2018/07/06 by Ilan Hirshberg, Hirshberg, Ilan, Jianchao Wu +1
Mathematics · #22A22 #46L35 (secondary) #54H20 (Primary) #Advanced Operator Algebra Research #Advanced Topics in Algebra #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #math.DS #math.OA #msc:22A22 #msc:46L35 #msc:54H20
paper · pdf · doi:10.48550/arxiv.1807.02246
46 pages; minor revisions; to appear in Advances in Mathematics
openalex publication_date 2018/07/06 · arxiv created 2021/05/10 · arxiv updated 2021/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that for any locally compact Hausdorff space Y with finite covering dimension and for any continuous flow ℝ \curvearrowright Y, the resulting crossed product C^*-algebra C0(Y) \rtimes 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 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.