For A a C*-algebra, E₁, E₂ two Hilbert bimodules over A, and a fixed isomorphism ϗ : E₁⊗_AE₂→ E₂⊗_AE₁, we consider the problem of computing the K-theory of the Cuntz-Pimsner algebra O_E2⊗AOE1 obtained by extending the scalars and by iterating the Pimsner construction.

The motivating examples are a commutative diagram of Douglas and Howe for the Toeplitz operators on the quarter plane, and the Toeplitz extensions associated by Pimsner and Voiculescu to compute the K-theory of a crossed product. The applications are for Hilbert bimodules arising from rank two graphs and from commuting endomorphisms of abelian C*-algebras.