A topological invariant for continuous fields of Cuntz algebras II

TL;DR

This paper introduces a new topological invariant for continuous fields of Cuntz algebras, establishing a bijection with matrix algebra fields over $ ext{O}_ ext{infinity}$, and offers a novel proof of a classification result.

Contribution

It constructs a continuous field of $ ext{M}_n( ext{O}_ ext{infinity})$ from that of $ ext{O}_{n+1}$ using the invariant, providing a new proof of classification of certain continuous fields.

Findings

Establishes a bijection between isomorphism classes of continuous fields of $ ext{O}_{n+1}$ and $ ext{M}_n( ext{O}_ ext{infinity})$ fields.

Provides a new proof for Dadarlat's classification of continuous fields from vector bundles.

Connects the invariant to the classification of continuous fields arising from vector bundles.

Abstract

We investigate an invariant for continuous fields of the Cuntz algebra $\mathcal{O}_{n+1}$ introduced in our previous work, and find a way to obtain a continuous field of $\mathbb{M}_n(\mathcal{O}_\infty)$ from that of $\mathcal{O}_{n+1}$ using the construction of the invariant. By Brown's representability theorem, this gives a bijection from the set of the isomorphism classes of continuous fields of $\mathcal{O}_{n+1}$ to those of $\mathbb{M}_n(\mathcal{O}_\infty)$. As a consequence, we obtain a new proof for M. Dadarlat's classification result of continuous fields of $\mathcal{O}_{n+1}$ arising from vector bundles, which corresponds to those of $\mathbb{M}_n(\mathcal{O}_\infty)$ stably isomorphic to the trivial field.