scholarly journals Minimal Dynamical Systems on Connected Odd Dimensional Spaces

2015 ◽  
Vol 67 (4) ◽  
pp. 870-892 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Abelian Group ◽  
Dynamical Systems ◽  
Finite Abelian Group ◽  
Finite Covering ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Group I ◽  
Tracial Rank ◽  
Minimal Dynamical Systems ◽  
Minimal Homeomorphisms

AbstractLet be a minimal homeomorphism (n ≥1). We show that the crossed product has rational tracial rank at most one. Let Ω be a connected, compact, metric space with finite covering dimension and with . Suppose that ,where Gi is a finite abelian group, i = 0,1. Let β:Ω→Ωbe a minimal homeomorphism. We also show that has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on X✗Ω, where X is the Cantor set.

scholarly journals Crossed products and minimal dynamical systems

2018 ◽  
Vol 10 (02) ◽  
pp. 447-469 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Metric Space ◽  
General Setting ◽  
Crossed Product ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Infinite Type ◽  
Mean Dimension ◽  
Tracial Rank ◽  
Minimal Dynamical Systems ◽  
Minimal Homeomorphisms

Let [Formula: see text] be an infinite compact metric space with finite covering dimension and let [Formula: see text] be two minimal homeomorphisms. We prove that the crossed product [Formula: see text]-algebras [Formula: see text] and [Formula: see text] are isomorphic if and only if they have isomorphic Elliott invariant. In a more general setting, we show that if [Formula: see text] is an infinite compact metric space and if [Formula: see text] is a minimal homeomorphism such that [Formula: see text] has mean dimension zero, then the tensor product of the crossed product with a UHF-algebra of infinite type has generalized tracial rank at most one. This implies that the crossed product is in a classifiable class of amenable simple [Formula: see text]-algebras.

scholarly journals On complexes in a finite abelian group, I

1988 ◽  
Vol 64 (7) ◽  
pp. 245-248 ◽  
Author(s):  
Tamás Szőnyi ◽  
Ferenc Wettl
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Group I

scholarly journals Dynamics of Endomorphism of Finite Abelian Groups

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Zhao Jinxing ◽  
Nan Jizhu
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Dynamical Systems ◽  
Automorphism Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Point System ◽  
Finite Abelian Groups

We study the dynamics of endomorphisms on a finite abelian group. We obtain the automorphism group for these dynamical systems. We also give criteria and algorithms to determine whether it is a fixed point system.

scholarly journals The structure of the C*-algebra of a locally injective surjection

2011 ◽  
Vol 32 (4) ◽  
pp. 1226-1248 ◽  
Author(s):  
TOKE MEIER CARLSEN ◽  
KLAUS THOMSEN
Keyword(s):  
Metric Space ◽  
Matrix Algebra ◽  
Finite Covering ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Classification Result ◽  
Full Matrix ◽  
C Algebra ◽  
Simple Quotient ◽  
The Ideal

AbstractIn this paper we investigate the ideal structure of the C*-algebra of a locally injective surjection introduced by the second-named author. Our main result is that a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is either a full matrix algebra, a crossed product of a minimal homeomorphism of a compact metric space of finite covering dimension, or it is purely infinite and hence covered by the classification result of Kirchberg and Phillips. It follows in particular that if the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is simple, then it is automatically purely infinite, unless the map in question is a homeomorphism. A corollary of this result is that if the C*-algebra of a one-sided subshift is simple, then it is also purely infinite.

MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

2011 ◽  
Vol 22 (01) ◽  
pp. 1-23 ◽  
Author(s):  
KAREN R. STRUNG ◽  
WILHELM WINTER
Keyword(s):  
Dynamical Systems ◽  
Compact Metric Space ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Minimal Dynamical Systems ◽  
Uniquely Ergodic ◽  
Stable Classification

Let X be an infinite compact metric space, α : X → X a minimal homeomorphism, u the unitary that implements α in the transformation group C*-algebra C(X) ⋊α ℤ, and [Formula: see text] a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in [Formula: see text]. Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra [Formula: see text], we prove that (C(X) ⋊α ℤ) ⊗ Mq∞ is tracially approximately [Formula: see text] if there exists a y ∈ X such that the C*-subalgebra (C*(C(X), uC0(X\{y}))) ⊗ Mq∞ is tracially approximately [Formula: see text]. If the class [Formula: see text] consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with [Formula: see text] for C*-algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either C(X) ⋊α ℤ or C*(C(X), uC0(X\{y})), nor on the dimension of X. The result is a key step in the classification of C*-algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C*-algebras of minimal dynamical systems where projections do not necessarily separate traces.

Crossed products of C∗-algebras C(X,A) and their applications

2016 ◽  
Vol 27 (03) ◽  
pp. 1650029
Author(s):  
Jiajie Hua
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Crossed Products ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
C Algebras ◽  
Finite Dimensional ◽  
Tracial Rank ◽  
Stably Finite ◽  
Unital Simple

Let [Formula: see text] be an infinite compact metric space with finite covering dimension, let [Formula: see text] be a unital separable simple AH-algebra with no dimension growth, and denote by [Formula: see text] the [Formula: see text]-algebra of all continuous functions from [Formula: see text] to [Formula: see text] Suppose that [Formula: see text] is a minimal group action and the induced [Formula: see text]-action on [Formula: see text] is free. Under certain conditions, we show the crossed product [Formula: see text]-algebra [Formula: see text] has rational tracial rank zero and hence is classified by its Elliott invariant. Next, we show the following: Let [Formula: see text] be a Cantor set, let [Formula: see text] be a stably finite unital separable simple [Formula: see text]-algebra which is rationally TA[Formula: see text] where [Formula: see text] is a class of separable unital [Formula: see text]-algebras which is closed under tensoring with finite dimensional [Formula: see text]-algebras and closed under taking unital hereditary sub-[Formula: see text]-algebras, and let [Formula: see text]. Under certain conditions, we conclude that [Formula: see text] is rationally TA[Formula: see text] Finally, we classify the crossed products of certain unital simple [Formula: see text]-algebras by using the crossed products of [Formula: see text].

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group

2021 ◽  
Author(s):  
Weidong Gao ◽  
Siao Hong ◽  
Wanzhen Hui ◽  
Xue Li ◽  
Qiuyu Yin ◽  
...  
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum
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