Let $\mathscr {A}$ be a topological Azumaya algebra of degree $mn$ over a CW complex X. We give conditions for the positive integers m and n, and the space X so that $\mathscr {A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees m and n. Then we prove that if $m<n$ and the dimension of X is higher than $2m+1$ , $\mathscr {A}$ may not have such decomposition.

中文翻译：

---

拓扑 Azumaya 代数的分解

令 $\mathscr {A}$ 是CW 复数X上的 $mn$ 次的拓扑 Azumaya 代数。我们给出了正整数m和n以及空间X的条件，使得 $\mathscr {A}$ 可以分解为度m和n的拓扑 Azumaya 代数的张量积。然后我们证明如果 $m<n$ 并且X的维度大于 $2m+1$ ， $\mathscr {A}$ 可能没有这样的分解。