Non-homogeneous extensions of Cantor minimal systems

Deeley, Robin J.; Putnam, Ian F.; Strung, Karen R.

doi:10.1090/proc/15342

Cited by 5 publications

(11 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that since we are only interested in the K-theory of W , we can and will assume W is connected. Finally, we recall a construction of minimal homeomorphisms on nonhomogeneous metric spaces from [16] which will allows us to show in Section 6.2 that there exist orbit-breaking subalgebras with interesting K-theory and many projections.…”

Section: Minimal Systemsmentioning

confidence: 99%

“…In [16], the authors further generalize the construction of Floyd by not only replacing the Cantor 3-odometer with more general minimal Cantor systems, but also by replacing the interval [0, 1] could be replaced by more complicated spaces, including spaces with arbitrarily large dimension. In particular, we have the following, which in Subsection 6.2 will allow us to find minimal equivalence relations by breaking an orbit at any arbitrary finite dimensional, compact, connected metric space Y .…”

Section: 2mentioning

confidence: 99%

“…In particular, we have the following, which in Subsection 6.2 will allow us to find minimal equivalence relations by breaking an orbit at any arbitrary finite dimensional, compact, connected metric space Y . We refer the reader to [16] for the details on this construction and in particular the proof of the next theorem.…”

Section: 2mentioning

confidence: 99%

“…The proof for the case that π −1 (x), x ∈ K, is either a single point or [0, 1] n is similar to the proof of [24,Theorem 4] (where n = 1). See also the proof in [16], where π −1 (x) can be higher dimensional cubes as well as more complicated spaces. Note that in [24] notation K(K, ϕ) is used to denote the 5-tuple (K 0 (A), K 0 (A) + , [1 A ], K 1 (A), T (A)) for A = C(K) ⋊ ϕ Z.…”

Section: Lemma 62 There Exists An Embeddingmentioning

confidence: 99%

See 3 more Smart Citations

Constructions in minimal amenable dynamics and applications to the classification of $\mathrm{C}^*$-algebras

Deeley,

Putnam,

Strung

2019

Preprint

View full text Add to dashboard Cite

We study the existence of minimal dynamical systems, their orbit and minimal orbit-breaking equivalence relations, and their applications to C * -algebras and K-theory. We show that given any finite CW-complex there exists a space with the same K-theory and cohomology that admits a minimal homeomorphism. The proof relies on the existence of homeomorphisms on point-like spaces constructed by the authors in previous work, together with existence results for skew product systems due to Glasner and Weiss. To any minimal dynamical system one can associate minimal equivalence relations by breaking orbits at small subsets. Using Renault's groupoid C * -algebra construction we can associate K-theory groups to minimal dynamical systems and orbit-breaking equivalence relations. We show that given arbitrary countable abelian groups G 0 and G 1 we can find a minimal orbit-breaking relation such that the K-theory of the associated C * -algebra is exactly this pair. These results have important applications to the Elliott classification program for C * -algebras. In particular, we make a step towards determining the range of the Elliott invariant of the C * -algebras associated to minimal dynamical systems with mean dimension zero and their minimal orbit-breaking relations.

show abstract

Section: Minimal Systemsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Lemma 62 There Exists An Embeddingmentioning

confidence: 99%

See 2 more Smart Citations

Constructions in minimal amenable dynamics and applications to the classification of $\mathrm{C}^*$-algebras

Deeley,

Putnam,

Strung

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Finally, we also apply our orbit-breaking technique to minimal homeomorphisms on non-homogeneous spaces constructed by the authors in [13], which are generalisations of systems constructed by Floyd [20] and Gjerde and Johansen [24]. In this case, thanks to the existence of a factor map onto a Cantor minimal system, the associated C * -algebras will always have real rank zero.…”

Section: Introductionmentioning

confidence: 99%

Classifiable $\mathrm{C}^*$-algebras from minimal $\mathbb{Z}$-actions and their orbit-breaking subalgebras

Deeley¹,

Putnam²,

Strung³

2020

Preprint

View full text Add to dashboard Cite

In this paper we consider the question of what abelian groups can arise as the K-theory of C * -algebras arising from minimal dynamical systems. We completely characterize the K-theory of the crossed product of a space X with finitely generated K-theory by an action of the integers and show that crossed products by a minimal homeomorphisms exhaust the range of these possible K-theories. Moreover, we may arrange that the minimal systems involved are uniquely ergodic, so that their C * -algebras are classified by their Elliott invariants. We also investigate the K-theory and the Elliott invariants of orbit-breaking algebras. We show that given arbitrary countable abelian groups G0 and G1 and any Choquet simplex ∆ with finitely many extreme points, we can find a minimal orbit-breaking relation such that the associated C * -algebra has K-theory given by this pair of groups and tracial state space affinely homeomorphic to ∆. We also improve on the second author's previous results by using our orbit-breaking construction to C * -algebras of minimal amenable equivalence relations with real rank zero that allow torsion in both K0 and K1. These results have important applications to the Elliott classification program for C * -algebras. In particular, we make a step towards determining the range of the Elliott invariant of the C * -algebras associated to étale equivalence relations.

show abstract

Classifiable $${\textrm{C}}^*$$-algebras from minimal $${\mathbb {Z}}$$-actions and their orbit-breaking subalgebras

2022

View full text Add to dashboard Cite

Non-homogeneous extensions of Cantor minimal systems

Cited by 5 publications

References 7 publications

Constructions in minimal amenable dynamics and applications to the classification of $\mathrm{C}^*$-algebras

Constructions in minimal amenable dynamics and applications to the classification of $\mathrm{C}^*$-algebras

Classifiable $\mathrm{C}^*$-algebras from minimal $\mathbb{Z}$-actions and their orbit-breaking subalgebras

Classifiable $${\textrm{C}}^*$$-algebras from minimal $${\mathbb {Z}}$$-actions and their orbit-breaking subalgebras

Contact Info

Product

Resources

About