K-theoretic duality for extensions of Cuntz-Krieger algebras

We introduce the notion of K-theoretic duality for extensions of separable unital nuclear $C^*$-algebras by using K-homology long exact sequence and cyclic six term exact sequence for K-theory groups of extensions. We then prove that the Toeplitz extension $\mathcal{T}_A$ of a Cuntz-Krieger algebra $\mathcal{O}_A$ is the K-theoretic dual of the Toeplitz extension $\mathcal{T}_{A^t}$ of the Cuntz-Krieger algebra $\mathcal{O}_{A^t}$ for the transposed matrix $A^t$ of $A$. A pair of isomorphic Cuntz--Krieger algebras $\mathcal{O}_A$ and $\mathcal{O}_B$ does not necessarily yield the isomorphic pair of $\mathcal{O}_{A^t}$ and $\mathcal{O}_{B^t}.$ However, as an application, we may show that two Toeplitz algebras $\mathcal{T}_A$ and $\mathcal{T}_B$ are isomorphic as $C^*$-algebras if and only if the Toeplitz algebras $\mathcal{T}_{A^t}$ and $\mathcal{T}_{B^t}$ of their transposed matrices are isomorphic.