A topological invariant for continuous fields of Cuntz algebras II

Abstract

We investigate an invariant for continuous fields of the Cuntz algebra O n + 1 \mathcal {O}_{n+1} introduced by Taro Sogabe [Math. Ann. 380 (2021), pp. 91–117], and find a way to obtain a continuous field of M n ( O ∞ ) \mathbb {M}_n(\mathcal {O}_\infty ) from that of O n + 1 \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 O n + 1 \mathcal {O}_{n+1} to those of M n ( O ∞ ) \mathbb {M}_n(\mathcal {O}_\infty ) . As a consequence, we obtain a new proof for M. Dadarlat’s classification result of continuous fields of O n + 1 \mathcal {O}_{n+1} arising from vector bundles, which corresponds to those of M n ( O ∞ ) \mathbb {M}_n(\mathcal {O}_\infty ) stably isomorphic to the trivial field.