2022/08/26 by Kengo Matsumoto, Matsumoto, Kengo · 1 citation
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2208.12476
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 TA of a Cuntz-Krieger algebra OA is the K-theoretic dual of the Toeplitz extension TAt 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.