A topological invariant for continuous fields of Cuntz algebras

Abstract

We wish to investigate continuous fields of the Cuntz algebras. The Cuntz algebras \({\mathcal {O}}_{n+1}, n\ge 1\) play an important role in the theory of operator algebras, and they are characterized by their K-groups \(K_0({\mathcal {O}}_{n+1})={\mathbb {Z}}_n\), the cyclic groups of order \(n\ge 1\). Since the mod n K-group for a compact Hausdorff space can be realized by the K-group of the trivial continuous field of \({\mathcal {O}}_{n+1}\) over the space, one can regard \({\mathcal {O}}_{n+1}\) as a noncommutative analogue of the Moore space of \({\mathbb {Z}}_n\), and classifying continuous fields of the Cuntz algebras is an interesting problem. M. Dadarlat classifies these fields which are constructed from the vector bundles of rank \(n+1\), and he also showed that not every continuous field comes from the vector bundle. For a continuous field of \({\mathcal {O}}_{n+1}\) over a finite CW complex, we introduce a topological invariant, which is an element in Dadarlat–Pennig’s generalized cohomology group, and prove that the invariant is trivial if and only if the field comes from a vector bundle via Pimsner’s construction.