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

Abstract

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 $C^{\ast }$-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.

arXiv commentv2: 25 pages. Several minor corrections implemented. A number of aspects of the exposition, particularly the proofs of Theorem 3.1 and Theorem 3.14 have been substantially improved by suggestions from a helpful referee. This version to appear in the Journal of Topology and Analysis