The nuclear dimension of $C^*$-algebras associated to topological flows and orientable line foliations. Academic Article uri icon

abstract

  • We show that if $Y$ is a locally compact metrizable space with finite covering dimension, then the crossed product $C^*$-algebra $C_0(Y) \rtimes \mathbb{R}$ associated to any continuous flow on $Y$ has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. This result is also analogous to the one we obtained earlier for possibly non-free actions of $\mathbb{Z}$ on $Y$. As an application, we obtain bounds for the nuclear dimension of $C^*$-algebras associated to orientable line foliations. Some novel techniques in this paper include the use of conditional expectations constructed from clopen subgroupoids, 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.

publication date

  • January 1, 2018