Dimension Groups and Their Affine Representations

1980 ◽  
Vol 102 (2) ◽  
pp. 385 ◽  
Author(s):  
Edward G. Effros ◽  
David E. Handelman ◽  
Chao-Liang Shen
Keyword(s):  
Dimension Groups ◽  
Affine Representations

Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows

2000 ◽  
Vol 20 (6) ◽  
pp. 1687-1710 ◽  
Author(s):  
RICHARD GJERDE ◽  
ØRJAN JOHANSEN
Keyword(s):  
Dimension Group ◽  
Orbit Equivalent ◽  
Bratteli Diagrams ◽  
Novel Approach ◽  
Dimension Groups ◽  
Substitution System ◽  
Zero Entropy ◽  
Factor Map ◽  
Cantor Minimal Systems ◽  
Uniquely Ergodic

We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum.Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$.Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors.The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.

Tensor Products of Dimension Groups and K0 of Unit-Regular Rings

1986 ◽  
Vol 38 (3) ◽  
pp. 633-658 ◽  
Author(s):  
K. R. Goodearl ◽  
D. E. Handelman
Keyword(s):  
Regular Ring ◽  
Hausdorff Space ◽  
Tensor Products ◽  
Matrix Algebras ◽  
Dimension Group ◽  
Grothendieck Groups ◽  
Dimension Groups ◽  
Direct Limits ◽  
Finite Products ◽  
Regular Rings

We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K0 of a unit-regular ring or even as K0 of a locally matricial algebra. If G is countable, this is well known to be true. Here we provide positive answers in case (a) the cardinality of G is ℵ1, or (b) G is an arbitrary infinite tensor product of the groups considered in (a), or (c) G is the group of all continuous real-valued functions on an arbitrary compact Hausdorff space. In cases (a) and (b), we show that G in fact appears as K0 of a locally matricial algebra. Result (a) is the basis for an example due to de la Harpe and Skandalis of the failure of a determinantal property in a non-separable AF C*-algebra [18, Section 3].

WEAK ORBIT EQUIVALENCE OF CANTOR MINIMAL SYSTEMS

1995 ◽  
Vol 06 (04) ◽  
pp. 559-579 ◽  
Author(s):  
ELI GLASNER ◽  
BENJAMIN WEISS
Keyword(s):  
Recent Work ◽  
Orbit Equivalence ◽  
C Algebra ◽  
Dimension Groups ◽  
Cantor Minimal Systems

This paper is a commentary on the recent work [4]. It has two goals: the first is to eliminate the C*-algebra machinery from the proofs of the results of [4]; the second, to provide a characterization of weak orbit equivalence of Cantor minimal systems in terms of their dimension groups.

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