The C*-algebra of a vector bundle

TL;DR

This paper characterizes the Cuntz-Pimsner algebra of a vector bundle over a compact space using K-theory invariants, providing classification results for certain continuous fields of Cuntz algebras.

Contribution

It introduces a K-theory-based invariant that determines the Cuntz-Pimsner algebra of a vector bundle up to isomorphism and establishes criteria for embeddings between such algebras.

Findings

The Cuntz-Pimsner algebra O(E) is classified by the ideal (1-[E])K(X).

Embedding conditions depend on divisibility of (1-[E]) by (1-[F]) in K-theory.

Classification of continuous fields over CW complexes with specific torsion conditions.

Abstract

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.