We show that for any locally compact Hausdorff space Y with finite covering dimension and for any continuous flow $\mathbb{R}\curvearrowright Y$, the resulting crossed product $C^{⁎}$-algebra $C_0(Y)⋊\mathbb{R}$ has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. As an application, we obtain bounds for the nuclear dimension of $C^{⁎}$-algebras associated to one-dimensional orientable foliations. This result is analogous to the one we obtained earlier for non-free actions of $\mathbb{Z}$. Some novel techniques in our proof include the use of a conditional expectation constructed from the inclusion of a clopen subgroupoid, as well as the introduction of what we call fiberwise groupoid coverings that help us build a link between foliation $C^{⁎}$-algebras and crossed products.

我们证明了对于任何具有有限覆盖维数的局部紧致 Hausdorff 空间Y和任何连续流$\mathbb{电阻}\curvearrowright 是$，得到的交叉产品 $C^{⁎}$-代数 $C_0(是)⋊\mathbb{电阻}$具有有限的核维数。这概括了先前的自由流动结果，其中使用 Rokhlin 维数技术证明了这一点。作为一个应用，我们获得了核维数的界限$C^{⁎}$-与一维可定向叶面相关的代数。这个结果类似于我们之前为非自由动作获得的结果$\mathbb{Z}$. 我们证明中的一些新技术包括使用由包含 cloopen subgroupoid 构建的条件期望，以及引入我们所谓的 Fiberwise groupoid 覆盖，帮助我们在 foliation 之间建立联系$C^{⁎}$-代数和交叉产品。