A topological invariant for continuous fields of Cuntz algebras II

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.

11pages