The extension class and KMS states for Cuntz-Pimsner algebras of some bi-Hilbertian bimodules

For bi-Hilbertian A-bimodules, in the sense of Kajiwara-Pinzari-Watatani, we construct a Kasparov module representing the extension class defining the Cuntz-Pimsner algebra. The construction utilises a singular expectation which is defined using the C*-module version of the Jones index for bi-Hilbertian bimodules. The Jones index data also determines a novel quasi-free dynamics and KMS states on these Cuntz-Pimsner algebras.