K-theoretic duality for extensions of Cuntz-Krieger algebras

Abstract

We introduce the notion of K-theoretic duality for extensions of separable unital nuclear $C^{\ast }$-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 $C^{\ast }$-algebras if and only if the Toeplitz algebras ${\mathcal{T}}_{A^t}$ and ${\mathcal{T}}_{B^t}$ of their transposed matrices are isomorphic.

25 pages, the paper is shortened for submitting it for publication