Minimal Dynamics and K-Theoretic Rigidity: Elliott’s Conjecture

By Andrew S. Toms, W. Winter. 2009

76 citations recorded by the source corpus.

Let X be a compact infinite metric space of finite covering dimension and α : X → X a minimal homeomorphism. We prove that the crossed product $${\mathcal{C}(X) \rtimes_\alpha \mathbb{Z}}$$ absorbs the Jiang–Su algebra tensorially and has finite nuclear dimension. As a consequence, these algebras are determined up to isomorphism by their graded ordered K-theory under the necessary condition that their projections separate traces. This result applies, in particular, to those crossed products arising from uniquely ergodic homeomorphisms.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c1fc4492339563fcd035c6b8b503d2a23e789912

How researchers see this paper

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

In Algebra and Number Theory

1. G'abor Szab'o
2. W. Winter
3. David Kerr
4. J. Zacharias
5. Jianchao Wu
6. N. Christopher Phillips
7. Andrew S. Toms
8. Huaxin Lin

In Geometry and Topology

1. G'abor Szab'o
2. J. Zacharias
3. David Kerr
4. W. Winter
5. N. Christopher Phillips
6. Jianchao Wu
7. Andrew S. Toms
8. Huaxin Lin

In Mathematical Physics

1. G'abor Szab'o
2. David Kerr
3. W. Winter
4. J. Zacharias
5. Jianchao Wu
6. N. Christopher Phillips
7. Andrew S. Toms
8. Eusebio Gardella

Citations in the corpus

• The Rokhlin dimension of topological ℤm‐actions (2013, background): “ms, Strung and Winter proved that crossed products of uniquely ergodic minimal homeomorphisms on infinite compact metrizable spaces with finite covering dimension are classified by ordered K-theory (see [21, 19]). The case of Zm-actions begins to gather more and more attention and interest as the next ”
• Nuclear dimension and -stability of pure C ∗ -algebras (2010, methodology): “The result has been particularly useful for Elliott’s classification program for nuclear C∗algebras; for example it has led to the complete classification of C∗-algebras associated to uniquely ergodic, minimal, finite-dimensional dynamical systems, in [33, 34].”
• Almost Finiteness and the Small Boundary Property (2018, background): “valence instead of a probability measure. While towers and tower decompositions have long played an important role in unraveling the structure of C∗-crossed products, notably in the case of Z-actions [35, 26, 25, 39, 13] and in the various theories of dimension for group actions [18, 36, 37, 21], th”
• Rokhlin dimension for actions of residually finite groups (2014, result): “Such nuclear dimension results were first obtained by Toms and Winter in [78] by different methods.”
• Large subalgebras (2014, methodology): “• The version with two points on different orbits has been used by Toms and Winter [39].”