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.