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

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 $\ast$-automorphisms.

30 pages

References in corpus (5)

• Dynamical Systems on Spectral Metric Spaces
• Wieler solenoids, Cuntz-Pimsner algebras and K-theory
• The Noncommutative Geometry of Graph $C^{\ast }$-Algebras I: The Index Theorem
• Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras
• The Cuntz-Pimsner extension and mapping cone exact sequences

Cited by in corpus (10)

• The $K$-theoretic bulk-edge correspondence for topological insulators
• Wieler solenoids, Cuntz-Pimsner algebras and K-theory
• Index theory and topological phases of aperiodic lattices
• Operator *-correspondences in analysis and geometry
• Constructing KMS states from infinite-dimensional spectral triples
• Boundaries, spectral triples and K-homology
• Untwisting twisted spectral triples
• The Cuntz-Pimsner extension and mapping cone exact sequences
• Toeplitz extensions in noncommutative topology and mathematical physics
• Gysin exact sequences for quantum weighted lens spaces