• DOI:10.1007/s00208-020-02101-6
• Corpus ID: 211132921

A topological invariant for continuous fields of Cuntz algebras

@article{Sogabe2020ATI,
  title={A topological invariant for continuous fields of Cuntz algebras},
  author={T. Sogabe},
  journal={Mathematische Annalen},
  year={2020},
  pages={1-27}
}

• T. Sogabe
• Published 17 February 2020
• Mathematics
• Mathematische Annalen

We wish to investigate continuous fields of the Cuntz algebras. The Cuntz algebras $${\mathcal {O}}_{n+1}, n\ge 1$$ O n + 1 , n ≥ 1 play an important role in the theory of operator algebras, and they are characterized by their K-groups $$K_0({\mathcal {O}}_{n+1})={\mathbb {Z}}_n$$ K 0 ( O n + 1 ) = Z n , the cyclic groups of order $$n\ge 1$$ n ≥ 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 $${\mathcal {O}}_{n+1}$$ O n + 1 over the space, one can regard $${\mathcal {O}}_{n+1}$$ O n + 1 as a noncommutative analogue of the Moore space of $${\mathbb {Z}}_n$$ 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 $$n+1$$ n + 1 , and he also showed that not every continuous field comes from the vector bundle. For a continuous field of $${\mathcal {O}}_{n+1}$$ 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.