Shift-tail equivalence and an unbounded representative of the Cuntz-Pimsner extension

We show how the fine structure in shift-tail equivalence, appearing in the non-commutative geometry of Cuntz-Krieger algebras developed by the first two listed authors, has an analogue in a wide range of other Cuntz-Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz-Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third listed author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz and Cuntz-Krieger algebras and for Cuntz-Pimsner algebras associated to vector bundles twisted by an equicontinuous *-automorphism