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

TL;DR

This paper explores the detailed structure of shift-tail equivalence in Cuntz-Pimsner algebras, providing a new unbounded representative of the algebra's extension, leading to novel spectral triples for these algebras.

Contribution

It introduces an unbounded representative of the Cuntz-Pimsner extension for a broad class of algebras, extending previous work and enabling new spectral triples.

Findings

Unbounded representative constructed for Cuntz-Pimsner extension.

New spectral triples for Cuntz and Cuntz-Krieger algebras.

Application to vector bundles twisted by automorphisms.

Abstract

We show how the fine structure in shift-tail equivalence, appearing in the noncommutative geometry of Cuntz-Krieger algebras developed by the first two 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 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 equicontinuous $*$-automorphisms.