-ALGEBRAS ASSOCIATED WITH TWO-SIDED SUBSHIFTS

2021 ◽  
pp. 1-34
Author(s):  
KENGO MATSUMOTO
Keyword(s):  
Compatibility Condition ◽  
Special Property ◽  
Crossed Product ◽  
Topological Conjugacy ◽  
Dimension Group ◽  
Bratteli Diagrams ◽  
K Theory ◽  
Shift Automorphism ◽  
Smale Spaces ◽  
Edge Labeling

Abstract This paper is a continuation of the paper, Matsumoto [‘Subshifts, $\lambda $ -graph bisystems and $C^*$ -algebras’, J. Math. Anal. Appl. 485 (2020), 123843]. A $\lambda $ -graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift $\Lambda $ , there exists a $\lambda $ -graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ with shift automorphism $\rho _{\mathcal {L}}$ from a $\lambda $ -graph bisystem $({\mathcal {L}}^-,{\mathcal {L}}^+)$ , and define a $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ by the crossed product . It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If $\lambda $ -graph bisystems come from two-sided subshifts, these $C^*$ -algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ and the K-theory formulas of the $C^*$ -algebras ${\mathcal {F}}_{\mathcal {L}}$ and ${\mathcal R}_{\mathcal {L}}$ . The K-group for the AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ is regarded as a two-sided extension of the dimension group of subshifts.

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.

Non-Commutative Deformations of C(T2) and K-Theory

1997 ◽  
Vol 08 (05) ◽  
pp. 555-571
Author(s):  
Cristina Cerri
Keyword(s):  
Positive Element ◽  
Crossed Product ◽  
C Algebra ◽  
Basic Properties ◽  
C Algebras ◽  
The Family ◽  
K Theory

For each α ≥ 0, let Bα be the universal C*-algebra generated by unitary elements uα, vα and a self-adjoint hα such that ||hα|| ≤ α and [Formula: see text]. In this work we prove that the family {Bα}α ∈ [0,∞[ extend the family of soft torus with the same basic properties, i.e., the field of C*-algebras {Bα}α ∈ [0,α0] is continuous and each Bα is a crossed product of a C*-algebra homotopically equivalent to C(S1) by Z. We then show that the K-groups of Bα are isomorphic to Z ⊕ Z. Applying results from the theory of rotation algebras we prove that every positive element (n,m) in K0(Bα) satisfies |m|α ≤ 2πn. It follows that these C*-algebras are not all homotopically equivalent to each other, although they have the same K-groups.

scholarly journals Homology for one-dimensional solenoids

2017 ◽  
Vol 121 (2) ◽  
pp. 219 ◽  
Author(s):  
Massoud Amini ◽  
Ian F. Putnam ◽  
Sarah Saeidi Gholikandi
Keyword(s):  
Finite Type ◽  
Homology Theory ◽  
One Dimensional ◽  
Axiom A ◽  
Group Invariant ◽  
Dimension Group ◽  
Basic Set ◽  
The One ◽  
Topological Dynamical Systems ◽  
Smale Spaces

Smale spaces are a particular class of hyperbolic topological dynamical systems, defined by David Ruelle. The definition was introduced to give an axiomatic description of the dynamical properties of Smale's Axiom A systems when restricted to a basic set. They include Anosov diffeomeorphisms, shifts of finite type and various solenoids constructed by R. F. Williams. The second author constructed a homology theory for Smale spaces which is based on (and extends) Krieger's dimension group invariant for shifts of finite type. In this paper, we compute this homology for the one-dimensional generalized solenoids of R. F. Williams.

scholarly journals The $K$-Theory of Some Reduced Inverse Semigroup $C^*$-Algebras

2015 ◽  
Vol 117 (2) ◽  
pp. 186 ◽  
Author(s):  
Magnus Dahler Norling
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
Inverse Semigroups ◽  
Crossed Product ◽  
Crossed Products ◽  
One Dimensional ◽  
C Algebras ◽  
Morita Equivalent ◽  
K Theory

We use a recent result by Cuntz, Echterhoff and Li about the $K$-theory of certain reduced $C^*$-crossed products to describe the $K$-theory of $C^*_r(S)$ when $S$ is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that $C^*_r(S)$ is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.

The cohomology and $K$-theory of commuting homeomorphisms of the Cantor set

1999 ◽  
Vol 19 (3) ◽  
pp. 611-625 ◽  
Author(s):  
ALAN FORREST ◽  
JOHN HUNTON
Keyword(s):  
Higher Order ◽  
Cantor Set ◽  
Crossed Product ◽  
Mapping Torus ◽  
C Algebra ◽  
Third Invariant ◽  
K Theory ◽  
Torsion Free ◽  
Order Continuous

Given a $\mathbb{Z}^d$ homeomorphic action, $\alpha$, on the Cantor set, $X$, we consider the higher order continuous integer valued dynamical cohomology, $H^*(X,\alpha)$. We also consider the dynamical $K$-theory of the action, the $K$-theory of the crossed product $C^*$-algebra $C(X)\times_{\alpha}\mathbb{Z}^d$. We show that these two invariants are essentially equivalent. We also show that they only take torsion free values. Our work links the two invariants via a third invariant which is based on topological complex $K$-theory evaluated on an associated mapping torus.

scholarly journals TENSOR-PRODUCT COACTION FUNCTORS

2020 ◽  
pp. 1-16
Author(s):  
S. KALISZEWSKI ◽  
MAGNUS B. LANDSTAD ◽  
JOHN QUIGG
Keyword(s):  
Tensor Product ◽  
Recent Work ◽  
Analogous Result ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Fixed Action ◽  
Product Action ◽  
K Theory

Recent work by Baum et al. [‘Expanders, exact crossed products, and the Baum–Connes conjecture’, Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. [‘Exotic crossed products and the Baum–Connes conjecture’, J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action $(C,\unicode[STIX]{x1D6FE})$ , then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact, and if $(C,\unicode[STIX]{x1D6FE})$ is the action by translation on $\ell ^{\infty }(G)$ , we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the $E$ -ization functor we defined earlier, where $E$ is a large ideal of $B(G)$ .

scholarly journals Ahlfors regularity and fractal dimension of Smale spaces

2021 ◽  
pp. 1-52
Author(s):  
DIMITRIS MICHAIL GERONTOGIANNIS
Keyword(s):  
Fractal Dimension ◽  
Markov Chains ◽  
Large Class ◽  
Metric Spaces ◽  
Topological Conjugacy ◽  
Compact Metric Spaces ◽  
Box Counting ◽  
Ahlfors Regularity ◽  
Smale Space ◽  
Smale Spaces

Abstract We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen’s construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry’s measure, which is Ahlfors regular. By extending Bowen’s construction, we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff, box-counting and Assouad dimensions of a large class of Smale spaces.

scholarly journals Compact operators and algebraic K-theory for groups which act properly and isometrically on Hilbert space

2018 ◽  
Vol 2018 (734) ◽  
pp. 265-292
Author(s):  
Guillermo Cortiñas ◽  
Gisela Tartaglia
Keyword(s):  
Hilbert Space ◽  
Approximation Property ◽  
Compact Operators ◽  
Crossed Product ◽  
K Theory

AbstractWe prove theK-theoretic Farrell–Jones conjecture for groups with the Haagerup approximation property and coefficient rings andC^{*}-algebras which are stable with respect to compact operators. We use this and Higson–Kasparov’s result that the Baum–Connes conjecture holds for such a groupG, to show that the algebraic and theC^{*}-crossed product ofGwith a stable separableG-C^{*}-algebra have the sameK-theory.

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