The nuclear dimension of C⁎-algebras associated to topological flows and orientable line foliations

We show that for any locally compact Hausdorff space Y with finite covering dimension and continuous flow R↷Y, the resulting crossed product C⁎-algebra C0(Y)⋊R has nuclear dimension. This generalizes previous results free flows, where this was proved using Rokhlin techniques. As an application, we obtain bounds of C⁎-algebras associated to one-dimensional orientable foliations. result is analogous one obtained earlier non-free actions Z. Some novel techniques in our proof include use a conditional expectation constructed from inclusion clopen subgroupoid, as well introduction what call fiberwise groupoid coverings help us build link between foliation products.