The C∗-algebra of a minimal homeomorphism of zero mean dimension

By G. Elliott, Z. Niu. 2014

90 citations recorded by the source corpus.

Let $X$ be an infinite compact metrizable space, and let $\sigma: X\to X$ be a minimal homeomorphism. Suppose that $(X, \sigma)$ has zero mean topological dimension. The associated C*-algebra $A=\mathrm{C}(X)\rtimes_\sigma\mathbb Z$ is shown to absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A\cong A\otimes\mathcal Z$. This implies that $A$ is classifiable when $(X, \sigma)$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang-Su algebra.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4a34adcbfe5bc16ea3e3b40602d56a224e0324b6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. G'abor Szab'o
2. David Kerr
3. Stuart White
4. Rufus Willett
5. Yasuhiko Sato
6. Guoliang Yu
7. W. Winter
8. E. Guentner

In Mathematical Physics

1. G'abor Szab'o
2. David Kerr
3. Rufus Willett
4. Stuart White
5. W. Winter
6. Guoliang Yu
7. Yasuhiko Sato
8. E. Guentner

Citations in the corpus

• Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and $$C^*$$C∗-algebras (2015, result): “We should note that there are many other results in the literature that give estimates on nuclear dimension or other C˚-algebraic regularity properties based on conditions on actions: see for example [36, 19, 37, 20, 11]. Many of these results go further than ours in at least some ways: for example,”
• Nuclear dimension and $$\mathcal Z$$Z-stability (2014, background): “In the uniquely ergodic case Elliott and Niu have shown that regularity is automatic [11] (a fact related to unique ergodicity implying mean dimension zero—regularity can fail for larger trace spaces).”
• Almost Finiteness and the Small Boundary Property (2018, background): “itions in the Toms–Winter conjecture are actually satisfied by all of the simple crossed products in the statement of Theorem 9.4. This is known to be the case when G = Zby the work of Elliott and Niu [13]. Remark 9.6. It is worth pointing out that the proof of Theorem 9.4 actually verifies a version ”
• Dimension, comparison, and almost finiteness (2017, methodology): “y and mean dimension are formally very different from either nuclear dimension or Z-stability and are more akin to slow dimension growth in inductive limit C∗-algebras, as demonstrated by the proof in [11], which employs arguments from an article of Toms on the equivalence of slow dimension growth an”