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

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.

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CONTINUOUS TRACE C*-ALGEBRAS, GAUGE GROUPS AND RATIONALIZATION

Journal of Topology and Analysis ◽

10.1142/s179352530900014x ◽

2009 ◽

Vol 01 (03) ◽

pp. 261-288 ◽

Cited By ~ 4

Author(s):

JOHN R. KLEIN ◽

CLAUDE L. SCHOCHET ◽

SAMUEL B. SMITH

Keyword(s):

Gauge Group ◽

Metric Space ◽

Principal Bundle ◽

Rational Homotopy ◽

Compact Metric Space ◽

Gauge Groups ◽

C Algebra ◽

C Algebras ◽

Rational Cohomology ◽

Rational Homotopy Type

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let Aζdenote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UAζ, the group of unitaries of Aζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

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Minimal Dynamical Systems on Connected Odd Dimensional Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2014-035-7 ◽

2015 ◽

Vol 67 (4) ◽

pp. 870-892 ◽

Cited By ~ 3

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.

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LIMITS OF HOMOMORPHISMS WITH FINITE-DIMENSIONAL RANGE

International Journal of Mathematics ◽

10.1142/s0129167x05003089 ◽

2005 ◽

Vol 16 (07) ◽

pp. 807-821 ◽

Cited By ~ 3

Author(s):

SHANWEN HU ◽

HUAXIN LIN ◽

YIFENG XUE

Keyword(s):

Metric Space ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Compact Metric Space ◽

C Algebra ◽

Finite Dimensional ◽

Necessary And Sufficient ◽

Unital Simple ◽

Dimensional Range

Let X be a compact metric space and A be a unital simple C*-algebra with TR (A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.

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Crossed products and minimal dynamical systems

Journal of Topology and Analysis ◽

10.1142/s1793525318500140 ◽

2018 ◽

Vol 10 (02) ◽

pp. 447-469 ◽

Cited By ~ 3

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.

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On the Radius of Comparison of a Commutative C*-algebra

Canadian Mathematical Bulletin ◽

10.4153/cmb-2012-012-9 ◽

2013 ◽

Vol 56 (4) ◽

pp. 737-744

Author(s):

George A. Elliott ◽

Zhuang Niu

Keyword(s):

Lower Bound ◽

Metric Space ◽

Cohomological Dimension ◽

Compact Metric Space ◽

C Algebra

Abstract.Let X be a compact metric space. A lower bound for the radius of comparison of the C*-algebra C(X) is given in terms of dimQ X, where dimQ X is the cohomological dimension with rational coefficients. If dimQX = dim X = d, then the radius of comparison of the C*-algebra C(X) is maxf0; (d/1)=2/1g if d is odd, and must be either d=2/1 or d=2/2 if d is even (the possibility d=2 - 1 does occur, but we do not know if the possibility d=2 - 2 can also occur).

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Isometric Embeddings of Pro-Euclidean Spaces

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2015-0019 ◽

2015 ◽

Vol 3 (1) ◽

Author(s):

Barry Minemyer

Keyword(s):

Riemannian Manifold ◽

Euclidean Space ◽

Metric Space ◽

Inverse Limit ◽

Isometric Embedding ◽

Embedding Theorem ◽

Euclidean Spaces ◽

Compact Metric Space ◽

Covering Dimension ◽

Isometric Embeddings

Abstract In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E2n+1. Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C0 version of) the famous Nash isometric embedding theorem from [10].

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Strong submeasures and applications to non-compact dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.132 ◽

2020 ◽

pp. 1-23

Author(s):

TUYEN TRUNG TRUONG

Keyword(s):

Metric Space ◽

Bounded Operator ◽

Continuous Functions ◽

Open Dense Subset ◽

Compact Metric Space ◽

Basic Properties ◽

Banach Theorem ◽

Complex Varieties ◽

Definition Of ◽

Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

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On equicontinuous factors of flows on locally path-connected compact spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.104 ◽

2020 ◽

pp. 1-18

Author(s):

NIKOLAI EDEKO

Keyword(s):

Lie Group ◽

Metric Space ◽

Betti Number ◽

Alternative Proof ◽

Simple Consequence ◽

Compact Metric Space ◽

Invariant Functions ◽

Compact Spaces ◽

Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

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