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.

中文翻译：

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Cuntz代数连续域的拓扑不变量

我们希望研究 Cuntz 代数的连续域。Cuntz 代数 $${\mathcal {O}}_{n+1}, n\ge 1$$ O n + 1 , n ≥ 1 在算子代数理论中起着重要作用，它们的特点是K-groups $$K_0({\mathcal {O}}_{n+1})={\mathbb {Z}}_n$$ K 0 ( O n + 1 ) = Z n ，$阶循环群$n\ge 1$$ n ≥ 1 。由于紧致 Hausdorff 空间的 mod n K-群可以通过 $${\mathcal {O}}_{n+1}$$ O n + 1 的平凡连续域的 K-群来实现，可以将 $${\mathcal {O}}_{n+1}$$ O n + 1 视为 $${\mathbb {Z}}_n$$ Z n 的摩尔空间的非交换类比，并且对 Cuntz 代数的连续域进行分类是一个有趣的问题。M.Dadarlat 对这些由秩为 $$n+1$$n + 1 的向量束构造的字段进行分类，他还表明，并非每个连续场都来自向量丛。对于有限 CW 复数上 $${\mathcal {O}}_{n+1}$$O n + 1 的连续域，我们引入了拓扑不变量，它是 Dadarlat–Pennig 广义上同调群中的一个元素，并证明不变量是微不足道的，当且仅当场通过皮姆斯纳的构造来自向量丛。