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 tau A of a Cuntz-Krieger algebra OA is the K-theoretic dual of the Toeplitz extension tau At of the Cuntz-Krieger algebra OAt for the transposed matrix At of A. A pair of isomorphic Cuntz-Krieger algebras OA and OB does not necessarily yield the isomorphic pair of OAt and OBt. However, as an application, we may show that two Toeplitz algebras TA and TB are isomorphic as C*-algebras if and only if the Toeplitz algebras TAt and TBt of their transposed matrices are isomorphic. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

We introduce the concept of K-theoretic duality for extensions of separable unital nuclear C*-algebras, using the K-homology long exact sequence and cyclic six term exact sequence for K-theory groups of extensions. We then prove that the Toeplitz extension tau A of a Cuntz-Krieger algebra OA is the K-theoretic dual of the Toeplitz extension tau At of the Cuntz-Krieger algebra OAt for the transposed matrix At of A. As an application, we show that if the Toeplitz algebras TAt and TBt of the transposed matrices are isomorphic, then the Toeplitz algebras TA and TB themselves are isomorphic as C*-algebras.