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

TL;DR

This paper constructs a Kasparov module for bi-Hilbertian bimodules' Cuntz--Pimsner algebras, introducing a new approach using Jones index data to define dynamics and KMS states.

Contribution

It introduces a novel construction of the extension class for Cuntz--Pimsner algebras of bi-Hilbertian bimodules using a singular expectation and Jones index.

Findings

Constructed a Kasparov module representing the extension class.

Defined a new quasi-free dynamics based on Jones index data.

Established KMS states on the Cuntz--Pimsner algebras.

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 $C^*$-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.