The C*-algebra of a vector bundle

We prove that the Cuntz-Pimsner algebra O(E) of a vector bundle E over a compact metrizable space X is determined up to an isomorphism of C(X)-algebras by the ideal (1-[E])K(X) of the K-theory ring K(X). Moreover, if E and F are vector bundles of rank >1, then a unital embedding of C(X)-algebras of O(E) into O(F) exists if and only if 1-[E] is divisible by 1-[F] in the ring K(X). We introduce related, but more computable K-theory and cohomology invariants for O(E) and study their completeness. As an application we classify the unital separable continuous fields with fibers isomorphic to the Cuntz algebra O(m+1) over a finite connected CW complex X of dimension d< 2m+4 provided that the cohomology of X has no m-torsion.

💡 Research Summary

The paper investigates the Cuntz‑Pimsner C*-algebra O(E) associated with a complex vector bundle E over a compact metrizable space X, and establishes a complete K‑theoretic classification of these algebras as C(X)-algebras. The central object is the ideal
 I_E := (1 −