category of topological spaces comes with a notion of duality, called Spanier-Whitehead duality. The KK-groups of C*-algebras form the morphisms in a triangulated category that is in many aspects the operator algebraic analogue of the stable homotopy category. It is straightforward to transfer the definition of duality to KK-theory. In joint work with Taro Sogabe we showed that this duality is compatible with the triangulated structure. As an application we study duality of extensions in the sense of Matsumoto. We were able to show that any extension of a separable unital nuclear UCT C*-algebra with finitely generated K-groups by the compact operators has a strongly K-theoretic dual. In the special case of Cuntz-Krieger algebras the dual of the Toeplitz extension is the Cuntz-Krieger algebra associated to the transposed matrix.
